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[email protected] of Milan, Department of Physics, via Celoria 16, I-20133 Milan, Italy Harvard-Smithsonian Center for Astrophysics, Mail Stop 72, 60 Garden Street, Cambridge, MA 02138 University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, AustriaIn this paper we present a novel method to identify and characterize stellar clusters deeply embedded in a dark molecular cloud.The method is based on measuring stellar surface density in wide-field infrared images using star counting techniques. It takes advantage of the differing H-band luminosity functions (HLFs) of field stars and young stellar populations and is able to statistically associate each star in an image as a member of either the background stellar population or a young stellar population projected on or near the cloud.Moreover, the technique corrects for the effects of differential extinction toward each individual star. We have tested this method against simulations as well as observations. In particular, we have applied the method to 2MASS point sources observed in the Orion A and B complexes, and the results obtained compare very well with those obtained from deep Spitzer and Chandra observations where presence of infrared excess or X-ray emission directly determines membership status for every star. Additionally, our method also identifies unobscured clusters and a low resolution version of the Orion stellar surface density map shows clearly the relatively unobscured and diffuse OB 1a and 1b sub-groups and provides useful insights on their spatial distribution. A new method to unveil embedded stellar clusters Marco Lombardi1, Charles J. Lada2, and João Alves3 Received ***date***; Accepted ***date*** ========================================================§ INTRODUCTIONEmbedded star clusters offer one of the best opportunities to understand star formation.Within these astrophysical laboratories hundreds of stars are formed in volumes below 1 ^3. Overall, it is estimated that 80%–90% of young stellar objects (YSOs) form in embedded clusters (; although the exact fraction is somewhat sensitive to the definition of a cluster, see ).It is now established that the fate of these objects is directly linked to the evolution of the molecular gas, which is responsible for most of the gravitational potential that binds the stars in the cluster.Therefore, it is important to study these objects in their early phases, before they are subject to infant mortality <cit.>.Operationally, clusters are often defined and identified as groups of stars whose stellar surface density exceeds that of field stars of the same physical type (see, e.g., ).[In this paper we focus on the optical identification of a cluster, and we explicitly ignore issues such as the gravitational stability of overdensities of stars.Therefore, we will call “cluster” any overdensity, irrespective of its boundness.] In this respect, embedded clusters pose special problems because they are buried in dust and gas, and therefore are often not even visible in optical images; moreover, their shape is often elongated and clumpy, reflecting the initial structure of the dense molecular gas <cit.>.However, even with infrared observations, discovering deeply embedded clusters and measuring their basic parameters (such as surface density and size) can still be a challenge since the associated dust extinguishes both the cluster members and the field stars behind the cloud (see, e.g., ). In fact, typical optical or near-infrared observations stellar fields around molecular clouds show underdensities at the location of the clouds because of the effects of extinction on the density of background stars. In such a situation the observed cluster stellar surface density can be comparable or even less than the unobscured field stellar surface density (see Fig. <ref>).Different authors have used different techniques to take into account the effects of extinction in the identification and characterization of star clusters. <cit.> built density maps in the direction of nearby molecular clouds using the 2MASS Point Source Catalog, and corrected the effect of dust extinction using publicly available CO maps (converted into infrared extinction with a constant X-factor). In his technique, the extinction correction is only applied to field stars: the cluster stellar density is obtained bysubtracting an extinction-corrected model of the field stellar density to the observed stellar density. Moreover, the use of CO maps and the largely uncertain X-factor represent a primary source of error.An opposite approach has been adopted by <cit.>, who studied star clusters embedded in the Rosette nebula using 2MASS data.In this case, the local extinction was determined directly from the stellar near-infrared colors, and a correction for the effects of extinction was applied to each star. In practice, as noted by the authors themselves, the correction applied would only be correct for background stars, while it is used for all stars (including the foreground, which should not be corrected, and embedded ones, which should be partially corrected).Things are further complicated because molecular clouds are know to have steep gradients in their column densities, and these, if undetected (because of resolution effects, which is surely the case of the Rosette cloud at the 2MASS resolution), would introduce further biases in the extinction map and in the correction to the cluster richness.<cit.> use yet another technique (similar to ) to study the properties of IC 348 in Perseus. The basic idea is that two independent measurements of extinction in molecular clouds are available, one from the star color excess (here applied to H - K color), and one from the star number counts. In absence of contaminants, both measurements should agree. The presence of a cluster, however, will affect both the color excess measurement (in a way that depends on the intrinsic color of the cluster members) and the star count method (in a way that depends on the location of the cluster within the cloud). The difference of the two extinction estimates is a proxy for the cluster density. The various assumptions made when applying this technique (in particular, the degree of embedding of the cluster within the cloud) needs to be resolved by calibrating it using independent measurements of cluster richness: this clearly limits its application.In this paper we present a new methodology to identify and characterize clusters in wide-field or all sky, multi-band, infrared imaging surveys.The method is based on the production of extinction-corrected maps of stellar surface density, and takes advantage of the different H-band luminosity functions of embedded clusters with respect to that of the background population. Additionally, in contrast to the methods described above, it is able to correct for cluster members unidentified because of extinction in a way that takes into account the position of the cluster within the cloud along the line of sight.The technique is based on a rigorous mathematical framework; this provides a clear advantage, in that we can perform a detailed statistical analysis of the method. But the detailed mathematical derivation of this method might not be of as much interest to astronomers as its implementation.However, we stress that those readers not interested in the detailed mathematical aspects of the derivation can still benefit from the method proposed here, because its application is simple and straightforward: Eq(<ref>), with optionally a second expression for the noise estimate, Eq. (<ref>).We also provide a pseudo-code for it in the appendix.This paper is organized as follows. In Sect. 2 we present the general framework and discuss the standard technique generally employed to identify clusters.In Sect. 3 we present our new method and we discuss its practical implementation in Sect. 4.Simple numerical tests used to validate the method are described in Sect. 5, while the results of an application to the Orion molecular complex are discussed in Sect. 6.Finally, we summarize the results obtained in this paper in Sect. 7.§ THE STANDARD TECHNIQUE Traditionally, star clusters are identified as overdensities in the distribution of stars.Although many different options are available to estimate the local angular density of stars σ (see, e.g., ), we consider in this paper the simple “moving average” estimate[Throughout this paper we will use “hats” to denote measured quantities.Hence, σ(x⃗) is the true star density at the angular position x⃗, while σ̂(x⃗) is the measured star density at the same position.]σ̂(x⃗) = ∑_n=1^N W(x⃗ - x⃗_n ) .Here n runs on the individual stars, {x⃗_n } are the locations of the various stars, and W is a window function, normalized to unity:∫ W(x⃗')^2 x' = 1 .So, by construction, W has a unit of one over area.As a simple example, suppose that W is taken to be a top-hat function:W(x⃗') =1 / (π r^2)if |x⃗'| < r ,0otherwise.In this case, the sum of Eq. (<ref>) runs just over the N_r stars within an angular distance r from the point of the estimate, and the observed estimates reduces toσ̂= N_r/π r^2.For a constant density of stars, this quantity is unbiased, in the sense that the mean (ensemble average) E of σ̂ is just the true density of stars:[ σ̂] ≡⟨σ̂⟩ = [N_r]/π r^2 = σ,and the associated variance is[σ̂] = [N_r]/(π r^2)^2 = σ/π r^2.This equation shows that the error associated with the measured star density decreases as N_r^-1/2: therefore, if it is known that the density of stars is constant, to determine its value it is sensible to use relatively large window functions W.More generally, if the star density is variable, the average measured density can be shown to be a convolution of the true density with the window function W:[ σ̂] (x⃗) = ∫ W(x⃗ - x⃗') σ(x⃗')^2 x' ,and the associated two point correlation function is[σ̂] (x⃗, x⃗') ≡[ ( σ̂(x⃗) - [σ̂] (x⃗) ) ( σ̂(x⃗') -[ σ̂] (x⃗') ) ]= ∫ W(x⃗ - x⃗”) W(x⃗' - x⃗”)σ(x⃗”)^2 x”.Equation (<ref>) shows that W sets the scale for the resolution of the density map; similarly, Eq. (<ref>) shows that points close enough in the density map will have correlated noise. Therefore, if one aims at finding changes in the star density (such as a star cluster), the window function should have a scale that is comparable or smaller than the typical size of the variations of the star density.However, this is in tension with the noise properties of Eq. (<ref>), because a small window function implies a large noise.In order to be able to detect a star cluster, the measured density of stars at the location of the cluster must differ from the average density much more than the standard deviation of σ̂.Hence, the quantity [σ̂](x⃗, x⃗) sets the sensitivity of the cluster finding algorithm.In general the true density σ = σ_field + σ_cluster is the sum of the field density and of the cluster density, and in some cases σ_cluster≪σ_field.In these conditions the error is dominated by the shot-noise due to the field star population.In reality, many other effects can prevent the discovery and characterization of star clusters: * Extinction by dark nebulæ, which reduces the surface density of background sources;* The galactic structure, which induces smooth large-scale variations;* Differences in the sensitivity across a mosaic image due to changes in the observational conditions;* Other systematical observational effects, such halos produced by bright objects within the image;§.§ Extinction correctionAmong the effects listed above, the first one is particularly important for young clusters, since these objects tend to be deeply embedded and thus can escape a detection; additionally, detected clusters are plagued by large uncertainties in their physical properties (number of stars, mass, and size). For this reason, <cit.> developed a technique to perform a simple correction of the results produced by Eq. (<ref>). They noted that the density of background stars observed through a molecular cloud decreases by a factor 10^-α A, where α is the exponential slope of the number counts and A is the extinction, both in the band used for the observation.Therefore, to account for the undetected stars one can just multiply the local density estimate σ̂(x⃗) by 10^α A(x⃗), where A(x⃗) is the local estimate extinction (i.e., the extinction derived from an extinction map at the location of the x⃗).The problem of this approach is that it uses the same correction factor for foreground stars, embedded objects, and background stars, which generally results in an overestimate of the local corrected star density. Additionally, the same correction is applied to young stellar objects (YSOs) and field stars, which however have number count distributions very different. This leaves a large uncertainty on the corrected σ̂(x⃗) and, ultimately, on the characterization of each cluster. § MAXIMUM-LIKELIHOOD APPROACH §.§ Constant extinction, no weightingAs discussed in the previous section, the error associated to σ̂ is often dominated by shot noise due to the field star population.However, as mentioned, YSOs have photometric properties that differ significantly from these of field stars: the former have a H-band luminosity function (hereafter HLF) that can be approximated by a Gaussian distribution <cit.>, while the latter have a HLF that follows, up to H ∼18mag, an exponential with slope α∼ 0.33 (see, e.g., ).This suggests that we might use suitable cuts in the H-band to reduce the number of field stars, while essentially retaining all YSOs, thus gaining in terms of noise. However, this naive procedure is difficult to apply, since both YSOs and field stars are also affected by extinction, which would change the shape and the peaks of the respective HLFs.Therefore, we are forced to use a more systematic approach.Let us first model the observed H-band luminosity function for a set of angularly close stars (i.e., in a patch of the sky).We suppose that the true, unextinguished HLF can be described as a mixture of L different HLFs, corresponding to different stellar populations (in the situation considered later on in this paper we will use just two components, one for the field stars and one for the YSOs, but deeper observations might require the inclusion of a third component corresponding to galaxies and following exponential number counts with a slope of 0.6; see ). The observed HLF differs from the true one because of extinction and because of incompleteness in the faint end; additionally, we will also have photometric errors, but generally these will be small compared to the typical width of the various HLF components, and for the sake of simplicity are therefore neglected here.[Photometric errors can be easily included in Eq. (<ref>) by replacing p_i(m - A) there with the convolution of this function with the photometric uncertainty.In presence of different uncertainties for each star, one will need to specialize Eq. (<ref>) to each star.A similar technique can be used to include the effects of errors on the extinction measurements.] We can thus write finally the observed HLF, measured in units of stars per area per magnitude bin, asσ(m) = c(m) ∑_i=1^L σ_i p_i(m - A) ,where A is the H-band extinction,[In this section we will take the extinction to be identical for a set of angularly close stars; we will then relax this assumption and use the individual extinction in the direction of each star.] c(m) is the completeness function (i.e. the probability to detect a star with apparent magnitude m), p_i(m) is the probability distribution for the i-th component of the HLF, and σ_i is a coefficient that indicates the predominance of the i-th component in the HLF.In order to give σ_i a simple interpretation, it is useful to take p_i suitably normalized.In particular, we assume that∫ c(m) p_i(m) m = 1 .With this choice, σ_i can be identified as the observed angular density of stars for the i-th component in absence of extinction.Our goal is to infer the parameters {σ_i } from a set of observed H-band magnitudes { m_n } in a patch of the sky of area S; we will then repeat this operation in different directions in order to build maps of densities for the various components.This will allow us to identify overdensities in each component, such as in the YSO one.To this purpose, we use Eq. (<ref>) to write the log-likelihood following the prescriptions of <cit.>.The entire analysis is done in the magnitude space, by using S σ as the probability distribution density for the magnitudes:lnℒ =∑_n=1^N ln S σ(m_n) - ∫ S σ(m) m =∑_n=1^N [ ln S c(m_n) ∑_i=1^L σ_i p_i(m_n - A) ]- ∫ S c(m) ∑_i=1^L σ_i p_i(m - A) m .This likelihood can be used in Bayes' theorem to infer a posterior probability for the densities {σ_i }.More simply, we can just find the most likely parameters {σ_i } using a maximum likelihood approach.To this purpose, we consider∂lnℒ/∂σ_i = ∑_n=1^N c(m_n) p_i(m_n - A)/σ(m_n) - S ∫ c(m) p_i(m-A) m .The maximum likelihood solution is given by the set of densities σ_i that maximize ℒ or, equivalently, lnℒ, i.e. by the zero of Eq. (<ref>); that is, we need to solve∑_n=1^N c(m_n) p_i(m_n - A)/σ(m_n) = S ∫ c(m) p_i(m-A) m .Unfortunately, the solution of this equation for L > 1 cannot be provided in analytic form, and must be obtained by numerical methods. We will discuss below in Sect. <ref> simple ways to find it.We can obtain an estimate of the errors associated to the measurements σ_i from the Fisher information matrix (see, e.g., ), that we recall is defined asI_ij = [ ∂lnℒ/∂σ_i∂lnℒ/∂σ_j] = - [ ∂^2 lnℒ/∂σ_i∂σ_j] .The Fisher information matrix is related to the minimum covariance matrix that can be attained by an unbiased estimator, as provided by the Cramér-Rao bound:[σ] ≥ I^-1.Since the maximum-likelihood estimator is asymptotically efficient (i.e. it attains the Cramér-Rao bound when the sample size tends to infinity) and the resulting errors on σ̂ tend to a multivariate Gaussian distribution, it is interesting to obtain an analytic result for the information matrix.The Fisher information matrix can be readily evaluated from our data using Eq. (12) of <cit.>:I_ij = S ∫c^2(m) p_i(m-A) p_j(m-A)/σ(m)m .This relation is interesting from several points of view.First, note that the Fisher matrix contains elements outside the diagonal, unless the probability distributions for the various components do not overlap, i.e. have non-intersecting support: this would basically mean that we could immediately tell to which component belongs a star from its unextinguished magnitude.Second, note that all elements of the matrix are non-negative and therefore, in case of L=2 components, [σ] ≃ I^-1 will have non-positive elements off-diagonal: in other words, the measurements of the two densities σ_1 and σ_2 will be affected by a negative correlation.This is expected and corresponds to a classification error: if p_1 overlaps with p_2, we cannot uniquely associate stars to each different population and in general an overestimate of one population is done at the expenses of an underestimate of the other population.It is instructive to consider the form of the information matrix in the special case where L=1.When just one component is used, then I is a scalar and it reduces toI = S/σ∫ c(m) p(m-A) m = S^2/[N],where [N] = S σ is the average number of stars observed in the area S.Its inverse, I^-1, is therefore [N] / σ^2, as expected from a simple Poisson statistics.With L ≥ 2, in principle we can encounter cases where the Fisher information matrix is singular.Given the analytic form of I, this happens in particular when the two components have the exact same probability distributions within the support of c(m), i.e. c(m) p_i(m) = c(m) p_j(m): in this case, the corresponding rows (or columns) of I are identical.In such a situation, clearly, it is virtually impossible to classify objects as one of the two degenerate components, and therefore the uncertainty on the respective densities σ_i and σ_j are infinite.For completeness, we also report the expected maximum value of the log-likelihood[ℒ] = L/2 + S ∫σ(m) [ ln S σ(m) - 1 ] m ,and the associated variance[ℒ] = S ∫σ(m) ln^2 S σ(m) m .These equations can be used to verify that the chosen model (<ref>) can account well for the data.§.§ Differential extinction and spatial weightSo far, we have assumed that all stars are subject to the same extinction A; moreover, we have not weighted stars depending on their angular position as done in Eq. (<ref>).In this section we intend to remove these two limitations and consider the full problem. The simpler approach to include the effects of differential extinction is to consider the joint density in magnitudes and positions. In this framework, we can rewrite Eq. (<ref>) for log-likelihood aslnℒ =∑_n=0^N lnσ(m_n, x⃗_n) - ∫ m ∫^2 x'σ(m, x⃗') .In this equation the quantity σ(m, x⃗') represents the predicted density of stars with magnitude m at the location x'. Similarly to Eq. (<ref>), we write this quantity as a mixture ofdifferent densities, corresponding to different stellar populations:σ(m, x⃗') = c(m) ∑_i=1^L σ_i(x⃗') p_i ( m - A(x⃗') ) ,where σ_i(x⃗') represents the density of stars of class i at the sky position x⃗'.In order to proceed, we need to model these densities.A simple and consistent way of doing this, is to suppose that lnσ_i(x⃗') can be written as the weighted sum of two terms: one, lnσ(x⃗), associated to the density at the point of interest x⃗ (the point where we intend to evaluate the local densities of stellar populations); and one, lnτ_i(x⃗'), which parametrizes the local changes of the densities. As a result, we writeσ_i(x⃗') = ( σ_i(x⃗) )^ω(x⃗ - x⃗')( τ_i(x⃗') )^1 - ω(x⃗ - x⃗').The function ω describes the spatial correlation between densities at different positions and plays a central role in identifying which stars contribute to the density estimate at x⃗.We can now insert Eqs. (<ref>) and (<ref>) in Eq. (<ref>) and find the maximum likelihood solution over σ_i(x⃗).Calling A_n ≡ A(x⃗_n) the extinction suffered by the n-th star, we find∂lnℒ/∂σ_i(x⃗) = ∑_n=1^N ω(x⃗ - x⃗_n) p_i (m_n - A_n) σ_i(x⃗_n) / σ_i(x⃗)/∑_j σ_j(x⃗_n) p_j(m_n - A_n) - ∫ m c(m) ∫^2 x'ω(x⃗') p_i( m - A(x⃗') ) σ_i(x⃗') / σ_i(x⃗) .We now make an important assumption.Because of the form of the parametrization (<ref>), a solution for the maximum likelihood can only be found if we specify the functional form of the functions τ_i(x⃗').However, these functions are truly unknown, since they parametrize the local changes of the various densities, which in turn depend on the local density map.Using a statistical approach, however, it is natural to assume that these functions have the same distribution of σ_i(x⃗), the quantity we are interested in.As a result, if we take an average of Eq. (<ref>), terms such as σ_i(x⃗') / σ_i(x⃗) cancel out:∂lnℒ/∂σ_i(x⃗) = ∑_n=1^N ω(x⃗ - x⃗_n) p_i (m_n - A_n)/∑_j σ_j(x⃗) p_j(m_n - A_n) - ∫ m c(m) ∫^2 x'ω(x⃗') p_i( m - A(x⃗') ) . Before proceeding, it is useful to consider the solution of the maximum likelihood approach in the simple case where there is a single population of stars and where the extinction vanishes, A(x⃗') = 0.We find in this case∂lnℒ/∂σ(x⃗) = ∑_n=1^N ω(x⃗ - x⃗_n)/σ(x⃗) - ∫^2 x'ω(x⃗') ,where we have used the normalization property (<ref>).The solution of this equation is immediately found asσ(x⃗) = ∑_n=1^N W(x⃗ - x⃗_n) ,whereW(x⃗) = ω(x⃗)/∫ω(x⃗')^2 x'We have therefore recovered Eq. (<ref>), with the correct normalization (<ref>) for the weight W.In the general case, the maximum-likelihood solution of Eq. (<ref>) must be obtained numerically.The errors associated to the solutions can be estimated using the Fisher matrix.In our case, we can obtain the Fisher matrix fromI_ij(x⃗) =∬ m^2 x'1/σ(m, x⃗')∂σ(m, x⃗')/∂σ_i(x⃗)∂σ(m, x⃗')/∂σ_j(x⃗)=∬ m^2 x' c^2(m) ω^2(x⃗ - x⃗') ×p_i ( m - A(x⃗') ) p_j ( m - A(x⃗') )/σ(m, x⃗')σ_i(x⃗') σ_j(x⃗')/σ_i(x⃗) σ_j(x⃗).As before, we can replace x⃗ to x⃗' in all arguments of σ_i, with the justification that the statistical properties of σ are invariant upon translation.We will discuss in the next section useful expressions to evaluate this quantity in practical cases. §.§ ImplementationThe algorithm proposed in this paper is essentially the search of the solutions of Eq. (<ref>) as a function of the star densities {σ_i } corresponding to the various components or star populations.The same procedure must be applied to different patches of the sky, so that maps of the star densities can be obtained. These, in turn, will allow us to identify and characterize star clusters, and in particular embedded ones.Although the practical implementation of the algorithm follows this schema, a number of technical and theoretical aspects must be correctly addressed in order to optimize the detection and make the technique efficient.First, we note that a simple way to obtain the (positive) solutions of Eq. (<ref>) is through the use of a recursive formula.To this purpose, multiply both members of this equation by σ_i(x⃗), and solve for this quantity, thus obtaining the expressionσ_i(x⃗) ←∑_n=1^N ω(x⃗ - x⃗_n) σ_i(x⃗) p_i(m_n - A_n)/∑_j=1^L σ_j(x⃗)p_j(m_n - A_n)/∫ m c(m) ∫^2 x'ω(x⃗') p_i( m - A(x⃗') ).Unfortunately, we are unable to use this equation because we only know the extinction at the locations of the stars A_n ≡ A(x⃗_n): this prevents us from evaluating the integral over x^2 in the denominator.We can, however, move the denominator inside the sum, and evaluate the second integral by replacing A(x⃗) with A(x⃗_n), the extinction at the direction of each star.Additionally, using the same argument that has been employed in Eq. (<ref>), that is the similarity between σ and τ, it is convenient to replace σ_i(x⃗) with σ_i(x⃗_n) in the numerator.This procedure leads to the iterationσ_i(x⃗) ←∑_n=1^NW(x⃗ - x⃗_n) σ_i(x⃗_n) p_i(m_n - A_n)/∑_j=1^L σ_j(x⃗_n) p_j(m_n - A_n)/∫ m c(m) p_i(m - A_n).Equation (<ref>) is the solution proposed in this paper to estimate the local density of stars.As indicated by the left arrow symbol, we can obtain the set of values {σ_i } by starting with some arbitrary (positive) values for these quantities, and then by calculating updated values of σ_i by applying Eq. (<ref>).The convergence is usually reached within a few tens of iterations.Note that Eq. (<ref>) has a simple interpretation.Let us ignore for a moment the weight W, i.e. let us assume that all stars have the same weight.The sum in Eq. (<ref>) is carried out over all stars in the patch of the sky, but each star is counted only partially (i.e., contributes with a term between zero and unity in the sum): precisely, each star contributes by the computed probability that the star be associated with the i-th component.The way this probability is computed is actually a simple application of Bayes' theorem, where p_i(m_n - A_n) plays the role of the likelihood, σ_i(x⃗_n) is proportional to the prior that the star is of class i, and the sum over j in the denominator is proportional to the evidence.The result of the sum of these terms is divided by the by the result of the integral: this is a correcting factor that takes into account the fact that, because of extinction and incompleteness, we will miss a fraction of stars.Note also that Eq. (<ref>) can be also considered as a special case of a K-means soft clustering algorithm where the only unknown quantities are the classes σ_i <cit.>.Before proceeding, it is useful to recall the hypotheses of this algorithm and its strengths.First, we assume that we have some knowledge of the H-band luminosity function for the various populations of stars that are likely to be present in the field. In practice, we will generally use two probabilities, one for the field stars, and one for the YSOs. Second, we assume that we have measured the extinction A_n of each star: note that this is not the average extinction at the location of the star, which might be very different because of perspective effects: for example, a foreground star in front of a cloud would have A_n ≃ 0, while the average extinction would be significant. This way, the algorithm can directly account for foreground contamination: foreground stars will not be corrected in their counts, since the integral in the denominator of Eq. (<ref>) will evaluate to unity. Similarly, stars within molecular clouds will be corrected only for the amount of material that is really in front of them.Finally, we stress that the iterative procedure proposed here only find positive solutions for the values σ_i.Although reasonable, nevertheless this choice inevitably introduces biases in the results: for example, in a region where no YSO is present, because of errors we will still measure small positive values for the density of YSOs.However, numerical tests have shown that the bias amount is limited; moreover, a reduction of the bias is associated to a large increase in the scatter.Therefore, we will force the σ_i to be positive and use Eq. (<ref>) for the solution.The uncertainties associated to Eq. (<ref>) can be computed from the Fisher matrix.For practical applications, it is convenient to rewrite Eq. (<ref>) by replacing the integrals over m and ^2 x' with a sum over the observed objects.This leads to the approximated Fisher matrix expressionI_ij = ∑_n=1^N ω^2(x⃗ - x⃗_n) c^2(m_n) p_i(m_n - A_n) p_j( m_n - A_n)/σ^2(m_n, x⃗_n).In this equation, we take the spatial function ω to be normalized such that∫ω(x⃗')^2 x' = ∫ω^2(x⃗')^2 x' ;that is, in terms of W,ω(x⃗) = W(x⃗)/∫ W^2(x⃗')^2 x'.Table <ref> reports a few common choices for the spatial function ω(x⃗) and the corresponding weight function W(x⃗), both correctly normalized.As usual, the covariance matrix associated with the measurements of the densities σ_i can be computed from the inverse of the Fisher matrix, I^-1. § SIMULATIONS In order to test our method and verify its robustness we have performed a set of simulations.We have considered a small patch of the sky with two different stellar populations: field stars, with exponentially increasing number counts p_1(m) ∝ 10^α m, with α = 0.33mag^-1; and YSOs, with Gaussian number counts p_2(m) ∝exp(-(m-m_0)^2 / 2 s^2 ), with m_0 = 12mag and s = 1.65mag.We have distributed both populations randomly in the small patch of the sky considered and we have added to each star a random extinction drawn from the probability distribution p_A(A) ∝ A^-2, in the range A ∈ [0.1, 2.0] mag.This choice is meant to simulate the effects of differential extinction for objects within molecular clouds.Finally, we have imagined that our observations are complete up to 15mag, and that no stars can be observed beyond that magnitude: in other words, we have modeled the completeness function as a Heaviside function c(m) = H(15 - m).This way, our final catalog contains, for each star, the angular position in the sky, the H-band magnitude, and the measured extinction.Note that the parameters used here are chosen to simulate a real situation corresponding to the sample application of Sect. <ref>, that is the Orion molecular cloud complex observed with 2MASS.We have used these data in our algorithm, represented by Eqs. (<ref>) and (<ref>).As weight function W(x⃗) we have used a Gaussian, and we have chosen the angular units so that∫ W^2(x⃗')^2 x' = [ ∫ W(x⃗')^2 x' ]^2 .This choice guarantees that the effective area of the weight function is unity, i.e. the effective number of stars used for the analysis, in absence of extinction, would be just σ_1 + σ_2. In reality, the presence of the extinction reduces this number by a factor that depends on the relative ratio between σ_1 and σ_2 (typically, by ∼ 20%).We have then performed different simulations, with various choices for σ_1 and σ_2, to verify the ability of the algorithm to recover the input densities.Figures <ref>–<ref> show the observed distributions of σ̂_1 and σ̂_2, together with the predicted ones (Gaussian distributions centered on the true values of σ_1 and σ_2, with the variances predicted from the Fisher matrix I).In general, we can note a few points: * When one of the input densities is below ∼ 7, there is a clear excess of the corresponding quantity for small measured values.This is a consequence of the fact that the proposed algorithm only returns positive values for the densities.* Except for the point above, the predicted distributions reproduce very well the measured data.The agreement is particularly good when the input densities are large.* Overall, the total density σ_1 + σ_2 is better constrained than the individual densities σ_1,2.Figures <ref> and <ref> show the biases and the errors on the measured densities σ̂_1,2 for σ_1 = 10 and σ_2 varying in the interval [0, 20].We can see that there is a measurable bias for σ_2 < 5, while the results are essentially unbiased above this value.Correspondingly, we observe in the same range σ_2 ∈ [0, 5] a measured scatter in measured densities that is significantly smaller than what predicted from the Fisher matrix.For larger values of the input density the error estimate gets closer to the measured errors, but still remains slightly above.This is actually good, because implies a small overestimate of the error which will make the entire method more robust for cluster detections (that is, it will decrease the number of false positive).In summary, all these simulations confirm that the method works and that the error estimate is quite accurate. § SAMPLE APPLICATION: ORION We have applied the method proposed in this paper to 2MASS data of the Orion A and B molecular cloud complexes.The regions selected ensure that we can verify the reliability of the algorithm proposed here using some of the best studied objects in the sky.In particular, the populations of embedded clusters for both clouds have been the targets of extensive observational campaigns using ground-based,near-infrared <cit.>) and millimeter-wave <cit.> surveys as well as space-based, mid-infrared Spitzer Space Telescope <cit.>, and Chandra X-ray<cit.> surveys.Additionally, the distance of these clouds ensures that the 2MASS H-band data are, in absence of extinction, complete for YSOs: that is, the cluster HLF at the distance of Orion is essentially entirely within the 2MASS H-band limiting magnitude (∼15mag).Figures <ref> and <ref> show the density of fiducial YSOs measured in Orion A and B.These maps have been produced by our pipeline together with Nicer <cit.> and Nicest <cit.> extinction maps from the 2MASS Point Source Catalogue (seefor details on the data selection and extinction map construction).The algorithm has been run in conditions similar to the simulations described above: that is, we have used two different stellar populations, one associated with the background and characterized by exponential number counts, and one associated with the YSOs and characterized by Gaussian number counts (with parameters consistent with the H-band luminosity function of ).Using an angularly close control field we also measured the distribution of intrinsic colors of stars and the shape of the completeness function: the latter has been modeled using an complementary error function erfc, as described in Appendix <ref>. This choice makes it possible to use the entire 2MASS catalogue without any magnitude cut (which would increase the noise in the final data products).The maps produced have a pixel size of 1.5arcmin and a weight function W ∝ω in turn proportional to a Gaussian with 𝐹𝑊𝐻𝑀 = 3arcmin.We have used a relatively large beam in these maps to increase the sensitivity of our algorithm and to minimize the effects of the biases shown in the simulations described in Sect. <ref>, while still retaining in most situations the ability to distinguish different clusters (i.e., avoid confusion effects at the distance of Orion).Since we have at our disposal the covariance map of these measurements, we have assessed the reliability of each density peak in these figures.The red contours in the figures show the areas (larger than 2 pixels) where the local YSO density exceeds 1.5stars/pixel, corresponding approximately to a signal-to-noise ratio larger than 3.5: note how some regions in black in Fig. <ref> do not reach the threshold because of the large error associated with them (mostly due to the high extinction values there).Table <ref> shows the YSO clusters identified in the Orion A and B areas using our simple prescription, together with the most relevant parameters. In some cases we clearly see that angularly close clusters appear as a single contour in our maps: the simple procedure used here to define clusters, the relatively coarse resolution used, and the cluster morphology itself prevent us from deblending some close objects. An extreme case of this situation might be the ISF (the Integral Shaped Filament) cluster, where the limitations due to angular resolution would make it difficult to resolve and separate smaller clusters if they exist in such a very populous region. We note that the ISF cluster encompasses M42, the Trapezium and ONC clusters as well as an extended population of YSOs along the ISF.The radius R reported in the table corresponds to the radius of a circle that would occupy the same area as the identified cluster, i.e. to the connected region of the sky where the inferred density of YSOs exceeds the background by 3σ.At the estimated distance of Orion, 413pc <cit.>, 1' corresponds to 0.12pc: therefore, the clusters identified have radii spanning from ∼2.4pc to ∼0.15pc.The well known clusters in these clouds are correctly identified by our procedure.It is interesting to compare Table <ref> with clusters identified independently using much more secure data.Among the ones at our disposal, the recent catalog of embedded YSOs obtained by <cit.> using the Spitzer Space Telescope and the Chandra observatory is probably the most secure and complete: we will therefore focus on this catalog. Since our definition of a cluster is based on an somewhat arbitrary parameters (signal-to-noise threshold, minimum number of pixels, no correction for the stars missed at the boundaries), and since different, more-or-less, arbitrary parameters are also used by <cit.>, we find it more appropriate and fair to make a comparison after we first homogenize the data. Specifically, we take <cit.> YSO list and we make out of it a density map using a Gaussian kernel of the same size of the one used for our map. Figures <ref> and <ref> show the results obtained for Orion A and B, which clearly compares very well with our own maps, derived purely from the 2MASS point source catalog. The most relevant structures are present in both maps and have very similar shapes; the only differences are the noise present in our maps (however at a relatively low level), and the limited coverange of the Spitzer derived density maps.The qualitative similarity of these maps can be quantified if compare clusters identified in both maps using the same criteria. Table <ref> shows a list of clusters identified in the smoothed <cit.> maps using a fix density threshold (set to 1.5stars/pixel). In this table we compare the number of Spitzer YSOs with the number of YSOs predicted from the integral of the σ_YSOs over the area of each cluster as defined from Megeath et al. density map, together with the computed 1-σ error. It is clear that in almost all cases we find an excellent agreement, although in many cases our estimates are slightly larger than the ones by Megeath et al. We can speculate that this is due to the presence of class III YSO, which likely would be missed by Spitzer. Indeed, a comparison of the two panels of Fig. <ref> shows that the bottom panel, corresponding to our density map, has spatially more extended clusters than the top panel, corresponding to Megeath et al. density map.As discussed earlier on, our algorithm is a statistical one and works best when it is applied to a sizeable number of stars. However, we can also push it and associate to each single star a probability of being a YSO: to this purpose, for the n-th star we can computeP_i = σ_i(x⃗_n) p_i(m_n - A_n)/∑_j=1^L σ_j(x⃗_n) p_j(m_n - A_n).Note how this quantity resembles the term within the outer sum of Eq. (<ref>).Figure <ref> shows the distribution of P_YSO (that is, the distribution in the probabilities assigned to each object to be a YSO) for the <cit.> YSO candidates and for all the other objects. It is clear how all the other objects have P_YSO that is concentrated around zero, while the YSO candidates have a much broader distribution that extends to unity.For these latter objects the distribution, in addition to a substal peak at P_YSO = 1, shows a long tail up to small values of P_YSO: this is not unexpected, since our identification is only statistical (and therefore we cannot identify unambigously YSOs). Note also how the relatively low values of P_YSO for some genuine YSOs in our algorithm are compensated by the small tail in the distribution of field stars (this works because there are many more field stars than YSOs, a fact that is accounted for in the algorithm). §.§ Sensitivity to the distributed populationRecently, , have identified a rich and well-defined stellar population of about 2 000 objects, mostly M stars without extinction or infrared-excesses, as the low-mass counterpart to the Orion OB Ib subgroup (the Orion belt population). This low-mass population is not obviously clustered but instead appears to be distributed across ∼ 10 square degrees and the authors speculate that it could represent the evolved counterpart of a Orion nebula-like cluster.While more data is needed to test this scenario, it is clear that much can be learned about the origin of stellar OB associations and the dispersal of clusters into the Galactic field if one is able to trace in a robust manner the distribution of the slightly older and more expanded populations surrounding star-forming molecular clouds.We now investigate the extent to which the technique proposed here is suitable for detection of looser, more expanded distributions of young stars, in particular the low-mass counterpart to the Orion OB association presented in .For this purpose, we have built a lower resolution map of the region, employing a FWHM of 30arcmin.Figure <ref> shows that, surprisingly, we are well able to recover the stellar over-density of the Ori Ib population, and for the first time, the stellar over-density of the Ori Ia population. These over-densities are likely to be created by low-mass stars as 2MASS is still sensitive to the peak of the IMF for the putative distance and age of these subgroups. An analysis of the substructure seen in the distributed population visible in Figure <ref> above the noise pattern is beyond the scope of this paper, but will best addressed once Gaia parallaxes are generally available. Of relevance for this paper is that the ability of the method to trace the dispersed population from existing all sky data opens a new window on the unsolved problem of the origins of OB association and cluster dispersal in to the field. § CONCLUSIONSThe following items summarize the main results presented in this paper: * We have developed a new method to discover and characterize deeply embedded star clusters.* The method is able to statistically classify objects as field stars or YSOs and corrects for the effects of differential extinction.* We have provided expressions for the covariance of the inferred densities and we have validated both the method and the analytic expression for the covariance with a set of simple but realistic simulations.* We have applied the new method to 2MASS point sources observed in the Orion A and B and we have shown that we can identify and characterize well protostellar clusters in these regions, as well as detect much looser associations such as the OB 1a and 1b subgroups. Finally, we note that the method proposed here can be easily extended to multi-band observations by using suitable probability distributions p_i(m⃗) for various populations as a function of the magnitude vector m⃗.Its implementation would be essentially identical, with minor modifications to take into account the different effects of extinction on different bands.The natural use of such an extension would be in the context of techniques such as the one proposed by <cit.> which are able to recover the extinction from a complex analysis of multi-band data. This research has made use of the 2MASS archive, provided by NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.e Additionally, this research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. § IMPLEMENTATIONIn this section we consider a few details and analytical expressions useful to evaluate some of the expressions needed to implement the method proposed here. §.§ AlgorithmThe algorithm proposed in this paper is essentially an iterative application of Eq. (<ref>) to the set of densities {σ_i (x⃗) }. A few notes, however, are necessary to better implement the algorithm: * The method assumes that the extinction values in the H band are available for each star used. In our implementation these values are obtained through the use of the Nicer algorithm. Other techniques can be used, as long as the extinction is refereed to the single star and is not an average extinction at the location of the star. In our implementation, this requirement limits the method to stars for which, in addition to the H-band measurements, have at least another band photometry.* Since the various densities are spatially variable, one needs to repeat the iteration of Eq. (<ref>) to each point in the map.* Equation (<ref>) itself needs to be iterated a few times (typically around ten, in some cases a little more) before reaching a good convergence.This generally suggests the use of simple analytical models for the various functions involved in this equation (the probability distributions p_i(m) and the completeness function c(m), discussed below). §.§ H-band probability distributions and completeness functionIn this paper we have considered two useful models for the H-band luminosity function: the exponential distribution and the normal (Gaussian) one.We parametrize the exponential distribution asp(m) = k β^β m∝ 10^α m,where k is a normalization constant that will be found lather on and β = αln 10.The normal distribution is parametrized asp(m) = k/√(2 π s^2)^-(m - m_0)^2 / 2 s^2,where again k is a normalization constant to be investigated later.Finally, we model the completeness function c(m) in terms of the complementary error function . This can justified from an empirical point of view since the error function has a shape that resemble the completeness function.It is also reasonable from a statistical point of view if the photometric errors are Gaussian: in this case, one can suppose that the probability that an object be detected is an integral over a relevant part of the Gaussian. Specifically, we assume for c(m) the following functional form:c(m) = 1/2( m - m_c/√(2 σ^2_c)) ,where m_c is the 50% completeness limit and σ_c sets the width of the completeness function. Figure <ref> shows that our simple model for the control field HLF, the product of p(m) c(m) with p(m) following an exponential distribution and c(m) described in terms of an erfc, reproduces well the data.§.§ Statistical errors and completeness functionAs noted above, statistical errors can be included in our algorithm by convolving the probability distributions p_i with appropriate kernels.We expect to have two main sources for statistical errors: photometric errors in the H-band magnitude measurements and errors in the extinction measurements.While the formers are relatively simple to characterize (typically, they will be provided in the star catalogue), the latter are a combination of different sources of errors: the scatter in the intrinsic color of sources and the photometric errors in each band used.If extinctions are derived using the Nicer algorithm, as assumed here, then the variance associated to each extinction measurement can be computed from the expression(A) ≡σ^2_A = 1/k⃗^T C^-1k⃗,where, following the notation of <cit.>, we have called k⃗ is the reddening vector and C the combined covariance matrix of the observed colors (including both the intrinsic scatter in the color of unextinguished stars and the individual photometric errors).Looking again at Eq. (<ref>), i.e. the main equation representing our method, we see that in reality the H-band magnitudes and the associated extinction measurements typically enter the problem through the combination m - A; moreover, while in the numerator the combination is m_n - A_n and involves therefore measurements of both m and A for individual stars, in the denominator the combination is m - A_n, with m integration variable. Therefore, in presence of errors these two parts of the expression must be computed separately.Let us consider first the denominator. Since there the only argument of p_i with associated errors is A_n, the measured extinction for the n-th star, we just need to convolve the magnitude distribution with a Gaussian kernel with variance provided by Eq. (<ref>). In the case of the numerator the situation is slightly more complicated, because there we find the combination m_n - A_n.Since A_n is computed from the observed magnitude of each star (including of course the H-band magnitude), m_n and A_n are correlated.A simple calculation shows that the variance associated to the combination m_n - A_n is(m_n - A_n) = k_H^2 σ^2_A + σ^2_H [ 1 - 2 (b_J-H - b_H-K) k_H ] .In this expression the last term is related to the correlation between A and m and is equal to σ^2_H, the square of the photometric error on the H-band magnitude. The factors b_J-H and b_H-K are quantities that can be computed from Eq. (12) of <cit.>, while k_H is the reddening vector for the H band, i.e. the ratio A_H / A. §.§ Convolution integralsThe convolution of the luminosity functions (<ref>) and (<ref>) with Gaussian kernels representing the statistical errors take simple forms.Calling σ^2_p the variance of the convolution kernel, given depending on the term by Eq. (<ref>) or by Eq. (<ref>), and using the tilde to represent the convolved H-band luminosity functions, we find for the exponential distributionp̃(m) = k β^β m + σ^2_p β^2 / 2.Similarly, for the Gaussian model we findp̃(m) = k/√(2 π (s^2+σ^2_p))^-(m - m_0)^2 / 2 (s^2 + σ^2_p),§.§ Completeness integralsFinally, we need to compute the integral in the denominator of Eq. (<ref>), involving the H-band luminosity function p(m) and the completeness function c(m). A simple change of variable casts this integral (similar to a convolution) into a more convenient form:∫ c(m) p(m-A) m = ∫ c(m+A) p(m) m ≡ K(A) .In this equation we have called K(A) the result of the integral, retaining its explicit dependence on A. We now consider the derivative K' of K:K'(A) = ∫ c'(m+A) p(m) m ,and note that, since c is the error function, c' is a Gaussian function. Since Eq. (<ref>) is also (essentially) a convolution, this allows us to compute K'(A) using formulae similar to the ones of the previous section: as a result, we see that K'(A) remains either an exponential or a Gaussian, depending on the function form of p(m).Finally, we need to integrate back K'(A) to obtain K. The result we obtain from this procedure is, in case of an exponential distribution,K(A) = k ^β (m_c - A) + σ^2_cβ^2 / 2.For the Gaussian distribution the result contains again the error function:K(A) = k/2(m_0 - m_c + A/√(2(σ_c^2 + s^2))) .If statistical errors have to be taken into account, these expressions simply change intoK(A) = k ^β (m_c - A) + (σ^2_c + σ^2_p) β^2 / 2,andK(A) = k/2(m_0 - m_c + A/√(2(σ_c^2 + s^2 + [b]σ^2_p))) .aa | http://arxiv.org/abs/1707.08594v1 | {
"authors": [
"Marco Lombardi",
"Charles J. Lada",
"Joao Alves"
],
"categories": [
"astro-ph.IM",
"astro-ph.GA"
],
"primary_category": "astro-ph.IM",
"published": "20170726180836",
"title": "A new method to unveil embedded stellar clusters"
} |
1]Ittai Abraham 2]Shiri Chechik 3]Michael Elkin 3]Arnold Filtser 3]Ofer Neiman[1]VMWare. Email:[2]Tel-Aviv University. Email:[3]Ben-Gurion University of the Negev. Email:Ramsey Spanning Trees and their Applications [============================================ The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor <cit.> devised the so called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset S⊆ V of a given graph G=(V,E), such that there exists a spanning tree of G that has small stretch for S. Applied iteratively, thisprovides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these treesserves as a special type of spanner, a tree-padding spanner. Weuse this spanner to devise the first compact stateless routing scheme with O(1) routing decision time, andlabels which are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem, and provide a new deterministic construction. We prove that for every k, any n-point metric space has a subset S of size at least n^1-1/k which embeds into an ultrametric with distortion 8k. This results improves the best previous result of Mendel and Naor that obtained distortion 128k and required randomization. In addition, it provides the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k, any n-vertex graph G=(V,E) has a subset S of size at least n^1-1/k, and a spanning tree of G, that has stretch O(k loglog n) between any point in S and any point in V.empty§ INTRODUCTIONInspired by the algorithmic success of Ramsey Type Theorems for metric spaces, in this paper we study an analogue Ramsey Type Theorem in a graph setting. The classical Ramsey problem for metric spaces was introduced in <cit.>, and is concerned with finding "nice" structures in arbitrary metric spaces. Following <cit.>, <cit.> showed that every n-point metric (X,d) has a subset M⊆ X of size at least n^1-1/k that embeds into an ultrametric (and thus also into Hilbert space) with distortion at most O(k), for a parameter k≥ 1. In fact, they construct an ultrametric on X which has O(k) distortion for any pair in M× X. Additionally, <cit.> demonstrated the applicability of their techniques, which they denoted Ramsey Partitions, to approximate distance oracles and ranking problems.We introduce a new notion that we call Ramsey Spanning Trees, which is a natural extension of the metricRamsey problem to graphs. We show that every graph G=(V,E) with n vertices admits a subset M⊆ V of size at least n^1-1/k, such that there exists a spanning tree of G that has stretch O(kloglog n) on all pairs in M× V. (The extra factor of loglog n in the stretch comes from the state-of-the-art result of O(log nloglog n) for low stretch spanning trees <cit.>. It is quite plausible that if that result is improved to the optimal O(log n), then the stretch in our result would be only O(k).)By applying this result iteratively, we can obtain a small collection of trees so that each vertex has small stretch to all other vertices in at least one of the trees. Let (u,v,G) denote the shortest path distance in the graph G between the vertices u,v∈ V, then our main result is the following.Let G=(V,E) be a weighted graph on n vertices, and fix a parameter k≥ 1. There is a polynomial time deterministic algorithm that finds a collection T of k· n^1/k spanning trees of G, and a mapping :V→ T, such that for every u,v∈ V it holds that (v,u,(v))≤ O(kloglog n)·(v,u,G). A spanner H with stretch t for a graph G, is a sparse spanning subgraph satisfying (v,u,H)≤ t·(v,u,G). Spanners are a fundamental metric and graph-theoretic constructions; they are very well-studied <cit.>, and have numerous applications <cit.>. thm:main-col can be viewed as providing a spanner which is the union of k· n^1/k spanning trees, such that every vertex has a tree with low stretch paths to all other vertices. We call such a spanner a tree-padding spanner of order k · n^1/k. To the best of our knowledge, no previous construction of spanners can be viewed as a tree-padding spanner of order o(n).Until now even the following weaker question was open: does there exist a spanner which is a unionof a sublinear in n number of trees, such that every pair of vertices has a low stretch path in one of these trees.Having a single tree that provides good stretch for any pair containing the vertex v, suggest that routing messages to or from v could be done on this one tree. Our main application of Ramsey spanning trees is a compact routing scheme that has constant routing decision time and improved label size, see sec:apps for more details.Deterministic Ramsey Partitions. As a first step towards our main result, which is of interest in its own right, we provide a new deterministic Ramsey ultrametric construction. In particular, we show a polynomial time deterministic algorithm, that given an n-point metric space (X,d) and a parameter k≥ 1, finds a set M⊆ X of size at least n^1-1/k and an ultrametric (M,ρ) with distortion at most 8k-2. That is, for each v,u∈ M,d(v,u) ≤ρ(v,u) ≤ (8k-2) · d(v,u) .Even though our construction is deterministic, it has smaller distortion than all previous constructions. The first result of this flavor was by Mendel and Naor <cit.>, obtaining distortion of 128k. Belloch et. al <cit.>showed that the (randomized) algorithm of <cit.> constructs an ultrametric with distortion 18.5k (they also provided a near-linear time implementation of it). Naor and Tao<cit.> declared that 16k is obtainable but that a Ramsey partition with distortion better than 16k-2 seems not to be possible with their current techniques. Moreover, <cit.> mention as a drawback that their solution is randomized (while <cit.> is deterministic).An application of our improved deterministic Ramsey ultrametric construction is a new distance oracle that has the best space-stretch-query time tradeoff among deterministic distance oracles. See sec:apps below.Techniques. Our construction of Ramsey ultrametrics uses the by-now-standard deterministic ball growing approach, e.g. <cit.>.In this paper we provide tighter and more parameterized analysis of these multi-scale deterministic region growing techniques. Our improved analysis of the deterministic ball growing technique of <cit.> obtains a similar type of improvement as the one obtained by the analysis of Mendel and Naor <cit.> on the randomized partition technique of <cit.>.Our construction of Ramsey spanning trees is based on combining ideas from our Ramsey ultrametric construction, with the Petal Decomposition framework of <cit.>. The optimal multi-scale partitions of <cit.> cannot be used in this petal decomposition framework, so we must revert to partitions based on <cit.>, which induce an additional factor of O(loglog n) to the stretch. In addition, the refined properties required by the Ramsey partition make it very sensitive to constant factors (these constants can be ignored in the <cit.> analysis of the average stretch, say). In order to alleviate this issue, we consider two possible region growing schemes, and choose between them according to the densities of points that can still be included in M. One of these schemes is a standard one, while the other grows the region "backwards", in a sense that it charges the remaining graph, rather than the cluster being created, for the cost of making a cut. See subsec:createPetal for more details.§.§ Applications Distance Oracles. A distance oracle is a succinct data structure that (approximately) answers distance queries. A landmark result of <cit.> states that any metric (or graph) with n points has a distance oracle of size O(k· n^1+1/k),[We measure size in machine words, each words is Θ(log n) bits.] that can report any distance in O(k) time with stretch at most 2k-1. A deterministic variant with the same parameters was given by <cit.>, and this was the state-of-the-art for deterministic constructions. The oracle of <cit.> has improved size O(n^1+1/k) and O(1) query time, but larger stretch 128k. This oracle was key for subsequent improvements by <cit.>, the latter gave a randomized construction of an oracle with size O(n^1+1/k), query time O(1) and stretch 2k-1 (which is asymptotically optimal assuming Erdos' girth conjecture).Similarly to <cit.>, our deterministic construction of Ramsey ultrametrics can provide a deterministic construction of an approximate distance oracle. For any metric space on n points, and any k>1, 0<ϵ<1, there is an efficient deterministic construction of a distance oracle of size O(n^1+1/k), that has stretch 8 (1+)k and query time O(1/). This is the first deterministic construction of an approximate distance oracle with constant query time and small size O(n^1+1/k).Moreover, our oracle is an essential ingredient towards de-randomizing the recent distance oracles improvements <cit.>. Specifically, if we construct <cit.> by replacing the distance oracle of Mendel and Naor <cit.> by our deterministic version, and replacing the distance oracle of Thorup and Zwick <cit.> by the deterministic version of Roditty, Thorup, and Zwick <cit.>, we immediately get a deterministic distance oracle of O(k · n^1+1/k) size, 2k-1 stretch and O(1) query time. This is a strict improvement over<cit.>. In addition, our oracle can be viewed as a first step towards de-randomizing the <cit.> oracle. A summary of all the previous and current results can be found at tab:DistanceOracle in sec:table. Routing with Short Labels and Constant Decision Time. A routing scheme in a network is a mechanism that allows packets to be delivered from any node to any other node. The network is represented as a weighted undirected graph, and each node can forward incoming data by using local information stored at the node, often called a routing table, and the (short) packet's header. The routing scheme has two main phases: in the preprocessing phase, each node is assigned a routing table and a short label. In the routing phase, each node receiving a packet should make a local decision, based on its own routing table and the packet's header (which may contain the label of the destination, or a part of it), where to send the packet. The routing decision time is the time required for a node to make this local decision. The stretch of a routing scheme is the worst ratio between the length of a path on which a packet is routed, to the shortest possible path. A routing scheme is called stateless if the routing decision does not depend on the path traversed so far.The classical routing scheme of <cit.>, for a graph on n vertices and integer parameters k,b> 1, provides a scheme with routing tables of size O(k· b· n^1/k), labels of size (1+o(1))klog_bn, stretch 4k-5, and decision time O(1) (but the initial decision time is O(k)). The stretch was improved recently to roughly 3.68k by <cit.>, using a similar scheme as <cit.>. With thm:main-col, we devise a stateless compact routing scheme with very short labels, of size only (1+o(1))log_bn, and with constant decision time, while the stretch increases to O(kloglog n) (and with the same table size as <cit.>).We wish to point out that our construction of a routing scheme is simpler in some sense that those of <cit.>. In both constructions there is a collection of trees built in the preprocessing phase, such that every pair of vertices has a tree that guarantees small stretch. Routing is then done in that tree. In our construction there are few trees, so every vertex can store information about all of them, and in addition, every vertex v∈ V knows its home tree, and routing towards v from any other vertex on the tree (v) has small stretch. In particular, the header in our construction consists of only the label of the destination. In the <cit.> scheme, however, there are n trees, and a certain process is used to find the appropriate tree to route on, which increases the initial decision time, and also some information must be added to the header of the message after the tree is found. Finally, our routing scheme is stateless, as opposed to <cit.>. (We remark that using ideas from <cit.>, one can devise a stateless routing scheme based on <cit.>, but this scheme seems to suffer from larger header and decision time at each node.) Given a weighted graph G=(V,E) on n vertices and integer parameters k,b> 1, there is a stateless routing scheme with stretch O(kloglog n) that has routing tables of size O(k· b· n^1/k) and labels of size (1+o(1))log_bn. The decision time in each vertex is O(1).Observe that choosing parameters 2k and b=n^1/(2k) for thm:route yields a routing scheme with stretch O(kloglog n) that has tables of size O(k· n^1/k) and labels of size only O(k). Another interesting choice of parameters is b=2 and k=100log n/loglog n, this provides a scheme with stretch O(log n) that has tables of size O(log^1.01n) and labels of size O(log n). Compare this to the <cit.> scheme, which for stretch O(log n) has tables of size O(log n) and labels of size O(log^2 n). §.§ OrganizationIn sec:Ramsey-Partitions we present our deterministic Ramsey partitions, that are used for Ramsey ultrametrics and distance oracles. In sec:Ramsey-spanning we show the Ramsey spanning trees, and the application to routing. Each section can be read independently.§ PRELIMINARIESLet G=(V,E) be a weighted undirected graph. We assume that the minimal weight of an edge is 1. For any Y⊆ V and x,y∈ Y, denote by (x,y,Y) the shortest path distance in G[Y] (the graph induced on Y). For v∈ Y and r≥ 0 let B(v,r,Y)={u∈ Y|(v,u,Y)≤ r}, when Y=V we simply write B(v,r). We may sometimes abuse notation and not distinguish between a set of vertices and the graph induced by them.An ultrametric (Z,d) is a metric space satisfying a strong form of the triangle inequality, that is, for all x,y,z∈ Z, d(x,z)≤max{ d(x,y),d(y,z)}. The following definition is known to be an equivalent one (see <cit.>). An ultrametric is a metric space (Z,d) whose elements are the leaves of a rooted labeled tree T. Each z∈ T is associated with a label ℓ(z)≥0 such that if q∈ T is a descendant of z then ℓ(q)≤ℓ(z) and ℓ(q)=0 iff q is a leaf. The distance between leaves z,q∈ Z is defined as d_T(z,q)=ℓ((z,q)) where (z,q) is the least common ancestor of z and q in T.§ RAMSEY PARTITIONS Consider an undirected weighted graph G=(V,E), and a parameter k≥ 1. Let D be the diameter of the graph and let = ⌈log(D+1)⌉. Let ρ_i = 2^i/(4k). We start by presenting a construction for a collection S of cluster partial partitions X_i satisfying the following key properties. [(G,U,k)-Fully Padded Strong Diameter Hierarchical Partial Partition] Given a graph G=(V,E), an index k and a set of nodes U⊆ V, a (G,U,k)-Fully Padded Strong Diameter Hierarchical Partial Partition () is a collection X_i of subsets of nodes X ∈ X_i with a center r(X) for 0 ≤ i ≤ with the following properties. * (i) For every 0 ≤ i ≤, the subsets in X_i are disjoint, namely, for every two different subsets X,X' ∈ X_i, X ∩ X' = ∅. * (ii) For every 0 ≤ i < and every subset X ∈ X_i, there exists a subset X' ∈ X_i+1 such that X ⊆ X'. * (iii) For every 0 ≤ i ≤ and every X∈ X_i and every v∈ X, (v,r(X),X) < 2^i. * (iv) There exists a set V̂⊆ U such that |V̂| ≥ |U|^1-1/k and for every v ∈V̂ and every i, there exists a subset X∈ X_i such that B(v,ρ_i) ⊆ X. For a node v and index i, we say that v is i-padded in S, if there exists a subset X ∈ X_i such that B(v,ρ_i) ⊆ X. We would like to maximize the number of nodes that are padded on all levels.Fully Padded Strong Diameter Hierarchical Partial Partition Construction:Let us now turn to the construction of the collection S of cluster partial partitions X_i given a set U.In the beginning of the algorithm, all nodes in U are set as marked. The algorithm iteratively unmarks some of the nodes. The nodes that will remain marked by the end of the process are the nodes that are padded on all levels. For a given graph H, let B_M(v,d,H) (M stands for marked) be the set of marked nodes at distance at most d from v in H.For a subgraph G' and a node v ∈ V(G'), let Z_i(v,G') = |B_M(v,2^i,G')|/|B_M(v,2^i-1,G')|. The construction given in alg-strong-ramsey. Let X(v) be a set constructed in line:X(v)def of alg-strong-ramsey, when partitioning X∈ X_i+1. We say that X is the parent of X(v). Let H(X(v)) denote the graph in alg-strong-ramsey just before X(v) was constructed (note that this is a graph induced on a subset of X). We say that B(v,2^i-1 + 2j(v) ρ_i, H(X(v))) is the interior part of X(v) . We also say that the set B_M(v,2^i-1 + 2(j(v)+1) ρ_i, H(X(v))) is the responsibility set of X(v), hereafter referred to as (X(v)). Note that every node u that is still marked after the processing of X is completed, belongs to exactly one set (X(v)) for X(v) ∈ X_i.We now define by induction the term i-surviving for 0≤ i ≤: All nodes in U are -surviving. We say that a node is i-surviving if it is (i+1)-surviving and it belongs to the interior part of some subset in X_i. Our goal in the analysis is to show that many nodes are 0-surviving, which is exactly the set V̂. For a subset X ∈ X_i, let (X) be the set of nodes in X that are 0-surviving. We now turn to the analysis.The next auxiliary claim helps in showing that property (iii) holds. Consider a subset X ∈ X_i centered at some node v = r(X). The index j(v) defined in line:RG of alg-strong-ramsey, satisfies j(v) ≤ k-1 Seeking contradiction, assume that for every 0≤ j'≤ k-1, |B_M(v,2^i-1 + 2(j'+1) ρ_i, H(X))| > |B_M(v,2^i-1 + 2j' ρ_i, H(X))| · |Z_i(v,H(X))|^1/k. Then applying this for j'=k-1,k-2,…,0 we get |B_M(v,2^i,H(X))| =|B_M(v,2^i-1+2kρ_i,H(X))| >|B_M(v,2^i-1+2(k-1)ρ_i,H(X))|·|Z_i(v,H(X))|^1/k> …>|B_M(v,2^i-1,H(X))|·|Z_i(v,H(X))|^k/k =|B_M(v,2^i,H(X))| , a contradiction.The next lemma shows that the collection S satisfies properties (i)-(iii). S satisfy properties (i)-(iii). Property (i) is straightforward from line:RemoveX(v). Property (ii) holds as each X(v) is selected from the graph H(X(v)), which is an induced graph over a subset of X (the parent of X(v)). Finally, property (iii) follows from claim:j-v, as the radius of X(v) is bounded by 2^i-1+(2(k-1)+1)ρ_i<2^i-1+2kρ_i=2^i. Next we argue that if a vertex is 0-surviving, then it is padded in all the levels. Suppose x∈(V), then x is padded in all the levels. Fix some x∈(V). To prove that x is i-padded, we assume inductively that x is j-padded for all i< j≤ (the base case i= follows as B(x,ρ_)⊆ V). Let X∈𝒳_i+1 such that x∈ X. Set B=B(x,ρ_i). By the induction hypothesis B⊆ B(x,ρ_i+1)⊆ X. Let X(v)∈𝒳_i such that x∈ X(v). First we argue that B⊆ H(X(v)). Seeking contradiction, let X(v')∈𝒳_i be the first created cluster such that there is u∈ B∩ X(v'). By the minimality of v', it follows that B⊆ H(X(v')). Thus (v,u,H(X(v')))=(v,u,G)≤ρ_i. Let j(v') the index chosen in line:RG of alg-strong-ramsey. Then X(v')=B(v',2^i-1+(2j(v')+1)ρ_i,H(X(v'))). Using the triangle inequality (x,v',H(X(v')))≤2^i-1+(2j(v')+2)ρ_i. Therefore x was unmarked in line:unMark, a contradiction. It remains to show that B⊆ X(v). Set j(v) s.t. X(v)=B(v,2^i-1+(2j(v)+1)ρ_i,H(X(v))). As x is part of the interior of X(v), it holds that (x,v,H(X(v)))≤ 2^i-1+2j(v)ρ_i. Therefore B⊆ B(v,2^i-1+(2j(v)+1)ρ_i,H(X(v)))=X(v).The next lemma bounds the number of surviving nodes. For every index i and every subset X = X(v) ∈ X_i, the 0-surviving nodes satisfy |(X)|≥|M_i(X)|/|B_M(r(X),2^i-1,H(X))|^1/k We prove the lemma by induction on i. Consider first the base case where X = X(v) ∈ X_0. As the subsets in X_0 contain a single node (their radius is less than 1), it holds that |(X)|=1= 1/1=|M_0(X)|/|B_M(r(X),1/2,H(X))|^1/k (observe that each cluster in X_i has at least 1 marked node, for all 0≤ i≤). Assume the claim holds for every subset X' ∈ X_i, and consider X ∈ X_i+1. Let v =r(X). Consider the children X_1,...,X_j' of X. For every 1≤ h ≤ j', set v_h=r(X_h). Note that by definition of j(v_h) in line:RG of alg-strong-ramsey and by the construction of X_h in line:X(v)def, we have that |M_i(X_h)|≥ |(X_h)|/Z_i(v_h,H(X_h))^1/k. Moreover, by the induction hypothesis we have that |(X_h)|≥|M_i(X_h)|/|B_M(v_h,2^i-1,H(X_h))|^1/k for every h. We claim that |B_M(v,2^i,H(X))|≥ |B_M(v_h,2^i,H(X_h))| for every 1≤ h ≤ j'. To see this, note that v is the node with maximal |B_M(v,2^i,H(X))|, hence |B_M(v,2^i,H(X))| ≥ |B_M(v_h,2^i,H(X))|. In addition, note that H(X_h) ⊆ H(X), hence |B_M(v_h,2^i,H(X))| ≥|B_M(v_h,2^i,H(X_h))|. It follows that |B_M(v,2^i,H(X))|≥ |B_M(v_h,2^i,H(X_h))|. Therefore the number of 0-surviving nodes in X_h is at least |(X_h)| ≥ |M_i(X_h)|/|B_M(v_h,2^i-1,H(X_h))|^1/k≥|(X_h)|/|Z_i(v_h,H(X_h))|^1/k/|B_M(v_h,2^i-1,H(X_h))|^1/k= |(X_h)|/|B_M(v_h,2^i,H(X_h))|^1/k·|B_M(v_h,2^i-1,H(X_h))|^1/k/|B_M(v_h,2^i-1,H(X_h))|^1/k≥ |(X_h)|/|B_M(v,2^i,H(X))|^1/k , we conclude that |(X)|=∑_h=1^j'|(X_h)|≥∑_h=1^j'|(X_h)|/|B_M(v,2^i,H(X))|^1/k=|M_i+1(X)|/|B_M(v,2^i,H(X))|^1/k .Using lem:frac-survival on V with i=, combined with lem:SurPaddedAll, implies property (iv). The number of marked nodesV̂ by the end of alg-strong-ramsey is at least |U|^1-1/k. Moreover, for every v ∈V̂ and every i, there exists a subset X∈ X_i such that B(v,ρ_i) ⊆ X. For every n-point metric space and k≥ 1, there exists a subset of size n^1-1/k that can be embedded into an ultrametric with distortion 8k-2. The hierarchical partial partition S={ X_i} naturally induce an ultrametric on V̂. The singleton sets of V̂ are the leaves, and each X∈ X_i for 0≤ i< will be a tree-node which is connected to its parent. Each set in X_i for i≥ 1 will receive the label 2^i+1(1-1/(4k)), while the leaves in X_0 receive the label 0 (recall def:ultra). Consider two nodes u,v ∈V̂. Assume the least common ancestor of u,v is X∈ X_i, for some 1≤ i≤. Hence (u,v,G) ≤ 2 · (2^i -2^i/4k) (they are both in the interior of X - a ball with radius ≤ 2^i-2ρ_i). Since this is the label of X, we conclude that distances in the ultrametric are no smaller than those in G. Next we argue that distances increase by a factor of at most 8k-2. Consider any u,v as above, and seeking a contradiction, assume that (v,u,G)<2^i+1(1-1/(4k))/8k-2 = ρ_i. Let P be the shortest path from v to u in G. As v was padded in X, necessarily P⊆ X. Consider the first time a vertex z∈ P was added to a cluster X'∈𝒳_i-1, then P⊆ H(X'). Let j be such that X'= B(r(X'),2^i-2 + (2j+1)ρ_i-1 , H(X')). Since P is a shortest path, at least one of u,v must be within distance less than ρ_i/2=ρ_i-1 from z, w.l.o.g assume (v,z,H(X'))≤ρ_i-1. This implies that v∈(X')=B(r(X'),2^i-2 + (2j+2)ρ_i-1 , H(X')), and as v is marked it must lie in the interior of X', which is B(r(X'),2^i-2 + 2jρ_i-1 , H(X')). But then the triangle inequality yields that u∈(X')∖ X', which is a contradiction to the fact that u∈V̂.§.§ Distance OracleWe show a distance oracle with O(n^1+1/k) size, (8+)k worst case stretch and O(1/) query time (which is O(1) for any fixed epsilon).For simplicity we start by showing a construction with O(k · n^1+1/k) size, 16k stretch and O(1) query time. We will later see how to reduce the size and stretch. Let D be the diameter of the graph.Our distance oracle is constructed as follows. enumerate* Set UV.* Construct the collection of cluster-partial-partitions S(U) on the graph G and the set U. Remove from U the set of nodes V̂ that were padded in all levels in S(U). Continue this process as long as U ≠∅.* Let M be the set of all collections S(U) that were constructed by this process.* For every S∈ M construct a cluster X( S) as follows.* Let S = { X_0,..., X_}. All nodes V are the leaves (recall that only nodes in V̂ are in X_0). For every index i and every set X ∈ X_i, add an intermediate node. Connect X to its parent set. Connect every node v ∈ V to the set X ∈ X_i of minimal i such that v ∈ X. This completes the construction of X( S).* In addition, we preprocess X( S) so that least common ancestor (LCA) queries could be done in constant time. In order to do that we invoke any scheme that takes a tree and preprocess it in linear time so that LCA queries can be answered in constant time (see <cit.>).* Finally, note that for every node v there exists a collection S∈ M, where v is padded in all levels. Denote this collection by (v). enumerate The query phase is done as follows. Given two nodes s and t. Let S = (s) and let S = { X_i | 1≤ i ≤}. Find the least common ancestor of s and t in X( S) and let i be its level. Namely, let μ∈ X( S) be the least common ancestor of s and t and let X be the cluster μ represents, the index i is the index such that X ∈ X_i. Return 2^i+1 (denoted by (s,t)). (s,t) ≤(s,t) < 16k·(s,t). Let d = (s,t,G) and let j be theindex such that 2^j-1<d ≤ 2^j. Let X_i ∈ X_i ∈ S=(s) be the i level subset such that s ∈ X_i. Recall that s is padded in all the subsets X_i for 0≤ i ≤. Note that X_i has diameter smaller than 2· 2^i (follows from property (iii)). Therefore t∈ X_i implies that (s,t,G)<2·2^i=2^i+1. In particular, t ∉ X_i for every i < j-1. Hence the least common ancestor is at least at level j-1. Hence the minimal distance returned by the algorithm is(s,t) ≥ 2^j ≥ d. It remains to show that (s,t) ≤ 16k· d. Let i be the level of s and t's least common ancestor. Note that t ∉ X_i-1. Also recall that s is padded in X_i-1 and thus B(s,ρ_i-1) ⊆ X_i-1, which implies d≥ρ_i-1 = 2^i-1/(4k) = (s,t)/(16k).Let U_i be the set U after constructing the first i collections. Note that |U_i+1| ≤ |U_i| - |U_i|^1-1/k. By resolving this recurrence relation one can show that the number of phases is O(k n^1/k) (see <cit.>) . Notice that for every S∈ M, T( S) is of size O(n). Hence, the size of our data structure is O(k n^1+1/k). Reducing the size of the data structure: We now show how to reduce the size of the data structure to O(n^1+1/k). We only outline the modifications to the algorithm and the analysis and omit the full details.Here we will use only the metric structure of the graph G, while ignoring the structure induced by the edges. Specifically, in line (2) of the algorithm, instead of the graph G, we will use the graph G_U which is the complete graph over U, where the weight of each edge {u,v} is equal to (u,v,G). This change allows us to remove the nodes V̂ from G_U after each iteration. The query algorithm, given two nodes s and t is as follows. Let S_s = (s) and S_t = (t), and assume w.l.o.g. that S_s was constructed before S_t. Find the least common ancestor of s and t in X ( S_s) and let i be its level. Return 2^i+1.Following the analysis of the previous construction we can show that properties (i)-(iv) are satisfied and that the stretch is bound by 16k.The size of the data structure is bounded by O(n^1+1/k) (see <cit.>, Lemma 4.2. Reducing the stretch to 8(1+)k: We now explain how to reduce the stretch to 8(1+)k. Note that we lose a factor of 2 in the stretch since we look on distances in multiplies of two. Recall that in the algorithm, for a pair of vertices s,t at distance d, we looked on the minimal index j such that d ≤ 2^j. It may happen that d is only slightly larger than 2^j-1. Note that by just considering all distances (1+)^i rather than all distances 2^i, we get that the number of nodes that are padded in all levels is a fraction of 1/n^1/( k) rather than 1/n^1/k, which is dissatisfying. So instead we construct O(1/) different copies of our data structure, one for each 1+ ℓ for 0≤ℓ < 1/. In the copy ℓ of the data structure we consider distances (1+ℓ)2^i for every 0≤ i ≤. Specifically, i-clusters have radius bounded by (1+ℓ)2^i, while the padding parameter is ρ_ℓ,i=(1+ℓ)ρ_i. We denote by _ℓ(s) the collection 𝒮, created for the ℓ's distance oracle, where s is padded in all levels. The distance estimation of the ℓ's copy (denoted by _ℓ(s,t)), will be (1+(ℓ+1))2^i_ℓ, where i_ℓ is the level of the least common ancestor of s and t in _ℓ(s).Set d=(s,t). For every ℓ, we haved>ρ_ℓ,i_ℓ-1=(1+ℓ)2^i_ℓ-1/4k=(1+ℓ)/(1+(ℓ+1))·_ℓ(s,t)/8k≥_ℓ(s,t)/8(1+)k .From the other hand, there exist indices ℓ',j such that(1+ℓ' )2^j-1 < d≤ (1+(ℓ'+1) )2^j-1. Following the analysis above, as t does not separated from s at level i_ℓ', it holds that i_ℓ'≥ j-1.Therefore_ℓ'(s,t)=(1+(ℓ+1))2^i_ℓ'≥(1+(ℓ+1))2^j-1≥ d .In the query phase we iterate over all O(1/) copies, invoke the query algorithm in each copy and return the largest distance. By equations (<ref>) and (<ref>), the stretch is 8(1+)k rather than 16k. The query time is O(1/) which is O(1) for every fixed .§ RAMSEY SPANNING TREES In this section we describe the construction of Ramsey spanning trees, each tree will be built using the petal decomposition framework of <cit.>. Roughly speaking, the petal decomposition is an iterative method to build a spanning tree of a given graph. In each level, the current graph is partitioned into smaller diameter pieces, called petals, and a single central piece, which are then connected by edges in a tree structure. Each of the petals is a ball in a certain metric. The main advantage of this framework is that it produces a spanning tree whose diameter is proportional to the diameter of the graph, while allowing large freedom for the choice of radii of the petals. Specifically, if the graph diameter is Δ, the spanning tree diameter will be O(Δ), and each radius can be chosen in an interval of length ≈Δ. For the specific choice of radii that will ensure a sufficient number of vertices are fully padded, we use a region growing technique based on ideas from <cit.>. §.§ Preliminaries For subset S⊆ G and a root vertex r∈ S, the radius of S w.r.t r,Δ_r(S), is the minimal Δ such that B(r,Δ,S)=S. (If for every Δ, B(r,Δ,S)≠ S, (this can happen iff S is not connected) we say that Δ_r(S)=∞.) When the root r is clear from context or is not relevant, we will omit it. Given a graph G=(V,E), a Strong Diameter Hierarchical Partition (SDHP) is a collection {𝒜_i}_i∈[Φ] of partitions of V, where each cluster in each partition has a root, such that: * 𝒜_Φ={V} (i.e., the first partition is the trivial one).* 𝒜_1={{v}_v∈ V} (i.e., in the last partition every cluster is a singleton). * For every 1≤ i<Φ and A∈𝒜_i, there is A'∈𝒜_i+1 such that A⊆ A' (i.e., 𝒜_i is a refinement of 𝒜_i+1). Moreover, Δ(A)≤Δ(A'). Given a graph G=(V,E) and a subset A⊆ V, we say that a vertex y∈ A is ρ-padded by a subset A'⊆ A (w.r.t A) if B(y,Δ(A)/ρ,G)⊆ A'. See fig:paddedVertex for illustration. We say that x∈ V is ρ-fully-padded in the SDHP {𝒜_i}_i∈[Φ], if for every 2≤ i≤Φ and A∈𝒜_i such that x∈ A, there exists A'∈𝒜_i-1 such that x is ρ-padded by A' (w.r.t A). §.§ Petal Decomposition Here we will give a concise description of the Petal decomposition algorithm, focusing on the main properties we will use. For proofs and further details we refer to <cit.>.The(see alg:h-petal) is a recursive algorithm. The input is G[X] (a graph G=(V,E) induced over a set of vertices X⊆ V), a center x_0∈ X, a target t∈ X, and the radius Δ=Δ_x_0(X).[Rather than inferring Δ=Δ_x_0(X) from G[X] and x_0 as in <cit.>, we can think of Δ as part of the input. We shall allow any Δ≥Δ_x_0(X). We stress that in fact in the algorithm we always use Δ_x_0(X), and consider this degree of freedom only in the analysis.]The algorithm invokesto partition X into X_0,X_1,…,X_s (for some integer s), and also provides a set of edges {(x_1,y_1),…,(x_s,y_s)} and targets t_0,t_1,…,t_s. Thealgorithm now recurses on each (G[X_j],x_j,t_j,Δ_x_j(X_j)) for 0≤ j≤ s, to get trees {T_j}_0≤ j≤ s, which are then connected by the edges {(x_j,y_j)}_1≤ j≤ s to form a spanning tree T for G[X] (the recursion ends when X_j is a singleton). See fig:HPDExample for illustration. Next we describe theprocedure, see alg:petal-d. Initially it sets Y_0=X, and for j=1,2,…,s it carves out the petal X_j from the graph induced on Y_j-1, and sets Y_j=Y_j-1∖ X_j (in order to control the radius increase, the first petal is cut using different parameters). The definition of petal guarantees that Δ_x_0(Y_j) is non-increasing (see <cit.>), and when at step s it becomes at most 3Δ/4, define X_0=Y_s and then theroutine ends. In carving of the petal X_j⊆ Y_j-1, the algorithm chooses an arbitrary target t_j∈ Y_j-1 (at distance at least 3Δ/4 from x_0) and a range [lo,hi] of size hi-lo∈{Δ/8,Δ/4} which are provided to the sub-routine .(One may notice that in line:edgeWeightReduce of theprocedure, the weight of some edges is changed by a factor of 2. This can happen at most once for every edge throughout the execution, thus it may affect the padding parameter by a factor of at most 2. This re-weighting is ignored for simplicity.) Bothandare essentially the algorithms that appeared in <cit.>. The main difference from their work lies in theprocedure, depicted in alg-pick-rad. It carefully selects a radius r∈[lo,hi], which determines the petal X_j together with a connecting edge (x_j,y_j)∈ E, where x_j∈ X_j is the center of X_j and y_j∈ Y_j. It is important to note that the target t_0∈ X_0 of the central cluster X_0 is determined during the creation of the first petal X_1. It uses an alternative metric on the graph, known as cone-metric:Given a graph G=(V,E), a subset X⊆ V and points x,y∈ X, define the cone-metric ρ=ρ(X,x,y):X^2→ℝ^+ as ρ(u,v)=|(d_X(x,u)-d_X(y,u))-(d_X(x,v)-d_X(y,v))|.(The cone-metric is in fact a pseudo-metric, i.e., distances between distinct points are allowed to be 0.)The ball B_(X,ρ)(y,r) in the cone-metric ρ=ρ(X,x,y), contains all vertices u whose shortest path to x is increased (additively) by at most r if forced to go through y.In thealgorithm, while working in a subgraph G[Y] with two specified vertices: a center x_0 and a target t, we define W_r(Y,x_0,t)=⋃_p∈ P_x_0t: d_Y(p,t)≤ rB_(Y,ρ(Y,x_0,p))(p,r-d_Y(p,t)/2) which is union of balls in the cone-metric, where any vertex p in the shortest path from x_0 to t of distance at most r from t is a center of a ball with radius r-d_Y(p,t)/2.The following facts are from <cit.>. Runningon input (G[X],x_0,t,Δ_x_0(X)) will provide a spanning tree T satisfying Δ_x_0(T)≤ 4Δ_x_0(X). If thepartitions X with center x_0 into X_0,…,X_s with centers x_0,…,x_s, then for any 0≤ j≤ s we have Δ_x_j(X_j)≤ (3/4)·Δ_x_0(X). We will need the following observation. Roughly speaking, it says that when thealgorithm is carving out X_j+1, it is oblivious to the past petals X_1,…,X_j, edges and targets – it only cares about Y_j and the original diameter Δ.Assume thaton input (G[X],x_0,t,Δ_x_0(X)) returns as output (X_0,X_1,…,X_s,{ y_1,x_1} ,…,{ y_s,x_s} ,t_0,…,t_s). Then runningon input (G[Y_j],x_0,t_0,Δ_x_0(X)) will output (X_0,X_j+1…,X_s,{ y_j+1,x_j+1} ,…,{ y_s,x_s} ,t_0,t_j+1…,t_s).§.§ Choosing a RadiusFix some 1≤ j≤ s, and consider carving the petal X_j from the graph induced on Y=Y_j-1. While the algorithm of <cit.> described a specific way to choose the radius, we require a somewhat more refined choice. The properties of the petal decomposition described above (in subsec:PetalDecompDesc) together with FactPetal3/4Radius and FactPetalTreeRadiusBound4,hold for any radius picked from a given interval. We will now describe the method to select a radius that suits our needs. Thealgorithm provides an interval [lo,hi] of size at least Δ/8, and for each r∈[lo,hi] let W_r(Y,x_0,t)⊆ Y denote the petal of radius r (usually we will omit (Y,x_0,t)). The following fact demonstrates that petals are similar to balls. For every y∈ W_r and l≥ 0, B(y,l,Y)⊆ W_r+4l.Note that FctW_rProp implies that W_r is monotone in r, i.e., for r≤ r' it holds that W_r⊆ W_r'.Our algorithm will maintain a set of marked vertices M⊆ V, and will update it in any petal creation. Roughly speaking, the marked vertices are those that are fully padded in the (partial) hierarchical partition generated so far by the algorithm. If initially |M|=m, we want that at the end of the process at least m^1-1/k vertices will remain marked. In the partition of X to X_0,…,X_s, some of the marked vertices will be ρ-padded by a petal X_j (w.r.t. X), and some of the others will be unmarked, by the following rule.FctW_rProp implies that if we choose a radius r when creating some petal X_j=W_r, then all marked vertices in W_r-4Δ/ρ will be ρ-padded by X_j, and thus remain marked. All the marked vertices in W_r+4Δ/ρ∖ W_r-4Δ/ρ are considered unmarked from now on, since their Δ/ρ ball may intersect more than one cluster in the current partition (note that some of these vertices can be outside X_j). Our algorithm to select a radius is based on region growing techniques, similar to those in alg-strong-ramsey, but rather more involved.Since in the petal decomposition framework we cannot pick as center a vertex maximizing the "small ball", we first choose an appropriate range that mimics that choice (see e.g. line:a-b in the algorithm below) – this is the reason for the extra factor of loglog n. The basic idea in region growing is to charge the number of marked vertices whose ball is cut by the partition (those in W_r+4Δ/ρ∖ W_r-4Δ/ρ), to those that are saved (in W_r-4Δ/ρ). In our setting we are very sensitive to constant factors in this charging scheme (as opposed to the average stretch considered in <cit.>), because these constants are multiplied throughout the recursion. In particular, we must avoid a range in [lo,hi] that contains more than half of the marked vertices, a constraint which did not exist in previous manifestation of this region growing scheme. To this end, if the first half [lo,mid] (with mid=(hi+lo)/2) is not suitable, we must "cut backwards" in the regime [mid,hi], and charge the marked vertices that were removed from M to the remaining graph Y_j, rather than to those saved in the created cluster X_j.[We remark that paying a factor of loglog(nW), where W is the maximum edge weight, might have simplified the algorithm slightly.] §.§ Proof of CorrectnessLet z∈ V be an arbitrary vertex, given a set M⊆ V, let T be the tree returned by callingon (G[V],z,z,Δ_z(V)) and marked vertices M. There is a natural Strong Diameter Hierarchical Partition (SDHP) X={𝒳_i}_i=1^Φ associated with the tree T, where 𝒳_i consists of all the clusters created in level Φ-i of the recursion (and 𝒳_Φ={V}). By FactPetal3/4Radius, the radius is always non-increasing. Hence X is indeed a SDHP. Denote by (M)⊆ M the set of vertices that remained marked throughout the execution of .Suppose x∈(M), then x is ρ-fully-padded in X. Fix any 2≤ i≤Φ. Let X∈ X_i be the cluster containing x in the (Φ-i)-th level of the recursion with Δ=Δ(X). Assume X was partitioned byinto X_0,…,X_s, and let X_j⊆ X be the cluster containing x∈ X_j. Assuming (inductively) that x was ρ-padded by X, we need to show that it is also ρ-padded by X_j, that is, B=B(x,Δ/ρ,G)⊆ X_j. (Note that B⊆ X since the radii are non-increasing, so x is padded in all higher levels.)First we argue that none of the petals X_1,…,X_j-1 intersects B. Seeking contradiction, assume it is not the case, and let 1≤ j'<j be the minimal such that there exists y∈ X_j'∩ B. By the minimality of j' it follows that B⊆ Y'=Y_j'-1, and thus (x,y,Y')=(x,y,G)≤Δ/ρ. Let r' be the radius chosen when creating the petal X_j'=W'_r', and FctW_rProp implies thatx∈ B(y,Δ/ρ,Y')⊆ W'_r'+4Δ/ρ=W'_r'+R/(4Lk) ,where we recall that Δ=8R and ρ=2^7Lk. This is a contradiction to the fact that x∈(X): clearly x∉ W'_r'-R/(4Lk) since it is not included in X_j'=W'_r' (and using the monotonicity of W'_r), so it should have been removed from M when creating X_j' (in line:M of the algorithm).For the case j=0 the same reasoning shows B does not intersect any petal X_1,… ,X_s and we are done. For j>0, it remains to show that B⊆ X_j, but this follows by a similar calculation. Let r be the radius chosen for creating the petal X_j=W_r, and Y=Y_j-1. We have B⊆ Y, and since x∈(X) it must be that x∈ W_r-R/(4Lk). Again by FctW_rProp we haveB=B(x,Δ/ρ,G)=B(x,Δ/ρ,Y)⊆ W_r-R/(4Lk)+4Δ/ρ = W_r=X_j . Consider a vertex v∈(M), then for every u∈ V, (v,u,T)≤ 8ρ·(v,u,G). Let X={𝒳_i}_i=1^Φ be the SDHP associated with T, and for 1≤ i≤Φ let A_i∈ X_i be the cluster containing v. Take the minimal 2≤ i≤Φ such that u∈ A_i (there exists such an i since u∈ A_Φ=V and u∉ A_1={v}). By lem:active-padded v is ρ-fully-padded, so we have that B(v,Δ/ρ,G)⊆ A_i-1, where Δ=Δ(A_i). But as u∉ A_i-1, it must be that (u,v,G)>Δ/ρ. Since both u,v∈ A_i, FactPetalTreeRadiusBound4 implies that the radius of the tree created for A_i is at most 4Δ, so that (u,v,T)≤ 2· 4Δ≤ 8ρ·(u,v,G) . |(M)|≥ |M|^1-1/k.We prove by induction on |X| that if a cluster X∈ X_i (for some 1≤ i≤Φ) has currently m marked vertices, then at the end of the process at least m^1-1/k of them will remain marked.The base case when X is a singleton is trivial. For the inductive step, assume we callon (G[X],x_0,t,Δ) with Δ≥Δ_x_0(X) and the current marked vertices M̂. Assume that thealgorithm does a non-trivial partition of X to X_0,…,X_s (if it is the case that all vertices are sufficiently close to x_0, then no petals will be created, and the will simply recurse on (G[X],x_0,t,Δ_x_0(X)), so we can ignore this case).Denote by M_j the marked vertices that remain in X_j (just before the recursive call on X_j), and recall that (X_j) is the set of vertices of M_j that remain marked until the end of thealgorithm. Then (X)=⋃_0≤ j≤ s(X_j), and we want to prove that |(X)|≥ m^1-1/k.Let X_1=W_r be the first petal created by thealgorithm, and Y_1=X∖ X_1. Denote by (X_1)=W_r+R/(4Lk)∩M̂ the responsibility set for X_1 (i.e. the marked vertices that are either in M_1 or were removed from M̂ when X_1 was created). Define M'=M̂∖(X_1), the set of marked vertices that remain in Y_1. By ob:delta, we can consider the remaining execution ofon Y_1 as a new recursive call ofwith input (G[Y_1],x_0,t_0,Δ) and marked vertices M'. Since |X_1|,|Y_1|<|X|, the induction hypothesis implies that |(X_1)|≥|M_1|^1-1/k and |(Y_1)|≥|M'|^1-1/k.We now do a case analysis according to the choice of radius in alg-pick-rad. * Case 1: w_mid≤ m/2 and w_lo+R/(2L)=0. In this case we set r=lo+R/(4L). Note that w_r+R/(4Lk)≤ w_lo+R/(2L)=0, so M'=M̂, and by the induction hypothesis on Y_1, the number of fully padded vertices is |(X)|= |(Y_1)|≥|M̂|^1-1/k=m^1-1/k, as required. * Case 2: w_mid≤ m/2 and w_lo+R/(2L)>0. In this case we pick a,b∈[lo,hi] so that b-a=R/(2L) andw_a>w_b^2/m ,and also choose r∈[a,b] such that w_r+b-a/2k≤ w_r-b-a/2k·(w_b/w_a)^1/k. As b-a/2k=R/4Lk and |M_1|=w_r-R/(4Lk) we have that|M_1|≥(X_1)·(w_a/w_b)^1/k .By the induction hypothesis on X_1 we have that |(X_1)|≥|M_1|/|M_1|^1/k(<ref>)≥|(X_1)|·(w_a/|M_1|· w_b)^1/k(<ref>)≥|(X_1)|·(w_b/m·|M_1|)^1/k≥|(X_1)|/m^1/k ,where in the last inequality we use that |M_1|= w_r-(b-a)/(2k)≤ w_b. Now by the induction hypothesis on Y_1 we get|(X)| =|(Y_1)|+|(X_1)|≥|M'|^1-1/k+|(X_1)|/m^1/k≥|M'|+|(X_1)|/m^1/k=|M̂|/m^1/k=m^1-1/k* Case 3: w_mid> m/2 and q_hi-R/(2L)=0. In this case we set r=hi-R/(4L). Note that q_r-R/(4Lk)≤ q_hi-R/(2L)=0 (recall that q_r is non-increasing in r, by FctW_rProp), so M_1=M̂, and by the induction hypothesis on X_1, |(X)|=|(X_1)|≥|M_1|^1-1/k=m^1-1/k, as required. * Case 4: w_mid> m/2 and q_hi-R/(2L)>0. In this case we pick a,b∈[lo,hi] so that a-b=R/(2L) andq_a>q_b^2/m ,and also choose r∈[b,a] such that q_r-b-a/2k≤ q_r+b-a/2k·(q_b/q_a)^1/k. In this case when we cut "backwards", we shift the responsibility for the vertices unmarked by the creation of X_1 to Y_1. This is captured by defining (Y_1)=M̂∖ M_1. Since |M'|=q_r+a-b/2k and |(Y_1)|= q_r-a-b/2k we have|M'|≥|(Y_1)|·(q_a/q_b)^1/k .By the induction hypothesis on Y_1 we have that |(Y_1)|≥|M'|/|M'|^1/k(<ref>)≥|(Y_1)|·(q_a/|M'|· q_b)^1/k(<ref>)≥|(Y_1)|·(q_b/m·|M'|)^1/k≥|(Y_1)|/m^1/k ,where in the last inequality we use that |M'|= q_r+(a-b)/(2k)≤ q_b. Now by the induction hypothesis on X_1 we get|(X)| =|(Y_1)|+|(X_1)|≥|(Y_1)|/m^1/k+|M_1|^1-1/k≥|(Y_1)|+|M_1|/m^1/k=|M̂|/m^1/k=m^1-1/k . From lem:paddedImpliesStretch and lem:SavedTerminals we derive the following theorem.Let G=(V,E) be a weighted graph, fix a set M⊆ V of size m and a parameter k≥ 1. There exists a spanning tree T of G, and a set (M)⊆ M of size at least m^1-1/k, such that for every v∈(M) and every u∈ V it holds that (v,u,T)≤ O(kloglog m)·(v,u,G). We conclude with the proof of our main result.Set M_1=V, and for i≥ 1 define M_i+1=M_i∖(M_i). We shall apply thm:main iteratively, where M_i is the set of vertices given as input to the i-th iteration, that has size |M_i|=m_i. Let T_i be the tree created in iteration i. By thm:main the sizes m_1,m_2,… obey the recurrence m_1=n and m_i+1≤ m_i-m_i^1-1/k, which implies that after k· n^1/k iterations we will have m_k· n^1/k+1<1 (see <cit.>), and thus every vertex is in (M_i) for some 1≤ i≤ k· n^1/k. For each v∈ V, let (v) be the tree T_i such that v∈(M_i). §.§ Routing with Short Labels In this section we prove thm:route. We first use a result of <cit.> concerning routing in trees. For any tree T=(V,E) (where |V|=n), there is a routing scheme with stretch 1 that has routing tables of size O(b) and labels of size (1+o(1))log_bn. The decision time in each vertex is O(1).Combining thm:main-col and thm:tree-routh we can construct a routing scheme. Let 𝒯 be the set of trees from thm:main-col. Each tree T∈𝒯 is associated with a routing scheme given by thm:tree-routh. Set L_T(x) be the label of the vertex x in the routing scheme of the tree T.In our scheme, the routing table of each vertex will be the collection of its routing tables in all the trees in T. Hence the table size is O(b)· |𝒯|=O(k· b· n^1/k). The label of each x∈ V will be ((x),L_(x)(x)), i.e., the name of the home tree of x and the label of x in that tree. The label size is 1+(1+o(1))log_bn=(1+o(1))log_bn.The routing is done in a straightforward manner, to route from y to x, we extract (x) from the given label of x, and simply use the routing scheme of the tree (x). Note that this process takes O(1) time, and is independent of the routing path traversed so far. Since all vertices store in their routing table the appropriate routing information for (x), the routing can be completed.alpha § PROOF OF CORRECTNESS FOR ALG-PICK-RADIn this section we prove that the choices made in theprocedure are all legal. In all the Lemmas that follow, we shall use the notation in alg-pick-rad. If w_mid≤m/2 and w_lo+R/2L≥1, then there is [a,b]⊆[lo,mid] such that b-a=R/2L and w_a≥ w_b^2/m. Seeking contradiction, assume that for every such a,b with b-a=R/2L it holds that w_b>√(m· w_a). Applying this on b=mid-i R/2L and a=mid-(i+1) R/2L for every i=0,1,…,L-2, we have that w_mid>m^1/2· w_mid-R/2L^1/2>… >m^1-2^-(L-1)· w_mid-(L-1)R/2L^2^-(L-1)≥ m·2^-1· w_lo+R/2L^1/(2log m)≥m/2 , where we used that 1+loglog m≤ L≤ 2+loglog m and mid=lo+R/2. In the last inequality we also used that w_a≥ 1, which follows since b=a+R/2L≥ lo+R/2L, thus w_b≥ 1, and in particular w_a≥ w_b^2/m>0. The contradiction follows. There is r∈[a,b] such that w_r+b-a/2k≤ w_r-b-a/2k·(w_b/w_a)^1/k. Seeking contradiction, assume there is no such choice of r, then applying this for r=b-(i+1/2)·b-a/k for i=0,1,…,k-1 we get w_b>w_b-b-a/k·(w_b/w_a)^1/k>⋯>w_b-k·b-a/k·(w_b/w_a)^k/k=w_a·w_b/w_a=w_b , a contradiction.The following two lemmas are symmetric to the two lemmas above. If w_mid>m/2 (implies q_mid≤m/2) and q_hi-R/2L≥1, then there is [b,a]⊆[mid,hi] such that a-b=R/2L and q_a≥ q_b^2/m. There is r∈[b,a] such that q_r-a-b/2k≤ q_r+a-b/2k·(q_b/q_a)^1/k.§ TABLE OF DISTANCE ORACLES | http://arxiv.org/abs/1707.08769v1 | {
"authors": [
"Ittai Abraham",
"Shiri Chechik",
"Michael Elkin",
"Arnold Filtser",
"Ofer Neiman"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170727080545",
"title": "Ramsey Spanning Trees and their Applications"
} |
Doubling]Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, IIN. Kapouleas]Nikolaos Kapouleas P. McGrath]Peter McGrathDepartment of Mathematics, Brown University, Providence, RI 02912 [email protected] [email protected] <cit.>new closed embedded smooth minimal surfaces in the round three-sphere 𝕊^3(1) were constructed,each resembling two parallel copies of the equatorial two-spherejoined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles,or the equatorial circle and the poles. In this sequelwe generalize those constructionsso that the catenoidal bridges can concentrate along an arbitrary number of parallel circles,with the further option to include bridges at the poles.The current constructions follow the Linearized Doubling (LD) methodology developed in <cit.> and the LD solutions constructed here can be modified readily for useto doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides <cit.>.In particular they allow us to develop in <cit.> doubling constructions forthe catenoid in Euclidean three-space, the critical catenoid in the unit ball,and the spherical shrinker of the mean curvature flow.Unlike in <cit.>, our constructions hereallow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed”, that is do not miss any open set ofin the limit.This in particular leads to interesting observations which seemto suggest that it may be impossible to construct embedded minimal surfaces with isolated singularitiesby concentrating infinitely many catenoidal necks at a point. [ [ December 30, 2023 ===================== § INTRODUCTION ab§.§ The general frameworkab This article is the second onein a series in which gluing constructions for closed embedded minimal surfaces in the round three-sphere ^3(1) by doubling the equatorial two-sphereare discussed.Doublings of the equatorial two-sphereare importantas a test case for developing the doubling methodology and alsobecause their area is close to 8π(the area of two equatorial two-spheres),a feature they share with the celebrated surfacesconstructed by Lawson in 1970 <cit.>. The classification of the low areaclosed embedded minimal surfaces in the round three-sphere ^3(1),especially of those of area close to 8πor less,is a natural open question. This is further motivated by the recent resolutions ofthe Lawson conjecture by Brendle <cit.>and the Willmore conjecture by Marques and Neves <cit.>where they also characterize the Clifford torus and the equatorial sphere as the only examples of area ≤2π^2.We refer to <cit.> for a survey of existence and uniqueness results for minimal surfaces in the round three-sphere.The general idea of doubling constructions by gluing methodswas proposed and discussed in<cit.>. Gluing methods have been applied extensively and with great success in Gauge Theories by Donaldson, Taubes, and others. The particular kind of gluing methods used in this article relate most closely to the methods developed in <cit.> and <cit.>, especially as they evolved and were systematized in <cit.>. We refer to <cit.> for a general discussion of this gluing methodologyand to <cit.> for a detailed general discussion of doubling by gluing methods.Roughly speaking, in such doubling constructions, given a minimal surface Σ, one constructs first a family of smooth, embedded, and approximately minimal, initial surfaces.Each initial surfaceconsists of two approximately parallel copies of Σ with a number of discs removed and replaced by approximately catenoidal bridges. Finally, one of the initial surfaces in the family is perturbed to minimality by Partial Differential Equations methods. Understanding such constructions in full generality seems beyond the immediate horizon at the moment. In the earliest such construction <cit.>where doublings of the Clifford torus are constructed,there is so much symmetry imposed,that the position of the catenoidal bridges is completely fixedand all bridges are identical modulo the symmetries. Moreover the bridges are uniformly distributed,that is when their number is large enough,there are bridges located inside any preassigned domain of Σ. Wiygul <cit.> has extended that construction to multiple doublingswith more that two copies of the Clifford torus involved (and some less symmetric doublings also),where the symmetries do not determine the vertical (that is perpendicular to Σ)position of the bridges.In a previous article <cit.> doubling constructions wherethe horizontal position of the bridges is not determined by the symmetries,or there are more than one bridge modulo the symmetries,were carried out for the first time. To realize such constructions an intermediate step in the construction was introduced. In this intermediate step singular solutions of the linearized equation,called linearized doubling (LD) solutions,are constructed and studied. This new approach which we call Linearized Doubling (LD), provides a systematic methodology for dealing with the obstructions involvedand also provides a detailed understanding of the regions further away from the catenoidal bridges. It can also be generalized to higher dimensions <cit.>.In <cit.> the conversion of suitable continuous families of LD solutions on into minimal surfaces whose catenoidal bridges “replace” the singularities of the LD solutions was realized in general as part of the LD methodology.This reduced the construction of the minimal surfaces to the construction and estimation of the LD solutions,which remained however a very difficult problem.In <cit.> the only LD solutions which were constructed had their singularities either on two parallelcircles symmetrically arranged around the equatorial circle of , orat the poles and the equatorial circle of . This way two (discrete) families of minimal surfaces were obtained,a family with the catenoidal bridges concentrating on two parallel circles symmetrically arranged around the equatorial circle ofand a family where there are two bridges at the poles and the rest concentrate on the equatorial circle of . §.§ Brief discussion of the resultsab In this article we study LD solutions with singularities on an arbitrary number (but ≥2) of parallelcircles of(subject to the same symmetries as in <cit.>),with the option to have singularities at the poles also. This leads to the construction of minimal doublings ofwith the catenoidalbridges replacing the singularities of the LD solutions(see Theorems <ref>, <ref>, <ref>, and <ref>) and thus concentrating along an arbitrary number ≥2 of parallel circles and optionally at the poles.In particular we obtain for the first time (unlike in <cit.>)sequences of minimal doublings ofwherethe number of the parallel circles tends to infinityand therefore the number of bridges contained in anyfixed in advance open subset oftends also to infinity,that is the catenoidal bridges become “densely distributed” in the limit.We observe then the interesting phenomenonthat the size of the catenoidal bridges on each minimal surface of such a sequencetends to become uniform, in the sense that lim_→∞ lim_→∞ τ_max/τ_min =1,whereis the number of the parallel circles,the number of bridges on each circle,and τ_max/τ_min the ratio of the maximum size over the minimum size of the bridges (see Remarks <ref> and <ref>).This happens even in the case where nearby catenoidal bridges experience very different geometries,as when the bridge at a pole is surrounded by a very large number of bridges on nearby parallel circles. This suggests that when there is a very large number of catenoidal bridges close to a point,even in asymmetric situations, the sizes of the bridges would tend to become uniform.This would imply a negative answer to the very important question on whether embedded minimal surfaceswith isolated singularities can be constructed by concentrating an infinite number of catenoidalbridges in the vicinity of a singular point. Finally we remark that the families of LD solutions we construct here find immediate application in <cit.>to constructions of complete minimal surfaces in the Euclidean space by doubling the catenoid,constructions of free boundary minimal surfaces in the unit ball by doubling the critical catenoid,and of self-shrinkers of the mean curvature flow by doubling the spherical self-shrinker:Although the LD methodology as applied in <cit.> and in the current paper assumesthat the minimal surface being doubled possesses a symmetry fixing it pointwise and exchangingits sides, asdoes,it is possible<cit.> to extend the methodology to the case of asymmetric sides,as in the case of the catenoid and the spherical self-shrinker. Because the catenoid is conformally isometric toand the spherical self-shrinker is roundwe can use the LD solutions we derive here with small modifications to carry out the constructionsin <cit.>. §.§ Outline of the approachab In this article, as in <cit.>, in order to construct a family of initial surfacesone of which is later perturbed to minimality by a fixed point theorem, we first construct a family of LD solutions(defined in <cit.> and <ref>).The LD solutions are converted tomatched linearized doubling (MLD) solutions(defined in <cit.> and <ref>), which are then converted to initial surfaces.In both articles the construction and estimation of the LD solutions relieson the rotational invariance of .Each LD solution φ is related to a solution ϕ obtained by averaging φ on the circles of latitude (see <cit.> and <ref>).ϕ then belongs to a class of rotationally invariant solutions which in this articlewe callrotationally invariant linearized doubling (RLD) solutions (see<ref>).The RLD solutions studied in this article are rather more complicated compared to the ones studied in <cit.>.Much of the progress achieved involves their detailed understanding.We summarize schematically the main steps in the construction in the following diagram(see Remark <ref> for a more detailed outline): We discuss now the main innovations of this paper.A large part of the effort is in understanding and estimating in detail the RLD solutions.Achieving this is helped by the observation that the class of LD solutions is invariantunder conformal changes of the intrinsic metric of the surface and therefore RLD solutions canbe considered as defined on the flat cylinder instead of the round sphere.We also employ a dimensionless flux F^ϕ,which amounts to the logarithmic derivative of the RLD solution ϕ,to carefully study the RLD solutions.Such a quantity was used also in <cit.>,but here we study F^ϕ much more carefullyand we establish that it satisfies a Riccati differential equation and has several useful monotonicity properties.The fact that unlike in <cit.> we are dealing with situations where there are more than one (modulo the symmetries) circles of latitude wherethe catenoidal bridges concentrate, makes the balancing and unbalancing questions for the RLD solutions harder to study.The fluxes F^ϕ are the main tool which allows us to study these questions systematically.We also have developed a careful and systematic approach to study the parameters related to unbalancing (see <ref>). The current article incorporates various improvements in the application of the LD methodology:First,we estimate our LD solutions on the flat cylinder instead of the equatorial two-sphere as in <cit.> by making use of the conformal invariance of the LD solutions.The uniformity and further symmetries of the cylinder allow us to obtain stronger estimates. Second, we improve the analysis of the relation between the LD solutions and the corresponding RLD solutions.In particular, the decompositionΦ=++ in <ref> of an LD solution Φ into a singular part ,a rotationally invariant part , and an error term ,is an improvement over the two decompositions <cit.> it replaces.Finally, we employ a simplified definition for [L] in <ref>.§.§ Organization of the presentationab In Section <ref>, we review definitions and notation from <cit.>relating to the elementary geometry of the objects we are interested inand catalog a useful conformal diffeomorphism (<ref>) between the cylinder and the twice-punctured two-sphere. We also discuss some special rotationally invariant solutions and Green's functions for the linearized equation. The main new features in this paper which refine the approach in <cit.> take place in Sections <ref> and <ref>. In Section <ref>, we define in <ref> a class of rotationally invariant solutions of the linearized equation (RLD solutions),establish appropriate criteria for their existence and uniqueness (see <ref>),and prove estimates governing their geometry in <ref>,and behavior under small perturbations of initial data in <ref>.In Section <ref>, we more generally study linearized doubling solutions (LD solutions) on the cylinder which have prescribed logarithmic singularities at L.More precisely in Lemma <ref> we convert RLD solutionsto corresponding LD solutions Φ.We introduce then in <ref> the decomposition Φ=++ of an LD solution Φ into a singular part ,a rotationally invariant part , and an error term . Much of the remaining work in the section is devoted to estimatingin <ref>.In Section <ref>, we convert the LD solutions we have constructed in Section <ref> to MLD solutions which satisfy the appropriate linear and nonlinear matching conditions. Using the earlier estimates, we prove in <ref> the estimates we need for these MLD solutions. In Section <ref>,we convert these families of MLD solutions into families of smooth initial surfaces with small mean curvature. by recalling work from <cit.>,and we then apply a fixed point theorem to perturbations of the families of initial surfaces constructed in Section <ref>to construct our minimal surfaces. Finally in Section <ref> we modify the construction to include catenoidal bridges at the poles or on the equatorial circle ofor both. §.§ Acknowledgmentsab The authors would like to thank Richard Schoen for his continuous support and interest in the results of this article.NK would like to thank the Mathematics Department and the MRC at Stanford University for providing a stimulating mathematical environment and generous financial support during Spring 2016. NK was also partially supported by NSF grant DMS-1405537.§ ELEMENTARY GEOMETRY AND NOTATION §.§ Hölder norms and cut-off functions abWe will find the following notation useful.We write a∼_c b to mean thata,b∈ are nonzero of the same sign,c∈(1,∞),and 1/c≤a/b≤ c.We use the standard notation u: C^r,β( Ω,g )to denote the standard C^r,β-norm of a function or more generally tensor field u on a domain Ω equipped with a Riemannian metric g. Actually the definition is completely standard only when β=0 because then we just use the covariant derivatives and take a supremum norm when they are measured by g. When β0 we have to use parallel transport along geodesic segmentsconnecting any two points of small enough distance and this may be a complication if small enough geodesic balls are not convex. In this paper we take care to avoid situations where such a complication may arise and so we will not discuss this issue further.We adopt the following notation from <cit.> for weighted Hölder norms. Assuming that Ω is a domain inside a manifold, g is a Riemannian metric on the manifold,r∈_0,β∈[0,1), u∈ C^r,β_loc(Ω)or more generally u is a C^k,β_loc tensor field(section of a vector bundle) on Ω,ρ,f:Ω→(0,∞) are given functions,and that the injectivity radius in the manifold around each point x in the metric ρ^-2(x) g is at least 1/10, we defineu: C^r,β ( Ω,ρ,g,f):= sup_x∈Ω u:C^r,β(Ω∩ B_x, ρ^-2(x) g) /f(x) ,where B_x is a geodesic ball centered at x and of radius 1/100 in the metric ρ^-2(x) g. For simplicity we may omit any of β, ρ, or f,when β=0, ρ≡1, or f≡1, respectively. f can be thought of as a “weight” function because f(x) controls the size of u in the vicinity of the point x. ρ can be thought of as a function which determines the “natural scale” ρ(x) at the vicinity of each point x. Note that if u scales nontrivially we can modify appropriately f by multiplying by the appropriatepower of ρ.Observe from the definition the following multiplicative property: u_1 u_2 : C^k,β(Ω,ρ,g,f_1 f_2 )≤ C(k) u_1 : C^k,β(Ω,ρ,g,f_1 )u_2 : C^k,β(Ω,ρ,g,f_2 ). Our arguments will require extensive use of cut-off functions, and it will be helpful to adopt the following. We fix a smooth function Ψ:→[0,1] with the following properties: * Ψ is nondecreasing. * Ψ≡1 on [1,∞) and Ψ≡0 on (-∞,-1]. * Ψ-1/2 is an odd function.Given a,b∈ with a b, we define smooth functions [a,b]:→[0,1] by[a,b]:=Ψ∘ L_a,b,where L_a,b:→ is the linear function defined by the requirements L(a)=-3 and L(b)=3.Clearly then [a,b] has the following properties: * [a,b] is weakly monotone. * [a,b]=1 on a neighborhood of b and[a,b]=0 on a neighborhood of a. * [a,b]+[b,a]=1 on .Suppose now we have two sections f_0,f_1 of some vector bundle over some domain Ω. (A special case is when the vector bundle is trivial and f_0,f_1 real-valued functions). Suppose we also have some real-valued function d defined on Ω. We define a new section [a,b;d](f_0,f_1):= [a,b]∘ d f_1 + [b,a]∘ d f_0.Note that [a,b;d](f_0,f_1) is then a section which depends linearly on the pair (f_0,f_1) and transits from f_0 on Ω_a to f_1 on Ω_b, where Ω_a and Ω_b are subsets of Ω which contain d^-1(a) and d^-1(b) respectively, and are defined byΩ_a=d^-1((-∞,a+1/3(b-a))), Ω_b=d^-1((b-1/3(b-a),∞)),when a<b, and Ω_a=d^-1((a-1/3(a-b),∞)), Ω_b=d^-1((-∞,b+1/3(a-b))),when b<a. Clearly if f_0,f_1, and d are smooth then [a,b;d](f_0,f_1) is also smooth. §.§ The parametrizations Θ andand the coordinates (,,) and (, θ)ab We consider now the unit three-sphere ^3(1)⊂^4. We denote by (x_1,x_2,x_3,x_4) the standard coordinates of ^4 and we define by:=^3(1)∩{x_4=0}an equatorial two-sphere in ^3(1). To facilitate the discussion we fix spherical coordinates (,,) on ^3(1) by defining a map Θ:^3→^3(1) byΘ(,,) =(coscoscos, cossincos, sincos,sin).Note that in the above notation we can think ofas the geographic latitude onand ofas the geographic longitude. We will also refer to_0:= ∩{x_3=0}= Θ({==0}), p_N:=(0,0,1,0)=Θ(π/2,,0), p_S:=(0,0,-1,0)=Θ(-π/2,,0),as the equator circle, the North pole, and the South pole ofrespectively.Clearly, the standard metric of ^3(1) is given in the coordinates of (<ref>) byΘ^* g = cos^2 ( d^2 +cos^2 d^2)+ d^2.Finally we define a nearest-point projection:^3(1)∖{(0, 0, 0, ± 1)}→ by(x_1,x_2,x_3,x_4)= 1/|(x_1,x_2,x_3,0)| (x_1,x_2,x_3,0).Clearly we have∘Θ(,,) = Θ(,,0). The study of the RLD and LD solutions can be simplified by the observation that ∖{p_N, p_S} is conformally equivalent to a flat cylinder ×^1.To be precise, let: = ×^1be the cylinder endowed with the flat product metric χ, where χ: = d^2 + dθ^2and (,θ) are the standard coordinates on .Consider the map :→∖{p_N, p_S} defined by (, θ) = ( cosθ, sinθ, tanh ) .Clearlyis a diffeomorphism and from now on we will use it to identify∖{p_N, p_S} with .and θ can then be considered as coordinates on∖{p_N, p_S} and by (<ref>) we have sin = tanh , cos = ,= log1+sin/cos,and = θ.Straightforward computations show g = (^2 ) χ, d /d= 1/cos,d/d = . We finally introduce some convenient notation.We use the shorthand notation {∈ (a, b)} to denote the annular region{ (, θ) ⊂: ∈ (a, b)} and similar abbreviations with regard to other types of intervals.For X a subset ofand h a Riemannian metric onor on ∖{p_N, p_S},we write ^h_X for the distance function from X, with respect to h.We define a tubular neighborhood of X byD^h_X(δ):={p∈:^h_X(p)<δ}. If X is finite we just enumerate its points in both cases.For example, ^g_q(p) is the geodesic distance between p and q and D^g_q(δ) is the geodesic disc inof center q and radius δ.§.§ The symmetries and the configurations ab To study the symmetries of the parametrization Θ, we first define reflections of its domain, _c, :=_0, and , and translations _c, where c∈, by (,,):=(-,,),_c(,,) :=(,2c-,),(,,):=(,,-),_c(,,) :=(,+c,).We also define corresponding reflections , _c, :=_0, andand rotations _c, of ^4 ⊃^3(1) by(x_1,x_2,x_3,x_4):= (x_1,x_2,-x_3,x_4),(x_1,x_2,x_3,x_4):= (x_1,-x_2,x_3,x_4),(x_1,x_2,x_3,x_4):= (x_1,x_2,x_3,-x_4),_c(x_1,x_2,x_3,x_4):= (x_1cos 2c + x_2sin 2c ,x_1 sin 2c- x_2 cos 2c,x_3, x_4)_c(x_1,x_2,x_3,x_4):= (x_1cos c - x_2sin c, x_1sin c + x_2cos c,x_3, x_4). Note that , , , and _c are reflections with respect to the 3-planes {x_3=0}, {x_2=0}, {x_4=0}, and _c({x_2=0}), respectively.exchanges the two sides ofwhich it fixes pointwise. Clearly _2π is the identity map. We record the symmetries of Θ in the following lemma:Θ restricted to:= (-π/2,π/2) ××(-π/2,π/2)is a covering map onto^3(1)∖{x_1=x_2=0}. Moreover the following hold: * The group of covering transformations is generated by _2π. * ∘Θ=Θ∘, _c∘Θ=Θ∘_c, ∘Θ=Θ∘, and _c∘Θ=Θ∘_c. The symmetry group of our constructions depends on a large number m∈ which we assume now fixed. We define =[m]⊂ to be the union of m meridians symmetrically arranged: =[m]:=Θ({(,,0):∈[-π/2,π/2],=2π i/m, i∈}).We denote byandthe groups ofisometries of ^3(1) andrespectively which fix [m] as a set. Clearlyis a finite group and is generated by the reflections , ,and _π/m, that is= ⟨, , , _π/m⟩,and moreovercan be identified with the subgroup ofgenerated by , and _π/m. The configuration of our constructions also depends on a number k∈ whose values are restricted in terms of m (see <ref> below).Given ∈ [0, ∞), we define L_par[] = { (, θ) ∈ := ±}.Furthermore, given:= (_1, …, _k)∈^k_+such that0 <_1<⋯<_k < ∞,we defineL_par[] = ⋃_i=1^k L_par[_i].Finally given alsoa domain Ω⊂,we will denote by Ω^ the“subdivision” of Ω by L_par[]:More precisely Ω^ is the abstract surfacewhich is the disjoint union of the Ω∩ A's, where A is the closure of any connected component(a disk or an annulus) of ∖ L_par[]. Clearly functions on Ω can be thought of as functions on Ω^ as well.Note for example that a function defined on Ω whichis in C^∞(Ω^) is also in C^0(Ω) but not necessarilyin C^1(Ω); it is “piecewise smooth” on Ω.For m as in <ref> and andas in <ref>,we defineL[ ; m]: =L_mer[m]∩ L_par[ ]=(,0),L = L[; m] :=L_mer[m]∩ L_par[] = ⋃_i=1^k L[_i;m].For i ∈{1,...,k} we define p_i:=(_i,0), L_i=L[_i;m],and given also a -symmetric function τ:L[; m]→,τ_i := τ(p_i). §.§ The linearized equation and the cylinderab A major step in the construction is to estimate solutions of the Jacobi equation u= 0, where= Δ_g + |A|^2 + (ν, ν)= Δ_g+ 2.We define also a version of ' usefulin the cylindrical picture by _χ := Δ_χ + 2^2= ^2', where Δ_χ := ∂^2/∂^2 +∂^2/∂θ^2.It will be easier to state some of our estimates if we use a scaled metriconand corresponding scaled coordinates ( ,) defined by :=m^2 g_, =m, =m.In the same fashion, we define a scaled metriconand scaled coordinates ( , ) defined by:= m^2 χ, = m , = m θ.We also define corresponding scaled linear operators_ : = Δ_ + 2m^-2 = m^-2,_: = Δ_+2m^-2^2= m^-2_χ. For future reference, we record the following global parametrization of a catenoid with unit waist size:_cat(, θ) = ( coshcosθ, coshsinθ, ).For τ>0,τ_cat parametrizes the catenoid with waist size τ. Let ∈_+.We define a shifted coordinate = [] by :=- m .For convenience we define δ := 1/(9m).Given ∈ [0, ∞), we havenested open sets D^χ_L[;m](3δ) ⊂Ω'[;m]⊂Ω[;m] whereΩ[; m] : = D^χ_L_par[](3/m),Ω'[; m] := D^χ_L_par[](2/m).Given a domain Ω⊂ which is invariant under the coordinate reflection through some ∈,we define a reflection operator _: C^0(Ω)→ C^0(Ω)by _ u(+', θ) = u(- ', θ), for (+ ', θ) ∈Ωand an antisymmetry operator _: C^0(Ω)→ C^0(Ω) by _ u = u - _ u. Let ∈(3/m, ∞).The following hold: * For all u, v ∈ C^0_(Ω[; m]),_[ u v] = u_ [v] + _ [u]_ [v]. * For all u ∈ C^2_(Ω[; m]),_[ _χ u]= _χ[ _ u] + 2_ [^2 ]_ [u],_[ _ u]= _[ _ u] + 2m^-2_ [^2 ]_ [u].** Let Ω⊂ be a domain.Then^2: C^j( Ω , χ) ≤ C(j) ^2: C^0(Ω, χ). * _[ ^2 ] : C^j( Ω[; m], χ )≤C(j)/m^2: C^0( Ω[; m], χ ). (i) follows from a straightforward computation, and (ii) follows from (i) and a similar computation, using the fact that Δ commutes with _.For (iii).(a), observe that for each j≥ 1,∂^j( ^2) is a polynomial expression in ^2 and tanh each term of which contains a factor of ^2.(iii).(b) is a discrete version of (iii).(a) which follows from the mean value theorem.More precisely, fix ∈_+ and let ∈ (-3, 3), where = [] is as in <ref>. By the mean value theorem, there exists ' ∈ (- , ) such that_[ ^2 ](+/m)= ^2(+/m) - ^2( - /m)= 4/m^2(+'/m)tanh(+'/m).Estimating the right hand side of the preceding, it follows that_[ ^2 ] : C^0( Ω[; m], χ )≤C/m^2 : C^0( Ω[; m], χ ). Estimates on higher order derivatives follow from the mean value theorem in a similar way. §.§ Rotationally invariant functions abWe call a function defined on a domain ofwhich depends only on the coordinatearotationally invariant function. The linearized equation _χϕ = 0 amounts to an ODE when ϕ is rotationally invariant.Motivated by this, we introduce some notation to simplify the presentation:Consider a function space X consisting of functions defined on a domain Ω⊂. If Ω is invariant under the action of(recall <ref>), we use a subscript “” to denote the subspace X_⊂ X consisting of those functionsf∈ X which are covariant under the action of . If Ω is a union of parallel circles, we use a subscript “” to denotethe subspace of functions X_ consisting ofrotationally invariant functions, which therefore depend only on . If moreover Ω is invariant under reflection with respect to { = 0}, we use a subscript “||” to denotethe subspace of functions X_||= X_∩ X_ consisting of those functions which depend only on ||. For example, we have C^0_||()⊂ C^0_() ⊂ C^0() and C^0_||() ⊂ C^0_(), but C^0_() is not a subset of C^0_(). We also have the following. Given a function φ on some domain Ω⊂, we define a rotationally invariant function φ_ onthe union Ω' of the parallel circles on which φ is integrable (whether contained in Ω or not), by requesting that on each such circle C, . φ_|_C :=_Cφ.We also define φ_ on Ω∩Ω' byφ_:=φ-φ_. If Ω⊂ is a domain and u ∈ C^0_(Ω) has one-sided partial derivatives at =,then we denote these partial derivatives by using the notation∂_+ u() :=. ∂ u/∂|_=+, ∂_-u() :=-. ∂ u/∂|_=-.If u is C^1, we use the abbreviation u := ∂ u/∂.In that case, ∂ u = ∂_+u = -∂_-u. If ϕ∈ C^∞_(), the equation _χϕ=0 (recall <ref>) is equivalent to d^2ϕ/d^2 + 2^2ϕ = 0.Define rotationally invariant functions ∈ C^∞_||() and ∈ C^∞_() by() = 1-tanh, () = tanh. andare even and odd inrespectively and satisfy _χ = 0 and _χ = 0. is strictly decreasing on [0, ∞) and has a unique root _root∈ (0, ∞). is strictly increasing.The Wronskian W[ , ] satisfiesW[, ](): = ()∂() - ∂() () = 1.Straightforward calculation using Definition <ref> and (<ref>).When written incoordinates (recall (<ref>)),and satisfy=sin, = 1- sin log1+sin/cos,and therefore our Definition<ref> is equivalent to <cit.>. It will be helpful to introduce the following auxiliary ODE solutions: Given ,∈, and ∈_+ we define= [,;]∈C^∞_({∈[0,∞)}) ⋂ C^0_||( ),=[;]∈C^∞_( {∈[,∞)} ) ⋂C^∞_({∈(0,]} ) ⋂C^0_||( ),by requesting they satisfy the initial data ()=, ∂ () = , ()=0, ∂_+()=∂_-()=m ,and the ODEs _=0 on {∈[0,∞)},and _=0 on {∈[,∞)} and on {∈[0,]}. Note thatdepends linearly on the pair (, )∈^2 anddepends linearly on ∈. Let ∈(3/m, ∞). The following estimates hold (recall <ref>). * [1, 0 ;] -1 : C_^j(Ω[; m] ,)≤C(j)/m^2. * [1;] - | |: C_^j(Ω[; m] ∖[],) ≤ C(j)/m^2. * _ [1, 0 ;] : C_^j(Ω[; m], )≤C(j)/m^3. * _ [1;] : C_^j(Ω[; m] ∖[] ,) ≤ C(j)/m^3.Denote in this proof = [1, 0; ].In -coordinates (where = [ ] is as in <ref>), the equation _ =0is equivalent to∂^2_ + 2/m^2^2( /m+ )= 0.Since ^2t is decreasing on (0, ∞), it is easy to see that for m≥ 6, >0 on Ω[; m], hence ∂^2_ + 2/m^2 > 0 onΩ[; m]. Integrating this differential inequality and matching the initial data implies1 -: C^0_(Ω[ ;m],)≤ 1 -cos( √(2)/m ) : C^0_(Ω[ ;m],)≤C/m^2.In conjunction with (<ref>), the C^0 bound above implies |∂^2_| < C/m^2 on Ω[, m].Integrating this bound with respect toimplies | ∂_|< C/m^2.Together, these bounds imply the C^2 bound in (i).Higher derivative bounds follow from the C^2 bound by differentiating (<ref>).For (ii), denote = [1; ].Arguing as above, we have on Ω[;m]∖[]∂^2_ + 2/m^2^2( /m+ )= 0 and∂^2_ + 2/m^2 > 0.A similar comparison argument establishes that-| | : C^0_ ( Ω[ ; m]∖[] ,) ≤m/√(2)sin( √(2)/m|| )-|| : C^0_( Ω[ ; m]∖[_i] ,)≤C/m^2.The C^0 bound above implies | ∂^2_| < C/m^2. For t∈ (-3, 3), we find by integrating this bound that| ∂_( ) - ∂_| | | = | ∂_(t) -∂_(0)|= | ∫_0^t∂^2_d| ≤C/m^2.This with the preceding bounds implies the C^2 bound in (ii).Estimates on the higher derivatives follow by differentiating (<ref>). By Lemma <ref>.(ii), _ _i satisfies the equation∂^2__ + 2/m^2^2 ( /m+ ) _ + 2/m^2_[ ^2 ] ( /m+ _i) _= 0.It follows from (i) that_ : C_^2(Ω[; m] ,)≤C/m^2.Using this bound and the estimate on _[ ^2 ] from Lemma <ref>.(iii) in (<ref>), we find | ∂^2__ | < C/m^3 on Ω[;m].Integrating this bound twice with respect totwice and using the fact that ∂_ _ () = _ () =0 establishes the C^2 bounds in (iii); as before, estimates on the higher derivatives follow from differentiating (<ref>).The proof of (iv) is almost exactly the same as the proof of (iii), so we omit it. §.§ Green's functions and the Green's function for _χab Let (Σ^2, g) be Riemannian, V∈ C^∞(Σ), and p∈Σ.If there exists a domain Ω⊂Σ containing p and G_p ∈ C^∞( Ω∖{p}) satisfying* ( Δ_g + V) G_p= 0 * For some neighborhood Ω'⊂Ω,G_p-log∘^g_p is bounded on Ω'∖{p},we sayG_p is a Green's function for Δ_g + V centered at p.For any (Σ, g), p, and V as above, standard theory guarantees the existence of a Green's function for Δ_g + V with center at p.Suppose G_p and _p are Green's functions for Δ_g + V, wherep, Ω, V, and (Σ, g) are as in Definition <ref>.Then G_p - _p has a unique extension to C^∞(Ω). In this proof, we denote = Δ_g + V. Definition <ref>.(ii) implies that u: = G_p - _p satisfies u : C^0( Ω∖{ p } , g) ≤ C.Since u = 0, the C^0 bound and Schauder estimates imply thatu : C^j( Ω∖{ p } , g) ≤ C(j). By standard regularity theory, to prove the existence of the extension,it suffices to prove that u solves u = 0 weakly in Ω.To this end, let φ∈ C^∞_c(Ω).Compute∫_Ω u φdμ_g =lim_ϵ↘ 0∫_Ω∖ D^g_p(ϵ)φ u dμ_g + lim_ϵ↘ 0∫_∂ D^g_p(ϵ) u ∂φ/∂η - φ∂ u /∂ηdμ_g = 0,where the last equality follows from (<ref>) and thatu = 0. Given two Green's functions G_p, _p as in <ref>, we abuse notation by writing G_p -_p for the smooth extension to C^∞(Ω) ofG_p - _p, whose existence and uniqueness is asserted in Lemma <ref>. For future reference, we recall <cit.> basic properties of the Green's function used in <cit.> to construct initial surfaces.There is a functionG^^2∈ C^∞((0,π)) uniquely characterized by (i) and (ii)and moreover satisfying (iii-vii) below. We denote bythe standard coordinate of ^+. * For smallwe have G^^2()=(1+O(^2)) log. * For each p∈ we have (G^^2∘^g_p) =0 where G^^2∘^g_p∈ C^∞(∖{p,-p}) (recall <ref>). * G^^2∘^g_p_N= (log 2-1)+ (recall <ref>). * G^^2(r)= 1+cos r( -1+log2sin r /1+cos r ).* ∂ G^^2/∂ r (r) = -sin rlog2sin r /1+cos r+ 1/sin r + sin r cos r /1+cos r. * G^^2 - cos rlog: C^j((0,1) ,r, dr^2,r^2 ) ≤C(j) .* G^^2: C^j((0,1) ,r, dr^2,|log r|) ≤C(j) . It will also be helpful to have Green's functions G_p for _χ well adapted to the cylinder. Given p = (, θ)∈, there exists G_p ∈ C^∞(D^χ_p(1/2)∖{p}) satisfying:* _χ G_p = 0 on D^χ_p(1/2)∖{p}.* For q near p, G_p(q) = log r + O(r^2|log r|), where r(q) = ^χ_p(q). * _ G_p : C^j(D^χ_p(1/2) ∖{ p }, r ,χ ,r )≤ C(j). As an auxiliary step, we consider solutions of the equationΔ_χ u + 2(^2) u = 0.Letr = ^χ_p be the polar radius on D^χ_p(1/2).When u = u(r) is radial, (<ref>) is equivalent to the ODEd^2 u/d r^2 + 1/rdu/d r + 2(^2) u = 0.The solution space to (<ref>) is spanned by{ J_0(√(2)()r),Y_0(√(2)()r)}, where J_0 and Y_0 are the Bessel functions (cf. <cit.>) defined by J_0(x)= ∑_j = 0^∞(-1)^j/4^j (j!)^2 x^2j,Y_0(x)= 2/π{(logx/2+ γ) J_0(x) + ∑_j=1^∞ (-1)^j+1(∑_l=1^j1/l)x^2j/4^j (j!)^2},and γ is the Euler constant. By a short computation, the function G'_p ∈ C^∞( D^χ_p(1/2) ∖{ p }) defined byG'_p :=π/2 Y_0( √(2)()r) - (γ+log/√(2)) J_0(√(2)()r)satisfiesΔ_χ G'_p + 2(^2 ) G'_p =0 and G'_p= log r + O(r^2| log r|).Let w_p ∈ C^2, β(D^χ_p(1/2)) be the unique solution of the Dirichlet problem_χ w_p= 2(^2- ^2 )G'_pon D^χ_p(1/2)w_p= 0 on∂ D^χ_p(1/2)and define (recall Definition <ref>)w'_p: = w_p + [ -w_p(p), ∂ w_p/∂( p); ].Finally, defineG_p = G'_p + w'_p.By combining (<ref>), (<ref>), and (<ref>) we find _χ G_p = 0, which yields (i). Now note that the right hand side of (<ref>) is in C^0, β(D^χ_p(1/2)) for any 0< β < 1.It follows that w'_p ∈ C^2, β(D^χ_p(1/2)).In light of (<ref>), w'_p(p) = .∇ w'_p|_p=0.Therefore, the second order Taylor series for w'_p and the Schauder estimates implyw'_p : C^j(D^χ_p(1/2))∖{ p}, r, χ, r)≤ C(j). (ii) then follows from combining (<ref>), (<ref>), and (<ref>). Note from (<ref>) that _ G_p =_ w'_p.(iii) then follows from this and (<ref>).The estimates in Section <ref>, and in particular the decomposition in <ref>, are adapted to the Green's functions G_p, for p∈ L. Since for such p, _p is defined (recall<ref>.(ii)) in terms of the Green's function G^^2∘^g_p, the following lemma will be useful.Fix ∈ and let p = (, 0)∈.We have (recall Convention <ref>)( G_p - G^^2∘^g_p)(p) = -log,d_p(G_p - G^^2∘^g_p)(p) = 1/2tanhd.Let q = (, 0), whereis close to .For convenience, in this proof denote r = ^χ_p(q).Recalling (<ref>), we have^g_p(q) = | ∫_^ t dt|= |(- ) + 1/2( - )^2 ( - tanh) + O( (-)^3)|=( )r ( 1- 1/2(- )tanh + O(( - )^2)).Consequently, recalling from <ref>.(i) that G^^2(t) = log t + O(t^2 |log t|) for small positive t, we haveG^^2∘^g_p (q) = log + log r - 1/2 (-)tanh + O(r^2| log r|). By Lemma <ref>.(ii),G_p(q)= log r + O(r^2|log r|) and hence( G_p - G^^2∘^g_p)(q) =- log + 1/2(-)tanh + O(r^2 |log r|).The conclusion then easily follows from (<ref>).§ ROTATIONALLY INVARIANT SOLUTIONS §.§ Basic facts and notation ab We will estimate our LD solutions by comparing with rotationally invariant solutions.It will therefore be useful to define a class of rotationally invariant solutions of the linearized equation.We begin with some notation.Let ^ = { (a_i)_i∈: a_i ∈} and _+^ = {(a_i)_i∈: a_i∈, a_i> 0}.For any k∈, we identify ^k with a subspace of ^ by the map(a_1, …, a_k) ↦ (a_1, …, a_k, 0, 0, …).We consider the normed space (ℓ^1(^), |· |_ℓ^1) defined byℓ^1(^) = { = (a_i)_i∈∈^ : ∑_i=1^∞ |a_i|< ∞},| |_ℓ^1 = ∑_i=1^∞ |a_i|.If = (σ_i)_i∈∈ℓ^1( ^) and = (ξ_i)_i∈∈ℓ^∞( ^) satisfies | |_ℓ^∞ < 1/10 and satisfye^σ_i = F_i+1+ + F_i+1-/F_i+ + F_i-, ξ_i = F_i+ - F_i-/F_i++F_i- i∈,for some positive numbers F_i±, i∈, then note that for any 1 ≤ j ≤ i < ∞, F_i+ = 1+ ξ_i /1 + ξ_j( e^∑_l=j^i-1σ_l) F_j+ = 1+ ξ_i /1 - ξ_j( e^∑_l=j^i-1σ_l) F_j-,F_i- = 1- ξ_i /1 + ξ_j( e^∑_l=j^i-1σ_l) F_j+ = 1- ξ_i /1 - ξ_j( e^∑_l=j^i-1σ_l) F_j-,and therefore sup{ F_i±}_i∈∼_exp(||_ℓ^1+3||_ℓ^∞)inf{ F_i±}_i∈. §.§ RLD solutions and the scale invariant fluxab IfΩ = {∈ (a, b)}⊂, ϕ∈ C^0_(Ω), andϕ is piecewise smooth and positive on Ω,we define F^ϕ_±:(a, b)→ by (recall <ref>) F^ϕ_±() = ∂_±ϕ()/ϕ().Note that F^ϕ_± = F^cϕ_±∀ c∈_+.Also, if ϕ is C^1 at =, then F^ϕ_+() = - F^ϕ_-().We say ϕ∈ C^0_||( ) is a rotationally invariant linearized doubling (RLD) solution if (recall <ref>)* ϕ >0. * There is k∈ and ^ϕ∈^k_+ as in <ref>such thatϕ∈ C^∞_|| ( ^^ϕ )and _χϕ =0 on ^^ϕ.* For i=1, …, k, F^ϕ_-(^ϕ_i)> 0 and F^ϕ_+(^ϕ_i)> 0.We call ^ϕ the jump latitudes of ϕ and L_par[^ϕ] the configuration of ϕ. If ϕ(0) = 1, we say ϕ is a unit RLD solution. If ϕ is extendible toC^∞_||(∖ L_par[])we say ϕ is smooth at the poles. If ^ϕ_i is a jump latitude of ϕ, note that <ref>.(iii) implies ∂ϕ is not defined at [_i^ϕ]. Thus, the jump latitudes of ϕ and their number are uniquely determined by ϕ. Let ϕ be an RLD solution. We define^ϕ := ( F^ϕ_1-, F^ϕ_1+, F^ϕ_2-, …, F^ϕ_k+) ∈^2k_+,^ϕ := ( F^ϕ_i)_i=1^k∈^k_+, ^ϕ := (σ^ϕ_i)_i=1^k-1∈^k-1,^ϕ := ( ξ^ϕ_i)_i=1^k ∈^k,where for i=1, …, kand j=1, …, k-1,F^ϕ_i± := F^ϕ_± (_i), F^ϕ_i := F^ϕ_+(_i)+F^ϕ_-(_i), e^σ^ϕ_j = F^ϕ_j+1/F^ϕ_j, ξ^ϕ_i = F^ϕ_i+ - F^ϕ_i-/F^ϕ_i+ + F^ϕ_i-. We define ^ϕ: = (^ϕ, ^ϕ) ∈^k-1×^k and call the entries of ^ϕ the flux ratios of ϕ.If ^ϕ =0 we call ϕ balanced. Finally we defineF^ϕ_avg : = 1/2k|^ϕ|_ℓ^1 =1/2k| ^ϕ|_ℓ^1.Using (<ref>) (see also Remark <ref>), we recover ^ϕ from F^ϕ_1 and ^ϕ: F^ϕ_1± = 1/2(1±ξ^ϕ_1)F^ϕ_1 , F^ϕ_i± =1/2(1±ξ^ϕ_i) ( e^∑_l=1^i-1σ^ϕ_l)F^ϕ_1, i>1.In Proposition <ref> we construct RLD solutions ϕ by prescribing F^ϕ_1- and ^ϕ.§.§ Existence, uniqueness, and estimatesab If ϕ is an RLD solution, it is immediate from (<ref>) that ∂ϕ (recall <ref>) is decreasing on any domain on which it is smooth.The scale invariant flux F^ϕ_± enjoys a similar monotonicity: Suppose ϕ∈ C^∞_({∈ (a, b)}), ϕ> 0 and _χϕ =0.For ∈ (a, b),d F^ϕ_-/d() = 2^2+ (F^ϕ_-())^2>0. Let t∈ (a, b) and Ω = {∈ (a, t)}.By the divergence theorem and that _χϕ =0, ∫_∂Ω⟨∇ϕ/ϕ, η⟩= ∫_ΩΔ( logϕ) =∫_Ω( Δϕ/ϕ- |∇ϕ|^2/ϕ^2)= -∫_Ω( 2^2+|∇ϕ|^2/ϕ^2).Using that η = ∂/∂ when = t and differentiating the above with respect to t yields the lemma.Suppose ϕ∈ C^∞_({∈ [a, b]}), ϕ>0, and _χϕ = 0. ThenF^ϕ_-(b) + F^ϕ_+(a) = 2( tanh b - tanh a) + ∫_a^b ( F^ϕ_-())^2d.Follows directly from integrating (<ref>) on (a, b). To study RLD solutions on domains {∈ (_i, _i+1)} between successive jumps, we introduce the following auxiliary functions.Given ∈ (0, ∞) and F> 0, we define functionsH^+= H^+[F; ] ∈ C^∞_(),H^-= H^-[F; ] ∈ C^∞_() by requesting that they satisfy the equations _χ H^+ = 0, _χ H^- =0 with initial data H^+() = 1,F^H^+_+()=F,H^-() = 1,F^H^-_-()=F. By straightforward computations (recall Lemma <ref>),H^+[F;]= H^-[-F;] and H^±() = ( F_+^() ∓ F) ()()+( - F_+^() ± F) ()().Note also that when ≥ 0, H^+[F; ]()= [1, F; ]() (recall <ref>). For any >, H^+ = H^+[F; ] satisfies* ∂ F^H^+_+/∂()=(2^2 +F^2)(H^+()/H^+())^2>0. * ∂ F^H^+_+/∂ F() =(H^+()/H^+())^2>0.Applying Lemma <ref>to H^+[F; ] (which we denote below by H for ease of notation) yieldsd F^H_+/d()+ (F^H_+())^2 = - 2 ^2 .Differentiating (<ref>) with respect toand switching the order of differentiation gives∂/∂( ∂ F_+^H/∂)+2(∂ F_+^H/∂)F_+^H= 0.After multiplying through by the integrating factor H^2, this is equivalent to ∂/∂(∂ F_+^H/∂ H^2) = 0,from which we conclude∂ F_+^H/∂() =∂ F^H_+/∂()(H()/H())^2.Finally, differentiating both sides of the equation F^H^+[F; ]_+() = F with respect toyields ∂ F^H_+/∂() = - ∂ F^H_+/∂() = 2^2 + F^2,where the last equality follows from <ref>.This completes the proof of (i). (ii) follows in an analogous way by differentiating (<ref>) with respect to F and observing that ∂ F_+^H/∂ F() = 1.Supposea>0,Ω = {∈ (-a, a)},∈^j_+ is as in <ref> with _j<a,andϕ is a function on Ω which satisfies ϕ∈ C^∞_||(Ω^), ϕ(0) = 1, ϕ>0, and _χϕ = 0 on Ω^.Finally, suppose thatF^ϕ_±(_i) = F_i± for some positive numbers F_i±, i∈{1, …, j}. Thenϕ={ A_0+B_0on{∈ [-_1, _1]}, A_1+B_1on{∈ [_1, _2]},…A_j + B_jon{∈ [_j, a)}, .,where the coefficients A_i, B_i, satisfy the recursive equations A_0 = 1 A_i = A_i-1- ϕ(_i)(F_i++F_i-)(_i),B_0=0 B_i = B_i-1 + ϕ(_i)(F_i++F_i-)(_i) ( 0< i ≤ k) . Clearly A_0 =1 and B_0 = 0, since ϕ(0) =1 and ϕ extends evenly across = 0.Now fix i∈{1, …, j}.By Remark <ref>,ϕ = ϕ(_i)H^-[ F_i-; _i] on {∈ [_i-1, _i]} and ϕ = ϕ(_i)H^+[ F_i+; _i] on {∈ [_i , _i+1]} and explicitly,ϕ(_i)H^+()= ( F_+^(_i)- F_i+)ϕ(_i) (_i)()+( - F_+^(_i)+F_i+)ϕ(_i) (_i)() ϕ(_i)H^-()= ( F_+^(_i)+ F_i-)ϕ(_i) (_i)()-(F_+^(_i)+F_i-)ϕ(_i) (_i)().Subtracting the second of these equations from the first yields (<ref>).Given _1 ∈ (0, _root) and = (, )=( (σ_i )_i=1^∞, (ξ_j)_j=1^∞ )∈ℓ^1(^) ⊕ℓ^∞( ^)satisfying | |_ℓ^∞ < 1/10,there is a unique unit RLD solution= [_1; ]satisfying the following.(a)._1^=_1.(b).^=. |_k wherek= k[_1; ]∈ is the number of jump latitudes of(recall <ref>) and . |_k := ( (σ_i )_i=1^k-1, (ξ_j)_j=1^k)∈^k-1×^k. Moreover the following hold. * k[_1; ] is a nonincreasing function of _1. Further, for eachas abovethere exists a decreasing sequence {a_0, := _root, a_1, , …} such that k[_1; ] = k if and only if _1 ∈ [a_k, , a_k-1, ).Moreover each a_k, depends only on. |_k (defined as above).* _2^, …, _k^ are increasing smooth functions of _1 for fixed . * [a; ] is smooth at the poles if and only if a = a_k, for some k≥ 1.* The restriction of [_1; ] on any compact subset of [0, ∞)depends continuously on _1 and .We first prove that there is at most one unit RLD solutionsatisfying (a) and (b).By the symmetries, it follows that = on {∈ [-_1, _1]}. Furthermore, by Remark <ref> and Lemma <ref>,has a unique local extension beyond _1. Next, the flux monotonicity—Lemma <ref>—and the requirementin Definition <ref>.(i) that >0,inductively determine all of the jump latitudesuniquely using <ref>. From this,is determined uniquely by <ref>. We next construct a family [_1; ] of RLD solutions,parametrized by _1 or equivalently F^_1, with flux ratios . By Lemma <ref>, the restriction . F^_-|_(0, _root): (0, _root)→ (0, ∞) is an orientation-preserving homeomorphism. Because any unit RLD solution coincides withon {∈ [0, _1]},it follows there is a 1-1 correspondence between choices of_1 ∈ (0,a_0, := _root) and F_1 = F_1(_1) : = 2/1- ξ_1 F^_-(_1)∈ (0, ∞).Let _1 ∈ (0, _root). By Remark <ref> and Lemma <ref>, there is a unique extension [_1; ] of. |_{∈ (-_1, _1)} to a maximal domain {∈ (-a, a) } such that the hypotheses of Lemma <ref> hold,where the coefficients F_i± (recall the notation of Lemma <ref>)are defined by (recall (<ref>))F_1± := 1/2(1±ξ_1)F_1 ,F_i± := 1/2 (1±ξ_i) ( e^∑_l=1^i-1σ_l) F_1,i>1. To show thatis an RLD solution, we must show that a = ∞ andhas finitely many jump latitudes. By Remark <ref> and Lemma <ref>,2 F_1∼_exp(||_ℓ^1+3||_ℓ^∞)( F_i+1- + F_i+)∼_exp(||_ℓ^1+3||_ℓ^∞)(2( tanh_i+1 - tanh_i) + ∫__i^_i+1( F^_-())^2d).This implies a lower bound on _i+1 - _i which is uniform in i.Therefore a = ∞.We next show there are finitely many jump latitudes by estimating an upper bound for _k.Supposehas a jump at _j+1.On {∈ (_j, _j+1)},coincides with(_j) H^+[F_j+; _j] = ( F^_+(_j) - F_j+) (_j)(_j) () +( - F^_+(_j) + F_j+) (_j)(_j) ().Sincehas a jump at _j+1, it follows that F^_+ (_j) > F_j+. Since F^_+ ()= ^2/tanh↘ 0 as→∞,and F_j+∼_exp(||_ℓ^1+3||_ℓ^∞) F^_-(_1), this implies an upper bound for _k-1 depending only on _1 and .This establishes the existence of the family of RLD solutions [_1; ].We next prove (i).From Definition <ref> and Lemma <ref>, it follows that f_1():=F^_+() - 1+ξ_1/1-ξ_1 F^_-() is monotone on (0, _root) and moreover satisfieslim__1↘ 0 f_1() = ∞, andlim__1 ↗ a_0,f_1(_1) = - ∞. We then define a_1, to be the unique root of f_1 in (0, a_0, ).By (<ref>), a_1, depends only on ξ_1.There are three cases.Case 1: _1> a_1,.It follows from Remark <ref>, (<ref>), and (<ref>) that A_1(_1)< 0.Since B_1>0 by Lemma <ref>, the monotonicity ofandimply that ∂ϕ>0 for all >_1. Hence k[_1; ] = 1.Case 2: _1 = a_1,.Then A_1(_1) = 0. By Lemma <ref>, ϕ coincides with a positive multiple ofon a maximal domain to the right of _1, which must be {∈ (_1, ∞)}since F^_+ is strictly decreasing and lim_→∞F^_+() = 0.Consequently [a_1, ; ] is smooth at the poles and k[a_1, ; ] =1. Case 3: _1 < a_1,.Then A_1 >0.Sinceis monotone and lim_↗∞∂()= -∞, there exists _1max∈ (_1, ∞) with the property that ∂()>0 when ∈ (_1, _1max) andattains a local maximum at _1max.In particular, F^_-(_1max) = 0.By the flux monotonicity, there exists _2 ∈ (_1max, ∞)such that (_1)H^+[F_1+; _1](_2) = F_2-. Sinceremains positive, _2∈ L_par andhas its second jump latitude at _2.We next prove _2(_1), _3(_1),… are increasing functions of _1.We first show this for _2(_1).By Lemma <ref>, F^_1 is an increasing function of _1.By <ref>, F^_2- = 1-ξ_2/2 e^σ_1 F^_1.By combining this with both parts of Lemma <ref>, it follows that _2(_1) is strictly increasing. Using this fact and replacing _1 by _2 and _2 by _3 in the preceding argument shows that _3(_1) is strictly increasing, and inductively in the same way _j(_1) is strictly increasing for 2< j≤ k.This proves (ii).By the discussion above, [a_1, ; ] is smooth at the poles and k[_1; ] = 1 for _1∈ I_1. By a straightforward inductive argument,there are unique a_2, >a_3, > …> 0,where a_j, depends only on the first j-1 entries ofand the first j entries of , and intervals[a_i, , a_i-1, ), j ∈ such that for each i≥ 1, [a_i, ; ] is smooth at the poles, k[a_i, ; ] = i, and k[_1; ] = i when _1∈ [a_i, , a_i-1, ). This concludes the proof of (i) and (iii). (iv) follows from the continuous dependence in of the coefficients in the conclusion of Lemma <ref> on _1 and the continuous dependence of the _2, _3, … on _1 from (ii).One construction in <cit.> produces surfaces whose configurations consist of pointson two parallel circles with latitudes whose absolute values are close to (in the notation of <cit.>) _balanced∈ (0, _root). Using (<ref>) to relateandcoordinates, we have that _root = cos_root and (a_1, ) = cos_balanced.If = [_1; ] is as in Proposition <ref>.(iii),recall from<ref>.(ii) that ^_i is an increasing function of _1. In Lemma <ref> and Corollary <ref> we estimate the derivatives precisely. Let = (, ),_1 ∈ (0, _root), = [_1; ],and = ^[ _1; ] be as in Proposition <ref>. Let k∈. has at least k jumps if and only if_1 ∈ (0, a_k-1, ).The kth jump latitude _k depends then only on _1 and . |_k. _k can be considered as a smooth function defined on the domain S_k⊂×^k-1×^k where S_k:= {(_1 , (σ_i )_i=1^k-1, (ξ_j)_j=1^k) :_1 ∈ (0, a_k-1, )and|ξ_j|<1/10forj=1,…,k }. Alternatively we can consider each _k as a smooth function ofF_1=F_1^ and . |_k and then we have for k=1 (where _1 is a function of F_1 and ξ_1 only) (2 ^2 _1 + ( F^_1-)^2)∂_1 /∂ F_1 = 1/2(1 - ξ_1),(2 ^2 _1 + ( F^_1-)^2)∂_1 /∂ξ_1 = - 1/2 F_1, and for k>1 the recursive formulas (note S_k⊂ S_k-1) ( 2^2 _k + (F^_k-)^2)∂_k/∂ F_1 =( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂ F_1( ϕ(_k-1)/ϕ(_k))^2++ ( ϕ(_k-1)/ϕ(_k))^21/2 (1+ ξ_k-1) ( e^∑_l=1^k-2σ_l)+1/2(1- ξ_k) ( e^∑_l=1^k-1σ_l) , ( 2^2 _k + (F^_k-)^2)∂_k/∂ξ_j =( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂ξ_j( ϕ(_k-1)/ϕ(_k))^2++( ϕ(_k-1)/ϕ(_k))^2δ_j(k-1)/2( e^∑_l=1^k-2σ_l)F_1-δ_jk/2( e^∑_l=1^k-1σ_l) F_1, ( 2^2 _k + (F^_k-)^2)∂_k/∂σ_j =( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂σ_j( ϕ(_k-1)/ϕ(_k))^2++( ϕ(_k-1)/ϕ(_k))^2F_1/2(1+ξ_k-1)∂/∂σ_j( e^∑_l=1^k-2σ_l)- F_1/2(1- ξ_k) ∂/∂σ_j( e^∑_l=1^k-1σ_l) .The first statement follows directly from Proposition <ref>.Below, we compute partial derivatives of _k with respect to F_1, _k-1, and the entries of .|_k, from which the smoothness claimed follows immediately.To this end, we recall from (<ref>) and <ref>, thatϕ = ϕ(_k-1) H^+[ 1/2(1+ ξ_k-1)( e^∑_l=1^k-2σ_l)F_1; _k-1] on{∈ [_k-1, _k]}. Denoting H^+ as in (<ref>) by H for simplicity and using (<ref>), we findF^H_-(_k) = 1/2(1-ξ_k)( e^∑_l=1^k-1σ_l)F_1.Items (<ref>)-(<ref>) then follow by using the chain rule to differentiate (<ref>) and Lemma <ref> and both parts of <ref> to calculate the partial derivatives of F^H. There is a constant _1>0 such that for all k∈, k>1 and all = (, ) ∈^k×^k+1 satisfying ||_ℓ^1+ ||_ℓ^∞< _1 and _1 ∈ (a_k+1, , a_k, ), = [_1; ] satisfies the following. * F^_ < C/k.* 1- () : C^0( {∈ [-_k, _k]})< C/klog k and | log(_i)/(_i-1)| < C/k for 2≤ i ≤ k.Using the monotonicity of the flux in lemma <ref> we conclude that the maximum of |F^_-| is achieved at the jump latitudes.Using also <ref> we conclude max_∈[0,∞) |F^_-()| =max_1≤ i≤ k+1F^_i±∼_exp(||_ℓ^1+3||_ℓ^∞) F^_avg∼_exp(||_ℓ^1+3||_ℓ^∞)min_1≤ i≤ k+1 F^_i±.By <ref> and the fact thathas k+1 jumps, it follows that F^_+(_k) = ^2 _k/tanh_k > F^_k+.From this and <ref> we estimateF^_avg < C ^2 _k.By using Lemma <ref> on the intervals {∈ [ 0, _1]}, …{∈ [_k-1, _k]} and summing, we findF^_1-+ F^_1++ ⋯ + F^_k- = 2tanh_k + ∫_0^_k( F^_-())^2 d.Next using <ref> and (<ref>) and (<ref>) to estimate (<ref>), we find(2k-1)F^_ < C(1+ _k ^2 _k)< C,from which we conclude (i).For (ii), we observe from <ref> that on any domain on whichis smooth, F^_+() = ∂( log).Integrating from _i-1 to _i, using the easy to see fact that for 2≤ i ≤ k, 0<_i-_i-1 <C, and estimating using part (i), | log( (_i)/(_i-1))| ≤∫__i-1^_i|F^_+()| d≤C/k∫__i-1^_i d≤C/k.Let = (, ), _1 ∈ (0, _root), = [_1; ], and = ^[_1;] be as in <ref> and suppose moreover that ||_ℓ^∞<_1/k and that(_1, . |_k+1) ∈ S_k+1 (recall <ref>).The following estimates hold. * ( 2^2 _k + (F^_k-)^2)∂_k/∂ F_1∼_C k.* | ( 2^2 _k + (F^_k-)^2)∂_k/∂σ_i | < C, i=1, …, k-1.* |( 2^2 _k + (F^_k-)^2)∂_k/∂ξ_j| < C/k, j=1, …, k.We first prove (i).To simplify notation in this proof, we denoteP_k:= ( 2 ^2 _k + ( F^_k-)^2) ∂_k/∂ F_1, Q_k-1 := ( (_k-1)/(_k))^2, R_k :=2 ^2 _k + ( F^_k+)^2/ 2 ^2 _k + ( F^_k-)^2,T_k-1 :=1/2 Q_k-1 (1+ ξ_k-1) ( e^∑_l=1^k-2σ_l)+1/2(1- ξ_k) ( e^∑_l=1^k-1σ_l).In this notation, (<ref>) from Lemma <ref> is equivalent to the equationP_k =R_k-1 Q_k-1P_k-1 + T_k-1from which we conclude by applying (<ref>) recursivelyP_k = P_1 ∏_i=1^k-1 Q_i R_i + ∑_i=1^k-1( T_i ∏_j=i+1^k-1 Q_j R_j).From (<ref>) it follows that P_1 = 1/2(1 - ξ_1).By <ref>, the assumption ||_ℓ^∞<_1/k, and <ref>, the following estimates hold:Q_i ∼_1+ C/k 1,R_i ∼_1+C/k,T_i ∼_C 1,i=1, …, k-1.Combining (<ref>) with (<ref>) completes the proof of (i).Proofs of (ii) and (iii) are similar and use respectively (<ref>) and (<ref>) in place of (<ref>), so we omit the details. We are particularly interested in RLD solutions which are smooth at the poles.Recall from <ref> that[a_k,;]is the unique smooth at the poles unit RLD solution with k jump latitudes and flux ratios . Moreover by <ref>.(i),[a_k,;]depends only on. |_k.This motivates the following. Given k, , and. |_k as in <ref>,we define [: k]:=[. |_k : k]:=[a_k,;],= [: k ] :=[. |_k : k ] :=^[: k] .Fixas in Proposition <ref> and let= [: k] be as in <ref>.Then* For fixed i, _i = _i[ :k ] is a decreasing function of k.* If =, and i is fixed, _k-i = _k-i[ : k] is an increasing function of k. (i) follows from parts (i) and (ii) of Proposition <ref>.(ii) follows from a dual version of Proposition <ref> where one builds [ : k] “from infinity" by taking the final flux F^_+(_k) as initial data rather than the first flux F^_-(_1) as in the proof of <ref>.We omit the details of the argument because we do not use part (ii) in the remainder of the paper.We next establish estimates for smooth at the poles RLD solutions with small flux ratios.Below, if we write f = O(w) for a number f and some w>0, we mean |f| ≤ C w, for some constant C independent of f and w. There is a constant _1>0 such that for all k∈ and all = (, )∈^k-1×^kwith ||_ℓ^1+ ||_ℓ^∞ < _1, =[: k] satisfies the following. * F^_avg = 1/k +O(1/k^3) and ^2 _k ∼_C 1/k.* * F^_1-=2tanh_1 +O(1/k^3). * F^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) +O(1/k^2(k-i+1)) for i=2, …, k. * F^_k+=2(1- tanh_k) + O(1/k^2). * 1-(): C^0() ≤C/klog k and|log(_i)/(_i-1)| ≤C/k for 2≤ i ≤ k.If =, then also 1>(_1)> ⋯ >(_k).* There is a constant C>0 depending only on _1 such that for any 1≤ i<j≤ k, _j - _i>(1+C/k) j-i/2kand^2 _i - ^2 _j ≤C(j-i)/k. The estimates in Proposition <ref> are substantive only when k is large.When k is small, by taking _1 small enough, using Proposition <ref>, and smooth dependence of ODE solutions on initial conditions, we maintain coarse control over the family [: k].More precisely we have the following alternatives for small k:given > 0 and taking _1 small enough in terms of , there exists a constant c = c()> 0 such that for all k∈, k <, (i). 0< 1/c<< c.(ii). 1/c< _1 and for i∈{2, …, k}, _i - _i-1∼_c 1. (iii). For i∈{1, …, k},F^_i±∼_c 1. Sinceis smooth at the poles,F^_k+= F^_+(_k) = ^2 _k/tanh_k.Using this with (<ref>) and estimating a uniform lower bound for tanh_k, we conclude ^2 _k ∼_Cexp (||_ℓ^1+||_ℓ^∞) F^_avg.By fixing _1 sufficiently small, we can ensure the constant C in (<ref>) is less than 10 and also the constants in the estimates in the statements (i)-(iii) are independent of k.Using Lemma <ref>, we have (where i=2, …, k)F^_1- =2tanh_1 +∫_0^_1(F^_-())^2d F^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) + ∫__i-1^_i(F^_-())^2d, F^_k+ =2(1-tanh_k) + ∫__k^∞(F^_-())^2d.Note ∫__i-1^_i^2 /^2 d = ∫__i-1^_i1/^2d( tanh)< C(tanh_i - tanh_i-1)/F^_ and also that ∫__k^∞( F^_-())^2 d = ∫__k^∞^4 /tanh^2d. Using (<ref>) and (<ref>) we estimate the integrals to obtain(where i=2, …, k)F^_1- =2tanh_1 +_1 O((F^_avg)^2) F^_i-+F^_i-1+ =2(tanh_i - tanh_i-1) +(tanh_i - tanh_i-1)O(F^_avg), F^_k+ =2(1-tanh_k) + O((F^_avg)^2).By summing the estimates in (<ref>) and dividing through by k,we find that F^_avg = 1/k - 1/kO(F^_avg), which implies that F^_avg = 1/k+O(1/k^2).Substituting this estimate for F^_avg into (<ref>), we find 2tanh_1 ∼_C 1/k, tanh_i - tanh_i-1∼_C1/k, i=2, …, k,hencetanh_i ∼_C i/k,i=1, …, k.Summing the estimates in (<ref>) with these improved bounds yieldskF^_avg = 1+O(1/k^2), and the first part of (i) follows.The second statement in (i) follows from the first and (<ref>). Substituting (i) into the first and last parts of (<ref>) yields (ii).(a) and (ii).(c).For (ii).(b), we substitute (i) into the error term of the second part of (<ref>) to getF^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) + 1/^2 _iO( 1/k^3).When i ≤ k/2, (ii).(b) follows immediately from (<ref>) because (<ref>) implies ^2 _i is bounded from below by a constant independent of k.On the other hand, when j≥ k/2, (<ref>) implies tanh_j ≥ C, so from this and (<ref>),^2 _j-1 - ^2 _j = (tanh_j - tanh_j-1)(tanh_j +tanh_j-1) ≥C/k.Using(<ref>) and summing (<ref>) and, we find for i≥ k/2^2 _i ≥C(k-i+1)/k.Substituting this bound in (<ref>) completes the proof of (ii).(b).We now prove (iii).From Definition <ref>, on any domain on whichis smooth,F^_+() = ∂( log).Integrating (<ref>) on intervals whereis smooth and adding, we find that for any ∈ [0, _k],|log() |≤∫_0^| F^_+()|d≤C/k∫_0^_kd≤C/klog k,where we have used (<ref>) in combination with part (i) above to estimate _k = O(log k).Sinceis smooth at the poles, it coincides with a multiple of = tanh on {∈ (_k, ∞)} and so in particular the estimate in the first part of (iii) holds on {∈ (_k, ∞)}.For the second statement, it is easy to see that for 2≤ i ≤ k, 0<_i-_i-1 <C.Then estimating in the same way as above, we get| log( (_i)/(_i-1))| ≤∫__i-1^_i|F^_+()| d≤C/k∫__i-1^_i d≤C/k.Next, suppose that =.Fix i∈{ 1, …, k-1}. Integrating (<ref>) from _i to _i+1, we find that log((_i+1)/(_i))= - ∫__i^_i+1 F^_-()d.Sinceis balanced, F^_i+1- = F^_i+:= F_0 and the restriction of F^_- to [_i, _i+1] satisfies.F^_-|_[_i, _i+1]: [_i, _i+1]→ [-F_0, F_0]anddF^_-/d = 2 ^2 + F^_-()^2>0.In particular, F^_-|_[_i, _i+1] is invertible.Reparametrizing the integral in (<ref>) by (F^_-|_[_i, _i+1])^-1, we havelog((_i+1)/(_i))= - ∫_-F_0^F_0Fd/dF dF= -∫_-F_0^0 F/2^2 +F^2dF-∫_0^F_0 F/2^2 +F^2dF<0,since the first integral term is positive, the second is negative, and ^2 is monotonic in F by Lemma <ref>.This completes the proof of (iii). For (iv), assume 1≤ i<j≤ k.By summing instances of (ii).(b) above, we findF^_i++⋯ +F^_j- = 2(tanh_j - tanh_i) + O( j-i/k^2(k-i+2)).Using (<ref>), we estimate(j-i) exp(-||_ℓ^1-3||_ℓ^∞)F^_≤(tanh_j - tanh_i) + O( j-i/k^2(k-i+2)).Applying the mean value theorem to the right hand side of the preceding gives(j-i) exp(-||_ℓ^1-3||_ℓ^∞)F^_ < _j - _i + O( j-i/k^2(k-i+2))and the first statement in (iv) follows from combining (<ref>) with (i) above and taking ||_ℓ^1 and ||_ℓ^∞. small enough.The second follows from combining (<ref>) with parts (i) and (ii) above.Proposition <ref> implies that the latitudes of L_par of RLD solutions which are close to being balanced arrange themselves in a regular way.Indeed, recalling from (<ref>) that tanh = sin, Proposition <ref>.(i) and (ii) together imply that when =,sin_1= 1/2k + O( 1/k^3), sin_i - sin_i-1 = 1/k + O( 1/k^2(k-i+1)),i=2, …, k-1, 1 - sin_k= 1/2k + O(1/k^2).Together with elementary geometric facts about spheres, this means that for large k * The (extrinsic ^3) distance between planes corresponding to adjacent circles in L_par is approximately 1/k. * For i=2, …, k, the area of the annulus {∈ (_i-1, _i) }⊂ is approximately 2π/k. Since the LD solutions we use to construct initial surfaces have configurations arising(in a way made precise later in Lemma <ref>)from RLD solutions which are approximately balanced,the circles of L_par in the minimal surfaces we ultimately construct are also approximately equally spaced. Let k∈ and suppose that = (, )∈^k-1×^k , '= (', ') ∈^k-1×^k satisfy ||_ℓ^1+ ||_ℓ^∞< _1/k and |'|_ℓ^1+ |'|_ℓ^∞< _1/k.Let = [ : k] and ' = [ ': k].There is a constant C>0 independent of k such that: * | ^' - ^|_ℓ^∞≤C/k(| ' - |_ℓ^1+ |' - |_ℓ^∞).* max_1≤ i ≤ k| tanh'_i - tanh_i | ≤C/k(| ' - |_ℓ^1+ |' - |_ℓ^∞).Observe that (i) follows from the estimate| F^'_1 - F^_1 | ≤C/k(|' -|_ℓ^1 + |' - |_ℓ^∞),because for any i∈{1, …, k} we may estimate (assuming (<ref>) for a moment)2| F^'_i+ - F^_i+|=| (1+ ξ'_i ) ( e^∑_l=1^i-1σ'_l)F^'_1 - (1+ ξ_i)( e^∑_l=1^i-1σ_l)F^_1| ≤|(1 + ξ'_i)( e^∑_l=1^i-1σ'_l)-(1 +ξ_i)( e^∑_l=1^i-1σ_l) | F^'_1 + (1 + ξ_i)( e^∑_l=1^i-1σ_l) | F^'_1 - F^_1| ≤C/k(|' -|_ℓ^1 + |' - |_ℓ^∞)and a corresponding bound for | F^'_i- - F^_i-| follows in an analogous way.We now prove (<ref>).Fix k∈, let _1 be as in <ref>, andconsider the mapdefined by( F_1,) = F^[ _1; ]_+(_k) - F^_+(_k),where [ _1; ] is as in <ref> and _1 is chosen so F^_1 = F_1.Clearly, ( F_1, ) = 0 if and only ifis smooth at the poles.Now let ∈^-1({0}) be arbitrary.It follows from Lemma <ref> and <ref> thatis C^1; below we estimate the partial derivatives ofat .Differentiating (<ref>) with respect to F_1 and using (<ref>) and <ref>, we compute. ∂/∂ F_1 |_ = 1/2(1+ ξ_k) ( e^∑_l=1^k-1σ_l)+ (2 ^2 _k + (F^_k+)^2) .∂_k/∂ F_1|_.By combining (<ref>) with Corollary <ref>, we estimate that for j∈{1, …, k} and i∈{1, …, k-1},.∂/∂ F_1|_∼_C k, | .∂/∂σ_i|_| ≤ C, |. ∂/∂ξ_j|_| ≤C/k,where C>0 is a constant independent of k.By the implicit function theorem, locally around (F_1, ) ∈^-1( {0}), ^-1({ 0 }) is a graph overand moreover (abusing notation slightly), for i∈{1, …, k-1} and j∈{1, …, k}.∂ F_1/∂σ_i|_ = - (. ∂/∂ F_1|_)^-1.∂/∂σ_i|_, .∂ F_1/∂ξ_j|_= - ( .∂/∂ F_1|_)^-1. ∂/∂ξ_j|_.(i) follows by combining this with the estimates (<ref>).The proof of (ii) is similar to the proof of (i), but we omit it since we will not use (ii) in the remainder of the paper.§ LINEARIZED DOUBLING (LD) SOLUTIONS §.§ LD solutionsabGiven a finite set L⊂ and a function τ:L→, we define a linearized doubling (LD) solution of configuration (L,τ) to be a functionφ∈ C^∞(∖ L) which satisfies the following conditions,where τ_p denotes the value of τ at p: * φ=0 on ∖ L. * ∀ p∈ L there is a smooth extension across {p},_p∈ C^∞( {p}∪(∖ (L∪{-p}) ) ),such that _p=φ-τ_p G^^2∘^g_pon∖ (L∪{-p}), where G^^2∘^g_pis as in <ref>. By Lemma <ref>,the class of LD solutions and each corresponding τ_p,do not depend on which Green's function we use in Definition <ref>.(ii) above.Note however thatand its domain do.Given L and τ as in Definition <ref>,and also a finite dimensional space [L] ⊂ C^∞(),we define a linearized doubling (LD) solution of configuration (L,τ,w) to be a functionφ∈ C^∞(∖ L) which satisfies the same conditions as in <ref>,except that condition (i) is replaced by the following:abc(i'). φ=w∈[L] ⊂ C^∞() on ∖ L. Note that LD solutions in the sense of Definition <ref> are LD solutions with w=0 in the sense of <ref>. Existence and uniqueness of -symmetric LD solutions modulo [L] is a consequence of the symmetries of the construction: We assume given -invariant L,τ,w,where L= L[;m] is as in <ref>, τ:L→, and w∈[L]. There is then a unique -invariant LD solution modulo [L]of configuration (L, τ, w), φ = φ[L, τ, w].Moreover, the following hold. * φ and _p each depend linearly on (τ, w). * φ_∈ C^0(∖(L∩{p_N,p_S}) ) and φ_ is smooth on()^where it satisfies the ODE φ_=0. If w=0, then we also write φ = φ[L; τ] andφ is the unique -invariant LD solution of configuration (L, τ) as in <ref>. This is Lemma 3.10 of <cit.>.§.§ Estimates for LD solutions on the cylinderab The next lemma asserts a balancing law for LD solutions φ which restricts their averages φ_avg. Suppose φ = φ[L[; m]; τ] is as in<ref>, where L[; m] and τ are as in <ref>. Then m τ_i = φ_avg(_i) F^φ__i,i=1, …, k. The proof is similar to the proof of <cit.>. Fix i ∈{1, …, k}. For 0<ϵ_1<<ϵ_2 we consider the domainΩ_ϵ_1,ϵ_2:= D^χ_L_par[_i](ϵ_2)∖D^χ_L_i(ϵ_1). By integrating _χφ=0 on Ω_ϵ_1,ϵ_2 and integrating by parts we obtain∫_∂Ω_ϵ_1,ϵ_2∂/∂ηφ+ 2 ∫_Ω_ϵ_1,ϵ_2( ^2 ) φ = 0,where η is the unit outward conormal vector field along ∂Ω_ϵ_1,ϵ_2.By taking the limit as ϵ_1→ 0 first and then as ϵ_2→ 0 we obtain using the logarithmic behavior near L that m τ_i = ∂_+ φ_avg(_i) + ∂_-φ_avg(_i). (<ref>) follows from(<ref>) after appealing to the definition ofF^φ_avg_± (recall <ref> and <ref>.(iii)). We assume from now on k, m∈ are fixed and m satisfies m > where = (k)>0 is a constant which can be taken as large as necessary in terms of k. In the next lemma we convert RLD solutionsto LD solutions Φ whose non-oscillatory partin the sense of <ref> is a multiple of .The overall scale of the LD solutions used is determined later in <ref>, while at the moment we only fix their scale by a suitable normalizing condition.The purpose of the rest of this section is to understand and estimate carefullyΦ by decomposing it to well described partsand ,and an error termwhich is estimated in Proposition <ref>. Given = (, )∈^k-1×^k as in <ref>and m as in <ref>,there is a unique -invariant LD solution (recall<ref>)Φ = Φ [ : k,m] := φ[L; τ'],characterized by the requirements that * ϕ = ϕ[: k,m]:=Φ_ is a multiple of [ : k] * L = L[[ : k ]; m ] (recall <ref>)* τ'_1 = 1 (normalizing condition). Moreover,the following hold.* For i∈{1, …, k}we haveτ_i' = ϕ(_i)/mF^ϕ_i.Moreover τ_i' is independent of m and satisfiesτ_i' = τ_i' [ : k] :=[: k](_i)/[: k ](_1)( e^∑_l=1^i-1σ_l).* ϕ[: k,m] =m/ [:k](_1)F^[:k]_1[: k ].* On Ω[_i;m], ϕ = _i + _i, where_i:= [ ϕ(_i), ; _i] =ϕ(_i) [1, 1/2(F^ϕ_+(_i) - F^ϕ_-(_i)) ; _i], _i:= [τ'_i/2; _i].Suppose Φ is as in (<ref>) and satisfies (a)-(c). Let c be such that ϕ=c andi∈{1, …, k}.Using Lemma <ref> to solve for τ'_i, we immediately conclude τ'_i = ϕ(_i) F^ϕ_i /m;furthermore using Lemma <ref>, (a)-(c) above, and (<ref>), we compute τ'_i= τ'_i/τ'_1 = ϕ(_i)/ϕ(_1)F^ϕ_i/ F^ϕ_1 = (_i)/(_1) F^_i/F^_1 = (_i)/(_1)( e^∑_l=1^i-1σ_l) ,1= τ'_1 = ϕ(_1)/m F^ϕ_1 =c (_1)/m F^_1.We conclude from these equations that (a)-(c) imply (i) and (ii). In particular, the second equation in (i) determines τ' and hence uniqueness follows from Lemma <ref>. To prove existence we define L by (b) and τ' by the second equation in (i).We then define Φ by (<ref>) and we verify that Φ_ = c, where c is defined by c (_1)F^_1 = m:Let i∈{1, …, k}. By Lemma <ref>, it follows that m τ'_i = Φ_(_i) F^Φ__i. By the definitions of τ'_i and c, we have m τ'_i = c (_i ) F^_i. Since F^_i = F^c_i, by equating the right hand sides of the preceding equations,we conclude that the function f := c - Φ_avg satisfies ∂_+f(_i)+∂_-f(_i) = 0,i=1, …, k.This amounts to the vanishing of the derivative jumps of f at each _i. Clearly f is smooth at the poles and satisfies _χ f = 0 in between the _i. Hence we conclude f∈ C^∞() and satisfies f = 0 everywhere. By the symmetries of f, we conclude f = 0.It remains to check (iii). The vertical balancing equation (<ref>) impliesm τ'_i =∂_+ ϕ(_i) + ∂_-ϕ(_i),so from the definition ofin <ref>,∂_+_i (_i)= ∂_-_i (_i)= ∂_+ϕ+∂_-ϕ/2(_i). Therefore, ϕ - _i satisfies∂_+ (ϕ - _i)(_i) = ∂_+ ϕ - ∂_- ϕ/2(_i) = - ∂_-(ϕ - _i)(_i).Hence, ϕ - _i∈C^1_(Ω[_i;m]) and _χ(ϕ-_i)= 0. By uniqueness of ODE solutions,ϕ - _i = _i.The second expression in (<ref>) follows from this by <ref>.For i=1, …, k we define numbers A_i: = τ'_i logδ (recall (<ref>)). We define ∈ C^∞_(∖ L) by requesting thatit is supported on D^χ_L(3δ) where it is defined by =[2δ, 3δ; ^χ_p_i](τ'_i G_p_i - [A_i,0; _i],0) on D^χ_p_i(3δ), for i=1, …, k,where the discs D^χ_p_i(3δ) are disjoint byAssumption <ref> and Proposition <ref>.The following hold. * : C_^j(∖ D^χ_L(δ) ,)≤ C(j) andis supported on D^χ_L(3δ)∖ L. * : C_^j( D^χ_L(3δ)∖ D^χ_L(2δ) , ) ≤ C(j)/m.The statement in (i) on the support follows from Definition <ref> and the corresponding estimate follows from Lemma <ref> and Definition <ref> (observe the scale is such that the cutoffhas all derivatives bounded in themetric).For (ii), becauseis supported on D^χ_L(3δ)∖ L, it will suffice to prove the estimate : C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ), )≤ C(j)/m.Recalling Definition <ref>, we have on D^χ_p_i(3δ)∖ D^χ_p_i(2δ)=[2δ, 3δ; ^χ_p_i](τ'_i G_p_i, 0) - [2δ, 3δ; ^χ_p_i]([A_i, 0; _i], 0):= (I)+(II).From Lemma <ref>.(iii) and the uniform bounds on the cutoff , we have (I):C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ) , ) ≤ C(j)/m.By Lemma <ref>.(iii), (II):C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ) , ) ≤ C(j) (log m) m^-3, and these estimates complete the proof of (ii).By Definition <ref> and Lemma <ref>,we have for each p_i ∈ L that = τ'_iG_p_i- [A_i, 0; _i] on D^χ_p_i(2δ). Using this,Definition <ref>.(ii) and (<ref>), we see that Φ- can be extended smoothly across L.In the next subsection, we will estimate Φ. The key tool is that Φ is well approximated by ϕ away from L. ϕ is well understood by the analysis in Section <ref>, in particular by <ref>, <ref> and <ref>. Lemma <ref> motivates the definition of a smooth rotationally invariant function ,which is a dominant term in the decomposition of Φ in <ref>. Note that Assumption <ref> and Proposition <ref> imply that Ω[_i;m] ∩Ω[_j;m] = ∅ when i≠ j. We define ∈ C^∞_||( ) by requesting that :=[ 2/m, 3/m; ^χ_[_i]] ( _i, ϕ) = ϕ - [ 2/m, 3/m; ^χ_[_i]] (_i, 0) on Ω[_i;m], i=1, …, k (recall (<ref>), (<ref>)) and that = ϕ on ∖ D^χ_L_par(3/m). We are now ready to define a decomposition Φ =++. The third termwe treat as an error term to be estimated (see Proposition <ref>). We define Φ', E'∈ C^∞_() by requesting that on ∖ L(recall <ref> and <ref>),Φ=++,E'=-_( + ).The decomposition in Definition <ref> in some way combines the decomposition Φ =+ Φ”<cit.> with the semilocal decomposition Φ” =+ [ϕ_1 - A_1, (_1); _1] <cit.>.There are the following differences: * Our definition oftransits to zero away from L, whereas thein <cit.> transits to A_1. * Here,is defined globally, whereas in <cit.>,is defined only on Ω_1[_1;m].§.§ Estimatingand ab We estimate the average and oscillatory parts of Φ separately.Taking averages of both sides of the equation Φ =+ + (recall Definition <ref>) we have _avg, _osc, E'_avg, E'_osc∈ C^∞_sym() and = _avg + _osc,E'=E'_avg+E'_oscon.Furthermore, since _χΦ vanishes by Definition <ref> and _χ is rotationally covariant,_ = E', __ = E'_, __ = E'_on. Because _χϕ =_χ = 0, it follows that E'_ = __ is supported on (D^χ_( 3/m) ∖ D^χ_( 2/m)) ⋃ D^χ_( 3δ). We have the following characterization of _avg. _ is supported on D^χ_(3/m).On Ω[_i;m], i=1, …, k,_avg = [ 2/m, 3/m; ^χ_[_i]] ( _i, 0 ) , onΩ[_i;m]∖Ω'[_i;m] _i -_, onΩ'[_i;m].By taking averages of the equation Φ =++ and rearranging, we find_ = ϕ -- _.Equation (<ref>) follows from this decomposition after substituting the expression forfrom <ref> and recalling that = 0 on Ω[_i;m]∖Ω'[_i;m].The decomposition Φ = ++ is designed so thatis small in comparison to(cf. Proposition <ref> below).To estimate , we will use that it satisfies the equation _ = E'.First we establish relevant estimates for E', E'_avg and E'_osc. E' vanishes on D^χ_L(2δ) and E'_ is supported on D^χ_(3δ). For each i∈{1, …, k}, the following hold. * * E':C_^j( Ω[_i;m] ,)≤ C(j)* E'_:C_^j(Ω[_i;m],)≤ C(j)* E'_:C_^j( Ω[_i;m],)≤ C(j) * * E' :C_^j(D^χ_[_i](3δ) , )≤ C(j)/m* E'_ :C_^j( D^χ_[_i](3δ), )≤ C(j)/m * E'_ :C_^j( D^χ_[_i](3δ), )≤ C(j)/m.The statements on the support of E' and E'_ follow from <ref>, <ref>, and <ref>. Next, note that parts (b) and (c) of items (i) and (ii) follow from part (a) of the respective items by takingaverages and subtracting, so it will suffice to prove part (a) of (i) and (iii). It follows from <ref> that on Ω[_i; m], E' = - _(+ ).On Ω[_i; m]∖Ω'[_i;m], it follows from this, <ref> and <ref> thatE' = _[ 2/m, 3/m; ^χ_[_i]] ( _i, 0 ).Thus, when restricted to Ω[_i; m]∖Ω'[_i;m], the bound in (i).(a)follows from <ref> and the uniform bounds of the cutoff.By <ref> and <ref>, E' vanishes on Ω[_i;m]∖ D^χ_[_i](3δ). On D^χ_[_i](3δ), note that _=0. Since _ = 0 onD^χ_L(2δ), when restricted to D^χ_[_i](3δ), the required bound in (i).(a) follows from Lemma <ref>.(i).For (ii).(a), consider D^χ_[_i](3δ).Applyingto both sides of (<ref>) yields that E' = - _ on D^χ_[_i](3δ).Since E' vanishes on D^χ_[_i](2δ), it is only necessary to prove the estimate on D^χ_[_i](3δ)∖ D^χ_[_i](2δ).Using <ref>.(ii) to switch the order of _ and , we findE' = - _ - 2m^-2 [ ^2 ]E' on D^χ_[_i](3 δ) ∖ D^χ_[_i](2δ).The estimate in (ii).(a) follows from this equation after using Lemma <ref>.(ii) above to estimate the first term and Lemma <ref>.(iii) and (i).(a) above to estimate the second.GivenE∈ C^0,β_() with E_≡ 0 and E supported on D^χ_[_i](3δ) for some i∈{1, …, k}, there is a unique u ∈ C^2,β_() solving _ u = E and satisfying the following. * u_=0.* u: C^2, β_sym(, , e^-m| || - _i | )≤ C E: C^0, β_sym(D^χ_[_i](3δ),). * u:C^2,β_(D^χ_[_i](3δ),)≤Cm^-3 E : C_^0,β(D^χ_[_i](3δ), ) +C E: C_^0,β(D^χ_[_i](3δ),).Since the kernel of _ : C^2,β_()→ C^0,β_() is trivial by the symmetries (recall Lemma <ref>), the existence and uniqueness of u is clear.Since _ is rotationally covariant, _ u_avg = E_avg = 0 and (i) follows. The equation _ u = E is equivalent to ( Δ_χ+ 2^2 ) u =m^2E.We use separation of variables to establish the estimate (ii) (see <cit.>for a similar technique). Recall {1, (cos n θ)_n∈,(sin n θ)_n∈} is a basis for L^2(^1).Define L^2-projectionsE_0()= 1/2π∫_0^2π E(, θ) dθE_n, even()= 2/π∫_0^2π E(, θ) cos (nθ) dθE_n, odd()= 2/π∫_0^2π E(, θ) sin (nθ) dθ.The assumption E_ = 0 implies E_0 ≡ 0.By the symmetries, E_n, odd≡ 0 and E_n, even≡ 0 for all n such that m does not divide n.The Fourier expansion of u then satisfies u(, θ) = ∑_q=1^∞u_mq, even() cos(mqθ)+∑_q=1^∞u_mq, odd() sin(mqθ)for appropriate functions u_n, even(), u_n, odd() described as follows.Separating variables leads us to consider the eigenspace {u ∈ C^∞_ (): _χ u = n^2 u}, which is spanned byu_± n = (± n - tanh) e^± n.From this, it can be checked that for any n≥ 2, _χ - n^2 has associated Green's function_n(, ξ) =1/2n(1-n^2) e^n( - ξ) (n+tanhξ)(n - tanh)for≤ξe^n(ξ - )(n - tanhξ)(n+ tanh) for≥ξ.For each q∈, we have then using that E is supported on Ω[_i;m]u_mq, even()=m^2 ∫_-∞^∞_mq(, ξ) E_mq, even(ξ) dξ= 2m^2/π∫__i - 3δ^_i+3 δ∫_0^2π_mq(, ξ) E(ξ, θ) cos(mq θ) dθ dξ , u_mq, odd()= 2m^2/π∫__i - 3δ^_i+3 δ∫_0^2π_mq(, ξ) E(ξ, θ) sin(mq θ) dθ dξ. A straightforward estimate of (<ref>) implies that for all q∈,| _mq, even| ≤C/q^2 E: C^0_sym(D^χ_[_i](3δ), ), | _mq, odd| ≤C/q^2 E: C^0_sym(D^χ_[_i](3δ), ). Therefore, u: C^0_sym(D^χ_[_i](3δ),)≤ C E:C^0_sym(Ω[_i;m],). Since _ u = E, the C^0 bound above implies with the Schauder estimates thatu: C^2, β_sym(D^χ_[_i](3δ), ) ≤ CE:C^0, β_sym(D^χ_[_i](3δ),). Combining (<ref>) with the exponential decay from (<ref>) yieldsu: C^2, β_sym(, , e^-m| || - _i |)≤ C E: C^0, β_sym(D^χ_[_i](3δ),).This proves (ii).Applyingto both sides of (<ref>) and using Lemma <ref>.(iii), we obtain_ [ u ]=- 2m^-2 [ ^2] u + E.Although E-2m^-2[ ^2] u is not supported on D^χ_[_i](3δ), it has average zero, so a straightforward modification of the argument leading up to (<ref>) by replacing the assumption that the inhomogeneous term is compactly supportedwith the assumption (from (ii) above) that the right hand side has exponential decay away from D^χ_[_i](3δ), we conclude thatu: C^2, β_(D^χ_[_i](3δ) ,) ≤ C E: C^0, β_(D^χ_[_i](3δ) ,)+ Cm^-2[ ^2]u: C^0, β_(D^χ_[_i](3δ) ,).(iii) follows after using (<ref>), Lemma <ref>.(iii), and part (i) above to estimate the last term.Recall that E'_ is supported on D^χ_L_par(3δ).With this and Lemma <ref> in mind, we make the following decompositions.For i=1, …, k, we define E'_, i∈ C^∞_() and _i ∈ C^∞_() by requesting that E'_,i is supported on D^χ_[_i](3δ) and that ∑_i=1^k E'_, i = E'_, ∑_i=1^k _, i = Φ'_, _ _, i = E'_osc, i. The functions Φ'_i are well-defined by Lemma <ref>.By combining Definition <ref> with Lemma <ref>, we get global estimates for _. The following estimates hold. * * _, i : C^j_( , , e^-m| || - _i|)≤ C(j). * _, i : C^j_(D^χ_[_i](3δ),)≤ C(j)/m. * _osc : C^j_( ,)≤ C(j). * _ : C^j_(D^χ_[_i](3δ),)≤ C(j)/m.(i) follows directly from applying Lemma <ref> to E'_, i, using Lemma <ref> and Schauder regularity for the higher derivative estimates. For small k, (ii) follows from (i).On the other hand, for k large enough in absolute terms, Lemma <ref>.(iv) implies that for all i, j ∈{1, …, k}, |_j - _i| > 3|j-i|/4k.Using this with part (i) above, we estimate_osc : C^j_( ,) ≤C(j) sup_∈∑_i=1^k e^-m| || - _i|≤ C(j) ∑_l=0^k-1 e^- 3m/4k l ≤ C(j),where we have used Assumption <ref>.This completes the proof of (ii).Now fix some i∈{1, …, k}. As in part (i), we may assume that for all i, j ∈{1, …, k}, |_j - _i| > 3|j-i|/4k.Using the definitions and (i) above, _ : C^r_(D^χ_[_i](3δ),) ≤_, i: C^r_(D^χ_[_i](3δ),)+∑_ j≠ i_, j : C^r_(D^χ_[_i](3δ),)≤C(r)/m + C(r)∑_ j≠ i_, j : C^r_(D^χ_[_i](3δ),)≤C(r)/m + C(r) ∑_j≠ i e^-m| _j - _i|≤C(r)/m+ C(r) ∑_l=1^k e^-3m/4kl ≤C(r)/m,where we have used Assumption <ref>.We culminate our understanding ofwith the following estimates. The following hold.* : C^j_( ,) ≤ C(j). * For i∈{ 1, …, k}, : C^j_( D^χ_[_i](3δ),)≤ C(j)/m.Because of the estimates on _ established in Proposition <ref>.(ii), it is enough to prove the estimate (i) forΦ'_.By Proposition <ref>, _ is supported on D^χ_L_par(3/m).Fix i∈{1, …, k}.We first establish the estimate on Ω[_i;m]. Equation (<ref>) shows that there,_ =[τ'_i/2; _i]-_. Note that the left hand side is smooth and the discontinuities on the right hand side cancel.Using that __ = E'_ from (<ref>), onΩ[_i;m] we have (where = [ _i] is as in <ref>)∂^2_ _ + 2/m^2^2( /m+ _i)_ =E'_.On a neighborhood of ∂Ω[_i;m], we have that _ = 0 from Definition <ref>.This combined with estimates onfrom Lemma <ref> implies that |_|<C and | ∂_Φ'_|<C on ∂Ω[_i;m].Using this as initial data for the ODE and bounds of the inhomogeneous term from Lemma (<ref>) yields the C^2 bounds in (i).Higher derivative estimates follow inductively from differentiating (<ref>) and again using Lemma <ref>.This establishes (i) on Ω[_i;m].The proof of the estimate (i)on Ω[_i;m]∖Ω'[_i;m] follows in a similar way using (<ref>) but is even easier since there _ = 0, so we omit the details.As in the proof of (i), Proposition <ref>.(iii) implies it is sufficient to prove the estimate in (ii) for _. By Lemma <ref>.(iii), _ satisfies∂^2__ +2/m^2^2( /m+ _i)_+ 2/m^2[ ^2 ] ( /m+_i) _ =E'_.The C^2 bounds in (ii) follow in a similar way as above, by using Lemma <ref>.(iii)-(iv) to estimate the initial data on ∂ D^χ_[_i](3δ), estimates on E'_ from Lemma <ref>.(iv), and estimates on [ ^2 ] and _ from Lemma <ref>.(iii) and (i) above.Higher derivative bounds follow inductively fromdifferentiating (<ref>) and using Lemma <ref>.(iii) and Lemma <ref>.(iv).§ MATCHED LINEARIZED DOUBLING (MLD) SOLUTIONS §.§ Mismatch and the spaces [L] and [L]abWe define a vector space [L] andgiven φ as in Definition <ref> with τ>0, the mismatch _L φ of φ, by_L φ : = ( _p φ)_p∈ L ∈ [L]:= ⊕_p∈ L[p],where _p φ : =( _p(p) + τ_p log (τ_p/2), d_p _p) ∈[p]:=⊕ T^*_p . Among all LD solutions modulo [L], we are mainly interested in the ones which are well matched: We define a matched linearized doubling (MLD) solution modulo [L] of configuration (L, τ, w)to be some φ as in <ref> which moreover satisfies the conditions_Lφ = and τ_p >0 ∀ p ∈ L.Given L = L[; m] as in <ref>,we define_[L]= span( (W_i)_i=1^k, ( W'_i )_i=1^k) ⊂ C^∞_(), _[L] =span( (V_i)_i=1^k, ( V'_i )_i=1^k) = {v∈ C^∞_(): v∈_[L] },where for i∈{1, …, k}and with δ as in (<ref>),V_i , V'_i, W_i, W'_i ∈C^∞_() are defined by requesting that they are supported on D^χ_L_i(2δ) andon D^χ_p_i(2δ) they satisfy V_i=V_i[_i, m]: = [ δ, 2δ; ^χ_p_i]([ 1, 0 ; _i], 0), W_i = W_i[_i,m]: = V_i, V'_i=V'_i[_i, m]: =[ δ, 2δ; ^χ_p_i]( [ 0, 1 ; _i ], 0),W'_i = W'_i[_i,m]: =V'_i.Note that the last equality follows from the symmetries imposed.Note also that _[L] and _[L] are both 2k-dimensionalwith corresponding bases ( W_i, W'_i)_i=1^k and ( V_i, V'_i)_i=1^k.We define a linear map _L : _[L] →_[L], where _[L] was defined in (<ref>) and _[L] is the subspace of [L] (recall <ref>) consisting of those elements which are invariant under the obvious action ofon [L], by requesting that_L (v) = (v(p),d_pv)_p∈ L∈_[L]. In Lemma <ref> below, we convert estimates established in Section <ref> for Φ on the cylinder into estimates for Φ on the sphere. Before doing this, we need the following lemma which compares the geometry induced by the metrics χ and g. Let i ∈{1, …, k}.There is a constant C>0—independent of m and k—such that:* For any a, b ∈ (_i - 3/m, _i +3/m),^2 a ∼_1+C/m^2 b. * When restricted to Ω[_i;m], ^g_p_i∼_1+C/m_i^χ_p_i.* For large enough m and any ϵ≤δ/2,D^χ_L_i(ϵ/2)⊂ D^g_L_i( ϵ_i) ⊂D^χ_L_i(2ϵ). * If f∈ C^j(Ω), where Ω⊂ is a domain such that sup_p∈Ω|(p)|≤_k+1 then f: C^j( Ω, g)∼_C k^j/2 f: C^j(Ω, χ). Fix i∈{1,…, k} and suppose _i - 3/m < a < b< _i + 3/m.Trivially, ^2 b< ^2 a.On the other hand, ∂( ^2 ) = -2 ^2 tanh, so by Grönwall's inequality,^2 a≤(^2 b) e^∫_a^b 2 tanh d≤^2 b( 1+ C/m).This completes the proof of (i).(ii) follows easily from (i) and that g = ( ^2 )χ (recall (<ref>)). (iii) follows from (ii) by taking m large enough.(iv) follows by using that g = ( ^2 )χ in combination with part (iii) above and the fact (the second part of Proposition <ref>.(i)) that _k ∼_C k^-1/2. For each i=1, …, k, V_i, V_i'∈ C^∞_() satisfy the following. * V_i: C^j_(, )≤ C(j) and V_i' : C^j_(,) ≤ C(j). * _L is an isomorphism and ^-1_L≤ C m^2+β k^2+β/2(recall <ref>), where ^-1_L is the operator norm of ^-1_L: _[L] →_[L] with respect to the C^2, β(, g) norm on the target and the maximum norm subject to the standard metric g of .(i) follows easily from the bounds onin Lemma <ref> and the uniform bounds on the cutoff in themetric.By the definitions above and (<ref>), it is easy to see that _L is invertible and that_L^-1( ( a_i , b_icosd)_i=1^k) =∑_i=1^k a_i V_i + ∑_i=1^k b_i V'_i.Combining (i) above with Lemma <ref>.(iv) to switch to the g metric (recall also (<ref>) and (<ref>)), we haveV_i: C^j_(, g)≤ C(j)k^j/2m^j,V_i' : C^j_(, g)≤ C(j)k^j/2m^j.Recall thatis computed with respect to the C^2,β(,g) norm on _[L] andthe maximum norm on _[L], subject to the standard metric g of . Then (ii) follows by combining (<ref>) and (<ref>).§.§ The family of MLD solutionsab In this subsection we convert the LD solutions we constructed and studied in section <ref> to MLD solutions.We first have to choose the scale of the LD solutions so that we have approximate matching.By a heuristic argument which we omit we find that the overall scaleshould be given by := 1/m e^ζ_1 e^-ϕ(_1) = 1/m e^ζ_1 e^- m/F^ϕ_1,where ζ_1 is an unbalancing parameter used to absorb error terms later.The continuous parameters of the construction are then:= (ζ_1, ) = (ζ_1, , ) ∈×^k-1×^kwhere we require|ζ_1| ≤_1, ||_ℓ^∞≤_1/m,| |_ℓ^∞≤_1/m,where _1>0 will be fixed later and which we assume may be taken as large as necessary depending on k but independently of m.With the overall scalehaving been chosen,we define the MLD solution φ=Φ+ for some ∈[L] uniquely determined by the matching condition _Lφ=:By the definitions _Lφ=_L(Φ)+_L().Using the invertibility of _L as in <ref>.(ii), the matching condition is equivalent then to = -^-1_L _L (Φ). To record in detail the dependence on the continuous parameters we have the following. We assumeis given as in (<ref>). Let ϕ=ϕ[: k,m], Φ = Φ[: k,m], and τ_i' = τ_i' [ : k]be as in <ref>. We define then= [;m]by <ref>,an MLD solutionφ = φ[ ; m]of configuration( L , τ,w ) (recall <ref>)by <ref> and <ref>,where L=L[; m], = [ : k ] (recall <ref>),τ= τ[ ; m] : L[; m] →_+is -invariant satisfying τ_i = τ_1 τ'_i for i∈{1, …, k },and w= w[ ; m] :=.Finally we define=[; m] = (μ_i[; m])_i=1^k∈^kand'='[; m] = (μ'_i[; m])_i=1^k∈^kby= ∑_i=1^k τ_i μ_i V_i + ∑_i=1^k τ_i μ'_i V'_iwhich also impliesw = ∑_i=1^k τ_i μ_i W_i + ∑_i=1^k τ_i μ'_i W'_i.Let φ be as in <ref>.The equation _L φ = is equivalent to the equations0= m/ F^ϕ_1( e^-∑_l=1^i-1σ_l - 1)+(p_i)/τ'_i+ μ_i +ζ_1 + log(9/2τ'_i)- log_i 0= 1/τ'_i∂/∂(p_i)+m/2ξ_i + μ'_i + 1/2tanh_ifor i=1, …, k.From Definition <ref>, the condition that _Lφ = is equivalent to the conditions that1/τ_p_i_p_i(p_i) + log( τ_p_i/2) = 0, 1/τ_p_i∂_p_i/∂(p_i)= 0,(i=1, …, k).By Definitions <ref>, <ref> and <ref>, (<ref>), and Lemma <ref>, we have on D^χ_p_i(3δ)1/τ_i_p_i = 1/τ'_iΦ - G_p_i+ μ_i V_i + μ'_i V'_i + ( G_p_i - G^^2∘^g_p_i).Expanding Φ =+ + and using expressions forandfrom Definitions <ref> and <ref> gives 1/τ_i_p_i =1/τ'_i_i - [ logδ, 0; _i] +1/τ'_i+μ_iV_i + μ'_i V'_i +( G_p_i - G^^2∘^g_p_i).Evaluating at p_i, using that V(p_i) = 1, adding log(τ_p_i/2) to both sides, using (<ref>)to see that ϕ(_i)/τ'_i = m/F^ϕ_i, and using Lemma <ref> shows that the vertical matching equation is equivalent to0 = m/F^ϕ_i+ Φ'(p_i) /τ'_i + μ_i + log( τ'_i τ_1/2δ) - log_i.Simplifying m/F^ϕ_i using (<ref>) and expanding the last term using (<ref>) gives (<ref>).Next, note using the definition of _i in <ref>, the vertical balancing equation and (<ref>)1/τ'_i∂_i/∂(_i)= ϕ(_i)/τ'_iF^ϕ_i+ - F^ϕ_i-/2= m/2F^ϕ_i+ - F^ϕ_i-/F^ϕ_i = m/2ξ_i.(<ref>) follows from this decomposition, (<ref>), and Lemma <ref>. [cf. <cit.>]We fix some α>0 which we will assume as small in absolute terms as needed.We also fix some β∈(0,1) and γ∈ (1,2)satisfying 1-γ/2>2α and (1-α) (γ-1)>2α, for example γ=3/2. We will suppress the dependence of various constants on β and γ.For each p∈ L we define δ_p'=τ_p^whereα is as in Convention <ref>.For each p∈ L, there exists a unique i∈{1, …, k} such that p∈ L_i. Define then δ_p = (_i ) δ. Define also τ_min =min_p∈ Lτ_p, τ_max=max_p∈ Lτ_p, δ_min = min_p∈ Lδ_p, δ_min'=min_p∈ Lδ_p'=τ_min^. For Φ as in <ref>, the following estimate holds.Φ : C^3, β_( ∖D^g_L(δ'_min), g)≤ C ( (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2).Recall from Definition <ref> that Φ =+ +.By Lemmas <ref> and<ref>, we have : C^3, β_( ∖ D^χ_L(δ'_min), χ)≤ C (δ'_min)^-3-β| logδ'_min | .Recall from <ref> that _ (+ ) = - _ andSince _ = 0 on D^χ_L(2δ), it follows from <ref>.(i) that_ (+ ):C^3, β_(), ) ≤ C.From <ref>.(i), <ref> and <ref>.(ii), we have also+:C^0_( ,) < Cmk,so it follows from the Schauder estimates that+: C^3, β_(, χ) ≤ C m^4+β k .Now let Ω: = ∩{ || ≤_k + 3/m}.After combining (<ref>) and (<ref>) and using Lemma <ref>.(iv), we concludeΦ : C^3, β_( Ω∖D^g_L(δ'_min), g)≤ C ( (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2).It remains to verify the estimate on ∖Ω.On ∖Ω, note that =0 (recall <ref>) and also that = c for c bounded independent of k, which follows from <ref> and <ref>.(iii). Using this and the exponential decay ofaway fromfrom <ref>, we conclude Φ : C^3, β ( ∖Ω, g) ≤ m^4+βk.The next lemma will be important in controlling certain error terms in the fixed point theorem. For Φ as in Definition <ref> and 1≤ j < i ≤ k, we haveτ'_i/τ'_j = ϕ(_i)/ϕ(_j)( e^∑_l=j^i-1σ_l)∼_1+C/k 1.The first equality follows from <ref>.(i). We have thenlogτ'_i/τ'_j = logϕ(_i)/ϕ(_j) + ∑_l=j^i-1σ_l=O(1/k) + O(k _1/m),where the estimates follow from Proposition <ref>.(iii), Definition <ref> and (<ref>).Given = (a_i)_i=1^k ∈^k, k≥ 2, we define ∈^k-1 by requesting that ()_i = a_i+1 - a_i, i=1, …, k-1. It is useful to think ofas a discrete derivative of .Letbe as in (<ref>) and φ = φ[; m] be as in <ref>. For m large enough as in <ref> (depending on _1), the following hold: * = [; m] and (, ') = ([; m], '[; m]) depend continuously on . * [; m] ∼_C(_1)[; m] and C(_1)>1 depends only on _1. * (Matching estimates) There is an absolute constant C independent of _1 such that * |ζ_1+μ_1 | ≤ C. * | - F^ϕ_1/m|_ℓ^∞≤ C/m. * |+2/m' |_ℓ^∞≤ C/m. * φ : C^3, β_(∖ D^g_L(δ'_min), g) ≤τ_min^8/9.* On ∖ D^g_L(δ'_min) we have τ_max^1+α/5≤φ.* For all p∈ L, (δ_p)^-2_p: C^2, β( ∂ D^g_p( δ_p),( δ_p)^-2 g)≤τ_p^1-α/9. The continuity of the parameter dependence ofandonthen follows from Definition <ref>,Proposition <ref> and Proposition <ref>.We next prove (ii).For convenience in this proof, denote ϕ =(Φ[: k, m])_ and = ( Φ[: k,m])_ and = [:k], ' = [': k] (recall <ref>).From Definition <ref>, τ_1[ ;m]/τ_1[; m] = e^ζ_1 e^ϕ'('_1) - ϕ(_1).From<ref>.(ii), Proposition <ref>, Proposition <ref>.(i), and <ref>, we have |ϕ'('_1) - ϕ(_1)| =m| F^ϕ_1- F^_1|/F^_1 F^ϕ_1≤ C _1.This establishes (ii). We next prove (iii). Taking i=1 in (<ref>) we obtain μ_1 + ζ_1=-log( 9/2) - (p_1) + log_1.In conjunction with Proposition<ref>.(ii), this proves (iii).(a). Now suppose i≥ 2. Subtracting the instance of (<ref>) evaluated at i from the instance evaluated at i-1 gives0= μ_i-1 - μ_i+m/ F^ϕ_1(σ_i-1 + O ( ^2/m^2) )- log(τ'_i/τ'_i-1)- (p_i)/τ'_i + (p_i-1)/τ'_i-1 + log_i/_i-1.Multiplying through by F^ϕ_1/m and rearranging, we findF^ϕ_1/m(μ_i-1-μ_i)+σ_i-1 =F^ϕ_1/m( logτ'_i/τ'_i-1 +(p_i)/τ'_i - (p_i-1)/τ'_i-1 - log_i/_i-1)+ O( _1^2/m^3).We estimate the right hand side of (<ref>): by Lemma <ref>, Proposition <ref>, and Proposition <ref>, F^ϕ_1/m|logτ'_i/τ'_i-1| ≤C/mk^2,F^ϕ_1/m| log_i/_i-1| ≤C/mk,F^ϕ_1/m| (p_i)/τ'_i- (p_i-1)/τ'_i-1|≤C/mk.This completes the proof of (iii).(b).Multiplying (<ref>) by 2/m and rearranging, we estimate that fori∈{1, …, k} we have|ξ_i+2/mμ'_i| ≤2/τ'_im|∂/∂(p_i)| +tanh_i/m.(iii).(c) then follows from <ref> with a constant C depending on k.Next, since (V_i)∩(V_j) = ∅ when i≠ j and likewise for the functions V'_i, we estimate∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i : C^3, β_( ,) ≤max_i=1, …, k( τ_iμ_i V_i : C^3, β_( ,) +τ_iμ'_iV'_i : C^3,β_( , )).It follows from (iii) above and (<ref>) that |τ_i μ_i| ≤ C _1 ,i=1, …, k.Therefore, using Lemma <ref> to estimate the norm of V_i, we find max_i=1, …, kτ_iμ_i V_i : C^3, β_( ,)≤ C _1. By (iii) above, we have |τ_i μ'_i | ≤ C _1 k^-2.Using Lemma <ref> to estimate the norm of V'_i, we getmax_i=1, …, kτ_iμ'_iV'_i : C^3, β_( ,)≤ C_1 .By combining (<ref>)and (<ref>) and switching to the g metric, we estimate using Lemma <ref>.(iv)∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i : C^3, β_( , g )≤ C _1 k^3+β/2 m^3+ β.Recalling from (<ref>) that φ = Φ+∑_i=1^kτ_iμ_i V_i + ∑_i=1^k τ_iμ'_i V'_i and combining (<ref>) with Lemma <ref> yieldsφ : C^3, β( ∖ D^g_L(δ'_min), g)≤ C( (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2+ _1 k^3+β/2m^3+β)τ_1. From Definition <ref> and Lemma <ref>, δ'_min = τ_min^α∼_C τ^α_1.(iv) then follows from the above by taking m large enough.For (v), from (<ref>), (<ref>), Lemma <ref>.(ii) and Definition <ref>,: C^0(∖ D^g_L(δ'_min)) ≤α Cm k ∑_i=1^k τ_iμ_i V_i + ∑_i=1^k τ_iμ'_i V'_i : C^0_sym(, g) ≤ C_1 τ_1: C^0( , g) ≤ C.It is easy to see from<ref>.(ii) and definition <ref> and that there is an absolute constant c>0 such that >cmk, so (v) follows from <ref> and <ref> by taking α small enough and m large enough.Finally, let i∈{1, …, k}.By <ref>, <ref> and <ref>, on D^g_p_i(δ_p_i), _p_i satisfies_p_i = τ_1(+ ) + τ_i' τ_1 ( G_p_i - G^^2∘^g_p_i) + ∑_i=1^k τ_iμ_i V_i + ∑_i=1^kτ_i μ_i' V'_i:= (I) + (II)+(III).By <ref> and <ref>, (I): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ Cτ_1 mk^2+β.By Lemma <ref>, G_p_i - G^^2∘^g_p_i satisfies the equation '( G_p_i - G^^2∘^g_p_i) = 0 and the C^0 estimate G_p_i - G^^2∘^g_p_i : C^0( D^g_p_i(δ_p_i), g) < C|log_i| ≤ C k,where the last estimate follows from <ref>.(i). Therefore, by the Schauder estimates, we have(I): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ C τ_1 k. Additionally, by (<ref>), it follows that(III): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ C _1 k^2+β/2τ_1. (vi) then follows by taking m large enough.§ MAIN RESULTS WITH NO NECKS AT THE POLES OR THE EQUATORIAL CIRCLE§.§ Initial surfaces from MLD solutionsab In this subsection we discuss the conversion ofthe MLD solutions (constructed in the previous subsection)toinitial surfaces. We also discuss the mean curvature and the linearized equation onthe initial surfaces constructed.These steps were carried out under generous assumptions for theMLD solutions in <cit.> and therefore we onlyquote the results here and confirm that our MLD solutionssatisfy the required conditions (see Lemma <ref>).For each p∈ L, the disks D^g_p(9δ_p) are disjoint for different points p∈ L. Follows from <ref>, Proposition <ref> and Assumption <ref> by taking m large enough.Let φ[;m] be as in <ref>.The following hold.* τ_max is small enough in absolute terms as needed. * ∀ p∈ L we have 9 δ_p' < τ_p^α/9< δ_p.* τ_max≤τ_min^1-/9. * ∀ p∈ L we have(δ_p)^-2 _p: C^2,β( ∂ D^g_p(δ_p),(δ_p)^-2 g ) ≤τ_p^1-/9. * φ:C^3,β_ (∖_q∈ LD^g_q(δ_q'),g )≤τ_min^8/9. * On ∖_q∈ LD^g_q(δ_q') we have τ_max^1+α/5≤φ. * _L: _[L]→_[L] is a linear isomorphism and δ_min^-4 τ_max^α≤ 1.(i)-(iii) follow from <ref> and <ref> by taking m large enough.(iv)-(vi) follow from Lemma <ref>.(v)-(vii).Finally, (vii) follows from <ref>.(ii). Given a -symmetric MLD solution φ of configuration (L, τ, w)asin <ref>, we modify it to∈ C^∞_(∖ L) as in <cit.>.By using cut-off functions then we define as in<cit.> φ_init = φ_init[L,τ,w]:∖_p∈ L D_p(τ_p)→[0,∞),and then as in<cit.>the initial smooth surface M[L,τ,w] as the union of the graphs of±.Recall however that our MLD solutions and their configurations(defined in <ref>)are parametrized by(which determines also k) and m.Because of this we introduce the notation M= M[[ ]] =M[[ ; m ]] := M[ L, τ, w],where L=L[[]] = L[[ ; m ]]:= L[[ : k ]; m], τ= τ[ ; m],and w= w[ ; m], are as in <ref>. Note that the double brackets are introduced to distinguish from earlier notation.Note also that as usual the value of k is implied byand we may not mention mwhen it is implied by the discussion. In the rest of this sectionwe assume throughout thatsatisfies (<ref>), (k,m) ∈^2 is as in Assumption <ref>,and m is large enough in terms of _1 as needed. Now that the initial surfaces have been defined,we need to discuss their mean curvature, the linearized operator on them, and the nonlinear terms in a small perturbation.All these have been studied in <cit.> and so we only need to quote a definition and three basic results.Note thatwas defined in (<ref>). Note also that Convention <ref> and Lemma <ref> implyall the requirements for the applicability of<cit.> so we do not mention them in <ref>. For k∈, ∈(0,1),∈, and Ω a domain inor an initial surface M as above,we define u_k,,;Ω := u:C^k,(Ω ,ρ,g,ρ^), whereρ:= ^g_L ∘when Ω⊂ or Ω⊂ M,and g is the standard metric on ,or the metric induced on M by the standard metric on ^3(1).The modified mean curvatureH-∘ on an initial surface M=M[[]] as aboveis supported on^-1(_p∈ L( D_p^g(3δ'_p) ∖ D_p^g(2δ'_p))). Moreover it satisfies the estimate H - ∘ _0,β,γ-2;M≤ τ_max^1 + α/3. If M=M[[]] is an initial surface as above, there exists a linear map_M: C^0,β_(M) → C^2,β_(M) ×_[L] such that if _M E:= (u,w) ∈ C^2,β_(M) ×_[L], then u= E+ w ∘. Moreover the following hold. * u_2,β,γ;M≤C(b) δ_min^-2-βE _0,β,γ-2;M. * w: C^0,β(,g)≤Cδ_min^γ-2-βE_0,β,γ-2;M. * _Mdepends continuously on the parameters of .Given ∈ C^1_(M), we define the normal perturbation of ,I_: M →^3(1),byI_(x)=exp_x( (x) ν(x) ),where ν:M→ T^3(1) is the unit normal to M.Ifis sufficiently small, the normal perturbation M_: = I_(M)is an embedded surface. Moreover, M_ is invariant under the action ofon the sphere ^3(1).We recall the following estimate (<cit.>) on the nonlinear terms of such a perturbation.If M is as in <ref> and∈ C^2,β_(M)satisfies _2,β,γ;M ≤ τ_max^1+α/4, then M_ is well defined as above and is embedded. IfH_ is the mean curvature of M_ pulled back to M by I_and H is the mean curvature of M, then we have H_-H -_0,β,γ-2;M ≤C_2,β,γ;M^2. §.§ The main theorem with no bridges at the poles or the equatorial circleabFor the purposes of the fixed point theorem, we will need to fix a reference initial surfaceand pull back perturbations of the initial surfaces we consider to the reference surface via appropriate diffeomorphisms:There exists a family of diffeomorphisms _ : M[[]] → M[[]],where= (0, …, 0 )∈^2k, satisfying the following:* _ depends continuously on .* _ is covariant under the action of .* For any u∈ C^2, β(M[[]]) and E∈ C^0, β(M[[]]) we have the following equivalence of norms: u∘_ _2,β,γ;M[[]]∼_2u _2,β,γ;M[[]], E∘_ _0,β,γ-2;M[[]]∼_2E _0,β,γ-2;M[[]].As a preliminary step, we construct a family of diffeomorphisms '_L_par:→ which depend smoothly on L_par and covariant under the action of . For ease of notation, denote the positive -coordinates of the circles L_par[[]] byand likewise the coordinates of the circles in L_par[[]] by '.We define '_L_par by requesting the following: * '_L_par is rotationally invariant in the sense that '_L_par((, θ)) depends only on . * On D^χ_L_par[_i](5 δ), we have '_L_par( (, θ) ) = ('_i - _i + , θ). * On ∖ D^χ_L[[]](5δ), '_L_par we have'_L_par( (, θ)) = (f_L_par(), θ)for a suitably chosen function f_L_par. By choosing f_L_par carefully, we can ensure that '_L_par depends smoothly on L_par, hence onand is close to the identity in all necessary norms.Next, we use '_L_par to define _ by requesting the following.* ∀ p∈ L[[]] we define_ to map Λ_: = M[[]] ∩Π^-1_( D^g_p_i(δ'_p_i)) ontoΛ_ : = M[[]] ∩Π^-1_( D^g_q_i(δ'_q_i))where q_i = '_L_par(p_i). On Λ_, _ satisfies_∘ Y_∘Π_, p_i = Y_∘Π_, q_i∘_,where Y_ (and similarly for Y_ is the conformal isometry from Π_, q_i(Λ_) equipped with the induced metric from the Euclidean metric. (τ[, m])^-2 g |_p_i, to{∈ [-ℓ_, ℓ_]}⊂ (, χ), and_:{∈ [-ℓ_,ℓ_ ] }→{∈ [-ℓ_,ℓ_ ]} is of the form _ (,θ) = (ℓ_/ℓ_ ,θ ), where the ambiguity due to possibly modifying the θ coordinate by adding a constant is removed by the requirement that _ is covariant with respect to the action of .* We define the restriction of _ on M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p)) to be a map onto M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p)) which preserves the sign of thecoordinate and satisfies Π_∘_ = '_L_par∘Π_. * On the region M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p))∖ M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (δ'_p)) we apply the same definition as in (2) but with '_L_par appropriately modified by using cut-off functions and ^g_L[[]] so that the final definition provides an interpolation between (1) and (2).By construction, _ satisfies (i) and (ii). To check the uniform equivalence of the norms,first observe from the parametrization of the catenoid (<ref>) and the uniform equivalence of the τs from Lemma <ref>.(ii), it follows that ℓ_∼_1+ C(_1)/mℓ_.Moreover, from Lemma <ref>.(ii),[; m] ∼_C(_1)[; m].From this, it follows from arguing as in Lemma 4.13 of <cit.> that when m is large enough in terms of _1, the estimates in (iii) hold. For the convenience of the reader and to motivate the fixed point map(see <ref>)used in the proof of the main theorem <ref>we recall the main steps of the constructionomitting smallness conditions and the precise ranges of the parameters:Step 1: Unit RLD solutions: Based on Lemmas <ref> and <ref> we construct in Proposition <ref> a unitRLD solution [_1; ] for given _1 ∈ (0, _root)and= (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^).Step 2: Unit RLD solutions with prescribed number of jumps and flux ratios: Based on Proposition <ref> we introduce Notation <ref> to describe theRLD solutions which are smooth at the poles.These solutions are denoted by[: k]and are determined uniquely by the number of jumps k andthe corresponding 2k-1 flux ratios= (, ) ∈^k-1×^k.Their jump latitudes are denoted by = [: k ]∈^k_+.Step 3: Normalized -symmetric LD solutions:Based on Lemmas <ref> and <ref> we construct in Lemma <ref>an LD solution Φ [ : k,m]of configuration (L= L[ [: k] ; m] ,τ' )for given k,m∈ and= (, ) ∈^k-1×^k.Φ = Φ [ : k,m] is normalized by the condition τ'_1=1 andis characterized by the form of the configuration (recall <ref>) and the requirement thatϕ = ϕ[: k,m]:=Φ_ is a multiple of [ : k].Note that theτ_i' = τ_i' [ : k]'sdo not depend on m. Step 4: -symmetric MLD solutions:In this step we first choose a suitable overall scale τ_1 in(<ref>) for our LD solutions, and then modify them by addingas in <ref> to obtain MLD solutionsφ[ ; m] (see <ref>),where:= (ζ_1, ) = (ζ_1, , ) ∈×^k-1×^k = ^2kinvolves an extra parameter ζ_1 used in unbalancing related to the overall scale τ_1. Step 5: -symmetric initial surfaces:In this step the MLD solutions are converted to initial surfaces by using the processdeveloped in <cit.>. Step 6: -symmetric minimal surfaces:In this remaining step we use a fixed point argumentto perturb to minimality one of the initial surfaces constructed earlier for given k,m.There is an absolute constant _1>0 such that given (k,m)∈^2 with m large enough in terms of _1 and k,there is ∈^2k satisfying (<ref>) such that φ[ , m] satisfies the conditions of Lemma <ref>and moreover there is ∈ C^∞(), where : = M[[ ; m ]],such that_2, β, γ, ≤_1^1+ α/4,and further the normal graph _ is a genus 2mk -1 embedded minimal surface in ^3(1)which is invariant under the action ofand has area Area(_)→ 8 π as m→∞.The proof is based on finding a fixed point for a map: B → C^2, β_sym(M[[]])×^2kwe will define shortly,whereB ⊂ C^2,β_( M[[]] )×^2kis defined byB :={v∈ C^2,β_(M[[]]):v_2,β,γ;M[[]] ≤ τ_1[;m]^1+α }× [-_1,_1]×[-_1/m,_1/m]^2k-1.To motivate the definition of , suppose (v, ) ∈ B. Use Proposition <ref> to define (u, w_H):= - _M[[]]( H - w∘Π_). Define also ∈ C^2, β_(M[[]]) by : = v∘^-1_ + u. We then have:* u + H = (w+w_H)∘Π_.* By Lemma <ref> and Proposition <ref>, w_H: C^0, β( , g) +_2, β, γ; M[[ζ]]≤^1+α/4.Using <ref> again, define (u_Q, μ_Q): = - _M[[]](H_- H - ).By definition,* u_Q+H_ = H ++w_Q ∘Π_.By <ref>, we have the following estimate on the quadratic terms:* w_Q : C^0, β( , g) +u_Q_2, β, γ; M[[]]≤^2+α/4.By combining the above, we see* ( u_Q - v∘^-1_) + H_ = (w+w_H+w_Q)∘Π_. We then define(v, ) = ( u_Q ∘_, ζ_1+_1, - F^ϕ_1/m , +2/m' ),where ∑_i=1^k τ_i_i W_i + ∑_i=1^kτ_i '_i W'_i = w+w_H+ w_Q. We are now ready for the fixed-point argument.Clearly B is convex.Let β' ∈ (0, β).By the Ascoli-Arzela theorem, the inclusion B ↪ C^2, β'_sym( M[[ ]]) is compact.Moreover, by <ref>.(iii) and item (4) above, when m is large enough, (B) ⊂ B. The Schauder fixed point theorem implies there is a fixed point (, ) of .By inspection of the defining formula for , this implies that _i = '_i = 0 for i=1, …, k, hence + _H+ _Q = 0 and also = _Q ∘ F_.By (5), we conclude the minimality of _.The smoothness follows from standard regularity theory and the embeddedness follows from Lemma <ref> and (4) above.The genus follows because we are connecting two spheres with 2km bridges.Finally, the limit of the area as m→∞ follows from the bound on the norm ofand from the estimates (cf. <cit.>) on the function φ_init[L[; m], , ] used to construct the initial surfaces. As k→∞, the sizes of the catenoidal bridges on each minimal surface in <ref> tends to become uniform in the following sense:given k∈ and ∈^2k as in (<ref>),it follows from Lemma <ref> thatlim sup_m→∞τ_max[]/ τ_min[] < C/kwhere C is a constant independent ofand k. Consequentlylim_k→∞lim_m→∞_max/_min = 1,where _max/_min := _max[]/_min[]is the ratio of the maximum size over the minimum size of the bridges associated with fixed point (, ) ofas in <ref>. For simplicity in this article we have not attempted to determine explicit boundsfor m in terms of k (recall <ref>)under which the conclusion of Theorem <ref> holds.It would be interesting to extend our results to include the cases that k≤mwith m large depending onbut independently of k.The case m small with large k would require new ideas as remarked in <cit.>.§ CONSTRUCTIONS WITH NECKS AT THE POLES OR THE EQUATOR ab In this section, we construct doublings ofwhose configurations L contain the poles,or points equally spaced along the equator circle of , or both,in addition to points distributed symmetrically along 2k parallel circles inas in the previous sections. §.§ The case with necks at the poles ab Given = (_1, …, _k) as in <ref> and L[;m] as in <ref>, define= [; m] = L[;m]∪ L_pol⊂,where L_pol = { p_N, p_S} =( p_N). Given also a -symmetric function τ:[; m]→, we denoteτ_i := τ(p_i) for i=1, …, k and := τ(p_N)= τ(p_S).Given = (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^) as in <ref>,k∈ and _1 ∈ (a_k+1, , a_k, ) (recall <ref>),we modify [_1; ] to [_1; ] by removing the last discontinuity in the sense that[_1; ] := [_1; ] on {∈ [0, _k+1]} and[_1; ] is smooth on {∈ [_k, ∞)}.By definingA_k = A_k[_1; ] and B_k = B_k[_1; ] by (<ref>) as in Lemma <ref>the last condition is equivalent to := A_k+ B_k on{∈ [_k, ∞)}. In Definition <ref>.(i), we required that RLD solutions ϕ be positive. This ensures that RLD solutions are meaningful for the linearized doubling apparatus when L consists of points equally spaced on 2k parallelcircles—in particular that each RLD solution ϕ is the average of an LD solutionΦ = φ[ L; τ] for some -invariant τ: L →_+ (recall Lemma <ref>). When L also contains the poles {p_N, p_S}, the presence of logarithmic singularities at {p_N, p_S} dictates thatthe expansion of the average of an -invariant LD solutionΦ = φ[ L ; τ] contains a positive multiple of(recall <ref> and <ref>.(ii)) on {∈ [_k, ∞)}. This is ensured by <ref>, (<ref>) and <ref>. There exists _2>0 such that given k∈ and = (, ) ∈ℓ^1 ( ^) ⊕ℓ^∞( ^) satisfying ||_ℓ^1 + ||_ℓ^∞<_1/k (recall <ref>), A_k[_1; ] is a strictly decreasing function of _1 on [ a_k,- _2/ k^2, a_k, ] which on this interval satisfies ∂ A_k [_1; ]/∂_1∼_C -k and A_k[ a_k, ; ] = 0, where C>0 is a constant independent of k. Let k∈, = (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^) satisfy ||_ℓ^1 + ||_ℓ^∞<_1/k, and [_1; ] be as in <ref>, where _1 ∈ (a_k+1, , a_k, ).By combining <ref> with <ref>we find thatF^_-(a_k, ) = F^[ a_k, ; ]_1-∼_exp(||_ℓ^1 + 3||_ℓ^∞) F^[ a_k, ; ]_ = 1/k + O ( 1/k^3).By a direct calculation, F^_-() =+ O(^2) for small .We we conclude from the preceding in combination with the assumption that ||_ℓ^1+||_ℓ^∞<_1/k that a_k,- a_k+1, > C/k^2. Assume now that _1 ∈ [ a_k, - _2/k^2, a_k, ] and that _2>0 is small enough that a_k, - _2/k^2> a_k+1,.From <ref> we have A_k[_1; ] = (_k)( F^_+(_k) - F^_k+)(_k).Differentiating (<ref>) with respect to _1, we find∂ A_k[_1; ] /∂_1 = (I) + (II) where(I) :=∂(_k)/∂_1( F^_+(_k) - F^_k+)(_k),(II):= (_k) ∂/∂_1[ ( F^_+(_k) - F^_k+)(_k)].Using (<ref>) and <ref> we find(II) = (_k)( - 2^2 _k (_k) ∂_k/∂_1 - ∂ F^_-(_1)/∂_11+ξ_k/1-ξ_1( e^∑_l=1^k-1σ_l) -F^_k+∂(_k) ∂_k/∂_1).Using Corollary <ref>, <ref>.(i), and that F^_-() =+ O(^2) for small , we see that (II) ∼_C -k. To estimate (I), first observe from the estimate on (II) and that F^_+(_k) - F^_k+ = 0 when _1 = a_k, that( F^_+(_k) - F^_k+) < Ck( a_k,- _1) < C _2/k. Recall now from <ref> that log( (_k)) = ∫_0^_k F^_+()d.Differentiating under the integral sign, we find1/(_k)∂(_k)/∂_1 = - F^_k-∂_k/∂_1 -∑_i=1^k-1 F^_i ∂_i/∂_1+ ∫_0^_k∂ F^_+()/∂_1 d.Using Corollary <ref>, we estimate ∂_i/∂_1 < C( 2 ^2 _i + ( F^_i-)^2)^-1i ,i=1, …, k,hence by <ref> and <ref>.(i)F^_k-∂_k/∂_1 + ∑_i=1^k-1 F^_i ∂_i/∂_1 < Ck^2.Using <ref>, when ∈ [_i, _i+1], i=1, …, k-1, we have∂ F^_+/∂_1 = ( (_i)/())^21/21+ ξ_i/1 - ξ_1( e^∑_l=1^i-1σ_l) ∂ F^_-/∂_1(_1) +( (_i)/())^2(2 ^2 _i + (F^_i+)^2)∂_i/∂ F_1Using <ref> and <ref>, we estimate |∂ F^_+/∂_1() |< k for ∈ [ 0, _k].From this and the estimate that _k < Clog k (recall <ref>), we find∫_0^_k∂ F^_+/∂_1() d < C k log k.Combining with (<ref>), we have by estimating (<ref>) that | ∂(_k)/∂_1| < C k^2.Combining this with (<ref>) we find that|(I)|< C k _2.The result then follows from the estimates on (I) and (II) by taking _2 small enough.Given k∈,as in <ref>, and ∈ [0, _3/k),where _3>0 is a constant depending only on _2 and the constant C>0 in Lemma <ref>, we define [ ,: k] := [ _1; ],where _1 is uniquely characterized (recall Lemma <ref>) by the properties that_1 ∈ [a_k,- _1/k^2, a_k, ] and A_k[_1; ] =. Note that by <ref> [ , 0 :k] = [: k]. We also define B_k,:= B_k[,: k] as in (<ref>).Given [, : k] as in <ref>, we define _k∈ (_k, ∞) to be the unique root of F^_+ in (_k, ∞).Note that _k is well defined by Lemma <ref>.There is a constant _1>0 such that for all k∈ and all (, )= (, , )∈ℓ^1(^)⊕ℓ^∞( ^)×_+ with ||_ℓ^1+ ||_ℓ^∞ < _1 and ∈ [0,_3/ k),=[, : k] satisfies the following. * F^_avg = 1/k +O(1/k^2) and ^2 _k ∼_C 1/k.* * F^_1-=2tanh_1 +O(1/k^3). * F^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) +O(1/k^2(k-i+1)) for i=2, …, k. * F^_k+=2(tanh_k- tanh_k) + O(1/k^2). * 1-(): C^0( {∈ [0, _k]}) ≤C/klog k and|log(_i)/(_i-1)| ≤C/k for 2≤ i ≤ k.If =, then also 1>(_1)> ⋯ >(_k). * There is a constant C>0 depending only on _1 such that for any 1≤ i<j≤ k, _j - _i>(1+C/k) j-i/2kand^2 _i - ^2 _j ≤C(j-i)/k. Since [_1; ] depends continuously on _1 on compact subsets ofand lim_↘ 0[ , : k] = [: k ], it follows from Proposition <ref> that forsufficiently small (depending on k), the estimates in <ref> hold.It remains to be seen that these estimates hold when ∈ [0, _3/k).To this end, note that on {∈ [_k, ∞)}, F^_+()= ∂ () + B_k, ∂()/() +B_k, ()= ^2 /tanh (I) + (II)where (I) :=B_k,- / B_k, +(- ),(II) := / B_k,+( - ).Evaluating (<ref>) at _k, taking ∈ [0, _3/k), and using that _k ∼_C log k, which follows from the fact that ^[: k]_k ∼_C log k (recall <ref>.(i)) by using the flux monotonicity and that _1<a_k,, we estimate that0<(I) < 2 and |(II)| < _3/k.Combining this with (<ref>) and (<ref>), it follows that ^2 _k ∼_Cexp(||_ℓ^1+ ||_ℓ^∞) F^_.The rest of the proof proceeds in the same way as the proof of <ref>, so we point out only the differences.The last equation in (<ref>) must be replaced with F^_k+ = 2( tanh_k - tanh_k ) + ∫__k^_k( F^_-())^2 d.It follows that tanh_k = 1 - O(1/k).Estimating, as in the proof of <ref>, we find F^_ = 1/k + O(1/k).The proofs of the remaining parts are nearly the same as the proof of <ref>, so we omit them.Let k∈ and = (, )∈^k-1×^k, '= (', ') ∈^k-1×^k.Suppose that ||_ℓ^1+ ||_ℓ^∞< _1/k, |'|_ℓ^1+ |'|_ℓ^∞< _1/k, and ' ∈ [0,_3/k), ∈ [0,_3/k).Let = [', : k] and ' = [ , ': k]. There is a constant C>0 independent of k such that: * | ^' - ^|_ℓ^∞≤C/k(| ' - |_ℓ^1+ |' - |_ℓ^∞+|'- |).* max_1≤ i ≤ k| tanh'_i - tanh_i | ≤C/k(| ' - |_ℓ^1+ |' - |_ℓ^∞+ |' - |). We first prove (i). As in the proof of <ref>, it suffices to prove| F^'_1 - F^_1 | ≤C/k(|' -|_ℓ^1 + |' - |_ℓ^∞).Fix k∈, let _1 be as in <ref>, and consider the map( F_1, , ) = (F^[_1; ]_+(_k) - F^_+(_k)) [_1; ](_k) (_k) + , where [_1; ] is as in <ref> and _1 is chosen so that F^_1 = F_1.Recall from Remark <ref> that = (_k)(_k) ( F^_+(_k) - F^_+(_k))+ (_k)(_k)( - F^_+(_k) + F^_+(_k))By combining (<ref>) with <ref>, we find that [_1; ] = [, : k] if and only if ( F_1, , ) = 0. Now let (F_1, , ) ∈^-1({ 0}) be arbitrary.Estimating the partial derivatives ofin a similar manner as in the proof of <ref> using <ref> to estimate the partial derivatives of _k, we estimate.∂/∂ F_1|_∼_C k, | .∂/∂σ_i|_| ≤ C, |. ∂/∂ξ_i|_| ≤C/k, . ∂/∂|_ = 1.By the implicit function theorem, locally around ∈^-1( { 0}), ^-1( { 0}) is a graph over ( , ) and moreover (abusing notation slightly),.∂ F_1/∂σ_i|_(, ) = - (. ∂/∂ F_1|_)^-1.∂/∂σ_i|_ .∂ F_1/∂ξ_j|_(, ) = - ( .∂/∂ F_1|_)^-1. ∂/∂ξ_j|_ .∂ F_1/∂|_(, ) = -( .∂/∂ F_1|_)^-1. ∂/∂|_.(<ref>) follows from this and the estimates (<ref>). (ii) follows from a similar, but omitted, calculation.Given (, ) = (, , ) ∈^k-1×^k ×_+ as in <ref> and m as in <ref>, there is a unique -invariant LD solution (recall <ref>)Φ = Φ[, : k, m] := φ[ ; τ'],characterized by the requirements that* ϕ = ϕ[ ,: k,m] : = Φ_ is a multiple of [, : k]* =[[ , : k]; m ] (recall <ref>)* τ'_1 = 1 (normalizing condition).Moreover, the following hold: * For i∈{1, …, k} we have τ'_i =ϕ(_i)/m F^ϕ_i.Moreover τ'_i is independent of m and satisfies τ'_i = [, : k](_i)/[, : k ](_1) e^∑_l=1^i-1σ_l. * ' = m/F^[, : k]_1 [, : k](_1).* ϕ = m/[, : k](_1) F^[, : k]_1[, : k]. The proof of the uniqueness part proceeds as in the proof of <ref>, so we only discuss the existence part.As in the proof of Lemma <ref>, it follows that f: = ϕ - Φ_ is in C^∞_||(^)∩ C^1_||(), hence therefore f = c_odd + c_even for some c_odd, c_even≥ 0.Since f∈ C^0_||(), c_odd =0. By the choice of τ' in (i)-(ii) above, it follows that f is smooth at the poles, hence c_even=0 and therefore ϕ= Φ_.The proof of (iii) is exactly the same as in the proof of <ref>. Let Φ[, :k, m] be as in <ref>.Define ∈ C^∞_sym( ∖ L), ∈ C^∞_||( ), and , E' ∈ C^∞_sym() using Definitions <ref>, <ref> and <ref> with Φ[, :k,m] in place of Φ[: k,m]. While the defining formula for [, : k, m] is the analogous to the one for [: k, m], note that [, : k, m] is not extendible to C^∞_||() since [, : k] is not smooth at the poles.Given [;m] as in <ref>, define , ∈ C^∞_() by requesting thatis supported on D^g_L_pol(2δ)∖ D^g_L_pol(δ) and onD^g_p_N(2δ)=[m] = [ δ, 2δ; ^g_p_N] ( cos∘^g_p_N, 0 ),and that = [m] :=.We define_ [ L_pol]= span( ) ⊂ C^∞_ ( ), _ [ L_pol]= { v∈ C^∞_() :v ∈_ [ L_pol]}⊂C^∞_ ( ),_ []= _[ L] ⊕_ [ L_pol]⊂ C^∞_ ( ), _[ ]= _[L] ⊕_[L_pol]⊂ C^∞_ ( ),where _[ L], _[L] are as in (<ref>). By the symmetries, clearly _ [ L_pol] is spanned by . For i=1, …, k, V_i, V'_i∈ C^∞_ () satisfy (i) of <ref>.Moreover,: C^j_( , )≤ C(j).Furthermore, _ is an isomorphism and ^-1_≤ C m^2+β k^2+β/2(recall <ref>), where ^-1_ is the operator norm of ^-1_: _[] →_[] with respect to the C^2, β(, g) norm on the target and the maximum norm subject to the standard metric g of .The estimates on V_i, V'_i follow exactly as in the proof of <ref>.(i).The estimate (<ref>) follows the uniform bounds of the cutoffin themetric.By the symmetries, _[L_pol] is one dimensional and _L_pol is an isomorphism. Moreover (abusing notation slightly) ^-1_L_pol( (1, 0)) = V_pol and it is easy to see that ^-1_L_pol≤ C m^2+β. It is clear that ^-1_ splits naturally, i.e. ^-1_ = ^-1_L + ^-1_L_pol, so the estimate on _ follows from the above and the estimate on _L established in <ref>. We now convert the LD solutions constructed above to MLD solutions.By heuristic arguments which we omit we find that the overall scale τ_1 should be given byτ_1:= 1/m e^ζ_1 e^-ϕ(_1) = 1/m e^ζ_1 e^- m/F^ϕ_1and that the scale for the bridges at the poles should be given by = [;m] : = e^F^_1/m B_k, 0,where ζ_1 andare unbalancing parameters used to absorb error terms later.The continuous parameters of the construction are then : = (ζ_1, , ) = (ζ_1, , , ) ∈×^k-1×^k×, where we require(ζ_1, ) satisfy (<ref>)and| | ≤_1 log m/m,where _1>0 will be fixed later and which we assume may be taken as large as necessary depending on k bud independently of m. With the overall scale τ_1 having been chosen, we define the MLD solutionφ = τ_1 Φ +for some ∈_[ ] uniquely determined by the matching condition _Lφ=:By the definitions _Lφ=_L(Φ)+_L().Using the invertibility of _L as in <ref>.(ii), the matching condition is equivalent then to = -^-1_L _L (Φ). To record in detail the dependence on the continuous parameters we have the following.We assumeis given as in <ref>.Let ϕ = ϕ[, : k, m], Φ = Φ[ , : k,m] and τ' = τ'[: k] be as in <ref>.We define then τ_1 = τ_1[ ; m] by (<ref>), an MLD solution φ = φ[; m ] of configuration ( , τ, w) (recall <ref>) by <ref> and <ref>, where = [;m], = [, : k] (recall <ref>), τ = τ[; m ] : [;m ] →_+ is -invariant satisfying τ_i = τ_1 τ'_i for i∈{1, …, k}, = τ_1 ', and w = w[ ; m]:=.Finally we define = [; m] = (μ_i)_i=1^k ∈^k, ' = '[; m] = (μ'_i)_i=1^k ∈^k, and ∈ by = ∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i + τ_pol, which also implies w = ∑_i=1^k τ_i μ_i W_i + ∑_i=1^k τ_i μ'_i W'_i +.Let φ be as in <ref>. The equation _φ = is equivalent to the system of (<ref>) and (<ref>)for i=1, …, k,and the condition that (recall(<ref>))0 = F^ϕ_1/m - 1 + F^ϕ_1/m( ζ_1 + log'/4m + 1) + F^ϕ_1/m B_k, /.By Lemma <ref>, the equation _Lφ= is equivalent to the system of (<ref>) and (<ref>). Recall that τ_p_N == τ_1 m/F^ϕ_1 (_1).By <ref>, = L ∪ L_pol. The condition that _L_polφ = is therefore equivalent to the conditions that1/_p_N(p_N) +logτ_1 '/2 = 0, d_p_N_p_N = 0.It is automatic from the symmetries that d_p_N_p_N = 0.By Definitions <ref> and <ref> and <ref>.(iii), it follows that on D^g_p_N(3 δ),1/_p_N = F^ϕ_1 (_1)/ mΦ -G^^2∘^g_p_N +.Expanding Φ = + + (recall <ref>), noting from <ref> thatvanishes on D^g_p_N(3 δ), using <ref> and <ref> to expand , and <ref>.(iii) to expand G^^2∘^g_p_N, we find that on D^g_p_N(3 δ),1/_p_N = B_k, / ++ F^ϕ_1 (_1)/m -( log 2 - 1)- + .Evaluating at p_N, and using that (p_N) = 1, and that (p_N) = 0, which follows from Lemma <ref>, we have 1/_p_N (p_N) = B_k, / - ( log 2 - 1)+Adding logτ_1 '/2 to both sides and using <ref> to expand τ_1, we find that (<ref>) is equivalent to0 = - m/F^ϕ_1+( ζ_1 + log'/4m + 1) +B_k, /.Multiplying through by F^ϕ_1/m completes the proof. Letbe as in (<ref>) and φ = φ[; m] be as in <ref>. For m large enough (depending only on _1), the following hold: * = [; m] and = [; m] depend continuously on . * [; m] ∼_C(_1)[; m] and C(_1)>1 depends only on _1. * (Matching estimates) There is an absolute constant C independent of _1 such that * |ζ_1+μ_1 | ≤ C. * | - F^ϕ_1/m|_ℓ^∞≤ C/m. * |+2/m' |_ℓ^∞≤ C/m. * |-F^ϕ_1/m| ≤ C log m/m. * φ : C^3, β_(∖ D^g_(δ'_min), g) ≤τ_min^8/9.* On ∖ D^g_(δ'_min) we have τ_max^1+α/5≤φ.* For all p∈ L, (δ_p)^-2_p: C^2, β( ∂ D^g_p( δ_p),( δ_p)^-2 g)≤τ_p^1-α/9.The bulk of the proof is nearly the same as that of Lemma <ref>, so we only prove the essentially new estimate, (iii).(d).Note that by <ref>, we have0 = F^ϕ_1/m - 1 + F^ϕ_1/m( ζ_1 + log'/4m + 1) + F^ϕ_1/m B_k, /. Using Lemma <ref>.(iii) to expand ' and , we find0 = F^ϕ_1/m - 1 + F^ϕ_1/m(-log m + ζ_1 + 1 ++ logF^_1 B_k, 0/F^ϕ_1 (_1)) + B_k,F^ϕ_1/B_k, 0 F^_1 e^- ζ_1.Expanding B_k,F^ϕ_1/B_k, 0 F^_1 e^- ζ_1 = 1 + B_k,F^ϕ_1 - B_k, 0 F^_1/B_k, 0 F^_1 -+ O(^2),we find the matching equation is equivalent toF^ϕ_1/m -= F^ϕ_1/m(log m + O(1)) -B_k,F^ϕ_1 - B_k, 0 F^_1/B_k, 0 F^_1 + O(^2).The estimate now follows from (<ref>), <ref>, and <ref>. There is an absolute constant _1>0 such that if (k, m)∈^2 with m large enough in terms of _1 and k, there is ∈^2k+1 satisfying (<ref>) such that φ[ ; m] satisfies the conditions of Lemma <ref> and moreover there is ∈ C^∞(), where : = M[[]] such that_2, β, γ, ≤^1+ α/4_1,and further the normal graph _ is a genus 2km+1 embedded minimal surface in ^3(1) which is invariant under the action ofand has area Area(_)→ 8 π as m→∞.Given = (ζ_1, , ) ∈×^2k-1× as in <ref>, we denote = (0, …, 0) ∈^2k+1 and M[[]] : = M[ [; m], τ[; m], w[; m]], [[]] := [; m].DefineB={v∈ C^2,β_(M[[]]):v_2,β,γ;M[[]] ≤ τ_1[; m]^1+α }× [-_1,_1]×[-_1/m,_1/m]^2k-1× [ - _1 log m/m, _1 log m/m]⊂C^2,β_( M[[]] )×^2k+1.The proof is the same in structure as the proof of <ref>; in particular, after carrying out steps (1)-(5) as in the proof of <ref>, we define(v, ) = ( u_Q ∘_, ζ_1+_1, - F^ϕ_1/m ,+ 2/m',-F^ϕ_1/m_pol),where ∑_i=1^k τ_i _iW_i + ∑_i=1^k τ_i '_i W'_i + _pol = w+w_H + w_Q.It follows from Lemma <ref>.(iii) and the estimates on the norms of w_H and w_Q as in the proof of <ref> that ( B) ⊂ B, so we may apply the Schauder fixed-point theorem as in the proof of <ref>. As k→∞, the sizes of the catenoidal bridges on each minimal surface in <ref> tends to become uniform in the same sense as in the case where there are no bridges at the poles (recall <ref>).To see this, let k ∈ and ∈^2k+1 be as in <ref>.By<ref> and<ref>, [;m]/τ_1[; m] = '[;m] = m/F^_1 (_1) = F^_1 B_k, 0/F^_1 (_1) e^,By <ref>, <ref>, and <ref>, it follows thatlim_k→∞lim_m→∞_pol/ _1 =1,where _pol/_1 is the ratio of the size of the bridges at the poles over the size of the bridges at the first jump latitude of a fixed point ofas in <ref>.Then arguing as in remark <ref> to compare with the sizes of the rest of the bridges away from the poles, we conclude lim_k→∞lim_m→∞_max/_min = 1,where _max/_min is the ratio of the maximum size over the minimum size of the bridges associated with a fixed point ofas in <ref>. §.§ The case with necks along the equator ab Here we construct doublings ofwherecontains the equator circle.Most of the construction is as before, so we outline the argument and describe in detail only those aspects which differ from the construction in Theorem <ref>.To begin, we define an expanded class of RLD solutions.Given k∈ and 0=_0 <_1<⋯<_k < ∞, we denote = (_0, _1, …, _k)∈^k+1.We say ϕ∈ C^0_||( ) is a rotationally invariant linearized doubling solution if ϕ is as in <ref>, except ^ϕ in <ref>.(ii) may be as in either <ref> or <ref>; in the latter case we require that <ref>.(iii) holds for i=0, …, k. From now on we assume ϕ is an RLD solution as in <ref> where ^ϕ is as in <ref>. Let ϕ be an RLD solution as in <ref>.Define^ϕ := ( F^ϕ_0-, F^ϕ_0+, F^ϕ_1-, …, F^ϕ_k+) ∈^2k+2_+,^ϕ := ( F^ϕ_i)_i=0^k∈^k+1_+, ^ϕ := (σ^ϕ_i)_i=0^k-1∈^k,^ϕ = ( ξ^ϕ_i)_i=1^k ∈^k,where for i=0, …, k,j=0, …, k-1, and l = 1, …, k, F^ϕ_i± := F^ϕ_± (_i), F^ϕ_i := F^ϕ_+(_i)+F^ϕ_-(_i), e^σ^ϕ_j = F^ϕ_j+1/F^ϕ_j, ξ^ϕ_l = F^ϕ_l+ - F^ϕ_l-/F^ϕ_l+ + F^ϕ_l-. We define ^ϕ: = (^ϕ, ^ϕ) ∈^k×^k and call the entries of ^ϕ the flux ratios of ϕ.If ^ϕ =0 we call ϕ balanced. Finally we defineF^ϕ_avg : = 1/2(k+1)|^ϕ|_ℓ^1 =1/2(k+1)| ^ϕ|_ℓ^1. Note that if ϕ is an RLD solution as in Definition <ref>, the assumption that ϕ∈ C^∞_||( ∖ L_par[]) implies that F^ϕ_0+ = F^ϕ_0-.For this reason, we did not define ξ^ϕ_0. We omit the proof of the next result, which is a straightforward modification of the proof of <ref>. Given F∈ (0, ∞) and = (, )=( (σ_i )_i=1^∞, (ξ_j)_j=1^∞ )∈ℓ^1(^) ⊕ℓ^∞( ^)satisfying | |_ℓ^∞ < 1/10, there is a unique unit RLD solution [F; ] satisfying the following.* ^ is as in <ref> and F^[F; ]_+(_0) = F.* ^ = . |_k where k is the number of jump latitudes of ϕ and . |_k := ( (σ_i)_i=0^k-1, (ξ_i)_i=1^k ). Moreover, the following hold.* k[F; ] is a nonincreasing function of F.Further, for eachas above there exists a decreasing sequence {b_0, : = ∞, b_1, , b_2, , …} such that k[F; ] = k if and only if F ∈ [b_k, , b_k-1, ).Moreover each b_k, depends only on . |_k (defined as above). * ^_1, …, ^_k are increasing smooth functions of F for fixed . * [F; ] is smooth at the poles if and only if F = b_k, for some k≥ 1.* The restriction of [F; ] on any compact subset [0, ∞) depends continuously on F and . We emphasize that the only substantial difference between RLD solutions = [F; ] as in <ref> and RLD solutions = [_1; ] as in <ref> is that while [_1;] coincides withon {∈ [ -^_1, ^_1]}, the expansion ofon {∈ [ -^_1, ^_1]} with respect to the basis {, } contains a positive multiple of .Given k,and .|_k as in <ref>, we define[: k] : = [ .|_k : k]: =[ b_k, ; ],= [: k] := [ .|_k :k] : = ^[: k].By modifying Definition <ref>, Lemma <ref>, and Definition <ref> we also define for all appropriately small >0 RLD solutions [ , : k] = [ _1; ] which satisfy A_k[ _1; ] =. Straightforward modifications of the respective statements and proofs of <ref> and <ref> provide versions of those results which hold for smooth at the poles RLD solutionsandas in <ref>.Furthermore, as in Lemma <ref> we convert each RLD solution(or ) as in <ref> into an LD solution whose non-oscillatory part is a multiple of(or ).The only important difference is that (recall <ref>.(c)) these LD solutions are normalized so that τ'(p_0):= τ'_0 = 1.These LD solutions are decomposed and estimated via obvious modifications of the machinery developed in Section <ref> used to estimate LD solutions Φ whose configurations L do not contain points on the equator circle of . For constructions where L contains points on the equatorial circle but not at the poles of , the parameters of the construction are _eq =( ζ_0, ) = (ζ_0, , ) ∈×^k×^k, where we require|ζ_0| ≤_1,||_ℓ^∞≤_1/m,||_ℓ^∞≤_1/m.In cases where L contains both points on the equator and at the poles, the parameters are _eq =( ζ_0, , ) = (ζ_0, , , ) ∈×^k ×^k × where we require that(ζ_0, ) satisfies(<ref>)and|| < _1 log m /m. Using the preceding, we modify the steps in Sections <ref> and <ref> construct MLD solutions φ[ _eq; m] and φ[ _eq;m] and in turn smooth initial surfaces M[[ _eq]] and M[[_eq]].For the sake of brevity, we omit the proofs of the following and leave the routine modifications to the reader.There is an absolute constant _1>0 such that if (k,m)∈^2 satisfies Assumption <ref>, there is _eq∈^2k+1 satisfying (<ref>) and moreover there is ∈ C^∞(), where : = M[[ _eq]] such that_2, β, γ, ≤_0^1+ α/4,and further the normal graph _ is a genus (2k+1)m-1 embedded minimal surface in ^3(1) which is invariant under the action ofand has area Area(_)→ 8 π as m→∞. There is an absolute constant _1>0 such that if (k,m)∈^2 satisfies Assumption <ref>, there is _eq∈^2k+2 satisfying (<ref>)and moreover there is ∈ C^∞(), where : = M[ [ _eq ]] such that_2, β, γ, ≤_0^1+ α/4,and further the normal graph _ is a genus (2k+3)m-1 embedded minimal surface in ^3(1) which is invariant under the action ofand has area Area(_)→ 8 π as m→∞. 10abramowitz Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. 0167642brendle Simon Brendle, Embedded minimal tori in S^3 and the Lawson conjecture, Acta Math. 211 (2013), no. 2, 177–190. 3143888Brendle:survey , Minimal surfaces in S^3: a survey of recent results, Bull. Math. Sci. 3 (2013), no. 1, 133–171. 3061135Cheng S.-Y. Cheng, Eigenfunctions and eivenvalues of the Laplacian, Proc. Symp. Pure Math. 27 (1975), 185-193haskins:kapouleas:invent Mark Haskins and Nikolaos Kapouleas, Special Lagrangian cones with higher genus links, Invent. Math. 167 (2007), no. 2, 223–294. 2270454kapouleas:high:doubling Nikolaos Kapouleas, Minimal hypersurfaces in the round n-sphere by doubling the equatorial (n-1)-sphere for any n>3, In preparation.kapouleas:annals , Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. 1043269kapouleas:wente:announce , Constant mean curvature surfaces constructed by fusing Wente tori, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 12, 5695–5698. 1165926kapouleas:wente , Constant mean curvature surfaces constructed by fusing Wente tori, Invent. 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LDa Nikolaos Kapouleas and Peter McGrath, Linearized doubling with asymmetric sides and applications, In preparation.kapouleas:clifford Nikolaos Kapouleas and Seong-Deog Yang, Minimal surfaces in the three-sphere by doubling the Clifford torus, Amer. J. Math. 132 (2010), no. 2, 257–295. 2654775L2 H. Blaine Lawson, Jr., Complete minimal surfaces in S^3, Ann. of Math. (2) 92 (1970), 335–374. 0270280neves Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. 3152944schoen Richard M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), no. 3, 317–392. 929283wiygul:t David Wiygul, Doubling constructions with asymmetric sides, Ph.D. thesis, Brown University, 2014.Wiygul:s , Minimal surfaces in the 3-sphere by stacking clifford tori, arXiv preprint arXiv:1502.07420 (2015).Yau:problems Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. 645762 | http://arxiv.org/abs/1707.08526v1 | {
"authors": [
"Nikolaos Kapouleas",
"Peter McGrath"
],
"categories": [
"math.DG"
],
"primary_category": "math.DG",
"published": "20170726162142",
"title": "Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II"
} |
Institute for Theoretical Solid State Physics, IFW-Dresden, Helmholtzstr. 20, D-01069 Dresden, GermanyInstitute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht,NetherlandsCNR-SPIN, I-84084 Fisciano (Salerno), Italy Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno I-84084 Fisciano (Salerno), Italy CNR-SPIN, I-84084 Fisciano (Salerno), Italy Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno I-84084 Fisciano (Salerno), Italy Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht,Netherlands Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno I-84084 Fisciano (Salerno), ItalyInstitute for Theoretical Solid State Physics, IFW-Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany We propose and analyze theoretically a one-dimensional solid-state electronic setup that operates as a topological charge pump in the complete absence of superimposed oscillating local voltages. The system consists of asemiconducting narrow channel with strong Rashba spin-orbit interaction patterned in a mesoscale serpentine shape.A rotating planar magnetic field serves as the external ac perturbation, and cooperates with the Rashba spin-orbit interaction, which is modulated by the geometric curvature of theelectronic channel to realize the topological pumping protocol originally introduced by Thouless in an entirely novel fashion.We expect the precise pumping of electric charges in our mesoscopic quantum device to be relevant for quantum metrology purposes.73.63.Nm, 73.21.Cd, 03.65.Vf, 73.43.-fA topological quantum pump in serpentine-shaped semiconducting narrow channels Carmine Ortix December 30, 2023 ==============================================================================Introduction – Precisely as in an Archimedean screw, where water is pumped by a rotating spiral tube, in a charge pump periodic perturbations induce a dc current <cit.> without an external bias.This phenomenon can be entirely adiabatic, as it occurs for instance in open quantum dots subject to a cyclic deformation of the confining potential <cit.>,with the system that always remains in its instantaneous ground state.In a topological charge pump <cit.> the zero bias dc current is precisely quantized, and the quantization is topologically protected against external perturbations <cit.>.Each pump cycle transports an integer number of electronic charges with the integer uniquely determined by a topological invariant: the Chern number 𝒞 of the quantum system <cit.>.The dc current generated by periodic variations of one parameter of the system Hamiltonian amounts indeed to I=e𝒞 ν where ν is the frequency of the variation <cit.>. Topological charge pumping can bealso understood as a dynamical analog of the integer quantum Hall effect <cit.>,with the charge pumped in each cycle that can be mapped exactly to the quantized Hall conductance of a “dual" two-dimensional electronic system.Such analogy is mathematically transparent considering the mapping between the Hofstadter model <cit.> – perhaps the most simple Hamiltonian to study quantum Hall physics in a two-dimensional lattice system – and theone-dimensional Aubry-André-Harper (AAH) model <cit.>, when the superlattice potential is assumed to rigidly slide in time <cit.>. Very recently, the advances in constructing optical lattice structures with precise control over lattice intensity and phases have enabled the realization of dynamically controlled optical superlattices, and consequently of topological quantum pumps inultracold atomic systems <cit.>.In one-dimensional electronic systems, instead, the creation of a dynamical superlatticepotential critically relies on the presence and control of superimposed oscillating local voltages <cit.>. This, in turns, severely hampers the possibility to bring topological charge pumping within reach in condensed matter experiments. In this Rapid Communication,we propose and validate theoretically an entirely novel solid-state system in whichtopological quantum pumping can be achieved even in the complete absence of superimposed voltage leads. The system consists of a Rashba spin-orbit coupledsemiconducting narrow channel with a serpentine shape at the mesoscopic scale [c.f. Fig.<ref>(a)] : It can be obtained either by processing a semiconducting quantum well lithographically <cit.>, or creating a “zigzag" nanowire network ofcrystalline quality <cit.>. To operate, the device makes use of an auxiliary external planar rotating magnetic field, which serves as the periodic (ac) perturbation driving the charge pumping [c.f. Fig.<ref>(b)] . The concomitant presence of the time-dependent Zeeman interaction and the spin-orbit coupling, which is effectively inhomogeneous due to the geometric curvature of thenanostructure <cit.> renders a sliding superlattice potential acting on the electronic charges, and ultimately yields a quantized dc current.We also show that as the strength of the rotating magnetic field is increased, the system undergoes a topological phase transition from a state where an even integer number of electronic charges are transported in each rotation period of the magnetic field, to a state where the rotating magnetic field does not pump any electronic charge. Theoretical model – We start out by expressing the effective k · p Hamiltonian for a planar one-dimensionalsemiconducting channel with curved geometry as <cit.> H_ k · p =- ħ^22 m^⋆∂^2_s - i α_Rτ_𝒩 (s) ∂_s + i α_R2κ(s) τ_𝒯(s)- ħ^28 m^⋆κ(s)^2.In the equation above, s is the arclength along thechannel measured from an arbitrary reference point,m^⋆ and α_Rindicate the effective mass of the carriersand the Rashba spin-orbit coupling constant respectively, and κ(s) is the local curvature explicitly entering in the so-called quantum geometric potential (QGP) <cit.>.We also introduced two local Pauli matrices, comoving with the electrons as they propagate along s, explicitly readingτ_𝒩(s) = τ·𝒩̂(s) and τ_𝒯(s) = τ·𝒯̂(s), where the τ's are the usual Pauli matrices.𝒯̂(s) and 𝒩̂(s), i.e. the local tangential and normal directions of the narrowchannel, can be expressed in Euclidean space by introducing an angle φ(s), in terms of which 𝒩̂(s) = {sinφ(s), cosφ(s), 0 } and 𝒯̂(s) = {cosφ(s), -sinφ(s), 0 }.Using the Frenet-Serretequation ∂_s 𝒯̂(s) ≡κ(s) 𝒩̂(s), it then follows that the angle φ(s)= - ∫^s κ(s^') d s^' is entirelydeterminedby the local curvature.To proceed further, we use that the undulating geometry of a serpentine-shape can be parametrized in the Monge gauge as y(x) = A cos(2 π x / λ) with Athe maximum departure of the channel from flatness, and λ the serpentine period.In the shallow deformation limit A / λ≪ 1, we can express the arclength of the channel s ≃ x while the geometric curvature isκ(s) ≃ - (2 π/λ)^2 A cos(2 π s / λ). This periodicity of the curvature transmutes the QGP in an effective superlattice potential <cit.>,and yields at the same time a periodic canting of the spin-orbit field axis <cit.>. In order to study their concomitant effect on the electronic properties of the quantumsystem, we next introduce a tight-binding model obtained by discretizing Eq. <ref> on a lattice. It can be written as H =-t∑_i,σc^†_i,σ c_i+1,σ- ħ^2/8 m^*∑_i,σκ(s_i)^2 c^†_i,σc_i,σ - i α_R/2∑_i,σ,σ^' c^†_i,σ[ τ_N^σ, σ^'(s_i) + τ_N^σ, σ^'(s_i+1) /2]c_i+1,σ^' + H.c.,where c^†_i , σ, c_i , σ are operators creating and annihilating, respectively, an electron at the ith site with spin projection σ= ↑, ↓ along the ẑ axis, and t = ħ^2 / (2 m^⋆ a^2) is the nearest-neighbour hopping amplitude.The atomic positions can be instead written as s_i / λ = p i / q + ϕ, with p and q coprimes whose ratio is p/q = a / λ (a being the tight-binding lattice constant), and ϕ a phase accounting for nonequivalent displacements of the atoms in one superstructure period.The Hamiltonian in Eq. <ref> can be seen as the combination of a conventional, commensurate diagonal AAH model with a spin-dependent off-diagonal AAH model <cit.>. Taken separately, both these models realize butterfly spectra <cit.> and are characterized by insulating states with non-trivial Chern numbers if the phase ϕ is assumed to vary in time.We will instead consider a constant ϕ value (taking ϕ≡ 0 for simplicity) and monitor the effect of an external magnetic field, which we account for adding the usual Zeeman term H_Z ≡∑_i,σ,σ^'c^†_i,σ[B⃗·τ⃗]_σ,σ^'c_i,σ^' to the Hamiltonian in Eq. <ref>.We first consider the ensuing electronic properties in the regime of negligible Rashba spin-orbit coupling, with preserved SU(2) spin symmetry,for the simple case of p/q=1/4.At filling fraction ν=1/4, the geometric curvature of thenanostructure yields, via the QGP, a metal-insulator transition (MIT) for strong enough external magnetic fields.In this regime indeed, the lowest energy bands are completely spin-polarized thereby allowing the spin-independent superlattice potential to open up a gap at the centre of the mini-Brillouin zone (mBZ) <cit.>. As explicitly proved in Fig. <ref>, for A / λ→ 0 the critical value of the magnetic field strength at which the MIT takes placeis B_c ≃ t, and it continuously flows to smaller values as the geometric curvature is increased. In the B<B_c region instead, two degeneracies between opposite spin states at unpinned points in the mBZcannot be removed by the action of the QGP thereby implying the existence of a semimetallic state at one-quarter filling.Having established the occurrence of a curvature-driven MIT in the presence of SU(2) spin symmetry, we now investigate the effect of the Rashba spin-orbit coupling.We first consider a magnetic field B= B ẑ, orthogonal to the plane in which the spin-orbit field lies.Contrary to a conventionalstraight channel, the canting of the Rashba field resulting from the geometric curvatureyields a finite gapat ν=1/4 filling independent of the strength of the external magnetic field [c.f. Fig. <ref>(a)]. This is because the degeneracies between opposite spin states encountered at unpinned momenta of the mBZ in the weak magnetic field regime are removed by the action of the curvature-induced modulated Rashba term.And indeed the gap increases monotonously with the Rashba coupling α_R for weak magnetic field strengths [c.f. Fig. <ref>(a)].Exactly the same featuresoccur by tilting the magnetic field direction toward the x̂ direction <cit.>,which corresponds to the direction in which the effective spin-orbit interaction averages to zero in one superlattice period.When the magnetic field points exactly in the latter direction, however, the situation changes drastically. As shown in Fig. <ref>(b),cranking up the magnetic field strength leads first to an increase of the quarter filling gap, which is subsequently followed by a substantial decreaseuntil a critical value where the gap undergoes a closing-reopening transition at the center of the mBZ <cit.>.Topological charge pumping – In the same spirit of Thouless seminal work <cit.>, we now consider a slow, adiabatic ac rotation of the magnetic field in the x̂ - ẑ plane. Since the system always remainsin its instantaneous ground state, we can interpret the angle θ that the magnetic field direction forms with the ẑ axis, as an additional quasi-momentum.With this dimensional extension, our system can be viewed as a two-dimensional insulator which, due to the absence of time-reversal symmetry, belongs to the class A of the Altland-Zirnbauer table <cit.>.Moreover, the presence of the gap closing-reopening discussed above suggests the existence of two insulating phases characterized bydifferent ℤ topological invariants.We have verified that our quarter-filled model supports an insulating phase with a non-trivial Chern number 𝒞≡ -2 in the weak magnetic field region,while a completely trivial 𝒞≡ 0 insulating phase is encountered in the strong-field regime.As shown in Fig.<ref>(a), the energy spectrum of a finite size system with open boundary conditions displaystwo chiral edge stateswithin the one quarter-filling gap for small enough magnetic field.They are related to each other by a π rotation of the magnetic field direction, and carry opposite spin content as can be shown with a symmetry argument <cit.>.In the opposite strong field regime, instead, the edge states do not connect the valence to the conduction band thereby implying a topologically trivial insulating state[c.f. Fig. <ref>(b)].We have additionally computed the Berry curvature of the insulating states using the method outlined in Ref. tak05. In the weak magnetic field regime, we find that the Berry curvature displays four peaks [c.f. Fig. <ref>(c)], each of which contributing ≃ -1/2 to the total Chern number. In the strong magnetic field regime, instead, the contributions coming from thetwo peaks located at the mBZ center for θ=±π/2[c.f. Fig. <ref>(d)], are identically cancelled by an almost homogeneous background with Berry curvature opposite in sign. The fact that our tight-binding model realises a 𝒞≡ -2 topological charge pump when subject to weak rotating magnetic fields is independent of the superlattice period. Specifically,for any superlattice period λ= 2 n a with n integer, and at 1 / ( 2 n) filling, all the discussion above remains unaltered. This, in turn, implies that also amesoscale serpentine-shaped quantum wirecan operate as a topological charge pump.Since the conditions for this effect to be observed are k_B T, h ν < E_g where k_B Tis the energy scale for temperature T,ν the magnetic field frequency, and E_g is the relevant energy gap, we analyzed the continuum k · p effective theory given by Eq. <ref> taking into accountthe Zeeman term H_Z= g^⋆μ_BB·τ, where g^⋆ is the effective Landé g factor,and μ_B the Bohr magneton.Fig. <ref> shows the corresponding topological phase diagrams with a map of the bulk gap when the first two Bloch minibands are fully occupied.By considering the serpertine period to be of the order ofλ≃ 0.5 μm, the effective mass of, e.g., InAs m^⋆≃ 0.023, and a strength of the Rashba spin-orbit interaction <cit.> up to0.2 eV Å, we find that the size of the bulk gapE_g≃ 20 μeV for A/λ=0.1 and a magnetic field strength≃ 60mT (the Landé g factor g_InAs≃ 15 in bulk crystals <cit.>).As a result, a topological charge pumping effect characterised by a quantized dc current can be observed for temperaturesT < 200mK and frequenciesω < 4 GHz. A rotating magnetic field with this freqency can be in principle obtained by running current pulses in two perpendicular conductors with a π/2 phase shift <cit.>. The resulting dc current I ≃ 1nA can be easily detected with present day experimental capabilities. Therefore, in this physical regime our solid-state device can be used to measure the fundamental electron charge e. Conclusions – To wrap up, we have shown that a Rashba spin-orbit coupledsemiconducting channel possessing a mesoscale serpentine shape, as obtained either by lithographic processing <cit.> or using “zigzag" nanowire networks <cit.>, can operate as a topological charge pump once subject to a relatively weak magnetic field rotating on a plane.Being topological in nature, the electronic pumping is robust against additional perturbations, e.g. tiny tilts of the magnetic field in the direction orthogonal to the plane.The existence of this phenomenon stems from theperiodic canting of the Rashba spin-orbit field, which is the prime physical consequence of the geometric curvature of ournanostructure, and provides a bridge between the Zeemaninteraction and the quantized charge flow in the wire.Adiabatic quantum pumping has been recently discussed in one dimensional semiconducting Rashba systems in the presence of local voltage leads or periodically arranged nanomagnets <cit.>. Our proposed set-up, instead, does not necessitate the usage of external local perturbationsince it only relies on the successfulgrowth of a one-dimensional channel with curved undulating geometry.Our findings therefore add a prime actor to the recently uncovered series of unique curvature-induced quantum effects in low-dimensional semiconducting systems <cit.>.Acknowledgements – We acknowledge the financial support of the Future and Emerging Technologies (FET) programme under FET-Open grant number: 618083 (CNTQC).C.O. acknowledges support from the Deutsche Forschungsgemeinschaft (Grant No. OR 404/1-1), and from a VIDI grant (Project 680-47-543) financed by the Netherlands Organization for Scientific Research (NWO). S. P. thanks Ulrike Nitzsche for technical assistance. N. S. thanks Guido van Miert for many fruitful discussions. S. P. and N. 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Ortix,10.1021/acs.nanolett.7b00426 journal journal Nano Letters volume 17,pages 3076 (year 2017)NoStop | http://arxiv.org/abs/1707.08773v2 | {
"authors": [
"Sudhakar Pandey",
"Niccolo' Scopigno",
"Paola Gentile",
"Mario Cuoco",
"Carmine Ortix"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727082238",
"title": "A topological quantum pump in serpentine-shaped semiconducting narrow channels"
} |
[ [ December 30, 2023 ===================== We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether asymmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically.The locus of cographic matroids arises from tropicalizing the Schottky–Igusa modular form. § INTRODUCTION The Schottky problem <cit.> concerns the characterization of Jacobians of genus g curvesamong all abelian varieties of dimension g. The latter are parametrized by theSiegel upper-half space ℌ_g, i.e. the set of complex symmetric g× g matrices τ with positive definite imaginary part. The Schottky locus𝔍_g is the subset of matrices τ in ℌ_g that represent Jacobians.Both sets are complex analytic spaces whose dimensions reveal that the inclusion is proper for g ≥ 4:dim(𝔍_g)= 3g -3 and dim(ℌ_g)=g+12. For g=4, the dimensions in (<ref>) are 9 and 10,so 𝔍_4 is an analytic hypersurface in ℌ_4.The equation defining this hypersurface is a polynomial of degree 16 inthe theta constants. First constructed by Schottky <cit.>, and further developed by Igusa <cit.>, this modular form embodies the theoretical solution (cf. <cit.>) to the classical Schottky problem for g=4.The Schottky problem also exists in tropical geometry <cit.>. The tropical Siegel space ℌ_g^ trop is thecone of positive definite g × g-matrices, endowed with the fan structure given by the second Voronoi decomposition. The tropical Schottky locus 𝔍_g^ trop is the subfan indexed by cographic matroids <cit.>. A detailed analysis for g≤ 5 is found in <cit.>. It is known, e.g. by <cit.>, that the inclusion𝔍_g^ trop⊂ℌ_g^ trop correctly tropicalizes the complex-analytic inclusion 𝔍_g ⊂ℌ_g. However,it has been an open problem (suggested in <cit.>) to find a direct link between the equations that govern these two inclusions.We here solve this problem, anddevelop computational tools for the Schottky problem, both classically and tropically. We distinguish between the Schottky Decision Problem and the Schottky Recovery Problem. For the former, the inputis a matrix τ in ℌ_g resp. ℌ_g^ trop, possibly depending on parameters, and we must decide whether τ lies in 𝔍_g resp. 𝔍_g^ trop. For the latter, τ already passed that test, and wecompute a curve whose Jacobian is given by τ. The recovery problem also makes sense for g=3, both classically <cit.> and tropically <cit.>. This paper is organized as follows. In Section <ref> we tackle the classical Schottky problem as a task in numerical algebraic geometry <cit.>. For g=4, we utilize the software abelfunctions <cit.> to test whether the Schottky–Igusa modular form vanishes. In the affirmative case,we use a numerical version of Kempf's method <cit.> to compute a canonical embedding into ^3. Our main results in Section <ref>are Algorithms <ref> and <ref>. Based on the work in <cit.>, these furnish acomputational solution to the tropical Schottky problem. Key ingredients arecographic matroids and the f-vectors of Voronoi polytopes.Section <ref> links the classical and tropical Schottky scenarios. Theorem <ref> expresses the edge lengths of ametric graph in terms of tropical theta constants, and Theorem <ref> explains what happens to the Schottky–Igusa modular form in the tropical limit. We found it especially gratifying to discover how the cographic locus is encoded in the classical theory.The software we describe in this paper is made available at the supplementary website<http://eecs.berkeley.edu/ chualynn/schottky>This contains several pieces of code for the tropical Schottky problem, as well as a more coherentprogramfor the classical Schottky problem that makes calls to abelfunctions. § THE CLASSICAL SCHOTTKY PROBLEM We fix g=4, and review theta functions andIgusa's construction <cit.> of the equation that cuts out 𝔍_4.For any vector m∈ℤ^8 we writem= (m',m”) for suitable m',m”∈ℤ^4. The Riemann theta functionwith characteristic m is the following function ofτ∈ℌ_4 and z∈ℂ^4:θ[m](τ,z)= ∑_n∈ℤ^4exp[πi (n+m'/2)^t τ (n+m'/2)+2πi(n+m'/2)^t(z+m”/2)].For numerical computations of the theta function one has to make a good choice of lattice points to sum over in order for this series to converge rapidly<cit.>. We use the software abelfunctions <cit.> to evaluate θ[m] for arguments τ and z with floating point coordinates.Up to a global multiplicative factor, the definition (<ref>) depends only on the image of m in (ℤ/2ℤ)^8. The sign of the characteristic m is e(m)=(-1)^(m')^t m”. Namely, m is even if e(m)=1 and odd if e(m) = -1. A triple {m_1,m_2,m_3}⊂ (ℤ/2ℤ)^8is called azygetic if e(m_1)e(m_2)e(m_3)e(m_1+m_2+m_3) = -1. Suppose that this holds. Then we choose a rank 3 subgroup N of (ℤ/2ℤ)^8 such that all elements of(m_1+N) ∪ (m_2+N) ∪ (m_3+N) are even. Weconsider the following three products of eight theta constants each:π_i =∏_m∈ m_i+Nθ[m](τ,0) for i=1,2,3.The function ℌ_4 →ℂ that takesa symmetric 4 × 4-matrix τ toπ_1^2+π_2^2+π_3^2-2π_1π_2-2 π_1π_3-2 π_2π_3is independent of the choices above. It vanishes if and only if τ lies in the closure of the Schottky locus 𝔍_4. We refer to the expression (<ref>) as the Schottky–Igusa modular form. This is a polynomial of degree 16 in the theta constantsθ[m](τ,0). Of course, the formula is unique only modulo the ideal that defines the embedding of the moduli space 𝒜_4 in the ^15 of theta constants. Our implementation uses the polynomial that is given by the following specific choices:m_1=[ 1 1; 0 0; 1 1; 0 0 ],m_2=[ 0 1; 0 0; 0 0; 1 0 ],m_3=[ 0 1; 0 0; 1 1; 1 1 ], n_1=[ 0 1; 0 1; 0 1; 1 0 ],n_2=[ 0 0; 0 0; 1 0; 1 1 ],n_3=[ 0 1; 0 0; 1 1; 0 1 ].The vectors n_1,n_2,n_3 generate the subgroup N in (ℤ/2ℤ)^8. One checks that the triple {m_1,m_2,m_3} is azygetic and that the three cosets m_i + N consist of even elements only. The computations to be described next were done with the Sage library abelfunctions <cit.>. The algorithm in <cit.> finds the Riemann matrix τ∈𝔍_g of aplane curve inℂ^2. It is implemented in abelfunctions.We first check that (<ref>)does indeed vanish for such τ.The plane curve y^5+x^3-1 = 0 has genus four. ItsRiemann matrix τ is [0.16913 + 1.41714i -0.81736 - 0.25138i -0.05626 - 0.44830i0.24724 + 0.36327i; -0.81736 - 0.25138i -0.31319 + 0.67096i -0.02813 - 0.57155i0.34132 + 0.40334i; -0.05626 - 0.44830i -0.02813 - 0.57155i0.32393 + 1.44947i -0.96494 - 0.63753i;0.24724 + 0.36327i0.34132 + 0.40334i -0.96494 - 0.63753i0.62362 + 0.73694i ] .Evaluating the 16 theta constants θ[m](τ,0) numerically with abelfunctions, we find thatπ_1^2+π_2^2+π_3^2 = -5.13472888270289 + 6.13887870578982i, 2(π_1π_2+π_1π_3+π_2π_3) = -5.13472882638710 + 6.13887931435788i.We trust that (<ref>) is zero, and conclude that τ lies in the Schottky locus 𝔍_4, as expected. Suppose now that we are given a matrixτ that depends on one or two parameters, so it traces out a curve or surface in ℌ_4. Then we can use our numerical method to determine the Schottky locusinside that curve or surface. Here is an illustration for a surface in ℌ_4. The following one-parameter family of genus 4 curves is found in <cit.>: y^6 = x(x+1)(x-t). This is both a Shimura curve and a Teichmüller curve. Its Riemann matrix isρ(t) = Z_2^-1 Z_1 where Z_1,Z_2 are given in <cit.>. Consider the following two-parameter familyin ℌ_4:τ(s,t) = s · diag(2,3,5,7) +ρ(t).We are interested in the restriction of the Schottky locus 𝔍_4 to the (s,t)-plane. For our experiment, we assume that the two parameters satisfys ∈ [-0.5,0.5] and λ^-1(t) ∈ [ i,i+1],where λ is the functionin <cit.>.Using abelfunctions, we computed the absolute value of the modular form (<ref>) at 6400 equally spaced rational points in the square [-0.5,0.5] × [i,i+1]. That graph is shown in Figure <ref>. For s different from zero, the smallest absolute value of (<ref>) is 4.3 × 10^-3. For s=0, all absolute values are below 2.9 × 10^-8. Based on this numerical evidence, we conclude that the Schottky locus of our family is the line s=0.We now come to the Schottky Recovery Problem. Our input is a matrix τ in 𝔍_4. Our task is to compute a curve whose Riemann matrix equals τ. We use the following result from Kempf's paper <cit.>. The theta divisor in the Jacobianℂ^4/(ℤ^4 + ℤ^4 τ ) is the zero locus Θ^-1(0) of the Riemann theta function Θ(z) := θ[0](τ,z). For generic τ this divisor is singular at precisely two points. These represent 3-to-1 maps from the curve to ^1. We compute a vector z^* ∈ℂ^4 that is a singular point of Θ^-1(0) by solving the system of five equationsΘ(z) =∂Θ/∂ z_1(z) = ∂Θ/∂ z_2(z) =∂Θ/∂ z_3(z) = ∂Θ/∂ z_4(z) =0.The Taylor series of the Riemann theta function Θ at the singular point z^* has the formΘ(z^*+x)= f_2(x)+f_3(x) +f_4(x)+ higher order terms,where f_s is a homogeneous polynomial of degree s in x = (x_1,x_2,x_3,x_4). The canonical curve with Riemann matrix τ is the degree 6 curve in ^3 that is defined by the quadratic equation f_2 = 0 and the cubic equation f_3 = 0. Thus our algorithm for the Schottky Recovery Problem consists of solving thefive equations (<ref>) for z^* ∈ℂ^4, followed by extracting the polynomials f_2 and f_3 in the Taylor series (<ref>). Both ofthese steps can be done numerically using the software abelfunctions <cit.>.Let τ∈𝔍_4 be the Riemann matrix of the genus 4 curve C = {x^3 y^3 +x^3 + y^3 = 1}.We obtain τ numerically using abelfunctions. We want to recover C from τ. To be precise, given only τ,we want to find defining equations f_2 = f_3 = 0 in ^3 of the canonical embedding of C. For that we use evaluations of Θ(z) and its derivatives in abelfunctions, combined with a numerical optimization routine in SciPy <cit.>. We solve the equations (<ref>) starting from random points z = u + τ v where u,v ∈^4 with entries between 0 and 1. After several tries, the local method in SciPy converges to the following solution of our equations:z^* = (0.55517+0.69801i,0.53678+0.26881i, -0.50000-0.58958i,0.55517+0.69801i) .Using (<ref>), we computed the quadric f_2, which is nonsingular, as well as the cubic f_3:f_2(x) =(-3.044822827 + 21.980542613 i)· x_1^2 +( - 237.95207224+ 252.54744634 i)· x_1 x_2 +(- 222.35552015+ 139.95612952 i)· x_1 x_3 +(- 200.66932133- 16.596272620 i)· x_1 x_4 + (- 191.16241727 - 85.22650070 i)· x_2^2 + (- 429.11449060 + 167.32094535 i)· x_2 x_3+(- 237.952072 + 252.54744632 i)· x_2 x_4 +(- 206.75896934 + 27.364814282 i)· x_3^2+(222.35552013 + 139.95612953 i)· x_3 x_4 + (- 3.0448227745 + 21.980542601 i)· x_4^2 f_3(x)= (441.375966486 + 61.14097461986 i)· x_1^3 +(2785.727151434 + 2303.609067429 i)· x_1^2 x_2 + ⋯⋯ +( 441.3759668263 + 61.14097402189 i)· x_4^3.As a proof of concept we also computed the 120 tritangent planes numerically directly from τ. These planes are indexed by the 120 odd theta characteristics m. In analogy to the computationin <cit.> of the 28 bitangents for g=3, their defining equations are∂θ[m](τ,z)/∂ z_1|_z=0· x_1+ ∂θ[m](τ,z)/∂ z_2|_z=0· x_2+ ∂θ[m](τ,z)/∂ z_3|_z=0· x_3+ ∂θ[m](τ,z)/∂ z_4|_z=0· x_4 = 0.We verified numerically that each such plane meets {f_2 = f_3 = 0} in three double points.On our website (<ref>), we offer a program in Sage whose input is a symmetric 4 × 4-matrix τ∈ℌ_4, given numerically. The code decides whether τ lies in 𝔍_4 and, in the affirmative case, it computes the canonical curve {f_2 = f_3 = 0} and its 120 tritangent planes. § THE TROPICAL SCHOTTKY PROBLEM Curves, their Jacobians, and the Schottky locus have natural counterparts in the combinatorial setting of tropical geometry. We review the basics from <cit.>. The role of a curve is played by a connected metric graph Γ = (V,E,l,w). This has vertex set V, edge set E, a length function l: E →ℝ_> 0, and a weight function w : V →ℤ_≥ 0. The genus of Γ isg =|E| -|V|+1+ ∑_v∈ V w(v).The moduli space ℳ_g^ trop comprises all metric graphs of genus g. This is a stacky fan of dimension 3g-3. See <cit.> for a colorful illustration. The tropical Torelli map ℳ_g^ trop→ℌ_g^ trop takes Γ to its (symmetric and positive semidefinite)Riemann matrix Q_Γ. Fix a basis for the integral homology H_1(Γ,ℤ) ≃ℤ^g.Beside the usual cycles in Γ, this group has w(v) generators for the virtual cycles at each vertex v. Let B denote the g × |E| matrix whose columns record the coefficients of each edge in the basis vectors. Let D be the |E| × |E| diagonal matrix whose entries are the edge lengths. The Riemann matrix of Γ isQ_Γ =B · D · B^t.One way to choose a basis is to fix an orientationand a spanning tree of Γ. Each edge not in that tree then determines a cycle with ± 1-coefficients. See <cit.> for details and an example. Changing the basis of H_1(Γ,ℤ) corresponds to the action of GL_g(ℤ) on Q_Γ by conjugation.The matrix Q_Γ has rank g - ∑_v∈ V w(v). We defined ℌ_g^ trop with positive definite matrices. Those have rank g. For that reason, we now restrict to graphs with zero weights, i.e. w ≡ 0.The tropical Schottky locus𝔍_g^ tropis the set of all matrices (<ref>), where Γ = (V,E,l) runs over graphs of genus g, and B runs over their cycle bases. This set is known as the cographic locus in ℌ_g^ trop, because the g × |E| matrix B is a representation of the cographic matroid of Γ.The Schottky Decision Problem asks for a test of membership in 𝔍_g^ trop. To be precise, given a positive definite matrix Q, does there exist a metric graph Γ such that Q = Q_Γ?To address this question, we need the polyhedral fan structures on 𝔍_g^ trop andℌ_g^ trop. Let G = (V,E) be the graph underlying Γ, with E = {e_1,e_2,…,e_m}. Fix a cycle basis as above. Let b_1,b_2,…,b_m be the column vectors of the g × m-matrix B. Formula (<ref>) is equivalent toQ_Γ= l(e_1) b_1b_1^t + l(e_2) b_2 b_2^t +⋯ +l(e_m)b_m b_m^t.The cone of all Riemann matrices for the graph G, allowing the edge lengths to vary, is σ_G,B=ℝ_>0{ b_1b_1^t,b_2 b_2^t,… ,b_m b_m^t} .This is a relatively open rational convex polyhedral cone,spanned by matrices of rank 1. The collection of all cones σ_G,B is a polyhedral fan whose support is the Schottky locus 𝔍_g^ trop.This fan is a subfan of the second Voronoi decomposition of the cone ℌ_g^ trop of positive definite matrices.The latter fan is defined as follows. Fix a Riemann matrix Q ∈ℌ_g^ trop and consider its quadratic form ℤ^g →ℝ,x ↦ x^t Q x. The values of this quadratic form define a regular polyhedral subdivision of ℝ^g with vertices at ℤ^g. This is denoted Del(Q) and known as the Delaunay subdivision of Q. Dual to Del(Q) is the Voronoi decomposition of ℝ^g. The cells of the Voronoi decomposition of Q are the lattice translates of the Voronoi polytope{ p∈ℝ^g:2 p^t Q x ≤ x^t Q x for allx∈ℤ^g}.This is the set of points in ℝ^g for which the origin is the closest lattice point, in the norm given by Q. If Q is generic then the Delaunay subdivision is a triangulation and the Voronoi polytope (<ref>) is simple. It is dual to the link of the origin in the simplicial complex Del(Q).The structures above represent principally polarized abelian varieties in tropical geometry. A tropical abelian variety is the torus ℝ^g/ℤ^g together with a quadratic form Q ∈ℌ_g^ trop. The tropical theta divisor is given by thecodimension one cells in theinduced Voronoi decomposition of ℝ^g/ℤ^g. See <cit.> for an introduction with many pictures and many references.We now fix an arbitrary Delaunay subdivision D of ℝ^g. Its secondary cone is defined asσ_D ={ Q∈ℌ_g^ trop|(Q) = D } .This is a relatively open convex polyhedral cone. It consists of positive definite matrices Q whose Voronoi polytopes (<ref>) have the same normal fan. The group GL_g(ℤ) acts on the set of secondary cones. In his classical reduction theory for quadratic forms,Voronoi <cit.> proved that the cones σ_D form a polyhedral fan, now known as the second Voronoi decomposition of ℌ_g^ trop, and that there are only finitely many secondary cones σ_D up to the action of GL_g(ℤ). The following summarizescharacteristic features for matrices in the Schottky locus 𝔍_g^ trop.Fix a graph G with metric D, homology basis B, and Riemann matrixQ = BDB^t. The Voronoi polytope (<ref>) is affinely isomorphic to the zonotope ∑_i=1^m [-b_i,b_i].The secondary cone σ_ Del(Q) is spannedby the rank one matrices b_i b_i^t: it equals σ_G,B in (<ref>).This can be extracted from Vallentin's thesis <cit.>. The affine isomorphism is given by the invertible matrix Q, as explained in item iii) of <cit.>. The Voronoi polytope being the zonotope ∑_i=1^m [-b_i,b_i] follows from the discussion on cographic lattices in <cit.>. The result for the secondary cone is derived from <cit.>. See <cit.> for many examples.We now fix g=4. Vallentin <cit.> listsall 52 combinatorial types of Delaunay subdivisionsof ℤ^4. His table contains the f-vectors of all 52 Voronoi polytopes. Precisely 16 of these types are cographic, and these comprise the Schottky locus 𝔍_4^ trop. These are described in rows 3 to 18 of the table in <cit.>. We reproducethe relevant data in Table <ref>. The following key lemma is found by inspecting Vallentin's list of f-vectors. The f-vectors of the 16 Voronoi polytopes representing the Schottky locus 𝔍_4^ trop are distinct from the f-vectors of the other 36 Voronoi polytopes, corresponding to ℌ_4^ trop\𝔍_4^ trop. This lemma gives rise to the following method for the tropical Schottky decisionproblem. |c|c|c|c|c|c|c| Graph G Riemann matrix Q_Γ f_0 f_1 f_2 f_3 Dimension of σ_D 0pt1.4 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (0.5,0.5) circle (1pt) (1,0.5) circle (1pt) (1.5,0) circle (1pt) (1.5,1) circle (1pt); (0,0)–(0,1)–(0.5,0.5)–(0,0) (0,1)–(1.5,1)–(1.5,0)–(0,0) (0.5,0.5)–(1,0.5) (1.5,0)–(1,0.5)–(1.5,1); [31 -10;1411; -114 -1;01 -13 ] 96 198 130 28 9[baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (2,0) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt) (2,1) circle (1pt);(0,0)–(0,1) (0,0)–(1,1) (0,0)–(2,1) (1,0)–(0,1) (1,0)–(1,1) (1,0)–(2,1) (2,0)–(0,1) (2,0)–(1,1) (2,0)–(2,1); [42 -2 -1;24 -1 -2; -2 -142; -1 -224 ] 102 216 144 30 9 0pt1.4 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt) (0.5,0.5) circle (1pt); (0,0)–(1,0) (1,1)–(0,1)–(0,0) (0,0)–(0.5,0.5)–(1,1) (0.5,0.5)–(0,1) (1,0) to [bend left] (1,1) (1,0) to [bend right] (1,1); [20 -10;0311; -114 -1;01 -13 ] 72 150 102 24 8 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt) (0.5,0.5) circle (1pt); (0,0)–(1,0)–(1,1)–(0,1)–(0,0) (0,0)–(0.5,0.5)–(1,1) (1,0)–(0.5,0.5)–(0,1); [321 -1;242 -1;1241; -1 -113 ] 78 168 116 26 8 0pt1.4 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt); (0,0)–(1,0) (0,0)–(0,1) (0,0)–(1,1) (1,0) to [bend left] (1,1) (1,0) to [bend right] (1,1) (0,1)–(1,1) (0,1)–(1,0);[31 -1 -1;1311; -1132; -1123 ] 60 134 98 24 7[baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt); (0,0)–(1,0) (1,0) to [bend left] (1,1) (1,0) to [bend right] (1,1) (0,1) to [bend left] (1,1) (0,1) to [bend right] (1,1) (0,0) to [bend left] (0,1) (0,0) to [bend right] (0,1);[20 -1 -1;02 -1 -1; -1 -143; -1 -134 ] 54 116 84 22 7 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0,1) circle (1pt) (1,0) circle (1pt) (1,1) circle (1pt); (0,0)–(1,0) (0,0)–(0,1) (0,0)–(1,1) (1,0) to [bend left] (1,1) (1,0) to [bend right] (1,1) (0,1) to [bend left] (1,1) (0,1) to [bend right] (1,1); [20 -10;020 -1; -1031;0 -113 ]54 114 80 20 7 [baseline=([yshift=1ex]current bounding box.center)] (0,0) circle (1pt) (0,0.8) circle (1pt) (0.8,0) circle (1pt) (0.8,0.8) circle (1pt); (0,0)–(0.8,0) (0,0)–(0,0.8) (0,0)–(0.8,0.8) (0.8,0) (0.8,0)–(0.8,0.8) (0,0.8)–(0.8,0.8) (0,0.8)–(0.8,0) (0.8,0) to [in=30,out=-60,distance=8mm] (0.8,0);[31 -10;1310; -1130;0001 ] 48 96 64 16 7 0pt1.4 [baseline=([yshift=-.5ex]current bounding box.center)] (0,1) circle (1pt) (1,1) circle (1pt) (0.5,0) circle (1pt); (0.5,0) to [bend left] (0,1) (0.5,0) to [bend right] (0,1) (0.5,0) to [bend left] (1,1) (0.5,0) to [bend right] (1,1)(0,1) to [bend left] (1,1) (0,1) to [bend right] (1,1); [20 -1 -1;0211; -1132; -1123 ] 46 108 84 22 6 [baseline=([yshift=-.5ex]current bounding box.center)] (0,1) circle (1pt) (1,1) circle (1pt) (0.5,0) circle (1pt); (0.5,0) to [bend left] (0,1) (0.5,0) to [bend right] (0,1) (0.5,0) to [bend left] (1,1) (0.5,0) to [bend right] (1,1)(0,1)–(1,1) (0.5,0)–(1,1);[2 -1 -1 -1; -1322; -1232; -1223 ] 42 94 72 20 6 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0.6) circle (1pt) (0.6,0.6) circle (1pt) (0.3,0) circle (1pt); (0.3,0) to [bend left] (0,0.6) (0.3,0) to [bend right] (0,0.6) (0.3,0) to [bend left] (0.6,0.6) (0.3,0) to [bend right] (0.6,0.6)(0,0.6)–(0.6,0.6) (0.3,0) to [in=-30,out=-150, distance=8mm] (0.3,0);[ 2 1 1 0; 1 3 2 0; 1 2 3 0; 0 0 0 1 ] 36 74 52 14 6 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (0.8,0) circle (1pt) (1.6,0) circle (1pt); (0,0) to [bend left=40] (0.8,0) (0,0) to [bend right=40] (0.8,0) (0,0)–(0.8,0); (0.8,0) to [bend left=40] (1.6,0) (0.8,0) to [bend right=40] (1.6,0) (0.8,0)–(1.6,0);[ 2 1 0 0; 1 2 0 0; 0 0 2 1; 0 0 1 2 ] 36 72 48 12 6 0pt1.4 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (1,0) circle (1pt); (0,0) to [bend left=20] (1,0) (0,0) to [bend left=60] (1,0) (0,0) to [bend right=20] (1,0) (0,0) to [bend right=60] (1,0) (0,0)–(1,0); [ 2 1 1 1; 1 2 1 1; 1 1 2 1; 1 1 1 2 ] 30 70 60 20 5 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt) (1,0) circle (1pt); (0,0) to [bend left=20] (1,0) (0,0) to [bend left=60] (1,0) (0,0) to [bend right=20] (1,0) (0,0) to [bend right=60] (1,0); (1,0) to[in=50,out=-50, distance=8mm,looseness=50] (1,0); [ 2 1 1 0; 1 2 1 0; 1 1 2 0; 0 0 0 1 ] 28 62 48 14 5 [baseline=([yshift=-.5ex]current bounding box.center)] (0.2,0) circle (1pt) (1,0) circle (1pt); (0.2,0) to [bend left=40] (1,0) (0.2,0) to [bend right=40] (1,0) (0.2,0)–(1,0); (1,0) to[in=90,out=0, distance=8mm] (1,0) (1,0) to[in=0,out=-90, distance=8mm] (1,0);[ 2 1 0 0; 1 2 0 0; 0 0 1 0; 0 0 0 1 ] 24 48 34 10 5 [baseline=([yshift=-.5ex]current bounding box.center)] (0,0) circle (1pt); (0,0) to[in=0, out=90,distance=8mm] (0,0) (0,0) to[in=90, out=180,distance=8mm](0,0) (0,0) to[in=180, out=270,distance=8mm] (0,0) (0,0) to[in=270, out=0,distance=8mm] (0,0); [ 1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] 16 32 24 8 4 The tropical Schottky locus for g=4 We implemented Algorithm <ref> using existing softwarefor polyhedral geometry, namely thepackage due to Dutour Sikirić <cit.>, as well as Joswig's polymake <cit.>. The first column of Table <ref> shows all relevant graphs G of genus 4. The second column gives a representative Riemann matrix. Here all edges have length 1 and a cycle basis B was chosen. Using (<ref>), we alsoprecomputedthe secondary conesσ_G,B for the 16representatives. Using thepackage<cit.> we compute the Voronoi polytope ofQ=[ 14 -9 110; -9 11 -21; 11 -2 21 11;01 11 14 ].Its f-vector is (62,142,104,24). This does not appear in Table <ref>. Hence Q is not in𝔍_4^ trop. We now address the Schottky Recovery Problem. The input is a matrix Q ∈𝔍_4^ trop. From Algorithm <ref> we know the f-vector of the Voronoi polytope. Using Table <ref>, this uniquely identifies the graph G. Note that our graphs G are dual to those in <cit.>. From our precomputed list, we also know the secondary cone σ_G,B for some choice of basis B. We implemented this algorithm as follows. Step 2can be done using<cit.>. This code computes the secondary coneσ_D containing a given positive definite matrix Q. The matrix X ∈ GL_4(ℤ) in Step 3 is also found by , but with external calls to the package due to Plesken and Souvignier <cit.>. We refer to <cit.> for details. For Step 4 we note that the rank 1 matrices b_1 b_1^t, …, b_m b_m^t are linearly independent <cit.>. Indeed, the two 9-dimensional secondary cones σ_G,B at the top ofTable <ref> are simplicial, and so are their faces. Hence the multipliers ℓ_1,…,ℓ_m found in Step 4 are unique and positive. These ℓ_i must agree with the desired edge lengths l(e_i), by the formula for Q= Q_Γ in (<ref>). Consider the Schottky Recovery Problem for the matrixQ =[ 17535;5 197 11;37 23 16;5 11 16 29 ].Using , we find that the f-vector of its Voronoi polytope is (96,198,130,28). This matches the first row in Table <ref>. Hence Q ∈𝔍_4^ trop, and G is the triangular prism. Usingand , we find a matrix thatmaps Q into our preprocessed secondary cone:X = [0001;1000;0110; -1 -100 ] ∈GL_4(ℤ) gives Q' = X^tQX = [ 269 -90;9 207 -2; -97 233;0 -23 17 ] ∈ σ_G,B.This Q' is the Riemann matrix of the metric graph in Figure <ref>, with basis cyclese_2+e_6-e_3, -e_1-e_4+e_7+e_2, -e_1-e_5+e_8+e_3, and e_4+e_9-e_5. These are the rows of the 4 × 9-matrix B. In Step 4 of Algorithm <ref> we compute D =diag(ℓ_1,…,ℓ_9) =diag( 7,9,9,2,3,8,2,4,12). In Step 5 we output the metric graph inFigure <ref>. Its Riemann matrix equals Q = BDB^t. It is instructive to compare Algorithms <ref> and <ref> with Section <ref>. Our classical solution is not just the abstract Riemann surface but it consists of a canonical embedding into ^3.Canonical embeddings also exist for metric graphs Γ, as explained in <cit.>. However, even computing the ambient space |K|, that plays the role of ^3, is non-trivial in that setting. For g=4 this is solved in <cit.>. An alternative approach is to construct aclassical curve over a non-archimedean field that tropicalizes to Γ. See <cit.> for first steps in that direction.Example <ref> explored the Schottky locus in a two-parameter family of Riemann matrices. In the tropical setting, it is natural to intersectℌ_g^ trop with an affine-linear space L of symmetric matrices. The intersectionℌ_g^ trop∩ L is a spectrahedron. By the Schottky locus of a spectrahedron we mean 𝔍_g^ trop∩ L. This is an infinite periodic polyhedral complex inside the spectrahedron. For quartic spectrahedra <cit.>, when g=4, this locus has codimension one. [The Schottky locus of a quartic spectrahedron] We consider the matrixQ = [ 1589 - 2922 s + 960 t789 - 1322 s -820 + 660 s - 1350 t -820 + 3260 s +2550 t;789 - 1322 s 1589 - 2922 s - 960 t-820 + 3260 s - 2550 t-820 + 660 s +1350 t; -820 + 660 s - 1350 t-820 + 3260 s - 2550 t 1665 + 450 s + 3120 t-25 - 2930 s; -820 + 3260 s +2550 t -820 + 660 s + 1350 t-25 - 2930 s 1665 + 450 s - 3120 t ].Here s and t are parameters. This defines a plane L in the space of symmetric 4 × 4-matrices. The left diagram inFigure <ref> shows the hyperbolic curve { det(Q)=0}. The spectrahedron ℌ_g^ trop∩ L is bounded by its inner oval. The right diagram shows the second Voronoi decomposition. The Schottky locus𝔍_g^ trop∩ Lis a proper subgraphof its edge graph.It is shown in red. Note that the graph has infinitely many edges and regions. We described some computations in GAP and in polymake that realize Algorithms <ref> and <ref>. The code for these implementationsis made available on our website (<ref>). § TROPICAL MEETS CLASSICALIn this section we present a second solution to the tropical Schottky problem. It is new and different from the one in Section <ref>, and it links directly tothe classical solution in Section <ref>.Let Q ∈ℌ_g^ trop be a positive definite matrix for arbitrary g.Mikhalkin and Zharkov <cit.> define the following analogue to the Riemann theta function in the max-plus algebra:Θ(Q,x):=max_λ∈ℤ^g{λ^t Q x-1/2λ^t Q λ}.This tropical theta function describes the asymptotic behavior of theclassical Riemann theta function with Riemann matrix t·τ when t goes to infinity, as long as there are no cancellations. This is made precise in Proposition <ref>. Here, the real matrix Q is the imaginary part of τ. Analogously, for u∈ℤ^g, we define the tropical theta constant with characteristic u to be Θ_u(Q):=2·Θ(Q,u/2)-1/4 u^t Q u. In the classical case, characteristics are vectors m = (m',m”) in ℤ^2g. But, only u=m'contributes to the aforementioned asymptotics.Note that Θ_u(Q) depends only on u modulo 2.For any v∈ℤ^g consider the following signed sum of tropical theta constants:ϑ_v(Q):=∑_u∈(ℤ/2ℤ)^g (-1)^u^t v·Θ_u(Q) . The theta matroid M(Q) is the binary matroid represented by the collection of vectors{v ∈(ℤ/2ℤ)^g: ϑ_v(Q)≠ 0 }.The tropical theta constants and the theta matroid are invariant under basis changes S∈GL_g(ℤ). We have ϑ_u(Q)=ϑ_S^-1 u(S^tQ S) for allu∈ℤ^g, and therefore M(Q) = M(S^tQS).Here is the promised new approach to theSchottky problem. If Q lies in the tropical Schottky locus thenM(Q) is the desired cographic matroidand (<ref>) furnishes edge lengths.If Q ∈𝔍_g^ trop then the matroid M(Q) is cographic.In that graph, we assign the length 2^3-g·ϑ_v(Q) to the edge labeled v. The resulting metric graph has Riemann matrix Q. This says, in particular, that ϑ_v(Q) is non-negative when Q comes from a metric graph. Since Q ∈𝔍_g^ trop, there exists a unimodular matrixB = (b_1,…,b_m) ∈{-1,0,+1}^g × m and a diagonal matrix D =diag(ℓ_1,…,ℓ_m) such that Q = B D B^t= ∑_i=1^m ℓ_i b_i b_i^t. We claim Θ_u (Q)=- 1/4 ·∑_b_i^t uis odd ℓ_i for all u∈^g. Here the ℓ_i are positive real numbers. First, we note that Θ_u(Q)=max_λ∈ℤ^g { -(λ+u/2)^t Q(λ+u/2) } ≤ ∑_i=1^m -ℓ_i·min_λ∈ℤ^g{( b_i^t· (λ+u/2) )^2 }. If b_i^t u is even, then b_i^t· (λ+u/2)=0 for someλ∈ℤ^g.Otherwise, the absolute value of b_i^t· (λ+u/2) is at least 1/2. This shows that Θ_u(Q)≤-1/4·∑_b_i^t u is oddℓ_i. To derive the reverse inequality, let I = { i:u_iis odd}⊂{1,…,g}.By a result of Ghouila-Houri <cit.> on unimodular matrices,we can find w∈ℤ^g with w_i=±1 if i∈ I and w_i=0 otherwise, such that b_i^t· w∈{0,±1} for all 1≤ i≤ m. The vector λ_0 = 1/2(w-u) lies in ℤ^g.One checks that-(λ_0+u/2)^t Q (λ_0+u/2)= ∑_i=1^m -ℓ_i· (b_i^t· (λ_0+u/2))^2= -1/4∑_i=1^m ℓ_i· (b_i^t· w)^2= -1/4·∑_b_i^t u is oddℓ_i. Therefore, we also have Θ_u(Q)≥-1/4·∑_b_i^t u is oddℓ_i. This establishes the assertion in (<ref>).We next claim that, under the same hypotheses as above,the function in (<ref>)satisfiesϑ_v(Q)= 2^g-3∑_b_i≡ v mod 2ℓ_ifor all v ∈ℤ^g .Indeed, substituting the right hand side of (<ref>) for Θ_u(Q) into (<ref>), we find thatϑ_v(Q)=-1/4·∑_u∈(ℤ/2ℤ)^g∑_b_i^t uis odd (-1)^u^t v·ℓ_i =-1/4·∑_i=1^m ℓ_i · (|E_i|-|O_i|),where E_i={u∈(ℤ/2ℤ)^g: b_i^t uodd,u^t veven} andO_i={u ∈(ℤ/2ℤ)^g: b_i^t uodd, u^t vodd}.If b_i≡ v mod 2 then E_i=∅ and |O_i|=2^g-1. Otherwise, |E_i|=|O_i|=2^g-2. This proves (<ref>).Since Q ∈𝔍_g^ trop, this matrix comes from a graph G. We may assume that G has no 2-valent vertices. This ensures that any pair is independent in the cographic matroid of G.The column b_i of the matrix B records the coefficients of the i-th edge in a cycle basis of the graph G. The residue class of b_i modulo 2 is unique. For v ∈ℤ^g with b_i ≡ v mod2, the sum in (<ref>) has only term ℓ_i, and we have ℓ_i = 2^3-gϑ_v(Q). If v ∈ℤ^g is not congruent to b_i for any i then ϑ_v(Q) = 0. This proves that the theta matroid M(Q) equals the cographic matroid of G, and the edge lengths ℓ_i are recovered from Q by the rule in Theorem <ref>. By Theorem <ref>, the non-negativity of ϑ_v(Q) is a necessary condition for Q to be in 𝔍_g^ trop.For the matrix Q in Example <ref>, we find ϑ_0001(Q)=-1/2.Hence Q ∉𝔍_4^ trop. This necessary (but not sufficient) condition translates into the following algorithm: Let Q be the matrixin Example <ref>. For each u ∈ (ℤ/2ℤ)^4, we list the theta constant Θ_u(Q), the weight 2^-1ϑ_u(Q)and the label of the corresponding edge in Figure <ref>: u 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 11001101 11101111 -Θ_u29/4 23/4 5 19/4 13/2731/4 17/4917/2 33/4 13/2 43/4 41/4 21/2 2^-1ϑ_u 9 7 9 8 2 0 4 12 0 0 0 0 2 0 3 Edgee_2 e_1 e_3 e_6 e_7-e_8 e_9- - - - e_4-e_5We now explain the connection between the classicaland tropical theta functions. In particular, we will show howthe process of tropicalization relates Theorems <ref> and <ref>. In order to tropicalize the Schottky–Igusa modular form, we must study the order of growth of the theta constants when the entries of the Riemann matrix grow. This information is captured by the tropical theta constants.The following proposition makes that precise.Fix Q ∈ℌ_g^ trop, and let P(t) be any real symmetric g × g-matrix that depends on a parameter t∈. For every m∈(/2)^2g there is a constant C ∈ such that0 ≤|θ[m](P(t)+t·i Q, 0)|/| exp (t ·π·Θ_m'(Q))|≤ Cfor all t≥0.Moreover, we can choose P(t) suchthat the ratio above does not approach zero for t→∞. Here θ[m](τ,0) is the classical theta constant from(<ref>), andΘ_m'(Q) is the tropical theta constantdefinedin (<ref>). We use the notation m = (m',m”) for vectors in ℤ^2g as in Section <ref>. Consider the lattice points λ where the maximum in (<ref>) for x = m'/2 is attained. The corresponding summands in (<ref>) with λ = nhave the same asymptotic behavior as exp (t ·πΘ_m'(Q)) for t→∞.The sum over the remaining exponentials tends to zero since it can be bounded by a sum of finitely many Gaussian integrals with variance going to zero for t→∞. We can choose thereal symmetric matrix P(t) in such a way that no cancellation of highest order terms happens. Then the expression in (<ref>) is bounded away from zero.On the Siegel upper-half space ℌ_g we have an action by the symplectic group Sp_2g(ℤ). Two matrices from the same orbit under this action correspond to the same abelian variety. However their tropicalizations may vary drastically. Consider for example the case g=1: [0 -1;1k ]∈Sp_2(ℤ)sends τ=i to a complex number with imaginary part 1/1+k^2.We now assume that g=4. For any subset M⊂ (/2)^8 we write M'={m':m∈ M} and similarly for M”. The following lemma concerns the possible choicesfor Theorem <ref>.For any azygetic triple {m_1,m_2,m_3} and any matching subgroup N ⊂ (ℤ/2ℤ)^8, (1) there exist indices 1 ≤ i < j ≤ 3 such that (m_i+N)'=(m_j+N)', and(2) if N' =3 and (m_1+N)'=(m_2+N)'≠ (m_3+N)', then m_1',m_2'∈ N'. This purely combinatorial statement can be proved by exhaustive computation. For instance, consider the specific choice of m_1,m_2,m_3,N made prior to Example <ref>. This has N'=2, and Lemma <ref> (1) holds with i= 2, j= 3. If we exchange the first four coordinates with the last four coordinates, thenN'=3, m_1',m_3'∈ N' and m_2'∉N'.Recall from Theorem <ref> that amatrix τ∈ℌ_4 is in the Schottky locus if and only if π_1^2+π_2^2+π_3^2-2(π_1π_2+π_1π_3+π_2π_3) vanishes. The tropicalization of this expression equals max_i,j=1,2,3( π^trop_i+π^trop_j), where π^trop_i=∑_m∈ m_i+NΘ_m'(Q)is the tropicalization of the product (<ref>), with Q= im(τ).The tropical Schottky–Igusa modular form (<ref>) defines a piecewise-linear convex function ℌ_4^ trop→. Its breakpoint locusis the set ofRiemann matrices Q for which the maximum in (<ref>) is attained twice. That set depends on our choice of m_1,m_2,m_3,N. That choice is called admissible if N⊂ (/2)^8 has rank three, the triple {m_1,m_2,m_3}⊂ (/2)^8 is azygetic, all elements of m_i+N are even,and the group N' ⊂ (/2)^4 also has rank three. We define the tropical Igusa locusin ℌ_g^ trop to be the intersection, over all admissible choices m_1,m_2,m_3,N, of the breakpoint loci of the tropical modular forms (<ref>). A matrix Q ∈ℌ_4^ trop lies in the tropical Igusa locus if and only if ϑ_v(Q)≥0 for all v∈^4. That locus contains the tropical Schottky locus 𝔍_4^ trop, but they are not equal.We are interested in how the maximum in (<ref>) is attained. By Lemma <ref> (1), after relabeling, π^trop_1=π^trop_2.The maximum is attained twiceif and only if π^trop_1≥π^trop_3.By Lemma <ref> (2),this is equivalent to ∑_u∈ N'Θ_u(Q)≥∑_u∉N'Θ_u(Q).Let v be the non-zero vector in (/2)^4 that is orthogonal to N'.Then (<ref>) is equivalent to ϑ_v(Q) =∑_u∈(/2)^4(-1)^u^t vΘ_u(Q) ≥0 .This proves the first assertion, if we knew thatevery v arises from some admissible choice. We saw in Theorem <ref> that ϑ_v(Q) ≥ 0for all v whenever Q ∈𝔍_4^ trop. Hence the tropical Schottky locus 𝔍_4^ trop is contained in the tropical Igusa locus. The two loci are not equal because the latter contains thezonotopal locus of ℌ_4^ trop. This consists of matrices Q = BDB^t where B represents any unimodular matroid, not necessarily cographic. By<cit.>,the second Voronoi decomposition of ℌ_4^ trop has a non-cographic 9-dimensional cone in its zonotopal locus. It is unique modulo GL_4(). We verified that all 16 tropical modular forms ϑ_v are non-negative on that cone. This establishes the last assertion in Theorem <ref>.To finish the proof, we still need that everyv ∈ (/2)^4\{0} is orthogonal to N' for some admissible choice m_1,m_2,m_3,N. By permuting coordinates, it suffices to show this for v∈ {(1,0,0,0)^t, (1,1,0,0)^t, (1,1,1,0)^t, (1,1,1,1)^t}. For v=(1,0,0,0)^t we takem_1=[ 0 1; 0 0; 0 0; 0 1 ],m_2=[ 1 1; 1 0; 1 1; 0 1 ],m_3=[ 0 1; 0 0; 1 0; 1 0 ], n_1=[ 0 1; 1 0; 0 1; 0 1 ],n_2=[ 0 1; 0 0; 1 0; 1 1 ],n_3=[ 0 0; 0 1; 1 0; 0 0 ]. For v=(1,1,0,0)^t we takem_1=[ 0 1; 0 0; 0 0; 0 1 ],m_2=[ 1 1; 1 0; 1 1; 0 0 ],m_3=[ 1 0; 0 0; 0 0; 0 1 ], n_1=[ 1 1; 1 1; 0 1; 1 0 ],n_2=[ 0 0; 0 1; 1 0; 0 0 ],n_3=[ 0 0; 0 1; 0 0; 1 1 ]. For v=(1,1,1,0)^t we takem_1=[ 1 0; 0 0; 0 0; 1 0 ],m_2=[ 0 1; 0 1; 0 1; 1 0 ],m_3=[ 0 0; 0 0; 0 0; 1 0 ], n_1=[ 0 0; 0 0; 0 0; 1 1 ],n_2=[ 1 0; 0 0; 1 0; 1 1 ],n_3=[ 1 0; 1 0; 0 0; 0 0 ].For v=(1,1,1,1)^t we takem_1=[ 1 0; 0 1; 0 1; 0 0 ],m_2=[ 0 1; 0 1; 1 0; 1 0 ],m_3=[ 0 1; 0 0; 0 1; 0 1 ], n_1=[ 1 0; 1 1; 0 0; 0 1 ],n_2=[ 1 0; 0 1; 0 1; 1 0 ],n_3=[ 1 1; 0 1; 1 1; 0 0 ]. This completes the proof of Theorem <ref>. We have shown that the tropicalization of the classical Schottky locus satisfies the constraints coming from the tropical Schottky–Igusa modular forms in (<ref>). However, these constraints are not yet tight. The tropical Igusa locus, as we have defined it, is strictly larger than the tropical Schottky locus. It would be desirable to close this gap, at least for g=4. One approach might be a more inclusive definition of which choices are “admissible”.Can the tropical Schottky locus 𝔍_4^ trop be cut out by additional tropical modular forms, notably those obtained in(<ref>) by allowing choices m_1,m_2,m_3,N with N' ≤ 2? The next question concerns arbitrary genus g. We ask whether just computing the theta matroid M(Q) solves the Tropical Schottky Decision problem. Note that we did not address this subtle issue in Algorithm <ref> because we had assumed that the input Q lies in 𝔍_g^ trop.Let Q be a positive definite g× g matrix such that the matroid M(Q) is cographic with positive weights. Does this imply that Q is in the tropical Schottky locus? If the answer is affirmative then we can use Tutte's classical algorithm <cit.> as a subroutine for Schottky Decision. That algorithm can decide whether the matroid M(Q) is cographic.We close with a question that pertains to classical Schottky Reconstructionas in Section <ref>. How to generalize the results in <cit.> from g=3 to g=4? Is there a nice tritangent matrix, written explicitly in theta constants, for canonical curves of genus four?Acknowledgments. We thank Riccardo Salvati Manni for telling us about Kempf's article <cit.>. We also had helpful conversations with Christian Klein and Emre Sertöz. Lynn Chua was supported by a UC Berkeley University Fellowship and the Max Planck Institute for Mathematics in the Sciences, Leipzig. Bernd Sturmfels received funding from the US National Science Foundation (DMS-1419018) and the Einstein Foundation Berlin. 10BBC B. Bolognese, M. Brandt, and L. Chua. From curves to tropical Jacobians and back. In Combinatorial algebraic geometry, volume 80 of Fields Inst. Commun., pages 21–45. Fields Inst. Res. Math. Sci., Toronto, ON, 2017.BMV S. Brannetti, M. Melo and F. Viviani: On the tropical Torelli map, Advances in Mathematics 226 (2011) 2546–2586.chan12 M. Chan: Combinatorics of the tropical Torelli map, Algebra and Number Theory 6 (2012) 1133–1169.bitangents F. Dalla Piazza, A. Fiorentino and R. Salvati Manni:Plane quartics: the universal matrix of bitangents,Israel J. Math. 217 (2017) 111–138.DHBvHS B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij and M. Schmies: Computing Riemann theta functions, Math. of Computation 73 (2004) 1417–1442.DvH B. Deconinck and M. van Hoeij: Computing Riemann matrices of algebraic curves, Advances in nonlinear mathematics and science, Physica D 152/153 (2001) 28–46.sikiric M. Dutour Sikirić: Polyhedral, a GAP package, mathieudutour.altervista.org/Polyhedral, 2013.sgsw M. Dutour Sikirić, A. Garber, A. Schürmann and C. Waldmann:The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices, Acta Crystallographica A 72 (2016) 673–683.polymake E. Gawrilow and M. Joswig: polymake: a framework for analyzing convex polytopes,Polytopes–combinatorics and computation (Oberwolfach, 1997), DMV Sem., 29, Birkhäuser, Basel, 2000,pp. 43–73.houri A. Ghouila-Houri: Caractérisation des matrices totalement unimodulaires, C. R. Acad. Sci. Paris 254 (1962) 1192–1194.gru S. Grushevsky: The Schottky problem, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., 59,Cambridge Univ. Press, 2012, pp. 129–164. GM S. Grushevsky and M. Möller. Explicit formulas for infinitely many Shimura curves in genus 4. Asian J. Math., 22(2):381–390, 2018.HMY C. Haase, G. Musiker and J. Yu: Linear systems on tropical curves, Math. Zeitschrift 270 (2012) 1111–1140. HS J. Hauenstein and A. Sommese: What is numerical algebraic geometry?J. Symbolic Computation 79 (2017) 499–507.igusa J. Igusa:On the irreducibility of Schottky's divisor, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28 (1981) 531–545.spicy E. Jones, P. Peterson,et al.: SciPy: Open source scientific tools for Python, 2001-, http://www.scipy.org/.kempf G. Kempf: The equations defining a curve of genus 4, Proc. Amer. Math. Soc. 97 (1986) 219–225.BoLin B. Lin: Computing linear systems on metric graphs, J. Symbolic Computation, in press.mz G. Mikhalkin and I. Zharkov: Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemporary Math., vol 465, Amer. Math. Soc., 2008, pp. 203–230.ORSVJ.C. Ottem, K. Ranestad, B. Sturmfels and C. Vinzant: Quartic spectrahedra,Mathematical Programming, series B 151 (2015) 585–612. ps95 W. Plesken and B. Souvignier:and , 1995,published under GPL licence at RSSS Q. Ren, G. Schrader, S. Sam and B. Sturmfels: The universal Kummer threefold, Experimental Mathematics 22 (2013) 327–362. Schottky1888 F. Schottky: Zur Theorie der Abelschen Funktionen von vier Variabeln, J. reine angewandte Mathematik 102 (1888) 304–352.SD C. Swierczewski and B. Deconinck: Computing Riemann theta functions in Sage with applications, Math. Comput. Simulation 127 (2016) 263–272. abelfunctions C. Swierczewski et. al.: Abelfunctions: A library for computing with Abelian functions,Riemann surfaces, and algebraic curves, github.com/abelfunctions/abelfunctions, 2016.tutte W.T. Tutte: An algorithm for determining whether a given binary matroid is graphic, Proc. Amer. Math. Soc. 11 (1960) 905–917.frank F. Vallentin: Sphere Covering, Lattices, and Tilings (in Low Dimensions), PhD thesis,TU München, 2003. voronoi G. Voronoi: Nouvelles applications des paramètres continus à la théorie des formes quadratiques (Deuxième mémoire: recherches sur les parallélloèdres primitifs), Journal für die reine and angewandte Mathematik 134 (1908) 198–287. Authors' addresses: Lynn Chua, UC Berkeley, [email protected] Mario Kummer, TU Berlin, [email protected] Bernd Sturmfels, MPI Leipzig,[email protected] UC Berkeley, | http://arxiv.org/abs/1707.08520v2 | {
"authors": [
"Lynn Chua",
"Mario Kummer",
"Bernd Sturmfels"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170726161351",
"title": "Schottky Algorithms: Classical meets Tropical"
} |
Shifted lattices and asymptotically optimal ellipses [ December 30, 2023 ====================================================We propose a new self-organizing hierarchical softmax formulation for neural-network-based language models over large vocabularies. Instead of using a predefined hierarchical structure, our approach is capable of learning word clusters with clear syntactical and semantic meaning during the language model training process. We provide experiments on standard benchmarks for language modeling and sentence compression tasks. We find that this approach is as fast as other efficient softmax approximations, while achieving comparable or even better performance relative to similar full softmax models.§ INTRODUCTIONThe softmax function and its variants are an essential part of neural network based models for natural language tasks, such as language modeling, sentence summarization, machine translation and language generation.Given a hidden vector, the softmax can assign probability mass to each word in a vocabulary. The hidden vector could be generated from the preceding context, source sentence, dialogue context, or just random variables. The model decides how the context is converted into the hidden vector, and there are several choices for this, including recurrent neural network <cit.>, feed forward neural network <cit.> or log-bilinear models <cit.>. In our experiments here, we use a long short-term memory (LSTM) model for language modeling, and a sequence-to-sequence model with an attention mechanism for sentence compression. Both models are simple but have been shown capable of achieving state-of-the-art results. Our focus is to demonstrate that with a well designed structure, the hierarchical softmax approach can perform as accurately as the full softmax, while maintaining improvements in efficiency.For word-level models, the size of the vocabulary is very important for higher recall and a more accurate understanding of the input. However the training speed for models with softmax output layers quickly decreases as the vocabulary size grows. This is due to the linear increase of parameter size and computation cost with respect to vocabulary. Many approaches have been proposed to reduce the computational complexity of large softmax layers <cit.>. These methods can largely be divided into two categories: * Approaches that can compute a normalized distribution over the entire vocabulary with a lower computational cost <cit.>. Normalized probabilities can be useful for sentence generation tasks, such as machine translation and summarization.* Methods that provide unnormalized values <cit.>. These methods are usually more efficient in the training process, but less accurate.In this paper, we propose a self-organized hierarchical softmax, which belongs in the first category. In contrast to previous hierarchical softmax methods which have used predefined clusters, we conjecture here that a hierarchical structure learned from the corpus may improve model performance. Instead of using term frequencies as clustering criteria <cit.>, we want to explore the probability of clustering words together considering their preceding context. The main contributions of this paper are as follows: * We propose an algorithm to learn a hierarchical structure during the language model learning process. The goal of this algorithm is to maximize the probability of a word belonging to its cluster considering its preceding context.* We conduct experiments for two different tasks: language modeling and sentence summarization. Results show that our learned hierarchical softmax can achieve comparable accuracy for language modeling, and even better performance for summarization when compared to a standard softmax. We also provide clustering results, which indicate a clear semantic relevance between words in the same cluster.* Empirical results show that our approach provides a more than a 3× speed up compared to the standard softmax.§ RELATED WORKRepresenting probability distributions over large vocabularies is computationally challenging. In neural language modeling, the standard approach is to use a softmax function that output a probability vector over the entire vocabulary. Many methods have been proposed to approximate the softmax with lower computational cost <cit.>. We briefly review the most popular methods below.§.§ Softmax-based approaches Hierarchical Softmax (HSM): <cit.> and its variants are the most popular approximations. In general, this approach organizes the output vocabulary into a tree where the leaves are words and intermediate nodes are latent variables, or classes. The tree structure could have many levels and there is a unique path from root to each word. The probability of a word is the product of probabilities of each node along its path. In practice, we could use a tree with two layers, where we want to organize words into simple clusters. In this case, the computational complexity reduces from O(|V|) to O(√(|V|)). If we use a deeper structure like the Huffman Tree, the computational complexity could be reduced to O(log|V|). In general, the hierarchical structure is built on frequency binning <cit.> or word similarities <cit.>. In this paper, we propose another word-similarity-based hierarchical structure. But, instead of performing k-means over pre-learned word embeddings, we propose a new approach that learns hierarchical structure based on the model's historical prediction during the language model learning process.Differentiated softmax (D-softmax): <cit.> is based on the intuition that not all words require the same number of parameters: The many occurrences of frequent words allows us to fit many parameters to them, while extremely rare words might only allow us to fit relatively few parameters. D-softmax assign different dimension of vector to words according to their frequency to speed up the training and save memory. Adaptive softmax <cit.> can be seen as a combination of frequency binning HSM andD-softmax.CNN-softmax: <cit.> is inspired by the idea that we could use convolution network to produce word embedding from a character level model. Aside from a big reduction in number of parameters and incorporating morphological knowledge from words, this method can also easily deal with out-of-vocabulary words, and allows parallel training over corpora that have different vocabulary size. But this method does not decrease the computational complexity compared to the standard full softmax <cit.>. §.§ Sampling-based approachesSampling based approaches approximate the normalization in the denominator of the softmax with some other loss that is cheap to compute. However, sampling based approaches are only useful at training time. During inference, the full softmax still needs to be computed to obtain a normalized probability. These approaches have been successfully applied to language modeling <cit.>, machine translation <cit.>, and computer vision <cit.>. Importance sampling (IS): <cit.> select a subset of the vocabulary as negative samples to approximate the softmax normalization.Originally unigram or bigram distribution of word in entire corpus are used for sampling negative samples <cit.>, but researchers found that sampling from a more carefully designed distribution could help achieve a better accuracy. Instead, two variants of n-gram distributions are proposed: * an interpolated bigram distribution and unigram distribution <cit.>, * a power-raised unigram distribution <cit.>.Noise Contrastive Estimation (NCE): Noise Contrastive Estimation (NCE) is proposed in <cit.> as a more stable sampling method than IS. NCE does not try to estimate the probability of a word directly. Instead, it uses an auxiliary loss that works to distinguish the original distribution from a noisy one. <cit.> showed that good performance can be achieved even without computing the softmax normalization.§ SELF-ORGANIZED HIERARCHICAL SOFTMAX §.§ Cluster-based Hierarchical SoftmaxWe employ a modified 2-layer hierarchical softmax to compute the distribution of next word in a sentence. Given vocabulary 𝐕 of size N, and pre-softmax hidden states 𝐡, we first project 𝐡 into a cluster vector 𝐡_c and a word vector 𝐡_w,[ 𝐡_c; 𝐡_w ] =Relu[ 𝐖_c 𝐡; 𝐖_w 𝐡 ]where 𝐖_c, 𝐖_w ∈ℝ^d × d. The cluster distribution can be expressed asP(c|𝐡 )=exp( 𝐡_c ^𝖳𝐔^𝒞_c) /∑ _c' ∈𝒞exp( 𝐡_c ^𝖳𝐔^𝒞_c')where 𝒞 is set of clusters, 𝐔^𝒞∈ℝ^|𝒞| × d is vector representation of clusters. The in-cluster probability function isP(w_t|𝐡, 𝒞(w_t))=exp( 𝐡_w ^𝖳𝐔^𝐕_w_t) /∑ _w' ∈𝒞(w_i)exp( 𝐡_w ^𝖳𝐔^𝐕 _w')where 𝐔^𝐕∈ℝ^|𝐕| × d is vector representation of words, 𝒞(w_t) is the cluster assigned to w_t in 𝒞. Thus, the final probability function isP(w_t|𝐡) = P(𝒞(w_t)|𝐡 ) P(w_t|𝐡, 𝒞(w_t) )If the number of cluster is in O(√(N)) and the maximum number of words in cluster is in O(√(N)), then the computational cost of normalization at each layers is only O(√(N)), (as opposed to O(N) for the standard softmax).Thus a large matrix dot product is transformed into two small matrix dot product, which are very efficient on a GPU <cit.>. §.§ Cluster PerplexityIn order to evaluate the quality of a clustering over words, we propose the cluster perplexity:ppl_cluster(𝒞) = 2^1/M∑_w_t - log_2 p(𝒞(w_t) | 𝐰_<t)where M is number of words in the dataset, 𝐰_<t is context preceding w_t, p(𝒞(w_t) | 𝐰_<t) is the probability that words in cluster 𝒞(w_t) appear behind 𝐰_<t. Given a word cluster 𝒞 and 𝐰_<t, this metric evaluate the difficulty to choose correct cluster. If words that share similar context have been successfully grouped together, the ppl_cluster should be small.In addition to ppl_cluster, we also propose the in-cluster perplexity:ppl_in-cluster ( 𝒞 ) = 2^1/M∑ - log_2 p(w_t | 𝐰_<t, 𝒞(w_t))where p(w_t | 𝐰_<t, 𝒞(w_t)) is the probability of word w_t appearing after 𝐰_<t given a subset of vocabulary 𝒞(w_t). If 𝒞(w_t) contains words that share the same context with w_t, ppl_in-cluster should be large. §.§ Optimizing Cluster PerplexityWith the definitions in Equations <ref> and<ref> established, our goal is to minimize ppl_cluster(𝒞):argmin_𝒞ppl_cluster(𝒞) = - argmax_𝒞1/M∑_w_tlog_2 p(𝒞(w_t) | 𝐰_<t) = - argmax_𝒞∑_w ∈𝐕n_w/M∑_w_t = wlog_2 p(𝒞(w) | 𝐰_<t)/n_w= - argmax_𝒞∑_w ∈𝐕 tf(w) g(w,𝒞(w))where tf(w)=n_w/M is term frequency of word w in the corpus, andg(w,𝒞(w))=1/n_w∑_w_t = wlog_2 p(𝒞(w) | 𝐰_<t)is the average of log_2 p(𝒞(w) | 𝐰_<t) over different preceding contexts 𝐰_<t that followed by word w.According to equation <ref>, we need a 𝒞 that maximize the weighted sum of g(w,𝒞(w)). While directly computing g(w,𝒞(w)) is intractable,the output of equation <ref> at training time can be considered a sample of p(c | 𝐰_<t). We propose to use exponential smoothing to estimate g(w,𝒞(w)):q_τ(w_t,c) =λ(w_t) q_τ-1(w_t,c) + (1 - λ(w_t)) log_2 P(c|𝐡_t)q_τ(c|w_t) is a weighted sum over all historical samples of p(c|𝐰_<t) under different context and parameters, previously seen in training.The smoothing factor λ(w_t) is defined asλ(w_t) = 1/f_w_twhere f_w_t is the raw count of w_t in the entire dataset. §.§ Greedy Word ClusteringIn practice, we assigned two constraints to each cluster: * The number of words in each cluster cannot exceed γ√(N), whereγ > 1 is a hyperparameter; and * The sum of term frequencies in each cluster should be smaller than a frequency budget f_b. This is known as the the frequency-budget trick <cit.>. These constraints prevent us from getting clusters that are either too large, which make computing in-cluster distributions very expensive, or too unbalanced in frequency, which will bias our word cluster distribution. The greedy algorithm <ref> is proposed to optimize the cluster perplexity. As each cluster has limited positions for words, some words w cannot be assign to their best cluster c = argmax_c q(w,c). If we assign words according to certain order, then words at the tail end of the sequence will be less likely to be assign to their best cluster. In the algorithm, we assign words to clusters in descending order of their term frequency tf(w). In this schema, high frequency words w have priority to choose clusters, because they have higher weight in equation <ref>.§.§ Training Language Model with Self-organized HSMIn the training phase, we start from a randomly initialized word cluster, and update parameters using gradient descent based optimization algorithms, updating word cluster every K iterations. K is a hyperparameter that is chosen based on dataset size and vocabulary size.This learning process can also be considered as an EM algorithm: In the E-Step, we update the clusters; in M-Step, we update parameters based on the new clusters.§ LANGUAGE MODELING EXPERIMENTLanguage Modeling (LM) is a central task in NLP. The goal of LM is to learn a probability distribution over a sequence of tokens from a given vocabulary set 𝐕. The joint distribution is defined as a product of conditional distribution of tokens given their preceding context. Given a sequence of word w_1,...,w_T ∈𝐕, the probability distribution can be defined as:P(w_1,...,w_T) = ∏_t=1^T P(w_t|w_1,...,w_t-1)To address this problem, much work has been done on both parametric and non-parametric approaches. In recent years, parametric models based on neural networks have became the standard method. In our experiment, we used the standard word-level Long-Short Term Memory (LSTM) model, since multiple works show it can obtain state-of-the-art performance on different datasets <cit.>. §.§ DatasetWe evaluate our method on the text8[http://mattmahoney.net/dc/textdata] dataset, and use the perplexity (ppl) as an evaluation metric. We also provide the training time for full softmax and our approach. Text8 is a standard compression dataset containing a pre-processed version of the first 100 million characters from Wikipedia in English. It has been recently used for language modeling (Mikolov et al., 2014) and has a vocabulary of 44k words. The dataset partitioned into a training set (first 99M characters) and a development set (last 1M characters) that is used to report performance <cit.>.§.§ ImplementationIn our experiments, we use the same setting as the one reported in<cit.>. A one-layer LSTM model is used. Both the dimension of hidden state and dimension of the input word embeddings is set to 512. LSTM parameters are regularized with weight decay (λ=10^-6). Batch size is set to 128. We use Adagrad <cit.> with learning rate 0.1, the norm of the gradients is clipped to 0.25, and a 20 steps gradient truncation is applied. For our model, we set the number of clusters to √(𝐕), and the maximum number of words in each cluster is γ√(𝐕) with γ=1.5, and frequency budget f_b is 0.1 as in <cit.>. We update the word clusters every K=1000 mini-batches. §.§ Baseline MethodsWe compare the proposed approach with (1) full softmax, (2) importance sampling <cit.>, (3) hierarchical softmax (HSM) with frequency binning <cit.>, (4) differentiated softmax <cit.>, and (5) adaptive softmax <cit.>. As we use the same implementation settings in <cit.>, we use their experiment results for baseline methods. Instead of using torch, we use theano <cit.> to implement our approach. Thus, in order to compare computation time, we implement another full softmax language model with theano. Our full softmax has the same perplexity on the development set as the one reported in <cit.>. §.§ Experimental resultsTable <ref> shows results on the text8 dataset. Our approach provides the best perplexity among all approximation approaches, nearly performing as well as a full softmax. Table <ref> shows that our approach is almost 4 times faster than a normal softmax, with the speed-up continuing to increase as the vocabulary size increases.Figure <ref> monitors learning process of our approach. At the beginning of training, we observe a high cluster perplexity, and very low in-cluster perplexity. Because we initialize clusters randomly, our model has difficulties to predict cluster given the preceding sequence context of the target word.As training continues, our cluster update algorithmconsiders the cluster word assignments based on the distribution given by Equation <ref>, resulting in the cluster perplexity decreasing rapidly. In contrast, the in-cluster perplexity first increases and then decreases slowly. Because our approach assigns similar words into the same cluster that are difficult to distinguish, the model has to explicitly learn small differences between words that share a similar context. In the end, the model reaches a balance between cluster and in-cluster perplexity.Table <ref> shows some examples of words that belong to the same cluster. We observe a strong syntactical similarity between these words, and semantic closeness. These examples show that our cluster update algorithm is capable of placing words with similar context into the same cluster. It is interesting to see that the unsupervised approach could learn word cluster with clear meaning. § ABSTRACTIVE SENTENCE SUMMARIZATION EXPERIMENTSummarization is an important challenge in natural language understanding. The aim is to produce a condensed representation of an input text that captures the core meaning of the original. Given a vocabulary 𝐕 and a sequence of M words x_1, ..., x_M ∈𝐕, a summarizer takes 𝐱 as input and outputs a shortened sentence 𝐲 of length N < M. Assuming that words in the output sentence also belong to 𝐕, we can express the output as y_1, ..., y_N ∈𝐕. The output sentence is then called a summary of the input sentence. Thus, the probability distribution of the summary can be defined as:P(y_1,...,y_N) = ∏_t=1^N P(y_t|𝐱, y_1,...,y_t-1)For an extractive summarization, the probability distribution P(y_t|𝐱, y_1,...,y_t-1) is on the set of input words, while for an abstractive summarization the distribution is on the entire vocabulary. In this experiment, we focus on the abstractive summarization task, which is more difficult and computationally expensive. §.§ DatasetWe trained our model on the Gigaword5 dataset <cit.>. This dataset was generated by pairing the headline of each article with its first sentence to create a source-compression pair. <cit.> provided scripts to filter out outliers, resulting in roughly 3.8M training pairs, a 400K validation set, and a 400K test set. We use the most frequent 69k words in the title as input and output vocabulary, which correspond to the decoder vocabulary size used in <cit.>. Out of vocabulary words are represented with a symbol <unk>.We evaluate our method on both the standard DUC-2004 dataset and the single reference Gigaword5 test set. The DUC corpus[http://duc.nist.gov/duc2004/tasks.html] 2004 corpus consists of 500 documents, each having 4 human generated reference titles. Evaluation of this dataset uses the limited-length Rouge Recall at 75 bytes on DUC validation and test sets. In our work, we simply run the models trained on Gigaword corpus as they are,without tuning them on the DUC validation set. The only change we have made to the decoder is to suppress the model from emitting the end-of-summary tag, and forced it to emit exactly 30 words for every summary. <cit.> provides a random sampled 2000 title-headline pair as test set. We acquired the exact test sample used by them to make a precise comparison of our models with theirs. Likeand , we use the full length F1 variant of Rouge[http://www.berouge.com/Pages/default.aspx] to evaluate our system. §.§ ImplementationIn this experiment, we use the standard Encoder-Decoder with attention architecture. Both encoder and decoder a consists of a single layer uni-direction LSTM model, and an attention mechanism over the source-hidden states and a softmax layer to output a distribution probability over an output vocabulary.The hidden state dimension and input embedding are both set to 512. All parameters are regularized with weight decay (λ=10^-6). Batch size is 128. We use Adam <cit.> with a learning rate of 0.001. No dropout or gradient clipping is used. At decode time, we use beamsearch of size 5 to generate summary. The maximum length of output summary is limited to 30.For our approach, we learn word clusters through training a language model on the titles of our training dataset. We then use these clusters as the fixed structure for hierarchical softmax in the summarization model. §.§ Baseline methodsWe compared the performance of our model with state-of-the-art models that are trained with teacher forcing and cross-entropy loss, including: (1) TOPIARY <cit.>, (2) ABS+ <cit.>, (3) RAS-Elman <cit.>, and (4) words-1vk5k-1sent <cit.>. We also include our implementation of normal softmax as a baseline method. There are some newly proposed summarization models that come up with different type of loss function including the reconstructive loss function <cit.>, and the minimum risk training (MRT) loss <cit.>. We did not compare with these methods, since this experiment is focused on evaluating our approach against other softmax-based approaches under similar implementations and learning settings. §.§ Experiment ResultsTable <ref> and Table <ref> show results of our approach comparing the different methods. Our approach not only outperforms the full softmax method, but also outperforms state-of-the-art methods on most of evaluation metrics. Table <ref> also shows that our approach is 3 times faster than standard full softmax, which is widely used in all kinds of different summarization models. Figure <ref> present examples of summaries generated by self-organized HSM in comparing with true headline and full softmax outputs.As the word cluster learned on Gigaword corpus shows similar in-cluster syntactical similarity and semantic closeness, we suggest that this hierarchical structure decomposes the difficult word generation task into 2 easier tasks. The first task is to decide correct syntactical role for next word, and the second task is to find semantically correct words in a subset of the vocabulary.§ CONCLUSIONIn this paper we have proposed a new self organizing variant of the hierarchical softmax. We observe that this approach can achieve the same performance as a full softmax approach for language modelling, and even better performance for sentence summarization. Our approximation approach is also as efficient as other hierarchical softmax approximation techniques. In particular in our experiments we observe that our self-organized HSM is at least 3 times faster than a full softmax approach.Our approach yields self organized word clusters which are influenced by the context of words. Examining word clusters produced by or approach reveals that our method groups words according to their syntactical role and semantic similarity. These results are appealing in that we have obtained a certain level of understanding of grammar rules without explicit part-of-speech tagged input. We therefore think that the use of this approach shows promise for other NLP tasks, including machine translation and natural language generation. emnlp_natbib | http://arxiv.org/abs/1707.08588v1 | {
"authors": [
"Yikang Shen",
"Shawn Tan",
"Chrisopher Pal",
"Aaron Courville"
],
"categories": [
"cs.CL",
"cs.LG"
],
"primary_category": "cs.CL",
"published": "20170726180132",
"title": "Self-organized Hierarchical Softmax"
} |
Impact of Correlation between Interferers on Coverage Probability and rate in Cellular Systems Suman KumarSheetal Kalyani Dept. of Electrical EngineeringIIT Madras,Chennai 600036, India{ee10d040,skalyani}@ee.iitm.ac.in December 30, 2023 =======================================================================================================================================================When the user channel experiences Nakagami-m fading, the coverage probability expressions are theoretically compared for the following cases: (i). The N interferers are independent η-μ random variables (RVs). (ii). TheN interferers are correlated η-μ RVs. It isanalytically shown thatthe coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in the presence ofindependent interfererswhen theshape parameter of the channel between the user and its base station (BS)is not greater than one. Further, rate is compared for the following cases: (i).The user channel experiences η-μ RV and the N interferers are independent η-μ RVs. (ii). TheN interferers are correlated η-μ RVs. It is analytically shown that the rate in the presence of correlated interferers is greater thanor equal to the rate in the presence of independent interferers.Simulation results are providedand these match with the obtained theoretical results. The utility of our results are also discussed.Majorization theory, Stochastic ordering, Gamma random variables, Correlation, Coverage probability, rate.§ INTRODUCTION Typically, in practical scenarios correlation exists among the interferers, as evidenced by experimental results reported in<cit.>.For example, in cellular networks when two base stations (BSs) from adjacent sectors act as interferers, the interferers are correlated and it is mandated that while performing system level simulation, this correlation be explicitly introduced in the system <cit.>. We refer thereader to<cit.> for a structured synthesis of the existing literature on correlation among large scale fading.Considering the impact of correlation in the large scaleshadowing component and the small scalemultipath component is also an essential step towards modeling the channel.The decorrelation distance in multipath components is lower when compared to shadowing components since shadowing is related to terrain configuration and/or large obstacles between transmitter and receiver <cit.>. Having said this, there is a need to analyse the performance of cellular system in the presence of correlation among interferers.Coverage probability[It is a probability that a user can achieve a target Signal-to-Interference-plus-noise-Ratio (SINR) T, and outage probability is the complement ofcoverage probability.] and rate areimportant metricfor performance evaluation of cellular systems. Coverage probability in the presence of interferers has been derived in<cit.> and references there in, and rate in the presence of interferers has been studied in <cit.>and references therein for the case of η-μ fading. Moreover, the correlation among interferers is assumed in <cit.>. Fractional frequency reuse and soft frequency reuse have been compared in the presence of correlation among interferers <cit.>. The impact of correlation among interferers on symbol error rate performance has been analysed in <cit.>. However, to the best of our knowledge, no prior work in open literature hasanalytically compared the coverage probability and rate for generalized fading when interferers are independent with the coverage probability and rate when interferers are correlated. In this paper, we compare coverage probability and rate using majorization theory and stochastic ordering theory, respectively. Majorization theory is an important theory for comparison of two vectors in terms of the dispersion of their components. Ithasbeen extensively used in various problems in information theory <cit.>. In particular, using the results of majorization theory, a new analysis on Tunstall algorithm is provided in <cit.> and the bounds ofthe average length of the Huffman source code in the presence of limited knowledge of the source symbol probability distribution is provided in <cit.>. Our analysis of comparison of coverage probability shows further evidence of the relevance of majorization theory. Stochastic ordering theory is used extensively for comparison ofrandom variables (RVs). Recently, it has been used in comparing various metrics in wireless communication <cit.>. Our analysis of comparison of rate shows another evidence of the relevance of stochastic order theory in wireless communication. In this work,we compare thecoverage probability when user experience Nakagami-m fading and interferers experience η-μ fading. In other words, we compare the coverage probability when the interferers are independent with the coverage probability when the interferers are positively correlated[If cov(X_i, X_j)≥ 0 then X_i and X_jare positively correlated RVs, where cov(X_i, X_j) denotes the covariance between X_i and X_j <cit.>.] using majorization theory. It is analytically shown that the coverageprobability in presence of correlated interferers is higher than the coverage probability when the interferersare independent, when the user channel's shape parameter is lesser than or equal to one. We also show that when the user channel's shape parameter is greater than one, one cannot say whether coverageprobability is higher or lower for the correlated case when compared to the independent case, and in some cases coverage probability is higher while in other cases it is lower.We then analytically compare the rate when the interferers are independentwith the rate when the interferers are correlated usingstochastic ordering theory. It is shown that the rate in the presence of positively correlated interferers is higher thanthe rate in the presence of independent interferers when both user channel and interferers experience η-μ fading. Our results show that correlation among interferers is beneficial for the desired user. We briefly discuss how the desired user can exploit this correlation among the interferers to improve its rate. Multi-user multiple input multiple output (MU-MIMO) system is also considered and it is shown that the impact of correlation is significant on MU-MIMO system. We have also carried out extensive simulations for both the independent interferers case and the correlated interferers case and some of these results are reported in the Simulation section. In all the cases, the simulation results match with our theoretical results. § SYSTEM MODELWe consider a homogeneous macrocell network with hexagonal structure with radius R as shown in Fig. <ref>. The Signal-to-Interference-Ratio(SIR)of a user located at r meters from the BS is given bySIR=η(r)= Pgr^-α/∑_i∈ψPh_id_i^-α= gr^-α/∑_i∈ψh_id_i^-α=S/Iwhere ψ denotes the set of interfering BSs andN=|ϕ| denotes the cardinality of the set ϕ. The transmit power of a BS is denoted by P.A standard path loss model r^-α is considered, where α≥ 2 is the path loss exponent.The distance between user to tagged BS (own BS) and the ith interfering BS is denoted by r and d_i, respectively. The user channel's powerand the channel power between i^th interfering BS and user are η-μ power RVs. The probability density function (pdf) f_g_η-μ(x) of the η-μ power RV g is given by <cit.>,f_g_η,μ(x)=2√(π)μ^μ+ 1/2h^μ x^μ- 1/2/Γ(μ)H^μ- 1/2e^-2μ h xI_μ- 1/2(2μ H x)where μ is shape parameter. Parameters H and hare given byH=η^-1-η/4,and h=2+η^-1+η/4.where 0<η< ∞ is the power ratio of the in-phase and quadrature component of the fading signal in each multipath cluster.The parameters of pdf of h_i are η_i and μ_i corresponding to η-μ power RV. The gamma RV is a special case of η-μpower RV with η=1 and μ=m/2 and the pdf f_g(x)of the gamma RV gis given byf_g(x)=m^m e^-mxx^m-1/Γ(m)where m and 1/m are the shape parameter and scaleparameter, respectively, and Γ(.) denotes the gamma function. When one assumes that there is correlation among interferers then h_i and h_j are correlated ∀ iand j. The coverage probability expression when signal of interest (SoI) experience Nakagami-m fading and interferers experience independent η-μ fading with equal μ, i.e.,μ_i=μ_c ∀ i is given by <cit.>C_p,η(r)=Γ(2Nμ_c+m)/Γ(2Nμ_c+1)1/Γ(m)∏_i=1^2N(1/Tr^αm λ_i +1)^μ_c× F_D^(2N)[1-m,μ_c,⋯,μ_c; 2Nμ_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_2N m+1] where λ_2i-1=d_i^-α/μ_c(1+η_i^-1) and λ_2i=d_i^-α/μ_c(1+η_i).Here F_D^(N)[a,b_1, ⋯, b_N;c;x_1,⋯, x_N] is theLauricella's function of the fourth kind <cit.>.The coverage probability expression when SoI experience Nakagami-m fading and interferers experience correlated η-μ fading with equal μ, i.e.,μ_i=μ_c ∀ i is given by <cit.>Ĉ_p,η(r)=Γ(2Nμ_c+m)/Γ(2Nμ_c+1)1/Γ(m)∏_i=1^2N(1/Tr^αm λ̂_i +1)^μ_c× F_D^(2N)[1-m,μ_c,⋯,μ_c; 2Nμ_c+1;1/Tr^αλ̂_1m+1,⋯,1/Tr^αλ̂_2N m+1]Here λ̂_is are the eigenvalues of𝐀_η=𝐃_η𝐂_η. 𝐃_η is the diagonal matrix with entries λ_i given in (<ref>) and 𝐂_η is the s.p.d. 2N× 2N matrix as given by𝐂_η=[ [10√(ρ_13)...0;010... √(ρ_22N);⋯⋯⋯⋱⋯;0 √(ρ_2N2)⋯⋯1 ]],here ρ_ij=0, when i+j=2n+1, where n is an integer. Now, we want to compare (<ref>) with (<ref>). However both involve N- fold infinite series making any comparison fairly difficult.In order to elegantly compare these two expressions we make use of Majorization theory. § PRELIMINARIES: MAJORIZATION ANDSTOCHASTIC ORDERING THEORYIn this section, we recall the basic notions of majorization andstochastic ordering theory, which are applicable to our context.Note that the majorization theory is used to compare deterministic vectors, whereas stochastic order is applicable on RVs <cit.>. We refer the reader to<cit.> and <cit.> as excellent references for the majorization theory and stochastic order theory, respectively.§.§ Majorization theorydefinitionDefinition Let A and B be m× n matrices with entries in C. The Hadamard product of A and B is defined by [A ∘ B]_ij=[A]_ij[B]_ij for all 1≤ i≤ m, 1≤ i≤ n.Let vector a=[a_1,⋯ a_n] and vector b=[b_1,⋯ b_n] witha_1≤, ⋯,≤a_n andb_1≤, ⋯,≤b_n then vector b is majorized by vector a, denoted by a≻b, if and only if ∑_i=1^kb_i≥∑_i=1^ka_i,k=1,⋯, n-1,and ∑_i=1^nb_i= ∑_i=1^na_i. lemmaLemmatheoremTheorem We will briefly state few well known results frommajorization theory which we use in our analysis.Let A and B be positive semidefinite matrices of size n. Let λ_1,⋯, λ_n be the eigenvalues of A ∘ B and let λ̂_1, ⋯, λ̂_n be the eigenvalues of AB. Then∏_i=k^nλ_i ≥∏_i=k^nλ̂_i, k=1,2,⋯,npropositionProposition If function ϕ is symmetric and convex, then ϕ is Schur-convex function. Consequently, x≻y implies ϕ(x)≥ϕ(y). For the details of this proof please refer to <cit.>.If ϕ_i is Schur-convex, i=1,⋯ k, and ϕ_i(x)≥ 0 for all i and x, then ψ(x)=∏_i=1^kϕ_i(x)is Schur-convex. For the details of this proof please refer to <cit.>. §.§ Stochastic order theoryIn this subsection, our focus will be on convex order. If X and Y are two RVs such that E[ϕ(X)]≤ E[ϕ(Y)]for all convex function ϕ: ℝ→ℝ,provided the expectation exist, then X is said to be smaller than Y in the convex order, denoted by X ≤_cx Y.Note that if (<ref>) holds thenY is more variable than X <cit.>. We now briefly state the theorems in convex order theorythat is relevant to this work. Let X_1, X_2,⋯, X_N be exchangeable RVs. Let a=(a_1,a_2,⋯,a_N) and b=(b_1,b_2,⋯,b_N) be two vectors of constants. If a≺b, then∑_i=1^Na_iX_i≤_cx∑_i=1^Nb_iX_i.The details of the proof is given in <cit.>. If X≤_cx Y and f(.) is convex, then E[f(X)]≤ E[f(Y)]. The details of the proof is given in <cit.>. § COMPARISON OF COVERAGE PROBABILITY In this section, we firstcompare the coverage probability in the independent case and correlated case, and analytically quantify the impact of correlation when bothSoI and interferers experience Nakagami-m fading. Then we analyse the impact of correlation on coverage probability for the scenario where SoI experiences Nakagami-m fading and interferers experience η-μ fading. The coverage probability expression when SoI experiences Nakagami-m fading and interferersexperience independent Nakagami-m fadingis given by <cit.>C_p(r)=Γ(∑_i=1^Nm_i+m)/Γ(∑_i=1^Nm_i+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ_i+1)^m_i× F_D^(N)[1-m,m_1,⋯,m_N; ∑_i=1^Nm_i+1;1/Tr^α mλ_1+1,⋯,1/Tr^αmλ_N+1]where λ_i=d_i^-α/m_iHere m_i is the shape parameter of the i^th interferer. Note that the coverage probability expression for the correlated case is derived in <cit.> when the interferers shape parameter are all equal, i.e., m_i=m_c ∀ i. The coverage probability expression whenSoI experiences Nakagami-m fading and interferersexperience correlated Nakagami-m fading is given by <cit.>Ĉ_p(r)=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ̂_i +1)^m_c× F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αmλ̂_1+1,⋯,1/Tr^αmλ̂_N+1]here λ̂_is are the eigenvalues of the matrix 𝐀=𝐃𝐂, where 𝐃 is the diagonal matrix with entries λ_i and 𝐂 is the symmetric positive definite (s.p.d.) N× N matrix defined by𝐂=[ [ 1 √(ρ_12) ... √(ρ_1N); √(ρ_21) 1 ... √(ρ_2N); ⋯ ⋯ ⋱ ⋯; √(ρ_N1) ⋯ ⋯ 1 ]],where ρ_ij denotes the correlation coefficient between h_i and h_j, and is given by,ρ_ij=ρ_ji=cov(h_i,h_j)/√(var(h_i)var(h_j)), 0≤ρ_ij≤ 1, i,j=1,2, ⋯, N.cov(h_i,h_j) denotes the covariance betweenh_i and h_j. 𝐂 is the s.p.d. N× N matrix given in (<ref>).Since 𝐀=𝐃𝐂, it is given by𝐀=[ [λ_1 λ_1√(ρ_12)⋯ λ_1√(ρ_1N); λ_2√(ρ_21)λ_2⋯ λ_2√(ρ_2N);⋮⋮⋱⋮; λ_N√(ρ_N1)⋯⋯λ_N ]]. Note that the coverage probability expression for the correlated case is derived when the interferers shape parameter are all equal and hence for a fair comparison we consider equal shape parameter for the independent case also, i.e., m_i=m_c ∀ i. Therefore, the coverage probability when interferers are independent and m_i=m_c ∀ i is given byC_p(r)=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αm λ_i +1)^m_c× F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_N m+1]Our goal is to compare the coverage probability in the presence of independent interferers, i.e., C_p(r) given in (<ref>) and the coverage probability in the presence of correlated interferers, i.e., Ĉ_p(r) given in (<ref>). We first start with the special case when user channel's fading is Rayleigh fading (i.e, m=1) and interferers experience Nakagami-m fading. When m=1, the coverage probability given in (<ref>) reduces toC_p(r)= ∏_i=1^N(1/Tr^αλ_i +1)^m_c F_D^(N)[0,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1+1,⋯,1/Tr^αλ_N+1]A series expression for F_D^(N)( .) involving N-fold infinite sums is given byF_D^(N)[a,b_1,⋯, b_N;c;x_1,⋯, x_N]=∑_i_1⋯ i_N=0^∞(a)_i_1+⋯+i_N(b_1)_i_1⋯(b_N)_i_N/(c)_i_1+⋯+i_Nx_1^i_1/i_1!⋯x_N^i_N/i_N!, max{|x_1|,⋯|x_N|}<1,where, (a)_n denotes the Pochhammer symbol which is defined as (a)_n=Γ(a+n)/Γ(a). With the help of series expressionandusing the fact that (0)_0=1 and (0)_k=0 ∀ k≥ 1,the coverage probability given in (<ref>) can be reduced to C_p(r)= ∏_i=1^N(1/Tr^αλ_i +1)^m_c Similarly, the coverage probability in correlated case when SoI experiences Rayleigh fading Ĉ_p(r) is given byĈ_p(r)= ∏_i=1^N(1/Tr^αλ̂_i +1)^m_cWe now state and prove the following theorem for the case where the SoI experiences Rayleigh fadingand interferers experienceNakagami-m fading and then generalize it to thecase where user also experiences Nakagami-m fading.The coverage probability in correlated case is higher than that of the independent case, when user'schannel undergoes Rayleigh fading, i.e.,∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_cwhere λ̂_isare the eigenvalues ofmatrix 𝐀 and λ_is are the diagonal elements ofmatrix 𝐀 given in (<ref>), and k=Tr^α is a non negative constant.Firstly, we show thatλ̂≻λ where λ̂=[λ̂_1, ⋯, λ̂_n] and λ=[λ_1, ⋯, λ_n]. Note that the Hadamard product of 𝐃 and 𝐂, i.e., 𝐃∘𝐂is defined as[Definition of Hadamard product is given in Definition <ref>] 𝐃∘𝐂=[ [ λ_1 0 ⋯ 0; 0 λ_2 ⋯ 0; ⋮ ⋮ ⋱ ⋮; 0 ⋯ ⋯ λ_N ]].Observe that 𝐃∘𝐂 is a diagonal matrix, the eigenvalues of the 𝐃∘𝐂 are λ_is. Since the eigenvalues ofmatrix 𝐀=𝐃𝐂 are λ̂_̂îs and 𝐃 and 𝐂 are the positive semi-definite matrices,hencefrom Theorem <ref> (given in Section II),∏_i=k^Nλ_i ≥∏_i=k^Nλ̂_i, k=1,2,⋯,NNote that λ̂_̂îs are the eigenvalues and λ_is are the diagonal elements of a symmetric matrix 𝐀=𝐃𝐂, Hence∑_i=1^Nλ_i=∑_i=1^Nλ̂_̂î If conditions given in (<ref>) and (<ref>) are satisfied then λ̂≻λ <cit.> . Now if it can be shown that ∏_i=1^N(1/1+kλ_i)^m_c ( and ∏_i=1^N(1/1+kλ̂_i)^m_c) is a Schur-convex function then by a simple application of Proposition <ref> (given in Section II) it is evident that ∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_c. To prove that ∏_i=1^N(1/1+kx_i)^m_c is a Schur convex function we need to show that it is a symmetric and convex function <cit.>. It is apparent that the function ∏_i=1^N(1/1+kx_i)^m_c is a symmetric function due to the fact that any two of its arguments can be interchanged without changing the value of the function. So we nowneed to showthat the functionf(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a is a convex function where x_i≥ 0, a > 0. Using the Theorem <ref>, it is apparent that if the function f(x)=(1/1+kx)^a is convex function then f(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a would be convex function. Note that f(x)=(1/1+kx)^a is convex function when x ≥ 0 and a>0 due to the fact that double differentiation of f(x)=(1/1+kx)^a is always non negative, i.e., f”(x)=a (a+1) k^2 (1/k x+1)^a+2≥ 0. Thus, f(x_1,⋯,x_n)=∏_i=1^N(1/1+kx_i)^a is a convex function.Since ∏_i=1^N(1/1+kx_i)^m_c is a convex function anda symmetric function therefore, it is a Schur-convex function. We have shown thatλ̂≻λand ∏_i=1^N(1/1+kx_i)^m_c is a Schur-convex function. Therefore, from Proposition 1,∏_i=1^N(1/1+kλ̂_i)^m_c≥∏_i=1^N(1/1+kλ_i)^m_c.Thus, the coverage probability in the presence of correlation among the interferersis greater than or equalto the coverage probability inthe independent case,when user channel undergoes Rayleigh fading and the interferers shape parameter m_i=m_c ∀ i. The same result was shown for the case of both Rayleigh user channel and Rayleigh interferers in <cit.> using Vieta formula. By exploiting the mathematical tool of Majorization we are able to provide a much simpler proof. Next, we compare the coverage probability for general case, i.e., when m is arbitrary. The coverage probability in the presence of the correlated interferers is greater than or equal to the coverage probability in presence of independent interferers, when user channel's shape parameter is less than or equal to 1, i.e., m ≤ 1. Whenm>1, coverage probability in the presence of independent is not always lesser than the coverage probability in the presence of correlated interferers. Please see Appendix.Summarizing, the coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in presence of independent interferers, when user channel's shape parameter is less than or equal to 1, i.e., m≤ 1. When m>1, one can not say whether coverage probability is better in correlated interferer case or independent interferer case. Note that when m≤ 1, usually the interferers m_i is also smaller than 1. However, the given proof holds for both m_i>1 and m_i<1.§.§ η-μ fading In this subsection, we analyse the impact of correlation on coverage probability for the scenario where SoI experiences Nakagami-m fading and interferers experience η-μ fading.Recently, theη-μ fading distribution with two shape parameters η and μ has been proposed to model a general non-line-of-sight propagation scenario <cit.>.It includes Nakagami-q (Hoyt), one sided Gaussian, Rayleigh and Nakagami-m as special cases. The coverage probability expression when SoI experience Nakagami-m fading and interferers experience independent η-μ fading with equal μ, i.e.,μ_i=μ_c ∀ i is given in (<ref>).The coverage probability expression when SoI experience Nakagami-m fading and interferers experience correlated η-μ fading with equal μ, i.e.,μ_i=μ_c ∀ i is given in (<ref>).Note that the functional form of the coverage probability expressions given in (<ref>) and (<ref>)are similar to the case when both SoI and interferers experience Nakagami-m fading.Hence, the same analysis holds and the coverage probability in the presence of correlated interferers is higher than the coverage probability in the presence of independent interferers whenthe user'schannel shape parameter is less than or equal to 1, and interfering channel experience η-μ fading. However, one can not conclude anything when the user'schannel shape parameter is higher than 1. § COMPARISON OF RATEIn this section, we analyse the impact of correlation on the rate for the scenario when both SoI and interferers experience η-μ fading with arbitrary parameters. In other words, we compare the rate when interferers are independent with the rate when the interferers are correlated usingstochastic ordering theory.The rate of a user at a distance r when interferers are independent is R=E[ln(1+S/I)], where S is the desired user channel power. For correlated case, the average rate of a user at a distance r is R̂=E[ln(1+S/Î)].Here I and Î are the sum ofindependent andcorrelated interferers, respectively.Using iterated expectation one can rewrite the rate asR=E_S[E_I[ln(1+S/I)|S=s]],and R̂=E_S[E_Î[ln(1+S/Î)|S=s]].Since theexpectation operator preserves inequalities, therefore if we can show that E_Î[ln(1+S/Î) |S=s]≥ E_I[ln(1+S/I)|S=s],then this implies R̂≥ R.The sum of interference power in the independent case can be written asI=∑_i=1^Nh_id_i^-αwhere h_i is η-μ power RV.It has been shown in <cit.> that the η-μ power RV can be represented as the sum of two gamma RVs with suitable parameters. In other words, if h_i is η-μ power RV then, h_i=x_i+y_iwherex_i∼𝒢(μ_c,1/2μ_c(1+η_i^-1))andy_i∼𝒢(μ_c,1/2μ_c(1+η_i))Now, the sum of interference power when interference experience η-μ RV can be written asI=∑_i=1^Nh_id_i^-α=∑_i=1^2Nλ_iG_iwithG_i∼𝒢(μ_c,1), λ_2i-1=d_i^-α/μ_c(1+η_i^-1) and λ_2i=d_i^-α/μ_c(1+η_i).Similarly, for correlated case,Î=∑_i=1^Nĥ_i=∑_i=1^2Nλ̂_iG_iRecall that these ĥ_iare correlated, andλ̂_is are the eigenvalues of the matrix A_η=D_ηC_η. In other words one can obtain a correlated sum of gamma variates by multiplying independent and identical distributed (i.i.d.) gamma variates with weight λ̂_is. Now, we use Theorem <ref> (given in Section III) to show that R̂ is always greater than equal to R. Note that a sequence of RVs X_1,⋯ X_N is said to be exchangeable if for all N and π∈ S(N) it holds that X_1⋯ X_N 𝒟= X_π(1)⋯ X_π(N) where S(N) is the group of permutations of {1,⋯ N} and 𝒟= denotes equality in distribution <cit.>. Furthermore, if X_is are identically distributed,they are exchangeable <cit.>. Hence G_is are exchangeable since they are identically distributed. It has already been shown thatλ̂≻λ in Section IV.Hence by a direct application ofTheorem <ref> (given in Section III), one obtains, I ≤_cxÎ.Note that ln(1+k/x) is a convex function when k≥ 0 and x ≥ 0 due to the fact that double differentiation of ln(1+k/x) is always non negative, i.e., ∂∂ln(k/x+1)/∂ x/∂ x=k (k+2 x)/x^2 (k+x)^2≥ 0. Note that S and I are non negative RVs, hence by a direct application of Theorem <ref> (given in Section III), one obtains E_Î[ln(1+S/Î)|S=s]≥ E_I[ln(1+S/I)|S=s]Since expectation preserve inequalities therefore, E_S[E_I[ln(1+S/I)]] ≤ E_S[E_Î[ln(1+S/Î)]]. In other words, positive correlation among the interferers increases the rate.Summarizing, the rate in the presence of the positive correlated interferers is greater than or equal to the rate in the presence of independent interferers, when SoI and interferers both experience η-μ fading. Now we briefly discuss the utility of our results in the presence of log normal shadowing.§.§ Log Normal ShadowingAlthough all the analysis so far(comparison of the coverage probability and rate) considered only small scale fading and path loss, the analysis can be further extended to take into account shadowing effects. In general, the large scale fading, i.e, log normal shadowing is modeled by zero-mean log-normal distribution which is given by,f_X(x)=1/x√(2π(σ_dB/8.686)^2)exp(-ln^2(x)/2(σ_dB/8.686)^2), x>0,where σ_dB is the shadow standard deviation represented in dB. Typically the value of σ_dB varies from 3 dB to 10 dB <cit.>,<cit.>. It is shown in <cit.> that the pdf of thecomposite fading channel (fading and shadowing) can be expressed using the generalized-K (Gamma-Gamma) model. Also in <cit.>, it has been shown that the generalized-K pdf can be well approximated by Gamma pdf 𝒢(β, γ) using the moment matching method, with β and γ are given by β=1/(1/m+1)exp((σ_dB/8.686)^2)-1=m/(m+1)exp((σ_dB/8.686)^2)-mand γ=1+m/mexp(3(σ_dB/8.686)^2/2)-exp((σ_dB/8.686)^2/2) Thus, SIR η_l of a user can now given by η_l(r)=Pkr^-α/∑_i∈ϕPl_id_i^-αwhere k∼𝒢(β, γ) and l_i∼𝒢(β_i, γ_i). Here β_i=m_i/(m_i+1)exp((σ_dB/8.686)^2)-m_i and γ_i=(1+m_i)/m_iexp(3(σ_dB/8.686)^2/2)-exp((σ_dB/8.686)^2/2)Further, the correlation coefficient between two identically distributed generalized-K RVs is derived in <cit.>, and it is in terms of correlation coefficientof the RVs corresponding to the short term fading component (ρ_i,j) and the correlation coefficient of the RVs corresponding to the shadowing component(ρ^s_i,j). The resultant correlation coefficient (ρ^l_i,j) is then given byρ^l_i,j=ρ_i,j/( exp(σ_dB/8.686)^2)-1)+ρ^s_i,j m_i+ρ_i,jρ^s_i,j/m_i+1/( exp(σ_dB/8.686)^2)-1)+1 Note that after approximation, the SIR expression in the presence of log normal shadowing is similar to the SIR expression given in (<ref>), where only small scale fading is present. Hence nowthe coverage probability andrate for the independent case and correlated case can be compared using the methods outlined in Section IV and Section V. In other words, it can be shown thatthe coverage probability in the presence of correlated interferers is greater than or equal to the coverage probability in presence of independent interferers, when user's shape parameter is less than or equal to 1, i.e., β≤ 1, in the presence of shadow fading. Also, the rate in the presence of positive correlated interferers is always greater than or equal to the rate in the presence of independent interferers, in the presence of shadow fading.§.§ Physical interpretation of the impact of correlated InterferersWe know that the Nakagami-m distribution considers a signal composed of n number of clusters of multipath waves. Within each cluster, the phases of scattered waves are random and have similar delay times. The delay-time spreads of different clusters are relatively large. More importantly, the non-integer Nakagami parameter m is the real extension of integer n. One of the primary reason of parameter m being real extension of n is the non zero correlation among the clusters of multipath components<cit.>. In other words, if there existcorrelation among theclusters, the shape parameter of Nakagami-m fading decreases.Now, in order to find the physical interpretation of the impact of correlated interferers, we consider a cellular system where the interferes are equidistant and also the shape parameters are identical for every interference. When all the interferers are independent, the shape parameter of the total interference to be mN where the shape parameter of each interfere is m and the total number of interference is N (since the sum of Gamma RV is Gamma RV). However, when there existscorrelation among the interferers, i.e., there exists correlation among the clusters of different interferers, the shape parameter of the total interferencedecreases as observed in <cit.>. For example, if the interferers are completely correlated, the shape parameter of total interference is only m, whereas it was mN, when the interferers were independent. The smaller the shape parameter more faster is the power attenuation of interferers. Hence, the rate increases when there exists correlation amonginterferers. In the next section, we will show simulation results and discuss how those match with the theoretical results. § NUMERICAL ANALYSIS AND APPLICATIONIn this section, we study theimpact of correlation among interferers on the coverage probability and rate using simulations and numerical analysis. For simulations, we have considered the classic 19 cell system associated with a hexagonal structure as shown in Fig. <ref>. For a user we generate the channel fading power corresponding to its own channel as well as that corresponding to the 18 interferersand then compute the SIR per user. For correlation scenario, we generate correlated channel fading powercorresponding to the 18 interferers and then compute SIR per user. Furthermore, based on the SIR, we find coverage probability and rate. Fig. <ref>depicts the impact of correlation among the interferers on the coverage probability for different values of shape parameter. Note that only small scale fading is considered in Fig. <ref>. Thecorrelation among the interferers is defined by the correlation matrix in (<ref>)withρ_pq=ρ^|p-q| where p,q=1,⋯ ,N <cit.>. From Fig. <ref>, it can be observed that form=0.5 and m=1, coverage probability in presence of correlation is higher than that of independent scenario (which match our analytical result).For example, at m=0.5, coverage probability increases from 0.16 in the independent case to 0.20 in the correlated case and at m=1, coverage probability increases from 0.148 to 0.216 when user is at normalized distance 0.7 from the BS. Whereas m=3, one cannot saythat coverage probability in presence of correlation is higher or lower than that of independent scenario. In other words, the coverage probability of independent interferers is higher than the coverage probability of correlated interferers when user is close to the BS. However, the coverage probability of independent interferers is significantly lower than the coverage probability of correlated interferers when the user is far from the BS.Fig. <ref> shows theimpact of correlation among the interferers on the coverage probability for different values of correlation coefficient. Here both small scale fading, i.e., η-μ fading and large scale fading, i.e., log normal shadowing are considered. The correlation among small scale fading is denoted by ρ_s and the correlation among the large scale fading is denoted by ρ_l. It can be seen that as correlation coefficient increases the coverage probability with increases for the correlated interferers case.§.§ How the User can Exploit Correlation among InterferersWe will now briefly discuss how the user in a cellular network can exploit knowledge of positive correlation among its interferers. Wecompare the coverage probability in the presence of correlated interferers for single input single output (SISO) network with the coverage probability in the presence of independent interferers for single input multiple output (SIMO) network to show that the impact of correlation is significant.For the SIMO network, it is assumed that each user is equipped with 2 antennas and both antennas at the user are used for reception since downlink is considered. A linear minimum mean-square-error (LMMSE) receiver <cit.> is considered. In order to calculate rate with a LMMSE receiver, it is assumed that the closest interferer can be completelycancelledat the SIMO receiver. Fig. <ref> plots the SISO rate in the presence of independent and correlated interferers case and the rate in the presence of independent interferers for a SIMO network. It can be seen that for ρ_s=0.6, ρ_l=0.6, the SISO rate for the correlated case[Thecorrelation among the interferers is defined by the correlation matrix in (<ref>)withρ_pq=ρ^|p-q| where p,q=1,⋯ ,N]is slightly higher than the SIMO rate for independent case. However, for ρ_s=0.7,ρ_l=0.9, SISO rate with correlated interferers is significantly higher than the SIMO rate with independent interferers. In other words, correlation among the interferers seemsto be as good as having one additional antenna at the receiver capable of cancelling the dominant interferer. Obviously, if one had correlated interferers in the SIMO system that would again lead to improved coverage probability and rate and may be compared to a SIMO system with higher number of antennas. In all three cases, it is apparent that if the correlation among the interferers is exploited, it leads toperformance results for a SISO system which are comparable to the performance of a 1× 2 SIMO system with independent interferers. We have now consider a MU-MIMO system and show that the impact of correlation on MU-MIMO is significant. It is assumed that each user and BSare equipped with 2 receive antennas and 2 transmit antennas, respectively. It is also assume that both transmit antennas at the BS are utilized to transmit 2 independent data streams to its own 2 users. A LMMSE receiver is considered and assume that user can cancel the closest interferer. Hence, in this MU-MIMO system, the user will experience no intra-tier interference coming from the serving BS.Fig. <ref> plots the MU-MIMO rate in the presence of independent and correlated interferers case. It can be seen that there is significant gain in rate for correlated case. For example, at normalized distance 0.6 from the BS, rate increases from 1.24 nats/Hz in the independent case to 2.55 nats/Hz in the correlated case when ρ_s=0.8 and ρ_l=0.95 and 1.985 nats/Hz in the correlated case when ρ_s=0.6 and ρ_l=0.6. The reason for significant gain in rate is due to the fact that interferers becomes double for 2× 2 MU-MIMO system. We would also likely to briefly point out that the impact of correlation among the interferers is like that of introducing interference alignment in a system. Interference alignment actually aligns interference using appropriate precoding so as reduce the number of interferers one needs to cancel. Here the physical nature of the wireless channel and the presence of co-located interferers also “aligns” the interferer partially. This is the reason one can get a gain equivalent to 1× 2 system in a 1× 1 system with correlated interferers provided the user knows about the correlation. Summarizing, our work is able to analytically shows the impact of correlated interferers on coverage probability and rate. This can be used by the network and user to decide whether one wants to use the antennas at the receiver for diversity gain or interference cancellation depending on the information available about interferers correlation. Note that interferers from adjacent sector of a BS will definitely be correlated <cit.>. We believe that this correlation should be exploited, since the analysis shows that knowledge of correlation will lead to higher coverage probability and rate.§ CONCLUSIONSIn this work, the coverage probabilityhave been compared analytically for following two cases: (a)User channel experiences Nakagami-m fading and interferers experience η-μ fading. (b) Interferers being correlated where the correlation is specified by a correlation matrix. We haveshown that the coverage probability in correlated interferer case is higher than that of the independent case, when the user channel's shape parameter is lesser than or equal to one, and the interferers have Nakagami-m fading with arbitrary parameters. Further, rate have been compared when both user channel and interferers experience η-μ fading. It has been shown that positive correlation among the interferers always increases the rate.We have also taken into account the large scale fading component in our analysis. The impact of correlation seems even more pronounced in the presence ofshadow fading. Moreover, MU-MIMO system is considered and it has been shown that the impact of correlation among interferers is significant on MU-MIMO. Our results indicate that if the user is aware of the interferers correlation matrix then it can exploit it since the correlated interferers behave like partially aligned interferers. This means that if the user is aware of the correlation then one may be able to obtain a rate equivalent to a 1× 2 system in a 1× 1 system depending on the correlation matrix structure. Extensive simulationswere performed and these match with the theoretical results. § PROOF OF THEOREM <REF>The coverage probability expressions for the scenario when interferers are independent and the scenario when interferers are correlatedare given as follows (given in (<ref>) and (<ref>), respectively). C_p(r)=K F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αλ_1m+1,⋯,1/Tr^αλ_N m+1] Ĉ_p(r)= K̂ F_D^(N)[1-m,m_c,⋯,m_c; Nm_c+1;1/Tr^αmλ̂_1+1,⋯,1/Tr^αmλ̂_N+1]where K=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αm λ_i +1)^m_c and K̂=Γ(Nm_c+m)/Γ(Nm_c+1)1/Γ(m)∏_i=1^N(1/Tr^αmλ̂_i +1)^m_c.From Theorem <ref> it is clear that K̂> K . Now, we need to compare the Lauricella's function of the fourth kind of (<ref>) and (<ref>). Here, for comparison we use the series expression for F_D(.). We expand theseries expression for the Lauricella's function of the fourth kind in thefollowing form:F_D^(N)[a,b,⋯, b;c;x_1,⋯, x_N]= 1+K_1,1∑_i=1^Nx_i+K_2,1∑_i=1^Nx_i^2+K_2,2∑_1≤ i<j≤ Nx_ix_j+K_3,1∑_i=1^Nx_i^3+K_3,2∑_i,j=1, s.t.i≠ j^Nx_i^2x_j+K_3,3∑_1≤ i<j<k≤ Nx_ix_jx_k+⋯ where K_1,1=(a)_1(b)_1/(c)_1 1!, K_2,1=(a)_2(b)_2/(c)_2 2!, K_2,2=(a)_2(b)_1(b)_1/(c)_21!1!, K_3,1=(a)_3(b)_3/(c)_3 3!, K_3,2=(a)_3(b)_2(b)_1/(c)_3 2!1!, K_3,3=(a)_3(b)_1(b)_1(b)_1/(c)_31!1!1! and so on.Hence the coverage probability for independent case given in (<ref>) can be written asC_p(r)=K[ 1+K_1,1∑_i=1^N(1/Tr^αm λ_i +1)+K_2,1∑_i=1^N(1/Tr^αm λ_i +1)^2+ K_3,1∑_i=1^N(1/Tr^αm λ_i +1)^3+ K_2,2∑_1≤ i<j≤ N(1/Tr^αm λ_i +1)(1/Tr^αm λ_j +1)+ K_3,2∑_i,j=1, s.t. i≠ j^N(1/Tr^αm λ_i +1)^2(1/Tr^αm λ_j +1)+ K_3,3∑_1≤ i<j<k≤ N(1/Tr^αm λ_i +1)(1/Tr^αm λ_j +1)(1/Tr^αm λ_k +1)+⋯]Similarly,for the correlatedcasethe coverage probability given in (<ref>) can be written asĈ_p(r)=K̂[ 1+K_1,1∑_i=1^N(1/Tr^αmλ̂_i +1)+K_2,1∑_i=1^N(1/Tr^αmλ̂_i +1)^2+ K_3,1∑_i=1^N(1/Tr^αmλ̂_i +1)^3+ K_2,2∑_1≤ i<j≤ N(1/Tr^αmλ̂_i +1)(1/Tr^αmλ̂_j +1)+ K_3,2∑_i,j=1, s.t. i≠ j^N(1/Tr^αmλ̂_i +1)^2(1/Tr^αmλ̂_j +1)+ K_3,3∑_1≤ i<j<k≤ N(1/Tr^αmλ̂_i +1)(1/Tr^αmλ̂_j +1)(1/Tr^αmλ̂_k +1)+⋯]HereK_1,1=(1-m)_1(m_c)_1/(N m_c+1)_1 1!, K_2,1=(1-m)_2(m_c)_2/(N m_c+1)_2 2!, K_2,2=(1-m)_2(m_c)_1(m_c)_1/(N m_c+1)_2 1!1!, K_3,1=(1-m)_3(m_c)_3/(N m_c+1)_3 3!, K_3,2=(1-m)_3(m_c)_2(m_c)_1/(N m_c+1)_3 2!1!, K_3,3=(1-m)_3(m_c)_1(m_c)_1(m_c)_1/(N m_c+1)_31!1!1! and so on. Note that here K_i,j are the same for both C_p(r) and Ĉ_p(r). Now, we want to show that each summation term in the series expression is a Schur-convex function.Each summation term in the series expression is symmetrical due to the fact that any two of its argument can be interchanged without changing the value of the function. We have already shown that ∏_i=1^N(1/1+kx_i)^a is a convex function ∀ x_i≥ 0 and ∀ a>0. Now, the terms in the summation terms in (<ref>) and (<ref>) are of the form ∏_i=1^M(1/1+kx_i)^a_i where M≤ N.Using Theorem <ref> (given in Section II), one can show that eachterm of each summation term is a convex function. Using the fact that convexity is preserved under summation one can show that each summation term is a convex function. Thus, each summation terminseries expression is a Schur-convex function.Now we consider following two cases.Case I when m< 1: Since m< 1, so 1-m > 0 and hence all the constant K_i, j> 0 ∀ i, j.Each summation term in series expression of coverage probability for correlated case is greaterthan or equal to the corresponding summation term in the series expression of coverage probability for independent case. Thus, if userchannel's shape parameter m< 1 then coverage probability ofcorrelated case isgreater than or equal tothe coverage probability for independent case.Case II when m> 1: Since m > 1, then 1-m < 0 and hence K_i,j<0 ∀ i ∈ 2|ℤ|+1and ∀ jwhere set ℤ denote the integer number, due tothe fact that (a)_N<0ifa<0andN ∈ 2|ℤ|+1.Whereas, K_i,j>0 ∀ i ∈ 2|ℤ|and ∀ j due of the fact that (a)_N>0ifa<0andN∈ 2|ℤ|. Thus, if m>1, we cannot state whether the coverage probability of one case is greater than or lower than the other case.IEEEtran | http://arxiv.org/abs/1707.08802v1 | {
"authors": [
"Suman Kumar",
"Sheetal Kalyani"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170727095149",
"title": "Impact of Correlation between Interferers on Coverage Probability and rate in Cellular Systems"
} |
1.11-.8mm l |0⟩ ⟨#|1⟨#1| |#⟩1| #1⟩ #1⟨#1 ⟩Tr adj𝕀id Idind DimEnd ResInd kerim sgnch STrSym 1/2 ⟩⟨ αβ̱ χ̧γ Γϵ ϕΦ ϕφ μμ νν øωØΩ ρ̊κ̨ κσ σΣ łλŁΛ ψψ χ∂̣ ∂†† α̇β̇ γ̇δ̇ α̇β̇ γ̇δ̇ θΘ θλ ϵγ zw Wi jk mn qQ x̂Δ Δ^†D M√(2) ℂ ℂℙℝ ℝℙℤ ℕℍ̋ / ∂/D/A/ #1#1#1#1 XX^† ĴĴ̅̂ VV^+ V^-R^† T^† Tr | http://arxiv.org/abs/1707.08578v1 | {
"authors": [
"Daniel W. F. Alves",
"Carlos Hoyos",
"Horatiu Nastase",
"Jacob Sonnenschein"
],
"categories": [
"hep-th",
"math-ph",
"math.MP",
"nlin.PS",
"physics.flu-dyn",
"physics.optics"
],
"primary_category": "hep-th",
"published": "20170726180004",
"title": "Knotted solutions, from electromagnetism to fluid dynamics"
} |
A search for FRB 121102-like persistent radio-luminous sources – Candidates and implications for the FRB rate and searches Eran O. Ofek1December 30, 2023 ========================================================================================================================== During the 125th European Study Group with Industry held in Limassol, Cyprus, 5-9 December 2016, one of the participating companies, Engino.netLtd, posed a very interesting challenge to the members of the study group. Engino.net Ltd is a Cypriot company, founded in 2004, that produces a series of toy sets – the Engino^ toy sets –consisting of a number of building blocks which can be assembled by pupils to compose toy models. Depending on the contents of a particular toy set, the company has developed a number of models that can be built utilizing the blocks present in the set, however the production of a step-by-step assembly manual for each model could only be done manually. The goal of the challenge posed by the company was to implement a procedure to automatically generate the assembly instructions for a given toy. In the present paper we propose a graph-theoretic approach to model the problem and provide a series of results to solve it by employing modified versions of well established algorithms in graph theory. An algorithmic procedure to obtain a hierarchical, physically feasible decomposition of a given toy model, from which a series of step-by-step assembly instructions can berecovered, is proposed. 62P30, 05C90, 68R10, 94C15. § INTRODUCTION Engino^ toy models are created by assembling small blocks or bricks together, with the purpose of helping pupils build technological models creatively and easily so that they can experiment and learn about science and technology in a playful way. Each of the toy sets produced by Engino.net Ltd has a specific number of blocks that can be assembled into many different models. It has been observed that the creative potential of each toy set increases exponentially as the number of blocks in the set increase. This is due to the patented design of the Engino^ blocks that allow connectivity on many directions in three-dimensional space simultaneously.To demonstrate the creative potential of its toy sets, the company has developed a large number of toy models that can be built using the contents of the set. The ingredients and the connections required to obtain each particular toy model has been recorded in a database system. Despite the detailed recording, the production of step-by-step instructions for the assembly of a particular toy model has been proved to be a tedious task that has to be accomplished manually. This is mainly due to the three-dimensional nature of the models and the complexity of the interconnections between the blocks, which in many cases impose a particular order in the steps that have to be taken to assembly the structure. The goal of the challenge posed by the company during the 125th European Study Group with Industry was the development of an automatic procedure able to produce step-by-step assembly instructions manual for every toy model that has been recorded in the company's databases. To accomplish this task, we propose a graph-theoretic approach. Given a toy model, we associate with it a directed graph whose vertices correspond to the building blocks of the model and whose edges represent physical connections between two blocks (see <cit.> and referencestherein). Moreover, in order to partially capture the actual geometry of thetoy model, every edge of the graph is labeled with a vector showing thedirection of the underlying physical connection in 3D space. This labelingof the edges provides an adequate description of the geometry of the model,for the purposes of our application. It should be however noted, that the exact geometry of the (possibly multiple) connections between individual blocks of a particular model has been recorded in full detail and this information is available at the final stage of the assembly instructions generation.With this setup, in order to produce the assembly instructions of a given model, we first apply the reverse process recursively. Given a description of a toy model, and hence its associated graph, we develop a method to break it apart into clusters of blocks in a manner that is physically possible. In what follows we call this procedure a Physically Feasible Decomposition (PFD) of the model. The result of such a decomposition is a collection of sub-models or components on which the method can be recursively applied until no further decompositions are possible. Thus, a characterization of PFD of a model is of fundamental importance in the decomposition procedure. The outcome of this separation process is a hierarchical tree structure of components, whose nodes can traversed in a postorder fashion, to generate an ordered sequence of nodes, which in turn dictate a series of step-by-step assembly instructions. The problem of determining a series of steps required to decompose a complex structure into its constituent components has been the subject of several studies dating back to the 1980s. This class of problems is termed disassembly sequencing and depending on the nature of the underlying structure, a number of different approaches have been employed (see <cit.> for an extensive survey). The motivation behind the study of disassembly sequencing originates mainly from the fact that by reversing the steps of a disassembly sequence, one can obtain an assembly procedure of the structure under study. In this respect, disassembly sequences are closely related to the automated generation of assembly instructions of complex structures (see for instance <cit.>). The procedure proposed in the present paper can be compared to the one presented in <cit.> for the computation of a hierarchical explosion graph. Contrary to the approach used in <cit.> for the construction of the explosion graph, which detaches individual parts one-by-one from the structure, and in turn applies a search strategy for the extraction of the hierarchy of components, our method produces directly a physically feasible decomposition into components along a given spatial direction. As shown in Section <ref>, a maximal physically feasible decomposition can be obtained using well known linear-time algorithms and the recursive application of this procedure results in a hierarchical decomposition which is comparable to the hierarchical explosion graph in <cit.>. The contents of the paper are organized as follows: In Section <ref>, we briefly recall some basic concepts and facts from graph theory required for the development of our results in the sequel. In the subsequent section, we present the proposed graph theoretic framework and through a series of motivating examples we introduce the notion of a Physically Feasible Decomposition (PFD) of a toy model. In the same section, we also define the Component Connectivity Graph (CCG) implied by the removal of a set of edges of the model's graph and show that such a removal gives rise to a PFD if and only if the corresponding CCG is a directed acyclic graph. In Section <ref>, we define maximal PFDs along a given direction and show that such decompositions can be obtained by applying well established, linear-time, algorithms used for the discovery of strongly connected components in directed graphs. In Section <ref> we outline an algorithmic procedureto obtain a hierarchical decomposition of a given toy model, using as intermediate steps for such a decomposition, maximal PFDs along appropriately chosen spatial directions.Moreover, at the end of section <ref>, the resulting hierarchicaldecomposition of the model is utilized to recover a series of assembly instructions. Finally, in Section <ref> we review and summarize our results. § GRAPH THEORY PREREQUISITES In this section, we review a number of definitions and facts from graph theory that will be instrumental in the sequel. Most of these definitions and results can be found in <cit.>.A directed graph G, denoted by G(V,E), is an ordered pair of sets (V,E) where: * V is the set vertices or nodes of G;* E is the set of directed edges consisting of directed pairs (u,v), where u, v ∈ V.Moreover, if E is allowed to be a multiset instead of a set, then G(V,E) is a directed multigraph. On the other hand, if pairs of the form (v,v), (called loops) are not allowed in E, then G(V,E) is a directed simple graph. Similar definitions can be given in case the edge set (multiset), has as elements undirected pairs of vertices. In such a case the (multi)graph is called undirected.A graph G_1(V_1,E_1) is a subgraph of a given graph G(V,E) if V_1⊆ V and E_1⊆ E consists exclusively of edges having both its endpoints in V_1. Moreover, for V_1⊆ V, we define the induced subgraph G[V_1] as the subgraph of G(V,E), whose vertex set is V_1 and its edge set is the set of all edges of E, having both their endpoints in V_1.In a directed graph G(V,E), a directed (resp. undirected) path of length k, starting from v_0 and ending to v_k, is a sequence of vertices v_0,v_1,…,v_k, such that(v_i,v_i+1)∈ E (resp.(v_i,v_i+1)∈ E or (v_i+1,v_i)∈ E), for all 0≤ i<k. In case v_0=v_k and k>0 the path is called a directed (resp. undirected) cycle. A vertex t∈ V is said to be reachable from s∈ V, if there exists a directed path from s and to t. A directed graph G(V,E) is set to be a Directed Acyclic Graph (DAG), if it contains no directed cycles, or equivalently, if there exists no vertex in V which is non-trivially reachable from itself. A topological ordering of the vertices of a directed graph G(V,E) is a total ordering of its vertices v_1,v_2,…,v_n, such that for all (v_i,v_j)∈ E, i≤ j holds. A directed graph G(V,E) is acyclic if and only if a topological ordering of its vertices exists.A directed graph G(V,E) is called strongly (resp. weakly) connected if for every pair of vertices u∈ V, v∈ V, there exists a directed (resp. undirected) path from u to v. A maximal strongly (resp. weakly) connected subgraph of a graph, i.e. a strongly connected subgraph which is not a proper subgraph of any other strongly connected subgraph, is called a strongly (resp. weakly) connected component. The condensation of a directed graph G(V,E) is a directed graph G_co(V_co,E_co), with: * V_co={C_i:C_i is a strongly connected component of G(V,E)}; * E_co={(C_i,C_j):∃(u,v)∈ E such that u∈ C_i,v∈ C_j}. The condensation of any directed graph G(V,E) is a directed acyclic graph. A tree is an undirected graph in which every pair of vertices is connected via a unique path. A rooted tree is a tree having one particular vertex designated as its root node. An ordered tree is a rooted tree in which an ordering is specified for the children of each vertex. A binary tree is a rooted tree in which every vertex has at most two children. A binary tree is full if every node has either zero or two children. § PHYSICALLY FEASIBLE DECOMPOSITION OF TOY MODELS We now present the proposed framework for the solution of the decomposition problem discussed above based on a graph-theoretic approach. Given a toy model ℳ, we associate to it a directed graph G(V,E), where: * V={v_1,v_2,…,v_n} is the vertex set of G with each vertex v_i corresponding to a block of ℳ;* E={(u,v): u, v∈ V} is the edge set of G with each directed edge representing a connection between two blocks of the model. Every physical connection between two blocks of the model can be aligned in space to one particular direction vector, chosen out of a finite collection of directions. For instance, if a model uses only perpendicular connections between its blocks in 3D space, we can identify three direction vectors î,ĵ,k̂ along which all connections can be aligned. A connection between two blocks of the model u,v, aligned to a particular direction d̂ in physical space, gives rise to a directed edge (u,v)∈ E, if the vector from u to v points towards the same direction as d̂. Assuming that all the connections of the model ℳ correspond to p, not necessarily orthogonal, distinct spatial directions d̂_̂î, we can partition the edge set E into a family of p mutually disjoint sets E_i, i=1,2,…,p, each of which contains the edges associated to connections sharing the same direction in space. It should be noted that the physical connections between the blocks of the toy model ℳ, are assumed to be fixed, meaning that the resulting construction is rigid and contains no moving or rotating parts. Thus, the only possible way to separate two connected blocks is to apply opposite forces along the physical direction d̂_̂î associated to the connection, provided that resulting the displacement is physically feasible in the sense described in the paragraph that follows. At first this may seem to be a rather restrictive assumption with respect to the types of toy models it allows to be constructed, as there are many actual toy compositions in Engino's collection involving moving or rotating parts. However, as discussed with representatives of the company during the 125th ESGI meeting, in most such cases the moving or rotating parts can either be considered as separate rigid submodels (e.g. a two wheel and axle submodel), or their connection to the rest of the model is non-fixed (e.g. a pinned joint), allowing them to be detached from it by pulling them along some non-blocking direction. In view of this setup, we propose the following principle to describe the conditions under which a disconnection of two blocks is physically possible.Physically Feasible Disconnection of two blocks:In order to disconnect two blocks corresponding to vertices v_1,v_2∈ V, connected via an edge (v_1,v_2)∈ E aligned to a given spatial direction d̂_̂î, the blocks v_1,v_2 must be able to be displaced along the directions -d̂_̂î,d̂_i respectively, when appropriate opposite forces are applied on the blocks.The idea behind the above principle is illustrated in the following example. Consider the blocks shown in Figure <ref>. In the left side, the blocks 1,2 can be disconnected using two opposite horizontal forces, since their application on the two blocks will result in displacements along the horizontal direction. If a third block is added as shown in right side of Figure<ref>, then the blocks 1,2 cannot be disconnected by applying on them opposite horizontal forces, since their displacement is blocked by their vertical connections to the block number 3. The idea of disconnecting two blocks of the model in a physically feasible manner can be easily generalized to describe the corresponding decomposition of a model into two submodels. In general, the removal of a set of edges along a given direction may result into a decomposition of the graph of the model into two or more weakly connected components. However, not all such removals can be actually applied on the physical model to decompose it into two or more submodels. This is due to the fact that in certain cases the physical displacement of the resulting weakly connected components of the model is blocked by other physical connections, due to the presence of edges not removed in the current phase.We can extend the principle of Physically Feasible Disconnection, introduced above, to the case of the separation of two weakly connected components.Physically Feasible Decomposition into two components: The removal of a set of edges, aligned to a particular space direction d̂_̂î, is physically feasible, if and only if the two resulting weakly connected components are able to be displaced along the directions -d̂_̂î,d̂_i respectively, when appropriate opposite forces are applied on these blocks.For brevity, in what follows, we shall call this decomposition a 2–PFD of the model. The above decomposition is equivalent to assuming that, during the separation process, each of the two weakly connected components behaves like a single block, but unlike the single blocks case, it is possible to have multiple parallel connections between them. Our next goal is to obtain a characterization of 2–PFD'sthat are possible along a given direction. In this respect it is instrumental to introduce the notion of the Component Connectivity Graph of a model ℳ, implied by the removal of a set of co-linear edges, which provides a higher level view of the decomposition. Let G(V,E) be the graph associated to a model ℳ and E̅_̅i̅⊆ E_i a non-empty set of edges, where E_i is the set of all edges of G(V,E) along the spatial direction d̂_i. The Component Connectivity Graph (CCG), implied by the removal of the edges of E̅_̅i̅, is a directed graph, G_C(V_C,E_C), whose vertices are the weakly connected components C_i, i=1,2,…,k, into which G(V,E) is partitioned with the removal of the edges of E̅_̅i̅. Two components C_i,C_j are connected via an edge (C_i,C_j)∈ E_C if and only if i≠ j and there exists an edge (v,u)∈E̅_̅i̅ with v∈ C_i and u∈ C_j. We should note that according to the above definition the CCG implied by the removal of a set of edges E̅_̅i̅⊆ E_i is a simple directed graph, since by construction it cannot contain neither loops nor parallel edges sharing the same source and target vertices. The above ideas are illustrated in the following example. Consider the model shown in Figure <ref>. The graph G(V,E) of the model is depicted in Figure <ref>, where d̂_1,d̂_2 are respectively the horizontal (left - right) and vertical (bottom - up) direction vectors.If we remove all edges along the horizontal direction, i.e. edges (2,3), (1,5) and (4,5), the graph is decomposed into two weakly connected components C_1={1,2,3,4} and C_2={5} as shown in Figure <ref>, and the implied CCG by this removal of edges is shown in Figure <ref>. Clearly, nothing prevents the displacement of the two components C_1,C_2 from moving towards -d̂_1,d̂_1 respectively, when appropriate horizontal forces are applied on them. Thus, the removal of all horizontal edges implies a 2–PFD of the model. On the other hand, if we choose to remove all edges along d̂_2, we end up with the weakly connected components C_1^',C_2^' shown in Figure <ref>, and the corresponding CCG is the one in Figure <ref>.Despite the fact that the removal of the four vertical edges separates the graph into two weakly connected components, it is clear that such a decomposition is not physically feasible. Obviously, the blocks 2,3 of C_1^' cannot be displaced vertically, because they are “trapped” between the components 1,4 of C_2^'. In view of the decomposition along the spatial direction d̂_1 shown in Example <ref>, it becomes apparent that not all the edges removed correspond to a physically feasible disconnection of two blocks. This is the case with the edge (2,3) in the graph of Example <ref>, which does not appear in Figure<ref> due to its removal. Despite the fact that this edge can be theoretically removed during the removal of all edges along d̂_1, the blocks 2,3 cannot be disconnected because the perpendicular connections with blocks 1,4 obstruct their horizontal displacement. On the other hand, the edges (1,5), (4,5) obviously contribute actively on the decomposition of the graph into two components C_1 and C_2, shown in Figure <ref>. The distinguishing property between these two types of edges is that the former has both its endpoints on the same weakly connected component after the removal of all edges along d̂_1, while each of the latter type of edges have their start and end points lying on distinct components. The edges that actively contribute to the formation of weakly connected components of a given CCG, will be called physically removable for the given CCG. A maximal subset of physically removable edges, along a given spatial direction, can be successfully computed using the technique presented in Section <ref>.Proceeding a step further, we can provide a characterization of 2–PFD's in terms of a particular property of the edges connecting the weakly connected components in the corresponding CCG. Let ℳ be a toy model with the associated directed graph G(V,E). Assume that the removal of a non-empty set of edges E̅_̅i̅⊆ E_i, where E_i is the set of all edges of G(V,E) along the direction d̂_i, gives rise to the CCG, G_C(V_C,E_C), where V_C={C_1,C_2}. Then, the removal of the edges of E̅_̅i̅ is a 2–PFD of ℳ if and only if E_C contains exactly one of the edges (C_1,C_2), (C_2,C_1). We first note that since E̅_̅i̅ is non-empty, so is E_C. Moreover, recall that G_C(V_C,E_C) is simple, so, E_C will either contain exactly one or both (C_1,C_2), (C_2,C_1). Assume now that E_C contains both (C_1,C_2) and (C_2,C_1). Then, due to the presence of (C_1,C_2), in order to separate C_1 from C_2 we should be able to displace C_1 towards -d̂_i and C_2 towards d̂_i, by applying appropriate opposite forces on C_1 and C_2. On the other hand, due to the presence of (C_2,C_1), in order to accomplish the same task, C_1 should be able to move towards d̂_i and C_2 towards -d̂_i, using again appropriate opposite forces. Obviously, neither C_1 nor C_2 can move simultaneously on both spatial directions -d̂_i, d̂_i. Thus, the removal of the edges of E̅_̅i̅, is not a 2–PFD, which proves the “only if” part of the lemma.Conversely, assume without loss of generality that E_C contains only (C_1,C_2). This means that, in physical space, the components C_1,C_2 are connected only on one side, leaving their externally exposed sides free (see Figure <ref>). Thus, removing the edges of E̅_̅i̅ connecting the vertices of C_1 to those of C_2, will result in a 2–PFD of the model, since C_1 can be displaced towards the direction of -d̂_i and C_2 towards that of d̂_i.Proceeding a step further we can generalize the idea of a Physically Feasible Decomposition into the case where the removal of a set of edges, along a particular spatial direction d̂_̂î, separates the model into more than two weakly connected components. Assume that after the removal of a set of edges E̅_̅i̅⊆ E_i, we end up with k>2 components. Such a decomposition is physically feasible if we can obtain it by applying a 2–PFD of the original model by removing an appropriate subset of edges of E̅_̅i̅, and in turn by repeating 2–PFD procedures on the resulting submodels, recursively. A PFD giving rise to k>2 components, that can be accomplished recursively by applying a series 2–PFD's, will be called a k–PFD. The above idea is formalized in the following definition.Let ℳ be a toy model and G(V,E) its associated directed graph. Assume that the removal of a non-empty set of edges E̅_̅i̅⊆ E_i, where E_i is the set of all edges of G(V,E) along the direction d̂_i, gives rise to the CCG, G_C(V_C,E_C), consisting of k≥2 weakly connected components. We say that the removal of the edges E̅_̅i̅ implies a k–PFD of the model ℳ, if there exists a set of edges E̅_i^0⊆E̅_̅i̅, whose removal implies a 2–PFD of ℳ into C_1,C_2, for which exactly one of the following is true: * C_1∈ V_C and C_2∈ V_C;* C_1∈ V_C and the removal of all edges of E̅_̅i̅\E̅_i^0 from C_2, implies its (k-1)–PFD;* C_2∈ V_C and the removal of all edges of E̅_̅i̅\E̅_i^0 from C_1, implies its (k-1)–PFD;* C_j∉ V_C, for j=1,2 and appropriate removal of edges of E̅_̅i̅\E̅_i^0 from each one of them, implies a k_1–PFD of C_1 and a k_2–PFD of C_2, such that k_1+k_2=k. If the removal of any set of edges E̅_̅i̅⊆ E_i, results in a CCG with only one weakly connected component, we say that we have a1–PFD or a non PFD of the model.The structure of a k–PFD of a model ℳ can be represented by a full, ordered, binary tree T, having as its root node the entire vertex set V_C. The internal nodes of T are subsets of V_C corresponding to weakly connected components of G_C resulting in each step of the recursive application of 2–PFD's. Finally, the leaves of T are the singletons of V_C, that is, the components of the CCG corresponding to the k–PFD. Clearly, by construction each node of T, will have either 0 or 2 children, thus T is full. Moreover, T can be assumed to be ordered, that is, we distinguish the left and the right child of each node. According to Lemma <ref>, every 2–PFD separates a weakly connected component into two child components, connected only in a single direction. In view of this property we assign to the left child of each node in T, the child component from which the edges originate, and to the right child of the node in T, the component to which the edges terminate.Our aim is to identify those subsets of edges E̅_̅i̅⊆ E_i, that is, sets of edges aligned to a spatial direction d̂_i, whose removal gives rise to a k–PFD of the model. The following theorem serves as a characterization of this property. Let ℳ be a toy model and its associated directed graph G(V,E). Let further G_C(V_C,E_C) be the CCG resulting after the removal of a non-empty set of edges E̅_̅i̅⊆ E_i, where E_i is the set of all edges of G(V,E) along the direction d̂_i. The removal of the edges E̅_̅i̅ implies a k–PFD, k≥2 of the model ℳ, if and only if G_C is a Directed Acyclic Graph (DAG). If G_C(V_C,E_C) is a DAG, then there exists a topological ordering of its vertices C_1, C_2,…, C_k, that is, an ordering such that for all (C_i,C_j)∈ E_C, i≤ j holds. In view of this fact, since C_1 is the first in this ordering, there will be only outgoing edges from the vertices of C_1, to those of V_C\{C_1}. Hence, according to Lemma <ref> removal of the edges originating from C_1 and terminating to V_C\{C_1} is a 2–PFD (see Figure <ref>).Now, if we denote by G_C^' the subgraph of G_C induced by V_C\{C_1}, we may note that C_2,…,C_k, is a topological order of its vertices. Hence, C_2 can be detached from G_C^' through a 2–PFD following a similar procedure as above. Thus, after k-1 recursive applications of 2–PFD's, utilizing appropriate subsets of E̅_̅i̅, we obtain a decomposition of the model ℳ into k weakly connected components C_1,C_2,…,C_k, which is a k–PFD.Conversely, assume that the removal of the set of edges E̅_̅i̅,implies a k–PFD of the model and let G_C(V_C,E_C) be the corresponding CCG. As explained in Remark <ref> a k–PFD of a model can be represented by a full, ordered, binary tree T. Moreover, in view of the way that the left and right children are assigned in each node of T, it is easy to verify that if (C_i,C_j)∈ E_C then C_i will appear on T, to the left of C_j. Hence, if we order the leafs of T starting from the leftmost one moving to the right, we get a total order C_1,C_2,…,C_k, which is clearly a topological ordering of G_C(V_C,E_C). Thus, G_C(V_C,E_C) is acyclic. § MAXIMAL PFD ALONG A SPATIAL DIRECTION In the previous section, a characterization of physically feasible decompositions along a particular spatial direction was given in terms of the absence of cycles on the implied CCG. In this section, we propose a method to derive such a maximal acyclic CCG, as the condensation of the graph resulting after making edges not aligned to the chosen direction, bidirectional. In this respect we introduce the following definitions.Let ℳ be a toy model and let G(V,E) be the associated directed graph.The removal of a set of edges E̅_i⊆ E_i, along a spatial direction d̂_i, implies a maximal PFD of the model along d̂_i, if the implied CCG is maximal, that is, any set of edges E̅_i^', such that E̅_i⊆E̅_i^'⊆ E_i, implies the same CCG with E̅_i.Let ℳ be a toy model, G(V,E) its associated directed graph and let E_i⊆ E be the set of all edges along the spatial direction d̂_i. We define the projection of G(V,E) along the direction d̂_i, to be the graph G_i(V,E∪ R_i), where R_i contains all the edges of G not in E_i, reversed, that is R_i={(u,v):(v,u)∈ E\ E_i}. We illustrate the above notion via the following example. Consider the model presented in Example <ref> and the corresponding graph G(V,E) shown in Figure <ref>. According to Definition <ref>, the projection of G(V,E) along d̂_1 and d̂_2 are shown in Figure <ref>.We proceed now to the main result of the present section. Let ℳ be a toy model, let G(V,E) be its associated directed graph andlet G_co^i(V_co^i,E_co^i) be the condensation of the projection G_i(V,E∪ R_i) of G(V,E), along d̂_i. Then, G_co^i(V_co^i,E_co^i) is a CCG corresponding to a maximal PFD along d̂_i. Define the set of edges whose endpoints lie on two distinct strongly connected components of G_i(V,E∪ R_i), that isE̅_i={(u,v)∈ E:u∈ C_k,v∈ C_l where C_k,C_l∈ V_co^i and k≠ l}.Note that if either (u,v)∈ E \ E_i or (u,v)∈ R_i, then u,v lie on the same strongly connected component of G_i(V,E∪ R_i), because there are edges connecting them in both directions. Thus, E̅_i⊆ E_i. If any two vertices u,v∈ V lie on the same strongly connected component of G_i(V,E∪ R_i), then there exists a directed path from u to v, whose intermediate vertices lie on the same strongly connected component with u,v. Every edge on the path which is in R_i, can be replaced by its “reverse”, which lies in E\ E_i⊆ E\E̅_i. The rest of the edges on the path, not in R_i, obviously cannot be in E̅_i, since the latter contains edges whose endpoints lie on two distinct strongly connected components of G_i(V,E∪ R_i). Hence, any two vertices u,v∈ V lying on the same strongly connected component of G_i(V,E∪ R_i), can be connected via an undirected path, which lies entirely on the same weakly connected component as u,v, using only edges from E\E̅_i. Thus, all vertices lying on the same strongly connected component of G_i(V,E∪ R_i), belong to the same weakly connected component of G(V,E\E̅_i). Conversely, if any two vertices u,v∈ V lie on the same weakly connected component of G(V,E\E̅_i), then there exists an undirected path from u to v, whose intermediate vertices are on the same weakly connected component with u,v. Our aim is to show that there exists a directed path from u to v in G_i(V,E∪ R_i). In this respect, the edges on the undirected path having the correct orientation, that is from u to v, can be used to form the directed path. If an edge on the undirected path belongs to E\ E_i and is oriented from v to u, then it can be replaced in G_i(V,E∪ R_i) by its “reverse” which belongs to R_i. On the other hand, if an edge on the undirected path belongs to E_i, then both its endpoints must lie in the same strongly connected component of G_i(V,E∪ R_i), otherwise this edge should be in E̅_i, whose elements have been removed from G(V,E\E̅_i). In view of this, if such an edge does not have the desired orientation (i.e. from u to v), we can find a directed path in G_i(V,E∪ R_i), with the correct orientation, to replace it. Thus, any two vertices lying on the same weakly connected component of G(V,E\E̅_i), belong to the same strongly connected component of G_i(V,E∪ R_i).In view of the above discussion, it is clear that the strongly connected components of G_i(V,E∪ R_i) coincide with the weakly connected components of G(V,E\E̅_i). Thus, V_co^i is the vertex set of the CCG implied by the removal of the edges of E̅_i from G(V,E). Further, it is straightforward to verify that the set of edges E_co^i are exactly the edges of the CCG implied by the removal of the edges of E̅_i from G(V,E). Thus, G_co^i(V_co^i,E_co^i) is a CCG corresponding to the removal of the edges of E̅_i. Since the condensation graph of any directed graph is a DAG, the removal of the edges of E̅_i, implies a k–PFD of the model, where k=|V_co^i|. To show that the removal of the edges of E̅_i, implies a maximal PFD along d̂_i, assume there exists a set of edges E̅_i^', such that E̅_i⊆E̅_i^'⊆ E_i, implying a PFD of the model along d̂_i. Consider an edge (u,v)∈E̅_i^'\E̅_i, whose endpoints lie on distinct weakly connected components C_u,C_v, in G(V,E\E̅_i^'), such that u∈ C_u and v∈ C_v. Clearly, since (u,v)∉E̅_i, it is present in G(V,E\E̅_i) and both u,v lie in the same weakly connected component of the latter. In this case, it is evident from the discussion above that u,v must lie on the same strongly connected component of G_i(V,E∪ R_i). Thus, there exists a directed path from v to u in G_i(V,E∪ R_i). Now, since u∈ C_u and v∈ C_v in G(V,E\E̅_i^'), there exists at least one edge (v^',u^')∈E̅_i^', in the directed path from v to u, such that u^'∈ C_u and v^'∈ C_v, otherwise C_u,C_v would not be distinct. Hence, the weakly connected components C_u,C_v are connected in the CCG implied by the removal of the edges of E̅_i^', via to opposite edges, which in turn implies that such a removal does not imply a PFD. Having arrived at a contradiction, we conclude that there exists no edge in E̅_i^'\E̅_i, thus E̅_i^'=E̅_i.Theorem <ref> essentially providesa method to obtain a maximal PFD of a given model along a spatialdirection d̂_i. According to the above result the CCGcorresponding to a maximal PFD along d̂_i coincides with the condensation, G_i(V,E∪ R_i), of G(V,E) along this particular direction. Thus, the components into which a maximal PFD decomposes the model,coincide with the strongly connected components of the corresponding projection. The computation of the strongly connected components can be accomplished inlinear time, using Kosaraju's algorithm <cit.>,Tarjan's strongly connected components algorithm <cit.> or Dijkstra'spath based strong component algorithm <cit.>. Moreover, Kosaraju'sand Tarjan's algorithms also compute a reverse topological ordering of thestrongly connected components of the graph on which it is applied. The topological ordering computed by these algorithms dictates the order under which the components detected can be detached from the model in the process of a step-by-step decomposition along the chosen spatial direction. Applying some strongly connected component computation algorithm on the projections of G(V,E) along d̂_1, d̂_2, given in Example <ref>, we get respectively the condensations shown in Figure <ref>.Clearly, the condensed graph corresponding to the projection along d̂_1, coincides with the CCG shown in Figure <ref> and clearly implies a 2–PFD of the model along this direction. On the other hand the condensed graph corresponding to the projection along d̂_2, consists of only one component, indicating that ak–PFD, for k≥2, along d̂_2 is not possible.§ HIERARCHICAL PFD OF TOY MODELS AND ASSEMBLY INSTRUCTIONS GENERATIONIn the present section an outline of the procedure to obtain a recursive, physically feasible decomposition of a given toy model ℳ, is proposed. The key step of the procedure presented in what follows, is based on both the theoretical analysis presented in Section <ref>, and the use of well established algorithmic tools for the detection of strongly connected components in directed graphs, as shown in Section <ref>. While each step of the procedure results in a flat collection of weakly connected components, corresponding to a maximal PFD along some given spatial direction, the outcome of the overall procedure will be a hierarchical model of components, i.e. a rooted tree, having as its top level component the toy model ℳ itself, and bottom level elements each of the constituent blocks of the model. Having obtained a hierarchical decomposition of the model, some appropriate tree traversal algorithm may be applied to reverse the decomposition process and produce a step-by-step assembly manual. This procedure is outlined at the end of this section.Using the setup of the previous sections, assume that G(V,E) is the directed graph associated to the model ℳ. Assume also that each directed edge in E is aligned to one of the p distinct spatial directions d̂_̂î, i=1,2,…,p. Finally, assume that MaxPFD(C,i) is a readily made function taking as its first argument a weakly connected component of G(V,E) and as its second argument an integer i=1,2,…,p. The function returns an ordered list of components C_1, C_2,…, C_k, k≥ 1, into which C can be decomposed as the result of a Maximal PFD along the direction d̂_̂î. According to the results of Section <ref> such a function can be implemented using well known, linear-time, strongly connected components detection algorithms. With this background we define the function HMaxPFD(C) which accepts as argument a weakly connected component of G(V,E), C, and returns a hierarchical decomposition of the model ℳ. The function HMaxPFD is outlined as follows:§.§.§ HMaxPFD(C) * Call MaxPFD(C,i) for i=1,2,…,p. * If for at least onei=1,2,…,p, the number of components C_1, C_2, …, C_k, returned by the respective MaxPFD, is greater than 1, then * For j = 1 … k * Call AppendChild(C, C_j) * Call HMaxPFD(C_j) In the above pseudocode the function AppendChild(C, C_j) is called, which is assumed to append the subcomponent C_j to C, as its child in the hierarchy of the intended decomposition. To implement this in practice, would require each of the discovered components to be able to maintain a list of pointers, pointing from each parent to its children components. The technical details of such an implementation are out of the scope of the present paper. Finally, when the argument of HMaxPFD is a single vertex v (which will necessarily be without edges), we define HMaxPFD(v) = v and the hierarchical operations terminate there, to then pass to the next branch (if any). To obtain the tree corresponding to the hierarchical PFD of ℳ, with the associated graph G(V,E), one has to invoke the function HMaxPFD, using the entire graph G as its sole argument. To provide a worst case analysis of the complexity of the HMaxPFD algorithm, we first take into account that each run of MaxPFD(C,i) is essentially a call of Tarjan's or a similar algorithm, whose time complexity is O(|V|+|E|), where|V|,|E| are the number of nodes and edges of the graph to which it is applied. Considering the worst case scenario, the MaxPFD will be called at most p times, until an actual decomposition, into two or more subcomponents is obtained. Moreover, at each level of the resulting hierarchical PFD tree, the total number of nodes (blocks) distributed along the components C_1, C_2, …, C_k, will be at most n, where n is total number of vertices in G (blocks in ℳ). Thus, if we denote by m the total number of edges in G, then the invocation of MaxPFD(C_j,i), for i=1,2,…,p, j=1,2,…,k will take at most O(p(n+m)) steps. Since the time complexity at each level of the tree is O(p(n+m)), the overall worst case complexity will occur on a PFD tree that has the maximum possible height, amongst all the PFD trees with n leaves in total and whose non-leaf nodes have at least two children. This becomes evident if we take into account the fact that the leaves of a PFD tree are exactly the components of G that can be no further decomposed, i.e. its individual blocks. In view of this, the maximum height PFD tree, will be a binary tree where every non-leaf node has exactly two children, out of which at least one is a leaf. The height of such a binary tree with n leaves can be easily seen to be n-1. Thus, the worst case time complexity is O(np(n+m)).We illustrate the above procedure in the following example. Consider the toy model of Example <ref> and its associated graph shown in Figure <ref>. InvokingHMaxPFD(G), the procedure will execute as follows: * Calling MaxPFD(G,1) returns two components C_1,C_2 where C_1,C_2 consist of the vertices {1,2,3,4} and {5} respectively. * Since MaxPFD returned more than one component for i=1, * For j=1, * C_1 is appended as a child of G. * HMaxPFD(C_1) is called. * MaxPFD(C_1,2) (d̂_2 is the only direction available) returns three components, C_11,C_12 and C_13, having as vertex sets {1}, {2,3} and {4} respectively. * Since MaxPFD returned more than one component for i=2, * For j'=1, * C_11 is appended as a child of C_1. * HMaxPFD(C_11) is called, returning C_11 since this is a single vertex. Recursion terminates. * For j'=2, * C_12 is appended as a child of C_1. * HMaxPFD(C_12) is called. * MaxPFD(C_12,1) (d̂_1 is the only direction available here) returns two components, C_121 and C_122, having as vertex sets {2} and {3} respectively. * Since MaxPFD returned more than one component for i=1, * For j”=1, * C_121 is appended as a child of C_12. * HMaxPFD(C_121) is called, returning C_121 since this is a single vertex. Recursion terminates. * For j”=2, * C_122 is appended as a child of C_12. * HMaxPFD(C_122) is called, returning C_122 since this is a single vertex. Recursion terminates. * For j'=3, * C_13 is appended as a child of C_1. * HMaxPFD(C_13) is called, returning C_13 since this is a single vertex. Recursion terminates. * For j=2, * C_2 is appended as a child of G. * HMaxPFD(C_2) is called, returning C_2 since this is a single vertex. Recursion terminates. The resulting hierarchical PFD of the model is depicted in figure <ref>. The assembly instructions for the model can be recovered by applying a depth - first traversal, starting from the root node of the tree.Having obtained a hierarchical decomposition of a toy model ℳ, which is essentially a tree structure like the one shown in figure <ref>, we can proceed to the composition of its nodes to reverse the PFD and produce the assembly instructions. This goal can be accomplished by employing a tree traversal algorithm, which respects the parent - child hierarchy, in the sense that each node is visited after its children. The necessity of the requirement regarding the priority of visits between parents and their children, emerges from the fact in order to assemble a component, from its constituent subcomponents, i.e. the children of the node in the tree, one has to assemble each child component first. An algorithm appropriate for this task could be a postorder traversal applied on the hierarchical PFD tree of ℳ. Preorder, inorder and postorder are well known traversal procedures that can be applied on ordered binary trees, i.e. rooted trees whose nodes have at most two children labeled as “left" and “right". A preorder traversal visits first the parent node, then traverses the left subtree and finally the right subtree. Respectively, the inorder traversal first traverses the left subtree of a node, then visits the node itself and finally traverses the right subtree. Postorder traversal, traverses first the left subtree of a node, then its right subtree and finally it visits the node itself. While inorder traversal may be ambiguous when applied to a general (non - binary) ordered tree, preorder and postorder traversals are well defined. Here we focus on the generalized version of the postorder traversal algorithm, which is applicable to non-binary trees. Given a node p in such a tree, the postorder traversal procedure can be defined recursively as follows: * Traverse the leftmost child of p, * Visit the node p, * Traverse the right sibling of p.The output of such an algorithm is a series of nodes ordered in such a way that parent nodes appear in the sequence after all their children. In view of this fact, the sequence generated by the postorder traversal can be used to generate the assembly instructions of the toy model ℳ. It should be noted that the leaf nodes of the hierarchical PFD tree represent individual toy blocks that require no assembly, thus they can be safely neglected in the instructions generation procedure. On the other hand, the internal nodes of the tree represent components of the toy model consisting of an assembly of individual blocks or other subcomponents, and thus they are the ones for which assembly instructions are needed. The ingredients required for the assembly of those components are no other than their child nodes in the hierarchy. If a subcomponent is used as a building block for a higher level component in the PFD hierarchy, then the former will precede the latter in the ordered sequence produced by the postorder traversal. Moreover, as noted in section <ref>, the exact geometry of the interconnections between any two blocks has been recorded beforehand. Thus, identifying the blocks from which a component is comprised, provides enough information to recover the exact geometric structure of each component. As a result, following the order dictated by the traversal, the assembly instructions of every subcomponent comprising a higher level component will appear earlier in the assembly procedure manual, leaving no room for inconsistencies in the flow of instructions.The procedure is illustrated in the following example. Given the hierarchical PFD of Example <ref>, we may apply the postorder traversal procedure on the tree shown in figure <ref>. The outcome of the algorithm is the following sequence of nodes: C_11, C_121, C_122, C_12, C_13, C_1, C_2, G As explained above the leaf nodes C_11, C_121, C_122, C_13, C_2 can be safely omitted, since they require no assembly. Doing so, the sequence of the remaining nodes consists only of C_12, C_1 and G, in that particular order. Thus, the assembly instructions manual in this case should consist of the following three steps: * Show how C_12 is assembled from its children nodes C_121, C_122. * Respectively, show how C_1 can be assembled from C_11, C_12, C_13. * Finally, show how G is assembled from using C_1, C_2. § CONCLUSIONS In this note we study the problem of automatically producing step-by-step assembly instructions for Engino^ toy models. The assembly manual of a toy model can be generated by reversing the decomposition process of the model to its constituent blocks. As explained in Section <ref> the disassembly process may under certain circumstances be blocked due to the presence of particular geometric structures in the interconnections between blocks. To avoid such situations we propose a graph theoretic framework for the analysis of the problem and provide a characterization of the decompositions that are physically feasible. Moreover, a procedure to obtain maximal physically feasible decompositions along a given geometric direction is presented, which can be implemented using well known, linear time, algorithms for the detection of strongly connected components in directed graphs. Based on these results, an algorithmicprocedure for the hierarchical decomposition of a given toy model, which takes into account the physical feasibility of the intermediate steps, is proposed.The final goal of generating a sequence of assembly instructions for the model is accomplished, by applying a postorder traversal of the hierarchical decompositiontree, from which a step-by-step series of instructions can be easily recovered.As for future extensions that could stem from our presented approach, and future challenges to be tackled, we remark the following:* First of all, notice that the connection principle for ENGINO blocks is mainly of binary type (just like those of LEGO and other toy systems), in the sense that even though some types of blocks have several male and/or female connectors, thus allowing for several ways of connecting two given blocks, any connection between two blocks is achieved by matching at least one pair of male-female connections, resulting in a finite set of possible relative spatial configurations between the blocks. An important exception to this is the freely-rotating pivoting connection, which allows for a continuous choice of the pivot angle, so the set of relative spatial configurations becomes infinite. In this paper we have focused on the binary type of connections because of the resulting finiteness of the set of possible spatial configurations, which allows us to tackle the problem by defining the connection directions d̂_i,i = 1, …, p. In future work, the feature of pivoting connections will be added to our Physically Feasible Decomposition, based on the fact that pivoting connections during the assembly process must be geometrically feasible, in the sense that small displacements associated with the rotation degree of freedom about the pivot point must be allowed to happen. The main difficulty lies in extending the current definition of the “fixed” connection directions d̂_i,i = 1, …, p, which will have to depend on the continuous “pivoting” degrees of freedom. * From the previous point it follows that our method can be applied to several toy systems, and even beyond that to industrial assembly processes <cit.> with binary-type connections as defined above. The main advantage of our method is that it requires very little geometrical and physical information about the connecting pieces. This is, at the same time, the main limitation of the method. For example, it does not apply to assemblies that require three or more hands <cit.>, and more generally it does not deal with cases when force and torque balances are relevant, as in the problems of grasping parts (form closure, force closure, etc.) <cit.>. However, this does not mean our method cannot be used in combination with these and other advanced assembly features. In fact, our method could be included as a complementary module in (dis-)assembly process planning for existing products in industry. For example, the feature of linearizability <cit.>, of practical importance in assembly lines, could be incorporated into our method because it is related to the distribution of internal nodes and leaves in our hierarchical PFD graphs. And, with a little bit of imagination, our method could potentially find its utility as a module in the recently discovered molecular assembly processes <cit.>, because these processes are characterized by constrained geometric arrangements, local interactions and reduced reactivity.This work originated from our participation in the 125th European Study Group with Industry (1st Study Group with Industry in Cyprus). We thank the Mathematics for Industry Network (MI-NET, www.mi-network.org), COST Action TD1409 for generous funding and support with the logistics of this first Study Group with Industry in Cyprus. MDB acknowledges support from Science Foundation Ireland under research grant number 12/IP/1491.We would also like to thank Costas Sisamos, founder and CEO of Engino Ltd, for the detailed exposition of the problem, his valuable insight on it and his comments on the present paper. Finally, we would like to thank the editor and the anonymous referees for taking time in reading and suggesting modifications to the paper. We highly appreciate it, as the comments have been very useful in improving the paper. 1Agrawala2003 Agrawala, M.(2003) Designing Effective Step-by-stepAssembly Instructions.ACM Transactions on Graphics (TOG) - Proceedings ofACM SIGGRAPH 2003 22, 828–37.BangGutin2008 Bang-Jensen, J. & Gutin, G. Z.(2008) Digraphs:Theory, Algorithms and Applications. Springer Publishing Company Inc. BondyAndMurty1976 Bondy, J. A. & Murty, U. S. R. (1976) Graph Theory with Applications. North Holland.CormenEtAl2001 Cormen, T. H., Leiserson, C. E., Rivest, R. L.& Stein, C. (2001) Introduction to Algorithms. Second Edition. MIT Press and McGraw-Hill.Dijkstra1976 Dijkstra, E. (1976) A Discipline of Programming. NJ: Prentice Hall.Hsu2011 Hsu, Y.-Y., Tai, P.-H., Wang, M.-W. & Chen, W.-C. (2011)A knowledge-based engineering system for assembly sequence planning. Int J Adv Manuf Technol 55, 763–82.Ka17 Kassem, S., Lee, A. T., Leigh, D. A., Marcos, V., Palmer, L. I. & Pisano, S. (2017) Stereodivergent synthesis with a programmable molecular machine. Nature 549, 374–378.Lambert2003 Lambert, A. J. D. (2003) Disassembly sequencing: A survey. International Journal of Production Research 41, 3721–59.Li2008 Li, W.,Agrawala, M., Curless, B. & Salesin, D. (2008)Automated generation of interactive 3D exploded view diagrams. ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH 2008 27, 101.Na88 Natarajan, B. K. (1988) On planning assemblies. In Proceedings of the fourth annual symposium on Computational geometry, ACM, 299–308.Peysakhov Peysakhov, M., Galinskaya, V. & Regli, W. C. (2000) Representation and evolution of lego-based assemblies. In AAAI/IAAI (p. 1089). Sharir1981 Sharir, M. (1981) A strong connectivity algorithm and its applications to data flow analysis. Computers and Mathematics with Applications 7, 67–72.Sn94 Snoeyink, J. & Stolfi, J. (1994) Objects that cannot be taken apart with two hands. Discrete & Computational Geometry 12, 367–384.Tarjan1972 Tarjan, R. E. (1972) Depth-first search and linear graph algorithms.SIAM Journal on Computing 1, 146–60. Wa09 Wang, L., Keshavarzmanesh, S., Feng, H. Y. & Buchal, R. O. (2009) Assembly process planning and its future in collaborative manufacturing: a review. The International Journal of Advanced Manufacturing Technology 41, 132–144.Wi94 Wilson, R. H. & Latombe, J. C. (1994) Geometric reasoning about mechanical assembly. Artificial Intelligence 71, 371–396. | http://arxiv.org/abs/1707.09040v3 | {
"authors": [
"Efstathios N. Antoniou",
"Adérito Araújo",
"Miguel D. Bustamante",
"Aviv Gibali"
],
"categories": [
"cs.DM",
"62P30, 05C90, 68R10, 94C15"
],
"primary_category": "cs.DM",
"published": "20170727205204",
"title": "Physically Feasible Decomposition of Engino$^{\\circledR}$ Toy Models: A Graph Theoretic Approach"
} |
[1]Institut de Mathématiques de Toulouse. CNRS UMR 5219. Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex 09.E-mail: [email protected] and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically William Oçafrain^1 December 30, 2023 ===================================================================================================================== We are interested in quasi-stationarity and quasi-ergodicity when the absorbing boundary is moving. First we show that, in the moving boundary case, the quasi-stationary distribution and the quasi-limiting distribution are not well-defined when the boundary is oscillating periodically. Then we show the existence of a quasi-ergodic distribution for any discrete-time irreducible Markov chain defined on a finite state space in the fixed boundary case. Finally we use this last result to show the quasi-ergodicity in the moving boundary case. Key words :Quasi-stationary distribution, quasi-limiting distribution, quasi-ergodic distribution, Q-process, periodic moving boundaries, random walk2010 Mathematics Subject Classification: 60B10; 60F99 47D03 , 60J10§ INTRODUCTION Let (Ω,A, ) be a probability space and let X = (X_n)_n ∈ be a Markov chain with a finite state space (E,E), E being the σ-algebra containing all the subset of E. Let _x be the probability measure such that _x(X_0 = x)=1 and, for any measure μ on E, define _μ = ∫_x dμ(x). Denote by _1(E) the set of probability measures defined on E. We define, for each time n ≥ 0, a subset A_n ⊂ E called killing subset at time n and we denote by E_n the complement of A_n called survival subset at time n. We will call (A_n)_n ∈ the moving killing subset or the moving killing boundary.We denote by τ the random variable defined as followsτ := inf{n ≥ 0 : X_n ∈ A_n}For any subset B ⊂ E, we define τ_B as followsτ_B := inf{n ≥ 0 : X_n ∈ B}and, to make the notation easier, for any m ∈, we denote by τ_m the random variable defined byτ_m := τ_A_m = inf{n ≥ 0 : X_n ∈ A_m} This chapter will deal with quasi-stationary, quasi-limiting and quasi-ergodic distributions that we define as follows.ν is a quasi-stationary distribution if for any n ≥ 0_ν(X_n ∈· | τ > n) = ν(·)ν is a quasi-limiting distribution if there exist some μ∈ M_1(E) such that_μ(X_n ∈· | τ > n) n →∞⟶ν(·)ν is a quasi-ergodic distribution or a mean-ratio quasi-stationary distribution if for any μ∈ M_1(E) and any bounded measurable function f_μ(1/nk=0n-1∑ f(X_k)| τ > n) n →∞⟶∫ f dν We will also be interested in the existence of a Q-process, which can be interpreted as the process X conditioned never to be absorbed by (A_n)_n ∈. In the case where the sequence (A_n)_n ∈ does not depend on the time, the existence of these probability measures was established under several assumptions. See for example <cit.> for a general review on the theory of quasi-stationary distributions. For modelling purpose, some recent works (see <cit.>) introduce some Markov processes absorbed by moving boundaries and the classical theory on quasi-stationary distributions does not allow anymore to describe the asymptotic behavior of the process conditioned not to be absorbed. Our purpose is therefore to check whether these probability measures are still well-defined when (A_n)_n ≥ 0 depends on the time or not.In all what follows, we will assume that for any x ∈ E_0,_x(τ < ∞) = 1and will also assume that the following hypothesis of irreducibility holds∀ n ∈, ∀ x,y ∈ E_n, ∃ m ∈, _x(X_m τ_n=y) > 0 Quasi-stationary distribution will be studied for general moving killing boundaries. However, in a significant part of this chapter we will deal with moving killing boundaries (A_n)_n ∈ which are γ-periodic with γ≥ 2. In this chapter, we will actually show that there are no quasi-stationary distributions and quasi-limiting distributions in the sense of Definitions <ref> and <ref> when the boundaries are moving periodically. However, we will show that the notion of quasi-ergodic distribution and Q-process still makes sense even when the boundary is moving. In particular, we will show the following statement.Assume that (A_n)_n ∈ is γ-periodic. Then, for some initial law μ∈ P(E) and under assumptions which will be spelled out later, there exists a probability measure η such that, for any bounded measurable function f,_μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶∫ f dη The proof is divided to several steps. First we reduce the problem to the study of quasi-stationary distribution in a non moving domain, but for a periodic Markov chain. Then we extend the result proved by Darroch and Senata (see <cit.>) in the aperiodic case to the periodic situation (γ∈^*). This chapter ends with an application of this theorem to random walks absorbed by 2-periodic moving boundaries.§ QUASI-STATIONARY DISTRIBUTION WITH MOVING KILLING SUBSET The following proposition shows that the notion of quasi-stationary distribution as defined in Definition <ref> is not relevant when the killing boundary is moving. Assume there exist l,m ∈ such that A_lA_m. Then there is no measure ν∈ M_1(E) satisfying the following property:∀ n ∈, _ν(X_n ∈· | τ > n) = ν(·) For any n ∈, denote by f_n :M_1(E) → M_1(E) the function defined byf_n : μ⟶ℙ_μ(X_1 ∈· | τ_n > 1)where τ_n is defined in (<ref>) and denote by μ_n the probability measure defined byμ_n = _μ(X_n ∈· | τ > n)By the Markov property, we have for any n ∈^*μ_n = _μ_n-1(X_1 ∈· | τ_n > 1) = f_n(μ_n-1)Thus, by induction, we obtain for any n ∈ℙ_μ(X_n∈· | τ > n) = f_n∘…∘ f_1(μ)We deduce from this equality that∀ n ∈, _ν(X_n∈· | τ > n) = ν(·) ⟺∀ n ∈, f_n(ν) = ν⟺∀ n ∈, _ν(X_1 ∈· | τ_n > 1) = ν(·)In other words, ν is a quasi-stationary distribution in the moving sense if and only if it is a quasi-stationary distribution in the non-moving sense for all the subsets A_n. In particular, if ν satisfies (<ref>),ν(·) = _ν(X_1 ∈· | τ_l > 1)and ν(·) = _ν(X_1 ∈· | τ_m > 1)where l and m have been mentioned in the statement of the proposition. However, since the assumption of irreducibility (<ref>) holds, the previous statement is impossible since the support of the quasi-stationary distributions are different.Proposition <ref> can be extended to any general Markov process defined on any space state. In particular, for continuous-time Markov processes defined on a metric space (E,d), we may replace the assumption of irreducibility (<ref>) by the following assumption∀ t ∈_+, ∀ x,y ∈ E_t, ∀ϵ > 0, ∃ t_0 ∈_+, _x(X_t_0 τ_t∈ B(y,ϵ)) > 0where B(y,ϵ) := {z ∈ E : d(y,z) < ϵ}. Notice moreover that we did not need any assumption about the behavior of (A_n)_n ∈. In all what follows, we consider that (A_n)_n ∈ is γ-periodic with γ≥ 2.§ QUASI-LIMITING DISTRIBUTION WHEN THE KILLING SUBSET IS MOVING PERIODICALLY We are now interested in knowing whether the definition of quasi-limiting distribution given in definition <ref> is meaningful when the killing subset is moving or not.In the usual case, it is well known (see <cit.> p. 345) that quasi-stationary distribution and quasi-limiting distribution are equivalent notions. This implies that the non-existence of a quasi-stationary distribution implies the non-existence of any quasi-limiting distribution. However, this equivalence does not hold anymore in the moving case. Consider for example(A_n)_n ≥ 0 such that there exists n_0 such that for any n ≥ n_0, A_n = A_n_0 and assume that there exists a quasi-stationary distribution ν_n_0 (in the non-moving sense) such that for any probability measure μ on E_n_0,_μ(X_n ∈· | τ_n_0 > n) n →∞⟶ν_n_0Thus, by the Markov property, for any μ∈ M_1(E) such that _μ(τ > k) > 0 for all k ∈ and any n ≥ 0,_μ(X_n+n_0∈· | τ > n+n_0) = _μ_n_0(X_n ∈· | τ_A_n_0 > n) n →∞⟶ν_n_0where μ_n is defined in (<ref>) for any n ∈.From now on, we will assume that (A_n)_n ∈ is periodic and will denote by γ its period. We will show that quasi-limiting distribution is not well defined when the killing subset is moving periodically. Assume (A_n)_n ∈ is γ-periodic and there exist 0 ≤ l,m ≤γ -1 such that A_lA_m. Then there is no ν∈ M_1(E) satisfying the following property:∃μ∈ M_1(E), _μ(X_n ∈· | τ > n) n →∞⟶ν(·) Consider again the functions f_m defined in (<ref>):f_m : μ⟶_μ(X_1 ∈· | τ_m > 1)Then using the periodicity of (A_n)_n ∈ and by the Markov property, for any k ∈{1, …, γ}, m ∈ℕ^* and μ∈ M_1(E)ℙ_μ(X_k + m γ∈· | τ > k+m γ) = g_k ∘ f^m(μ)with g_k = f_k∘…∘ f_1f = f_γ∘…∘ f_1 Assume that there exists μ∈ M_1(E) such that the sequence (ℙ_μ(X_m∈· | τ > m))_m ∈ℕ converges to its limit ν. Thenν = m →∞limℙ_μ(X_mγ∈· | τ > mγ) = m →∞lim f^m(μ)So for any k ∈{1, …, γ}ν = g_k(ν) = ℙ_ν(X_k∈· | τ > k)In other words, for any k ∈{1, …, γ},ν = f_k(ν)We thus may conclude in the proof of proposition <ref>. The previous statement implies therefore that the quasi-limiting distribution as defined in Definition <ref> is not well-defined when the moving killing subset is periodic. However, according to the proof of the previous proposition, it seems that the sequence of these conditioned probabilities could have some limit points.The following proposition allows us to pass from a moving problem to a non-moving problem. The existence of limit points will be therefore a consequence of the existence of classical quasi-stationary distributions for the transformed Markov chain. For any 0 ≤ m ≤γ-1 and μ∈ M_1(E), there is a Markov chain (X^(m)_n)_n ∈ℕ such that ℙ_μ((X_m, X_m+γ, …, X_m + nγ) ∈· | τ > m + nγ) = ℙ_μ_m((X^(m)_0, X^(m)_1, …, X^(m)_n) ∈· | τ^(m)_m > n)where μ_m is defined in (<ref>) and τ^(m)_m = inf{n ∈ : X^(m)_n ∈ A_m}.According to the Markov property, it is enough to show that for any γ-periodic sequence of subsets B = (B_n)_n ∈ and any measure μ∈ M_1(E), there exists a Markov chain (Z_n)_n ∈ℕ such thatℙ_μ((X_γ, …, X_nγ) ∈· | τ(B) >nγ) = ℙ_μ((Z_1, …, Z_n) ∈· | τ̃_B_0 > n)where τ(B) = inf{m ≥ 0 : X_m ∈ B_m} and τ̃_B_0 = inf{n ∈ : Z_n ∈ B_0}. Denote F_0 the complement of B_0.For any x ∈ F_0 define p(x,·) byp(x,A) = ℙ_x(X_γ∈ A, τ_B > γ), ∀ A ⊂ F_0p(x,B_0) = 1-p(x,F_0)and we denote by (Z_n)_n ∈ℕ the Markov chain for which the transition kernel is p. We will show by induction that, for any ϕ_1, …, ϕ_n bounded measurable functions, 𝔼_μ(ϕ_1(X_γ) …ϕ_n(X_nγ) 1_τ(B) > nγ) = 𝔼_μ(ϕ_1(Z_1) …ϕ_n(Z_n) 1_τ̃_B_0 > n)By definition of (Z_n)_n ∈, for any probability measure μ and any bounded measurable function ϕ,_μ(ϕ(Z_1)_τ̃_B_0 > 1) = _μ(ϕ(X_γ) _τ(B) > γ)which entails the base case. Now assume that the equality for n-1 is satisfied. Let ϕ_1, …, ϕ_n be some bounded measurable functions. Then𝔼_μ(ϕ_1(X_γ) …ϕ_n(X_nγ) 1_τ(B) > nγ)= 𝔼_μ(ϕ_1(X_γ) 1_τ(B) > γ𝔼_X_γ(ϕ_2(X_γ) …ϕ_n(X_(n-1)γ) 1_τ(B) > (n-1)γ)) =𝔼_μ(ϕ_1(Z_1) 1_τ̃_B_0 > 1𝔼_Z_1(f_2(Z_1) …ϕ_n(Z_n-1) 1_τ̃_B_0 > (n-1))) = 𝔼_μ(ϕ_1(Z_1) …ϕ_n(Z_n) 1_τ̃_B_0 > n)This concludes the proof. § EXISTENCE OF QUASI-ERGODIC DISTRIBUTION WITH PERIODIC MOVING KILLING SUBSETSIn this section, our aim is to show the existence of a quasi-ergodic distribution as defined in Definition <ref> when the boundary is moving periodically. This section will be split into three parts :* We will first study quasi-ergodicity in the non-moving case (when A_n = A_0, ∀ n ∈) for irreducible Markov chains.* Then we will use the results obtained in the first part to deduce quasi-ergodicity for general Markov chains (irreducible or not), but still considering non-moving boundaries* Finally we will show the existence of quasi-ergodic distribution when (A_n)_n ∈ is moving periodically.§.§ Quasi-ergodic distribution in the classical non-moving sense for irreducible periodic Markov chains In this subsection we will study the quasi-ergodicity of one irreducible Markov chain Y = (Y_n)_n ∈ in the classical non-moving sense, that is when the killing edge does not move. Without loss of generality, assume Y is defined in the state space E_0 ={0, …, K} and that the cemetery is {0}. In this subsection and the following, τ will be defined as (<ref>) but refering to Y, that isτ = inf{n ≥ 0 : Y_n = 0} Denote by P the transition matrix of Y. Since 0 is an absorbing state for Y, P has the following formP = [ 1 0; v Q ]where Q is the sub-transition matrix. We will assume that Q is irreducible (i.e. ∀ x,y ∈ E_0, ∃ n ∈, Q^n(x,y) > 0). As a result we can define T_x the period of x asT_x := gcd{n ∈ :_x(Y_n = x, τ > n) > 0}where gcd refers to the greatest common divisor. By irreducibility of Q, all the x have the same period and we denote by T this common period.The existence of quasi-ergodic distributions has already been proved by Darroch and Seneta in <cit.> when T=1. However we will see that this result is not enough for our purpose and we need to extend it to periodic Markov chains.Due to the periodicity of Q, there exist (C_i)_0 ≤ i ≤ T-1 a partition of E_0 such that if the support of the initial distribution μ is included in C_0, then for any n ∈ and 0 ≤ k ≤ T-1,_μ(Y_k+nT∈ C_k, τ > k+nT) = 1 Without loss of generality, we construct (C_i)_0 ≤ i ≤ T-1 such that 1 ∈ C_0. Formally (C_i)_0 ≤ i ≤ T-1 are defined byC_0 := {y ∈ E_0 : ∃ n ∈^*, _1(Y_nT = y,τ > nT)> 0}∀ 1 ≤ i ≤ T-1, C_i := {y ∈ E_0 : ∃ x ∈ C_i-1, _x(Y_1 = y)>0}For each j ∈{0, …, T-1} and any v ∈^K, we will denote by v^(j) the sub-vector of v restricted to C_j. It is well known by the Perron-Frobenius theorem that the spectral radiusρ := max{|λ| : λ∈ Sp(Q)}is a simple eigenvalue of Q and that one can find a left-eigenvector ν = (ν(j))_1 ≤ j ≤ K and a right-eigenvector ξ = (ξ(j))_1 ≤ j ≤ K for ρ (i.e. ν Q = ρν and Qξ = ρξ) such that ν(j) > 0 and ξ(j) > 0 for all j ∈{1, …, K}. We may choose ν and ξ such that<ν,> = <ν,ξ> = 1 where <·,·> is the usual Hermitian product on ^K. Moreover, since Q is T-periodic, {λ_k := ρ e^2ikπ/T : 0 ≤ k ≤ T-1}⊂ Sp(Q)and each λ_k is simple. For each λ_k we can obtain a left eigenvector v_k and a right-eigenvector w_k from ν and ξ with the following transformation∀ j ∈{0, …, T-1}, v_k^(j) = e^-i2π j k/Tν^(j) andw_k^(j) = e^i2π j k/Tξ^(j)See Theorem 1.7 in [<cit.>,p.23-24] for more details. The vectors (v_i)_0 ≤ i ≤ T-1 are linearly independent. We can complete this family into a basis V = (v_i)_0 ≤ i ≤ K-1 such that v_i ∈ Span^⊥(v_0, …, v_T-1) for all T ≤ i ≤ K-1 whereSpan^⊥(v_0, …, v_T-1) = {v ∈^K : <v,v_i> = 0, ∀ i ∈{0, …, T-1}} Let us denote by R the matrix representing the change of basis from the canonical basis to V. Then we have the following decompositionQ = R [ λ_0 ⋱λ_T-10;0Q_0 ] R^-1where Q_0 is a (K-T) × (K-T) matrix. We define the matrix Q' byQ' = R [ 0 0; 0 Q_0 ] R^-1 Let f : {1, …, K}→ be a bounded measurable function. Then for any x ∈{1, …, K} and n ∈^*,_x(1/nk=0n-1∑ f(Y_k) _τ > n) = ρ^n φ(f) l=0T-1∑ e^-2inlπ/T<w_l,δ_x> <v_l,> + o(ρ^n)whereφ(f) = i=1K∑ f(i)ν(i) ξ(i) . Let f:{1, …, K}→ be a bounded measurable function. In this proof we will consider probability measures on {1, …, K} and functions defined on {1, …, K} as K-vectors. Thus for any x ∈{1, …, K} we can say_x(f(Y_n)_τ > n) = <δ_x Q^n, f>where δ_x is the Dirac measure on x. For any x ∈{1, …, K}, define (α_k(x))_0 ≤ k ≤ T-1 the unique family of ^K such that there is μ_x ∈ Span^⊥(v_0, …, v_T-1) such that δ_x = k=0T-1∑α_k(x) v_k + μ_xWe will use the following lemma whose proof is postponed after the proof of the theorem. For any 0 ≤ k ≤ T-1,(α_k(x))_x ∈ E_0 = w_kwhere w_k is defined in (<ref>) Thus we can writeδ_x = k=0T-1∑ w_k(x) v_k + μ_xSo, using (<ref>) and (<ref>), for any n ∈𝔼_x(f(X_n)1_τ > n)= <k=0T-1∑ w_k(x) v_k Q^n, f> + <μ_x Q^n,f> = k=0T-1∑λ_k^n w_k(x) <v_k, f> + <μ_x(Q')^n,f> Now, using the Markov property, for any k ≤ n, 𝔼_μ(f(Y_k)1_τ > n)= 𝔼_μ(1_τ > k f(Y_k) ℙ_Y_k(τ > n-k)) =𝔼_μ(1_τ > kg_k,n(f)(Y_k))where, for any y ∈ E_0, g_k,n(f)(y) = f(y)_y(τ > n-k)Then,g_k,n(f)(y)= f(y) <δ_y Q^n-k, 1> = m=0T-1∑λ_m^n-k f(y) w_m(y)<v_m,1> + f(y)<μ_y (Q')^n-k,>Define, for any k ∈{0, …, T-1} and n ∈,g_k(f):y → f(y)w_l(y)w_n(f) : y → f(y) <μ_y(Q')^n,> Then, using (<ref>), for any k ≤ n,𝔼_x(f(Y_k)1_τ > n)=<δ_x Q^k, g_k,n(f)>= l=0T-1∑λ_l^k w_l(x) <v_l, g_k,n(f)> + <μ_x(Q')^k,g_k,n(f)> = A_k,n+B_k,n+C_k,n+D_k,nwhereA_k,n = l=0T-1∑m=0T-1∑λ_l^k λ_m^n-kw_l(x)<v_l,g_m(f)><v_m,> B_k,n = l=0T-1∑λ_l^k w_l(x) <v_l, w_n-k(f)> C_k,n = m=0T-1∑λ_m^n-k <v_m,1> <μ_x(Q')^k,g_m(f)> D_k,n = <μ_x(Q')^k,w_n-k(f)>Hence for any n ∈^*k=0n-1∑𝔼_x(f(Y_k)1_τ > n) = k=0n-1∑ A_k,n +k=0n-1∑ B_k,n +k=0n-1∑ C_k,n +k=0n-1∑ D_k,ni) Study of k=0n-1∑ A_k,nFor any n ∈^*,k=0n-1∑ A_k,n =l=0T-1∑m=0T-1∑(k=0n-1∑λ_l^k λ_m^n-k)w_l(x)<v_l,g_m(f)><v_m,> = l=0T-1∑ n λ_l^n w_l(x) <v_l,g_l(f)><v_l,> + lm∑λ_m(λ_l^n - λ_m^n/λ_l-λ_m) w_l(x)<v_l,g_m(f)><v_m,>On the one side,l=0T-1∑ n λ_l^n w_l(x) <v_l,g_l(f)><v_l,> = nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,>On the other side, for any 0 ≤ lm ≤ T-1,λ_m(λ_l^n - λ_m^n/λ_l-λ_m)= ρe^-2imπ/T(ρ^ne^-2inlπ/T -ρ^ne^-2inmπ/T/ρe^-2ilπ/T-ρe^-2imπ/T) = ρ^ne^-2imπ/T(e^-2inlπ/T -e^-2inmπ/T/e^-2ilπ/T-e^-2imπ/T)(e^-2imπ/T(e^-2inlπ/T -e^-2inmπ/T/e^-2ilπ/T-e^-2imπ/T))_n ∈ is bounded, hence 1/n× e^-2imπ/T(e^-2inlπ/T -e^-2inmπ/T/e^-2ilπ/T-e^-2imπ/T) n →∞⟶ 0We deduce that, for any 0 ≤ lm ≤ T-1,ρ^ne^-2imπ/T(e^-2inlπ/T -e^-2inmπ/T/e^-2ilπ/T-e^-2imπ/T) = o(nρ^n)and thereforelm∑λ_m(λ_l^n - λ_m^n/λ_l-λ_m) w_l(x)<v_l,g_m(f)><v_m,> = o(nρ^n)since this is a finite sum. Hencek=0n-1∑ A_k,n =nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,> + o(nρ^n)ii) Study of k=0n-1∑ B_k,nFor any y ∈ E, n ∈ and 0 ≤ l ≤ T-1k=1n-1∑λ_l^k w_n-k(f)(y)= f(y) <μ_y (k=0n-1∑λ_l^k (Q')^n-k),> = f(y) <μ_yQ'(λ_l I_K - Q')^-1(λ_l^n I_K-(Q')^n),>where I_K is the K × K-identity matrix. For any 0 ≤ l ≤ T-1 and n ∈,λ_l^n I_k - (Q')^n = ρ^n (e^2iπ nl/T I_k - ρ^-n(Q')^n)and (e^2iπ nl/T I_k - ρ^-n(Q')^n)_n ∈ is bounded since the spectral radius of Q' is smaller than ρ. Hence 1/n(e^2iπ nl/T I_k - ρ^-n(Q')^n) n →∞⟶ 0where 0 is understood as the zero matrix, and we deduce that <μ_yQ'(λ_l I_K - Q')^-1(λ_l^n I_K-(Q')^n),> = o(nρ^n)As a result, for any n ∈,k=1n-1∑λ_l^k w_n-k(f)(y) = o(nρ^n)Hence for any n ∈k=0n-1∑ B_k,n = l=0T-1∑w_l(x) <v_l,k=0n-1∑λ_l^k w_n-k(f)> = o(nρ^n)iii) Study of k=0n-1∑ C_k,nIn the same way as k=0n-1∑ B_k,n,k=0n-1∑ C_k,n = ∑_k=0^n-1∑_m=0^T-1λ_m^n-k <v_m, > <μ_x (Q')^k, g_m(f)> = ∑_m=0^T-1 <v_m,> <μ_x ( ∑_k=0^n-1λ_m^n-k (Q')^k), g_m(f)>For any 0 ≤ m ≤ T-1 and n ≥ 1,∑_k=0^n-1λ_m^n-k (Q')^k = λ_m × (λ_m I_K - Q')^-1 (λ_m^n I_K-(Q')^n)We already showed that for any 0 ≤ m ≤ T-1 and n ≥ 1 (λ_m I_K - Q')^-1 (λ_m^n I_K-(Q')^n) = o(n ρ^n)Finally,k=0n-1∑ C_k,n = o(nρ^n)iv) Study of k=0n-1∑ D_k,nFinally, let us denote by (q')_i,j^(n), for i,j ∈{1, …, K-T} and n ∈, the coefficient of (Q')^n located at the ith row and the jth column. Then for any n ∈k=0n-1∑ D_k,n = i,j,l,m∑μ_x(j) f(i) μ_i(m) (k=0n-1∑(q')_m,l^(n-k)(q')_i,j^(k))Let i,j,l,m ∈{1, …, K}. By definition of the matrix Q', the spectral radius of Q' is strictly smaller than ρ. We deduce from this (q')^(n)_i,j = o(ρ^n), (q')^(n)_m,l = o(ρ^n)In particular there is a positive number C such that for any n ∈ and m,l ∈{1, …, K},(q')^(n-k)_m,l≤ C ρ^n-kHence,k=0n-1∑(q')_m,l^(n-k)(q')_i,j^(k) ≤ Ck=0n-1∑ρ^n-k(q')_i,j^(k)= C n ρ^n (1/nk=1n-1∑ρ^-k(q')_i,j^(k))However, by (<ref>), ρ^-nq_i,j^(n)→ 0 when n tends to infinity and using Cesaro's lemma,1/nk=0n-1∑ρ^-k(q')_i,j^(k)n →∞⟶ 0Thus using (<ref>) and (<ref>), we deduce thatk=0n-1∑ D_k,n = o(nρ^n)Hence, gathering all these results and using (<ref>),k=0n-1∑𝔼_x(f(Y_k)1_τ > n) = nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,> + o(nρ^n)However we have for any l ∈{0, …, T-1}<v_l, g_l(f)>= j=1K∑ f(j) v_l(j) w_l(j)= j=0T-1∑x ∈ C_j∑ f(x) v_l(x) w_l(x)= j=0T-1∑x ∈ C_j∑ f(x) e^-i2π j l/Tν(x) e^i2π j l/Tξ(x)= <v_0, g_0(f)>As a result,_x(k=0n-1∑ f(Y_k) _τ > n) = nρ^n <v_0,g_0(f)> l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,> + o(nρ^n) Now we prove Lemma <ref> quoted in the previous proof.Let us start by proving that α_l is a right-eigenvector associated to λ_l. Since Q is a real matrix, it is equivalent to show that α_l is a right-eigenvector associated to λ_l. First remind that α_l is defined by the relationsδ_k = l=0T-1∑α_l(k) v_l + δ_k'for any k ∈ E_0 and with δ_k' ∈ Span^⊥(v_0, …, v_T-1). This implies for any k <δ_k, v_m> = l=0T-1∑α_l(k) <v_l,v_m>or, in other words,[ <δ_k, v_0>;⋮; <δ_k, v_T-1> ] = [<v_0,v_0>…<v_T-1,v_0>;⋮⋱⋮; <v_0, v_T-1>… <v_T-1, v_T-1> ][ α_0(k);⋮; α_T-1(k) ]Denote by A the matrixA = [<v_0,v_0>…<v_T-1,v_0>;⋮⋱⋮; <v_0, v_T-1>… <v_T-1, v_T-1> ]A is simply the Gram's matrix of the basis (v_i)_0 ≤ i ≤ T-1. Thus the determinant (A) is positive and for any x ∈ E_0α_l(x) = 1/(A) <v_0,v_0>…<δ_x, v_0> …<v_T-1,v_0>⋮ ⋱ ⋮ ⋱ ⋮<v_0,v_T-1>…<δ_x, v_T-1>…<v_T-1,v_T-1>where the column (<δ_x, v_0>, …, <δ_x,v_T-1>)^T is the l-th columns of the matrix. We want to show now that α_l is a right-eigenvector associated to λ_l, that is∀ v ∈^K, <v,Qα_l> = λ_l <v,α_l>In fact it is enough to show (<ref>) when v is one of left-eigenvectors and when v ∈ Span^⊥(v_0, …, v_T-1).In the case where v = v_k for k ∈{0, …, T-1}<v_k, α_l>= j=1K∑v_k(j)1/(A) <v_0, v_0>…<δ_j, v_0>… <v_T-1,v_0>⋮ ⋱ ⋮ ⋱ ⋮<v_0, v_T-1>…<δ_j, v_T-1>…<v_T-1,v_T-1> = 1/(A) <v_0, v_0>…<v_k, v_0>… <v_T-1,v_0>⋮ ⋱ ⋮ ⋱ ⋮<v_0, v_T-1>…<v_k, v_T-1>… <v_T-1,v_T-1> ={[ 1 l=k; 0 ].We deduce from this <v_k, Q α_l> = λ_l <v_k, α_l>, ∀ 0 ≤ k ≤ T-1Finally, for any v ∈ Span(v_0, …, v_T-1)^⊥,<v,α_l> = 1/(A) <v_0, v_0>…0… <v_T-1,v_0>⋮ ⋱ ⋮ ⋱ ⋮<v_0, v_T-1>…0… <v_T-1,v_T-1>=0Thus we have<v,Qα_l> = 0 = λ_l <v_k, α_l>because ^tQv ∈ Span(v_0, …, v_T-1)^⊥. Hence for each k ∈{0, …, T-1}, there is β_k ∈ such that α_k = β_k w_k (where w_k is defined at the beginning of the subsection). We will show that β_k = β_0 = 1 for any k. Remark that A can be written as i=1T∑ a_i-1 P_σ_i where P_σ_i is the permutation matrix of σ_i where σ_i = (i i+1 … i-2 i-1) and a_0 > 0 and a_1, …, a_T-1∈. In other words, A is of the following shapeA = [ a_0 a_1 a_2 … a_T-1; a_T-1 a_0 a_1 … a_T-2; ⋮ ⋮ ⋱ ⋮; a_1 a_2 a_3 … a_0 ]with a_0 > 0 and a_1, …, a_T-1∈^T-1. Moreover, since 1 ∈ C_0, <δ_1, v_l> = <δ_1, v_0> = ν_1 for any l ∈{0, …, T-1}. As a result, for any l ∈{0, …, T-1},(A) α_l(1)=a_0… ν_1… a_T-1 ⋮ ⋱ ⋮ ⋱ ⋮a_1… ν_1… a_1 = ν(1) a_l+1 … … a_l-1 ⋮ ⋮ ⋱ ⋱ ⋮ ν(1) a_l+2 … … a_l = ν(1) a_1 … … a_T-1 ⋮ ⋮ ⋱ ⋱ ⋮ ν(1) a_2 … … a_0 =(A) α_0(1)Indeed, from (<ref>) to (<ref>), we applied a circular permutation for the columns in order to put the vector ^t(ν(1), …, ν(1)) at the first column, and the determinant stays the same after this transformation. From (<ref>) to (<ref>), we did a circular permutation on the rows, which does not affect either the determinant. We deduce from this equality that β_k = β_0 for any k ∈{0, …, T-1} because w_k(1) = w_0(1). Concerning the fact that β_0 = 1, remark that i=1K∑ν(i) α_0(i) = i=1K∑v_0(i) α_0(i)= 1/(A) <v_0,v_0>…<v_T-1,v_0>⋮ ⋱ ⋮<v_0, v_T-1>…<v_T-1, v_T-1>=1And i=1K∑ν(i) α_0(i) = β_0 i=1K∑ν(i) ξ(i) = 1 The statement of Theorem 1 is meaningful provided the coefficient of the leading term ρ^n is not equal to 0. In the following proposition we prove that this coefficient is actually not 0. For any n ∈ and any x l=0T-1∑ e^-2inlπ/T <w_l,δ_x> <v_l,>0Let x ∈ E_0. Then there exists k ∈{0, …, T-1} such that x ∈ C_k. Thus, for any n ∈,l=0T-1∑ e^-2inlπ/Tw_l(x)<v_l,>= l=0T-1∑ e^-2i(n+k)lπ/Tξ(x)(j=0T-1∑y ∈ C_j∑ e^2iπ l j/Tν(y))=j=0T-1∑y ∈ C_j∑ξ(x) ν(y) ( l=0T-1∑ e^-2iπ (n+k-j)l/T)=TT | n+k-j∑y ∈ C_j ∑ξ(x) ν(y) +T ∤ n+k-j∑y ∈ C_j ∑ξ(x) ν(y) e^iπ(n+k-j)(T-1)/Tsin(π(n+k-j))/sin(π(n+k-j)/T)_=0=TT | n+k-j∑y ∈ C_j∑ξ(x) ν(y) > 0 §.§ Quasi-ergodic distribution for the classical non-moving sense in the general case Now assume that the sub-transition matrix Q is not necessarily irreducible. For each x ∈{1, …, K}, we denote by D_x the subset of {1, …, K} defined byD_x := {y ∈{1, …, K} : ∃ n,m ∈, _x(Y_n = y) > 0and _y(Y_m = x) > 0}It is well-known that (D_x)_x ∈{1, …, K} are equivalence classes. Note that, for each x, the restriction of Y on D_x is irreducible. Thus we can associate, for each D_x, a period T_x.We can also associate to D_x a spectral radius ρ_x and some left and right-eigenvectors (v_x,l)_0 ≤ l ≤ T_x-1 and (w_x,l)_0 ≤ l ≤ T_x-1 constructed in the same way as in the subsection <ref>. Particularly, for every x ∈{1, …, K}, ν_x := v_x,0 and ξ_x := w_x,0 are vectors whose all the components are positive and such that <ν_x,>=<ν_x,ξ_x>=1. We define also, for any x,φ_x : f →j=1|D_x|∑ f(j)ν_x(j) ξ_x(j) where |D_x| is the number of elements in D_x.Now fix μ∈ M_1({1, …, K}). Denote by Supp(μ) the support of μ. Then we can define B = {x ∈{1, …, K }: Supp(μ) ∩ D_x ∅} ρ_max = x ∈ Bmax ρ_x and we define B_max as followsB_max = {x ∈ B : ρ_x = ρ_max}We set the following hypothesis There exists x_max∈{1, …, K} such thatB_max = D_x_maxUnder this hypothesis, the following notation will be usedν_max = ν_x_max ξ_max = ξ_x_max φ_max = φ_x_max In all what follows, we have to keep in mind that the definition of B_max implicitly depends on the initial distribution μ (more precisely on the support of μ). The following statement explains therefore that the quasi-ergodic distribution exists if the Hypothesis <ref> holds. Let μ∈ M_1({1, …, K}). Then, if the Hypothesis <ref> holds, the following convergence holds for any measurable bounded function f :{1, …, K}→,_μ(1/nk=0n-1∑ f(Y_k) | τ > n) n →∞⟶φ_max(f) According to Proposition <ref>, giving the fact that Y is irreducible on each D_x, we have for any x ∈{1, …, K}_x(1/nk=0n-1∑ f(Y_k) _τ > n) = ρ_x^n φ_x(f) l=0T-1∑ e^-2inlπ/T_x<w_x,l,δ_x> <v_x,l,> + o(ρ_x^n)Thus, for any μ∈ M_1(E)_μ(1/nk=0n-1∑ f(Y_k) | τ > n) = j=1K∑μ(j) _j(1/nk=0n-1∑ f(Y_k) _τ > n)/j=1K∑μ(j) _j(τ > n)=j=1K∑μ(j) ρ_j^n φ_j(f) l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(ρ_j^n)/j=1K∑μ(j) ρ_j^n l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(ρ_j^n)=j ∈ B_max∑φ_j(f)μ(j) l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(1)/j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(1)=φ_max(f)j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(1)/j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j<w_j,l,δ_x> <v_j,l,> + o(1)where the Hypothesis <ref> was used for the last equality, implying that φ_j(f) = φ_max(f) for all j ∈_max. Note moreover that this hypothesis is useful only to make this equality right. Then using Proposition <ref>, we can conclude_μ(1/nk=0n-1∑ f(Y_k) | τ > n) n →∞⟶φ_max(f) §.§ Quasi-ergodic distribution with periodic moving killing subsetIn this subsection we are interested in the quasi-ergodicity of the chain X defined in the Introduction considering that the boundaries are moving γ-periodically. We denote by Y = (Y_n)_n ∈ the Markov chain defined on E ×/γ by Y_n = (X_n,n)where n̅ is the residue of n, modulo γ.Y is therefore a Markov chain defined on a finite space state, which is irreducibleif and only if gcd(T(X),γ) = 1, where T(X) is the period of (X_n)_n ∈. If the chain Y is actually irreducible, the associated period is T = LCM(T(X),γ)where LCM(·, ·) refers to the least common multiple.Moreover we haveτ = inf{n ≥ 0 : X_n ∈ A_n} = inf{n ≥ 0 : Y_n ∈∂}with∂ := {(x,k): x ∈ A_k} Remark that ∂ is a non moving killing subset for the chain Y. Thus we can apply Theorem <ref> to the process Y which yields the following theorem Let μ∈ M_1(E_0). Assume that (A_n)_n ∈ is periodic andY defined in (<ref>) satisfies the Hypothesis <ref>. Then, for any measurable bounded function f,_μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶(x,y) ∈ E ×/γ -∑ f(x) ν_max(x,y) ξ_max(x,y)where ν_max and ξ_max are the probability measures defined in (<ref>) and (<ref>) relatively to Y.We can also give the following corollary which requires assumptions on X and (A_n)_n ∈. Assume that (A_n)_n ∈ is γ-periodic and that gcd(T,γ)=1 (where T is the period of X). Then there exists η∈ M_1(E) such that, for any μ∈ M_1(E_0) and any f bounded measurable, _μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶∫ f dηIt is enough to apply Theorem <ref> to the chain Y defined on (<ref>) and to deduce the results on X thanks to the following equality𝔼_μ(1/nk=0n-1∑ f(X_k) | τ > n) = 𝔼_μ⊗δ_0(1/nk=0n-1∑f̃(Y_k) | τ > n), ∀ n ≥ 1where f̃ is the projection on the first component. § EXISTENCE OF Q-PROCESS WITH BOUNDARIES MOVING PERIODICALLYIn this section, we are interested in the Q-process, which can be interpreted as the law of the process X conditioned never to be killed by the moving boundary. As before, we still consider that the boundary moves periodically period γ. To show the existence of the Q-process, we will consider again the Markov chain Y defined in (<ref>), that is defined byY_n = (X_n,n), ∀ n ∈and we take back the notation introduced in subsection <ref> associated to Y. The following statement ensures the existence of a Q-process even when the boundary moves. However, it is interesting to observe that we lose the homogeneity of the Q-process because of the movement of the killing boundary. For any n ∈ and any x ∈ E_0, the probability measure ℚ_x defined byℚ_x(X_1 ∈·, …, X_n ∈·) = m →∞lim_x(X_1 ∈·, …, X_n ∈· | τ > m)is well-defined and, under the probability ℚ_x, (X_n)_n ∈ is a time-inhomogeneous Markov chain such that for any n ∈, for any (y,z) ∈ E_n-1× E_nℚ_x(X_n = z | X_n-1 = y) = ξ_x(z,n)/ρ_x ξ_x(y,n-1)_y(X_1 = z, τ_A_n > 1)For any m,n ∈, for any f_1, …, f_n measurable bounded functions and for any x ∈ E_0,_x(f_1(Y_1) … f_n(Y_n) | τ > n+m)= _x(f_1(Y_1) … f_n(Y_n) _τ > n+m)/_x(τ > n+m)=_x(f_1(Y_1) … f_n(Y_n) _τ > n_Y_n(τ > m)/_x(τ > n+m))According to the equality (<ref>) applied to the function equal to 1, for any y ∈ E × Z/γ - $̣ andn ∈,_y(τ > n) = ρ_y^n l=0T_y-1∑ e^-2inlπ/T_y <w_y,l, δ_y> <v_y,l,> + o(ρ_y^n)Thus, using this in (<ref>),_x(f_1(Y_1) … f_n(Y_n)| τ > m+n) = _x(f_1(Y_1)… f_n(Y_n) _τ > nρ_Y_n^m l=0T_Y_n-1∑ e^-2imlπ/T_Y_n <w_Y_n,l,δ_Y_n> <v_Y_n,l,> + o(ρ_Y_n^m)/ρ_x^n+ml=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n+m)) = _x(f_1(Y_1)… f_n(Y_n) _τ > nρ_x^m l=0T_x-1∑ e^-2imlπ/T_x <w_x,l,δ_Y_n> <v_x,l,> + o(ρ_x^m)/ρ_x^n+ml=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n+m)) = _x(f_1(Y_1)… f_n(Y_n) _τ > nl=0T_x-1∑ e^-2imlπ/T_x <w_x,l,δ_Y_n> <v_x,l,> + o(1)/ρ_x^nl=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n))The passage from (<ref>) to (<ref>) is due to the fact that, for anyn ∈,Y_n ∈D_xalmost surely and the quantitiesT_x,ρ_x,w_x,landv_x,ldepends only onD_x. Since the restriction of the chainYonD_xis irreducible, we can construct as in the subsection <ref> some clusters(C_j)_0 ≤j ≤T_x-1such thatx ∈C_0and_x(Y_k+nT_x∈ C_k, τ > k+nT_x) = 1, ∀ k ∈{0, …, T_x-1}, ∀ n ∈ For anyy ∈D_x, denote byj(y)the integer such thaty ∈C_j(y). Then we deduce from the equality (<ref>) in the subsection <ref> that for anyy ∈E ×/γ- $̣ and n ∈, e^-2inlπ/T_x <w_x,l,δ_y> = e^-2iπ(n+j(y))l/T_xξ_x(y)Thus, according to (<ref>) and the previous equality,_x(f_1(Y_1) … f_n(Y_n)| τ > m+n) = _x(f_1(Y_1)… f_n(Y_n) _τ > nξ_x(Y_n) (l=0T_x-1∑ e^-2iπ(m+j(Y_n))l/T_x<v_x,l,> + o(1))/ρ_x^nξ_x(x) (l=0T_x-1∑ e^-2iπ(m+n+j(x))l/T_x<v_x,l,> + o(1)))However, for any n ∈, j(Y_n) = j(x) + nT_x, a.s.and for any m,n ∈,l=0T_x-1∑ e^-2iπ(m+n+j(x))l/T_x<v_x,l,>0 Since the state space E × / γ is finite, we may first consider function f_i(y) = _y = x_i, so that quantities in the ratio except _τ > n are fixed. This justifies that we can exchange the expectation and the limit as n →∞ in the previous expression. We deduce that,_x(f_1(Y_1) … f_n(Y_n)| τ > m+n) m →∞⟶_x(f_1(Y_1)… f_n(Y_n) _τ > nξ_x(Y_n)/ρ_x^nξ_x(x,0))The statement on X is obtained using projection functions and we can deduce from it the transition kernel of the Q-process. § EXAMPLE : DISCRETE-TIME RANDOM WALK We shall illustrate theprevious results by looking at a discrete-time random walk. Let p ∈ ]0,1[. We denote by (M_n^p)_n ∈ℕ the Markov chain defined onsuch that(M_n+1^p = M_n^p + 1 | M_n^p) = 1-p (M_n+1^p = M_n^p - 1 | M_n^p) = pBefore dealing with the quasi-ergodicity with moving boundaries, let us recall some properties about quasi-stationarity concerning random walks. For any K ≥ 1 we defineT_K = inf{n ≥ 0 : M_n^p ∈ (-∞,0] ∪ [K+1, ∞) }The sub-Markovian transition matrix associate to (M_nT_K^p)_n ∈ is the matrix Q_K∈ M_K(ℝ) defined by :Q_K = [ 0 1-p 0 … 0 0; p 0 1-p … 0 0; 0 p 0 … 0 0; ⋮ ⋮ ⋮ ⋱ ⋮ ⋮; 0 0 0 … 0 1-p; 0 0 0 … p 0 ]For any K ≥ 1, denote by P_K(X) the characteristic polynomial of Q_K. Using standard algebraic manipulations, one can show that for any K ≥ 1, the following recurrence relation is satisfied P_K+2(X) = -X P_K+1(X) - p(1-p) P_K(X)with P_1(X) = -X and P_2(X) = X^2 - p(1-p). We set P_0(X) = 1.For any K ≥ 0, defineU_K(X) = (-1/√(p(1-p)))^K P_K(2√(p(1-p))X)Then the following equation is satisfiedU_K+2(X) = 2X U_K+1(X) - U_K(X)for which U_0(X) = 1 and U_1(X) = 2X. In other words, the sequence (U_K)_K ≥ 0 are the Chebyshev's polynomials of the second kind and we have for any θ∈ℝU_K(cos(θ)) = sin((K+1)θ)/sin(θ)The set of roots of U_K, hence of P_K, is thus well-known. It followsSp(Q_K) = {λ_j := 2√(p(1-p))cos(j π/K+1) : j ∈{1, …, K}}We are interested now in the eigenvectors of Q_K. Let K ≥ 1. Then, for any j ∈{1, …, K}, Ker(Q_K - λ_j I_k) = Span(x_j) wherex_j(i) = (-1/1-p)^i-1 P_i-1(λ_j) = (√(p/1-p))^i-1sin(ijπ/K+1)/sin(jπ/K+1), ∀ i ∈{1, … , K}Let λ∈ Sp(Q_K). We want to find all the eigenvectors x = (x(i))_1 ≤ i ≤ K associated to λ such that x(1)=1. We will prove the proposition by double induction.Base case: According to the relation Q_K x = λ x, we have λ x(1) = (1-p) x(2)Having x(1)=1, we will have therefore x(2) = λ/1-p = -1/1-p P_1(λ), which conclude the base case Inductive step: Let i ∈{3, …, K-1}. We assume that the equality is satisfied for i-1 and i-2, so we have x(i-2) = (-1/1-p)^i-3 P_i-3(λ) x(i-1) = (-1/1-p)^i-2 P_i-2(λ)Using λ x = Q_K x, λ x(i-1) = px(i-2) + (1-p)x(i)Sox(i)= 1/1-p( λ x(i-1) - px(i-2)) =1/1-p( λ(-1/1-p)^i-2 P_i-2(λ) - px(i-2)) = (-1/1-p)^i-1(-λ P_i-2(λ) - p(1-p) P_i-3(λ)) =(-1/1-p)^i-1 P_i-1(λ)which concludes the proof.The previous proposition gives us left and right eigenvectors of Q_K : if we denote by (v_i)_1 ≤ i ≤ K (respectively (w_i)_1 ≤ i ≤ K) the left (respectively right) eigenvectors satisfying v_iQ_K = λ_i v_i (respectively Q_K w_i = λ_i w_i), thenv_i(j) = (√(1-p/p))^j-1sin(ijπ/K+1)/sin(iπ/K+1)w_i(j) = (√(p/1-p))^j-1sin(ijπ/K+1)/sin(iπ/K+1)In particular, considering the spectral radius λ_1, the quasi-stationary distribution ν and the right-eigenvector ξ associated to λ_1 satisfying <ν,ξ>=1 are as follows:ν(j) = (√(1-p/p))^j-1sin(jπ/K+1)/k=1K∑(√(1-p/p))^k-1sin(kπ/K+1)ξ(j) = k=1K∑(√(1-p/p))^k-1sin(kπ/K+1)/k=1K∑sin^2(kπ/K+1)(√(p/1-p))^j-1sin(jπ/K+1)We are interested now in moving boundaries. Let N ≥ 1and consider the simplest case where (A_n)_n ∈ is defined byA_n = {[ (-∞,0] ∪ [2N,∞)if n is even; (-∞,1] ∪ [2N-1,∞) if n is odd ].Recall the previous notation Y^p_n = (M^p_n τ_0,n) with n∈/2. The chain is not irreducible (if M^p_0 is even, then for any n, M^p_n have the same parity as n). It admits exactly two irreducible subsets: * P = {(x,y) ∈ E : x+yis even}* I = {(x,y) ∈ E : x+yis odd}But, as we can see in Figure <ref>, the chain Y^p behaves as a random walk on each irreducible subsets: * On P, Y^p has the same behavior as a random walk on ℤ starting from [2,2N-2] absorbed by {1,2N-1}.* On I, Y^p has the same behavior as a random walk on ℤ starting from [1,2N-1] absorbed by {0,2N}.Denote by Y^p_ P (respectively Y^p_ I) the Markov chain such that for any μ∈ M_1( P) (respectively M_1( I))_μ(Y^p_1 ∈·) = _μ((Y^p_ P)_1 ∈·)(respectively _μ(Y^p_1 ∈·) = _μ((Y^p_ I)_1 ∈·) )Let μ∈ M_1(E ×/2). Then there are λ∈ [0,1] and μ_ P, μ_ I∈ M_1( P) × M_1( I) such thatμ = λμ_ P + (1-λ)μ_ IHence we see that two cases are possible* if λ = 1, B_max =P. Then ρ_max = 2√(p(1-p))cos(π/2(N-1)),and_μ(1/nk=0n-1∑ f(M^p_k) | τ > n) n →∞⟶j=22N-2∑ f(j) sin^2((j-1)π/2(N-1))/k=12N-3∑sin^2(kπ/2(N-1)) * if λ 1, B_max =I. Then ρ_max = 2√(p(1-p))cos(π/2N),and_μ(1/nk=0n-1∑ f(M^p_k) | τ > n) n →∞⟶j=12N-1∑ f(j) sin^2(jπ/2N)/k=12N-1∑sin^2(kπ/2N) | http://arxiv.org/abs/1707.08419v4 | {
"authors": [
"William Oçafrain"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170726130858",
"title": "Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically"
} |
Lecture Notes in Computer Science Authors' Instructions The Internet of Hackable ThingsNicola Dragoni1,2, Alberto Giaretta2 and Manuel Mazzara3 DTU Compute, Technical University of Denmark, Denmark Centre for Applied Autonomous Sensor Systems, Örebro University, Sweden Innopolis University, Russian Federation December 30, 2023 ==================================================================================================================================================================================================================================== The Internet of Things makes possible to connect each everyday object to the Internet, making computing pervasive like never before. From a security and privacy perspective, this tsunami of connectivity represents a disaster, which makes each object remotely hackable. We claim that, in order to tackle this issue, we need to address a new challenge in security: education.§ THE IOT TSUNAMIIn the last decade, we all have witnessed a turmoil of interest around the Internet of Things (IoT) paradigm. It has been claimed that such a paradigm may revolution our daily lives and pervasive applications are behind the corner both in the civil and military complex. Such a strong hype on pervasive technologies requires a step back to consider the potential threat on security and privacy. First of all, What exactly is the IoT? Accordingly to the Online Oxford Dictionary it is the “interconnection via the Internet of computing devices embedded in everyday objects, enabling them to send and receiving data”. To get a grasp of the dimension of this phenomenon, according to Evans Data Corporation the estimated population of IoT devices in June 2016 was 6.2 billion <cit.>, number that according to several predictions will grow as up as 20 billion in 2020 <cit.>. Projections and data are not so straightforward to analyse since some firms take into account devices like smartphones, while others do not count them, therefore it is quite hard to make comparisons. Nonetheless, the growing trend is confirmed by every analyst, to the point that by 2025 the IoT market could be worth $3.9 trillion to $11 trillion per year <cit.>. On the academic front, this ongoing excitement and interest in all the IoT world has given rise to an increasing number of related conferences, research projects andresearch centres (like the recently formed IoT Center in Denmark, <http://iotcenter.dk>). As a matter of fact, even though IoT refers to an ample variety of different devices, these devices all share a common architecture. First of all, any IoT device usually connects to the Internet through a more powerful gateway, which could be a smartphone or a tablet. Then data flow is elaborated by (and eventually hosted into) the cloud, enabling the end user to remotely connect to the device and control it. Figure <ref> shows how this IoT architecture looks like in a generic scenario.IoT applications span from industrial automation to home area networks and personal (body) area networks. In particular, Smart homes will heavily rely upon IoT devices to monitor the house temperature, eventual gas leakages, malicious intrusions and several other parameters concerning the house and its inhabitants. Another growing area of interest is represented by pervasive healthcare applications, which use IoT devices to perform continuous biological monitoring, drug administration, elderly monitoring and so on. Last, but not least, in the recent years wearable devices gained a huge popularity (e.g., fitness trackers), to the point that in the span of just a year sales grew 18.4% in 2016 <cit.>. §.§ A Security and Privacy Disaster From a security perspective, this ongoing excitement for IoT is having tremendous consequences, so that it's not an exaggeration to talk about a security and privacy disaster. Indeed, if the fundamental IoT axiom states that “everything can be connected to the Internet (becoming, in this way, an IoT device)”, its security corollary is somehow catastrophic:“everything that can be connected to the Internet can be hacked" <cit.>. This is particularly critical if we consider that, by means of the various kinds of devices connected to the Internet, people are sharing more and more information about themselves, often without being aware of that. This means that the amount of data available online is going to increase unrelentingly, literally given away to cybercriminal eager to take control of our devices, and thus of our life. In the early days of the “IoT shift”, researchers highlighted how much critical security would be in a real IoT context <cit.> and gave some hints about what should be done to defend our devices and our privacy. This message has clearly not been listened.To put things in perspective, in July 2014 HP Security Research <cit.> analysed 10 of the most popular IoT devices on the market revealing a generally alarming situation: * 90% of devices collected at least some information via the device;* 80% of devices, along with their cloud and mobile components, did non require a password complex enough;* 70% of devices, along with their cloud and mobile components, enabled an attacker to identify valid user accounts through enumeration;* 70% of devices used unencrypted network services;* 6 out of 10 devices that provided user interfaces were vulnerable to a range of weaknesses, such as persistent XSS[Cross site scripting (XSS) is an attack that injects malicious code into a Web application.] and weak credentials.To make matters worse, security in a IoT scenario is even harder than expected for a number of reasons <cit.>, such as: * It implies complex and distributed systems, with a huge variety of different operating sistems, programming languages and hardware;* Even developing a simple application for a IoT device can be non-trivial;* Securing the applications is even less easy, because the attack surface is enormous (any device could be a possible entry point) and defining beforehand all the potential threats is extremely challenging;* The contained data are sensitive and highly valuable for the market, nowadays, which entails huge potential gains for any successful attacker and high attractiveness. Given that providing security for the IoT is still a really hard thing to do, the atavistic problem with exciting new technologies is that companies are in a hurry and most of them ignore quite at all any kind of security issues, postponing the matter as much as possible. Just to give some numbers, Capgemini Consulting in 2015 highlighted some critical aspects <cit.>, such as: * Only 48% of organizations focus on security of their devices from the beginning of the development phase;* Only 49% of organizations provide remote updates for their devices;* Only 20% hire IoT security experts;* Only 35% invite third parties (like hackers) to identify vulnerabilities in their devices. As a rule of thumb, we could depict the prevalent approach of manufacturers to IoT security with the following “insecurity practice” rule <cit.>: Development Rush + Hard to Develop ⇒Skip (or Postpone) Security At this point it should be quite easy to detect the reasons why hackers actually love the on-going IoT outburst. In the following Sections, we will show plenty of examples about this vast attention, with focus on two of the most promising IoT contexts: smart homes (Section <ref>) and pervasive healthcare (Section <ref>).§ SMART HOME... OF HORROR! Smart homes and, in general, smart buildings are one of the current trends for IoT devices, and probably the most active one.Our team is also currently engaged in a project on microservice-based IoT for smart buildings <cit.>. Everyday things are being transformed into much more powerful and smart objects, in order to meet customers' increasing needs. But availability of connected things could come with a high price in terms of privacy and security issues, in light of the fact that at the present moment too many things are too easily hackable.Few years ago some irons imported from China included a wireless chip that was able to spread viruses by connecting to unprotected Wi-Fi networks, while some other hidden chips were able to use companies networks to spread spam on the Internet. Researchers achieved to hack the remote firmware update of a Canon Pixma printer, which makes possible to do funny things, like installing an old-school videogame such as Doom, and not so funny other ones, like installing a crippling malware that could even force the device to destroy itself.Smart light bulbs, which enable the owners to remotely control and adjust their home light through an app or a web interface, are another fitting example of IoT devices. Some of these bulbs, such as the popular Philips Hues, have been compromised and researchers showed how easy is to set up a car, or even a drone, that drives in a residential area aiming to infect as much bulbs as possible with a crippling malware. This malware is able to shut them down or even force them to flicker on and off at desired speed <cit.>.Smart TVs sales are constantly growing all over the world. Smart TVs provide a combination of a traditional TV and a Internet-connected personal computer, blending the two worlds into a single device. Usually these devices are equipped with various components, such as microphones and webcams, aiming to give the user the fullest experience possible. Clearly enough, if security is badly managed in these kind of devices, hackers could easily eavesdrop and peek at our lives without us even noticing that. An attack that could likely be struck is a HTML5 browser-based attack, therefore the devices resilience should always be assessed by using some penetration testing frameworks, such as BeEF <cit.>.Talking about spying, there are other devices that have been hacked with the specific intent to gather information about us. For instance, baby monitors are very unsafe devices, since that manufacturers generally equip them with default passwords easily guessable by attackers, passwords that usually are never changed by the customers. New York's Department of Consumer Affairs (DCA) issued a public statement <cit.> to inform people about the issue, even reporting that some parents walked in their child's room and heard some stranger speaking to them down the monitor.Another perfect candidate to become a common IoT device in our smart home is the thermostat. Being able to remotely choose and monitor our house temperature can greatly benefit our wellness and comfort. Nonetheless, issues can arise too as shown by researchers at Black Hat USA, which demonstrated that a Nest thermostat (a popular device in the USA) could be hacked in less than 15 second if physically accessible by a hacker. The violated thermostat could be used to spy the residents, steal credentials and even infect other appliances. Recently, other researchers made a proof-of-concept ransomware that could remotely infect the aforementioned thermostat and shut down the heating, until the victim gives in to blackmail <cit.>. Similar vulnerabilities have been found in many other smart home devices, where connectivity has been “embedded” in the device without considering anysecurity protection. Even more serious is the threat posed by the lack of security in top-selling home alarm systems, which unveiled weaknesses are critical to such an extent that a malicious attacker could easily control the whole system, suppressing the alarms or creating multiple false alarms. In fact, some of these systems do not encrypt nor authenticate the signals sent from the sensors to the control panel, easily enabling a third party to manipulate the data flow. Life-threatening vulnerabilities have been found even in smart cars. Security researchers at Keen Security Lab were able to hack a Tesla Model S, achieving to disrupt from a distance of 12 miles various electronically controlled features of the car, such as the brakes, the door locks and the dashboard computer screen <cit.>.Last but not least, we have seen a proliferation of wearable health trackers in the last couple of years. In order to provide the user its monitoring features, a fitness tracker is an embedded system which collects sensitive data about the wearer and communicates it to a mobile application by means of a Bluetooth Low Energy (BLE) protocol, hence enabling the user to access the gathered information. Moreover, nowadays most of the mobile applications sync the collected data to a cloud service, whenever an Internet connection is available (see Figure <ref>). Researchers conducted some deeper investigations about this whole system <cit.>, evaluating the security of the implemented protocols in two of the most popular fitness trackers on the market. The research highlighted how vulnerable these devices are to several kinds of attacks, from Denial of Service (DoS) attacks that can prevent the devices from correctly working, to Man-In-The-Middle (MITM) attacks based on two fake certificates <cit.> resulting in a disclosure of sensitive data. Worryingly, the implemented attacks can be struck by any consumer-level device equipped with just bluetooth and Wi-Fi capabilities (no advanced hacking tools have been required).If you think that escaping from a hacked smart home to find some peace in a hotel room is a temporary solution, well you might be wrong. Recently, guests of a top-level hotel in Austria were locked in or out of their rooms because of a ransomware that hit the hotel's IT system. The hotel had no choice left except paying the attackers.§ PERVASIVE HEALTHCAREIf the so-far depicted Smart Home scenario is already scary, things can even get worse when we look at the pervasive healthcare context, for example the the infrastructure to support elderlies developed by our team <cit.>. Indeed, when we talk about security in healthcare we inherently talk about safety, since malfunctioning, attacks and lack of service could endanger many lives, as we will show in the following.§.§ eHealth: How to Remotely Get Big Data Duo Security highlighted how security is badly managed in healthcare corporations, showing that the density of Windows XP computers is 4 times greater than the density of machines running the same OS found, for instance, in finance. Given that Microsoft ended the support to Windows XP since 2014, this means that an enormous quantity of devices has not been updated for 2 years, at least. Not only obsolete operating systems, even additional (and most of the times, useless) software can become a problem: many healthcare endpoints and healthcare customers' terminals have Flash and Java installed, entailing a huge risk of vulnerabilities exploitation.To get an idea of how much valuable eHealth data is, and consequently how critical the related security is, the global information service Experian estimated that on the black market health records are worth up to 10 times more than credit card numbers. Particularly, a single eHealth record (which comprises social security number, address, kids, jobs and so on) can be priced as high as $500.For the sake of clarity, we are definitely talking about risks which are far from theoretical: healthcare industry suffers estimated costs of $5.6 billion per single year because of data thefts and systems malfunctioning. According to <cit.>, in February 2015 78.8 million of Anthem customers were hacked. In the same year, according to the Office of Civil Rights (OCR), more than 113 million medical records were compromised. Earlier last year Melbourne Health's networks got infected with a malware capable of keylogging and stealing passwords. In February 2016 Hollywood Presbyterian Medical Centre was struck with a devastating ransomware, conveyed by simple Word document in an email attachment. The most recent demonstration of hackers interest about eHealth data is a massive sale of patients records on the dark web, where more than 650.000 tags were auctioned off to the highest bidder.What strikes the most is that we are dealing with a huge amount of data weakly defended, easily accessible and highly valuable to malicious third parties. People tend to link security to tangible money stored in bank accounts, but we've witnessed a radical shift about what's valuable in the black market, in the last decade. Hackers do not just want our credit cards, they want the patterns of our life.§.§ IoT Medical Devices: How to Remotely Kill You The IoT revolution is particularly relevant for a number of healthcare fields of application, since networked devices make possible to monitor and deliver necessary treatments to any remote patient, meaning that day-to-day and even life-saving procedures can be promptly performed. Nowadays, devices like insulin pumps, cochlear implants and cardiac defibrillators are used on a daily basis to deliver remote assistance to a lot of patients. Furthermore, in the last years bigger devices like blood refrigeration units, CT scan systems and X-ray systems are connected to the Internet, in order to check remotely their operational state and make whatever adjustment is needed (e.g., lower the blood unit inside temperature).Keeping in mind that, as we stated in Section <ref>, when something is connected to the Internet it is inherently not secure, the other side of the coin is that the IoT-based healthcare exposes the aforementioned life-saving procedures to the public domain. Therefore, this exposure entails that “if it isn't secure, it isn't safe" <cit.>. For the sake of clarity, Capgemini Consulting conducted an investigation in February 2015 <cit.> where firms executives were asked about the resilience of IoT products in general, in their own opinion. Results shown in Figure <ref> show that medical devices are critically at the bottom of the survey, with only a 10% of executives that believe that IoT devices are highly resilient to cybercriminals. Indeed, various life-threatening vulnerabilities have been found in a number of IoT devices. At least 5 models of intravenous drug pumps manufactured by Hospira, an Illinois firm that administers more than 400.000 devices all over the world, recently showed critical vulnerabilities that could allow a malicious attacker to alter the amount of drugs delivery to patients. Medtronic, one of the world's largest standalone medical technology development company, manufactures an insulin pump that enables patients to autonomously manage their blood glucose levels; sadly, the system does not encrypt the commands sent to the pumps by patients, nor do authenticate the legitimacy of the user. Such an uncontrolled system means that unauthorized third parties could intercept a legitimate command and replace it, delivering a deadly insulin dose to the patient. Some companies that produce Implantable Cardioverter Defibrillators (ICDs), used to deliver shocks to patients going into cardiac arrest, use a Bluetooth stack to test their devices after the first implantation, but they use default and weak passwords which makes their product easily hackable. Similar problems have been found in blood refrigeration units, protected only by a hardcoded password that could be deciphered by malicious attackers and used to alter the refrigeration unit temperature, consequently wrecking the blood provision. Another attack could be struck by targeting CT scanning equipments and altering the radiation exposure limits, killing a patient by administering a huge amount of radiation. Even some X-ray systems have been proved to be vulnerable, as they do not provide any kind of authentication when patients' X-rays are backed up in centralized storage units, nor log who views the images.Bad security can be as dangerous as lack of security, as seen in May 2016 when Merge Hemo, a medical equipment used to supervise hearth catheterization procedures, crashed due to a scan triggered by the antivirus software installed: installing antivirus e antimalware software is not only insufficient, sometimes it can even be hazardous if superficially done.§ ON THE NEED OF DEVELOPING A SECURITY CULTUREToday technology is so sophisticated that counteracting outside threats requires a high level of knowledge and a vast set of skills. This becomes even more challenging if security is mostly unheeded as it happens today, treated as a postponable aspect of a product instead than a inherent and essential trait. And while firms struggle to keep on track, hackers keep on gaining competence and resources: as an example, ransomware victims receive easy and detailed instructions on how to unlock their devices, and sometimes hackers themselves provide 24/7 call centres, in case their targets should run into any kind of technical difficulty. Shockingly, the support victims get from hackers is better than the support they get from their own Internet Service Provider.So, what are the recommendations that should be followed in designing more secure IoT devices? How can we mitigate, if not solving, this security and privacy disaster? We believe that, to provide an answer, we first need to step back to the basic question: what is the nature of the problem? Is it technological? Rephrasing, do we have a lack of proper technology to protect IoT systems? Do we need new security solutions? Our (probably provocative) answer is no, we do not need technological innovation. Or better, of course we do need that, as we do need government regulation, but these are not the priority. The priority is instead education. Indeed, what we actually miss is to develop an effective security culture, raising the levels of awareness and understanding of the cyber risk and embedding “security-aware” values and behaviours in our everyday life. Security and trust are indeed also matter of education and method. For example, in social networks algorithm to compute users trust exist <cit.>, still people need to rely on their own experience and understanding and should not blindly follow computer suggestions. It is the integration of human understanding and algorithms that always offer the best solutions. To support the above argument, consider all the examples of IoT devices mentioned in this paper (a summary is given in Table <ref>). It is noteworthy to highlight that all the described vulnerabilities have the common characteristic of being possible thanks to the naive approach that manufactures adopted in the design phase of their products, approach that clearly shows how security is merely sketched out or even neglected at all. Following basic and well known security practices, it would have been possible to protect these devices against all those cyber-attacks. This is something extremely important to understand. For instance, just to provide another example supporting our argument, let us consider the Mirai malware that operated in October 2016, achieving the largest Distributed Denial of Service (DDoS) attack ever, approximately hitting the targets with 1.2 Tbps of requests <cit.>. Mirai simply scans the Internet, looking for vulnerable IoT devices to attack with a simple dictionary approach and, once that access is gained, the device becomes a bot of a huge network ready to strike a massive DDoS attack. Noticeably, the dictionary used by Mirai is filled with a tiny number of entries, around 50 combinations of username/password, which gives an idea of how little effort is put by firms into designing security for their IoT devices, at the present moment. Again, what was the key issue making this huge cyber attack possible? Was it a lack of technological innovation, for instance a stronger authentication mechanism? Or a lack of a basic security culture, so that we do not apply the technology we already have and that could actually solve most of nowadays security vulnerabilities? Security best practices recommend that a detailed risk analysis should be done, in order to have a clear view of what are the actual cyber threats and consequently choose the right approach to secure the devices. Moreover, device security should be designed as an essential part of the product lifecycle and not as a one-time issue. Once that the right path has been chosen for the new products, already existing devices should be thoroughly tested, following a fairly simple schedule like: automated scanning of web interfaces, reviewing of network traffic, reviewing the need of physical ports (e.g., USB ports), reviewing authentication and authorization processes, reviewing the interaction of devices with cloud and mobile application counterparts (an example for health trackers is given in <cit.>).In the end, what we have learned by this excursus is that the main problem and concern with IoT security is that a security culture is nearly non-existent in our society. It should sound obvious that the more the technology develops and becomes pervasive in our lives, the more the security awareness should be growing. But this is not happening, or it is happening at a too slow pace. Indeed, while the concept of “computing” has rapidly and significantly evolved in the last decades (from mainframes to personal computing to mobile and then pervasive computing), the development of security has not followed the same evolution. Nowadays, kids are able to use almost any mobile device like smart phones, laptops, tablets, wearable devices and so on. On the other hand, they have no concept of “security” or “privacy”. With the explosion of IoT, computing has become pervasive like never before. It's time that also security becomes so pervasive, starting from the development of a new security culture. This is surely a long term goal that has several dimensions: developers must be educated to adopt the best practices for securing their IoT devices within the particular application domain; the general public must be educated to take security seriously, too, which among other things will fix the problem of not changing default password. This education effort, however, will surely need the support of both innovation and government regulations, in order to enforce security when education is not enough. We are strongly convinced that education is the key to tackle a significant number of today IoT security flaws. Therefore, if we raise the levels of cyber risks understanding, both in the corporations and in the general end-users, maybe what future holds would not look as daunting as it looks today. We call the research community to this new exciting challenge.1E16 Press Releasem, Thirty-four Percent Rise in IoT Development, June 22, 2016, <https://evansdata.com/press/viewRelease.php?pressID=237>.GartIoT15 Press Release, Gartner, November 10, 2015, <http://www.gartner.com/newsroom/id/3165317>.M15 J. 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Weingarten, “IoT Goes Nuclear: Creating a ZigBee Chain Reaction," 2016, <http://iotworm.eyalro.net/iotworm.pdf>.BeEF BeEF, The Browser Exploitation Framework, <http://beefproject.com/>.DCA Consumer Alert: Consumer Affairs Warns Parents to Secure Video Baby Monitors, January 2016, <http://www1.nyc.gov/site/dca/media/pr012716.page>.Thermostat Thermostat Ransomware: a Lesson in IoT Security, <https://www.pentestpartners.com/blog/thermostat-ransomware-a-lesson-in-iot-security/>Tesla Keen Security Lab of Tencent, “Car Hacking Research: Remote Attack Tesla Motors”, <http://keenlab.tencent.com/en/2016/09/19/Keen-Security-Lab-of-Tencent-Car-Hacking-Research-Remote-Attack-to-Tesla-Cars/>GDS16 Rohit Goyal, Nicola Dragoni, and Angelo Spognardi. 2016. “Mind the tracker you wear: a security analysis of wearable health trackers”. In Proceedings of the 31st Annual ACM Symposium on Applied Computing (SAC '16). ACM, New York, NY, USA, 131-136.CDG13 M. Conti, N. Dragoni, and S. Gottardo, “MITHYS: Mind the hand you shake - Protecting mobile devices from SSL usage vulnerabilities,” in Security and Trust Management. New York, NY, USA: Springer-Verlag, 2013.Nalin2016 M. Nalin, I. Baroni, and M. Mazzara, “A Holistic Infrastructure to Support Elderlies' Independent Living”, In Encyclopedia of E-Health and Telemedicine, Chapter: 46, Publisher: IGI-Global, Editors: Maria Manuela Cruz-Cunha, Isabel Maria Miranda, Ricardo Martinho, Rui Rijo, pp.591-605.Anthem Anna Wilde Mathews, “Anthem: Hacked Database Included 78.8 Million People”, The Wall Street Journal, February 24, 2015, <https://www.wsj.com/articles/anthem-hacked-database-included-78-8-million-people-1424807364>NB16 K. Netkachova and R. E. Bloomfield, “Security-Informed Safety" in IEEE Computer, vol. 49, no. 6, pp. 98-102, June 2016.MBGDMQN2013 M. Mazzara, L. Biselli, P.P. Greco, N. Dragoni, A. Marraffa, N. Qamar and S. de Nicola, “Social Networks and Collective Intelligence: A Return to the Agora", In Social Network Engineering for Secure Web Data and Services, Publisher: IGI-Global, Editors: Luca Caviglione, Mauro Coccoli and Alessio Merlo, 2013.INSERT17 M. De Donno, N. Dragoni, A. Giaretta, A. Spognardi, “Analysis of DDoS-Capable IoT Malwares”, In Proceedings of 1st International Conference on Security, Privacy, and Trust (INSERT), 2017, IEEE. | http://arxiv.org/abs/1707.08380v1 | {
"authors": [
"Nicola Dragoni",
"Alberto Giaretta",
"Manuel Mazzara"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170726112123",
"title": "The Internet of Hackable Things"
} |
Économétrie & Machine LearningArthur CharpentierUniversité de Rennes 1 & CREM7 Place Hoche, 35065 Rennes Cedex, [email protected] Emmanuel FlachaireAix-Marseille Université, AMSE, CNRS & EHESS5 bd Maurice Bourdet, CS 50498, 13205 Marseille Cedex 01, [email protected] et Antoine LyUniversité Paris-Est5, boulevard Descartes, 77454 Marne-la-Vallée cedex, [email protected] L'économétrie et l'apprentissage machine semblent avoir une finalité en commun: construire un modèle prédictif, pour une variable d'intérêt, à l'aide de variables explicatives (ou features). Pourtant, ces deux champs se sont développés en parallèle, créant ainsi deux cultures différentes, pour paraphraser Breiman. Le premier visait à construire des modèles probabilistes permettant de décrire des phénomèmes économiques. Le second utilise des algorithmes qui vont apprendre de leurs erreurs, dans le but, le plus souvent de classer (des sons, des images, etc). Or récemment, les modèles d'apprentissage se sont montrés plus efficaces que les techniques économétriques traditionnelles (avec comme prix à payer un moindre pouvoir explicatif), et surtout, ils arrivent à gérer des données beaucoup plus volumineuses. Dans ce contexte, il devient nécessaire que les économètres comprennent ce que sont ces deux cultures, ce qui les oppose et surtout ce qui les rapproche, afin de s'approprier des outils développés par la communauté de l'apprentissage statistique, pour les intégrer dans des modèles économétriques.JEL Code: C18; C52; C55Key-words: apprentissage; données massives; économétrie; modélisation; moindres carrés;Mars2018 § INTRODUCTION L'utilisation de techniques quantitatives en économie remonte probablement au 16ème siècle, comme le montre Morgan. Mais il faudra attendre le début du XXième siècle pour que le terme « économétrie » soit utilisé pour la première fois, donnant naissance à l'Econometric Society en 1933. Les techniques de machine learning (apprentissage machine) sont plus récentes. C'est à Arthur Samuel, considéré comme le père du premier programme d'auto-apprentissage, que l'on doit le terme « machine learning » qu'il définit comme « a field of study that gives computer the ability without being explicitly programmed ». Parmi les premières techniques, on peut penser à la théorie des assemblées de neurones proposée dans Hebb (qui donnera naissance au perceptron dans les années 1950, puis aux réseaux de neurones) dont Widrow montreront quinze ans plus tard les liens avec les méthodes des moindres carrés, aux SVM (support vector machine) et plus récemment aux méthodes de boosting. Si les deux communautés ont grandi en parallèle, les données massives imposent de créer des passerelles entre les deux approches, en rapprochant les « deux cultures » évoquées par Breiman, opposant la statistique mathématique (que l'on peut rapprocher de l'économétrie traditionnelle, comme le note Aldrich) à la statistique computationnelle, et à l'apprentissage machine de manière générale. §.§ La Modélisation économétrique L'économétrie et les techniques d'apprentissage statistique supervisé sont proches, tout en étant très différentes. Proches au départ, car toutes les deux utilisent une base (ou un tableau) de données, c'est à dire des observations { (y_i,_i) }, avec i=1,⋯,n, _i∈𝒳⊂ℝ^p et y_i∈𝒴. Si y_i est qualitative, on parlera d'un problème de classification[Nous utiliserons ici le terme « classification » lorsque 𝒴 est un ensemble de classes, typiquement une classification binaire, 𝒴={0,1}, ce cas correspondant à la réalisation d'une variable indicatrice, 1_Y_t≤ 0, ou 1_Y∈𝒜, par exemple. Ce terme est moins daté que « discrimination » par exemple, et plus général que la constitution d'un « score » (qui est souvent une étape intermédiaire). Il ne doit pas être confondu avec la classification non-supervisée (comme la « classification ascendante hiérarchique ») qui est la constitution de classe homogène à partir d'une mesure de similarité (on utilisera parfois, dans ce cas, le terme de « constitution de classes », ou de « clusters »).], et dans le cas contraire, d'un problème de régression. Proches à l'arrivée, car dans les deux cas, on cherche à construire un « modèle », c'est à dire une fonction m:𝒳↦𝒴 qui sera interprétée comme une prévision.Mais entre le départ et l'arrivée, il existe de réelles différences. Historiquement, les modèles économétriques s'appuient sur une théorie économique, avec le plus souvent des modèles paramétriques. On a alors recours aux outils classiques de l'inférence statistique (comme le maximum de vraisemblance, ou la méthode des moments) pour estimer les valeurs d'un vecteur de paramètres , dans un modèle paramétrique m_(·), par une valeur . Comme en statistique, avoir des estimateurs sans biais est important car on peut quantifier une borne inférieure pour la variance (borne de Cramér-Rao). La théorie asymptotique joue alors un rôle important (développements de Taylor, loi des grands nombres, et théorème central limite). En apprentissage statistique, en revanche, on construit souvent des modèles non-paramétriques, reposant presque exclusivement sur les données (i.e. sans hypothèse de distribution), et les méta-paramètres utilisés (profondeur de l'arbre, paramètre de pénalisation, etc) sont optimisés par validation croisée.Au delà des fondements, si l'économétrie étudie abondamment les propriétés (souvent asymptotiques) de(vu comme une variable aléatoire, grâce à la représentation stochastique sous-jacente), l'apprentissage statistique s'intéresse davantage aux propriétés du modèle optimalm^⋆(·) (suivant un critère qui reste à définir), voire simplement m^⋆(x_i) pour quelques observations i jugées d'intérêt (par exemple dans une population de test). Le problème de choix de modèle est aussi vu sous un angle assez différent. Suivant la loi de Goodhart (« si une mesure devient un objectif, elle cesse d'être une mesure »), les économètres utilisent des critères de type AIC ou BIC pour choisir un modèle optimal (pénalisant la qualité d'ajustement d'un modèle par sa complexité, ex-post, lors de la phase de validation ou de choix), alors qu'en apprentissage statistique, c'est la fonction objectif qui tiendra compte d'une pénalisation, comme pour le lasso, ressemblant à une forme de pénalisation ex-ante. §.§ Applications Avant de revenir sommairement sur l'évolution des modèles économétriques, c'est à Francis Galton que l'on doit le terme « régression », comme le rappelle KoenkerGalton. Si le terme est parfois devenu synonyme de « modèle économétrique », il avait été introduit dans le contexte de « regression towards mediocraty inhereditary stature », pour reprendre le titre de l'article paru en 1886. Galton utilisait un modèle linéaire pour modéliser la taille moyenne d'un garçon (à l'âge adulte) en fonction de la taille de son père. Si cette technique de régression était connue par les économistes, il a fallu attendre les années 1930 pour voir surgir le concept de « modèle » économique. Comme le note Debreu, la première étape a été de formuler des affirmations économiques dans un language mathématique. Les différentes grandeurs sont vues comme des variables, et dans les années 1930, on verra apparaître les « statistical demand curves », pour reprendre la terminologie d'Henry Schultz. Cette approche statistique permettra d'aller plus loin que les travaux pionners de Engel qui étudiait empiriquement la relation entre la consommation et le revenu des ménages, par exemple, dans une approche uniquement descriptive.Les modèles économétriques se sont développés en parallèle des modèles macro-économiques. Les premiers travaux de la Commission Cowles ont porté sur l'identificationdes modèles économiques, et l'estimation de modèles à équations simultanées. Ces développements vont aboutir à un âge d'or de l'économétrie, dans les années 1960, où les modèles économétriques seront utilisés afin d'améliorer les prévisions macroéconomiques. On va alors voir apparaître tout un ensemble de « lois » qui sont souvent traduites comme des relations linéaires entre des grandeurs agrégées, telle que la « loi de Okun » introduite dans Okun qui postule une relation linéaire entre le variation du nombre de demandeurs d'emploi et de la croissance du PIB,ΔChômage_t=β_0+β_1Croissance_t+ε_t,quand on étudie ces grandeurs au cours du temps (t), ou la loi de « Feldstein-Horioka » introduite dans FH qui suppose une relation linéaire entre les taux d'investissement et d'épargne, relativement au revenu national,investissement_i/revenu national_i=β_0+β_1 épargne_i/revenu national_i+ε_iquand on modèlise les liens entre les allocations investissement-épargne pour plusieurs pays (i). Cet âge d'or correspond aussi à un questionnement profond, suite à la critique de Lucas, s'interrogeant sur l'inefficacité de ces outils à expliquer et à prévoir des crises. La principale explication était alors le manque de fondement micro-économiques de ces modèles, ce qui donnera un second souffle aux modèles micro-économétriques. On pourra rappeler que cette critique dite « de Lucas » avait été formulée dans Orcutt, qui avançait l'idée que les données macroéconomiques posaient des problèmes insolubles d'identification. La solution passait forcément par de l'économétrie sur données individuelles (au lieu de données obtenues par aggrégation), ce qui sera reformulé quelques années plus tard par Koopmans. Malheureusement, les modèles micro-économétriques sont généralement plus complexes, car ils se doivent de tenir compte d'une éventuelle censure dans les données, avec par exemple le modèle introduit par Tobin, d'erreurs sur les variables (qui pourront être corrigées par des instruments avec une technique initiée par Reiersol) ou avoir été collectée avec un biais de sélection, avec les techniques proposées par Heckman. On notera que les économètres se sont beaucoup interrogés sur la manière dont les données étaient construites, et ne se sont jamais contentés de « construire des modèles ». Un exemple peut être l'évaluation des politiques publiques, largement détaillé dans Givord. Dans ce cas, en effet, deux écoles se sont opposées (initiant un débat que l'on verra resurgir tout au long de l'article sur les méthodes d'apprentissage statistique). La première, dite « structuraliste », cherchera à construire un modèle complet afin de décrire le comportement des agents économiques. La seconde, souvent qualifiée d'« empiriste », vise à tester l'effet d'une mesure sans pour autant expliciter les mécanismes sous-jacents. C'est ce qu'explique AngristKrueger, en affirmant « research in a structuralist style relies heavily on economic theory to guide empirical work [⋯] An alternative to structural modeling, [⋯] the `experimentalist' approach, [⋯] puts front and center the problem of identifying causal effects from specific events or situations ».On peut aussi souligner que si l'approche de la Commission Cowles était très exigeante, en supposant le modèle connu, toute une littérature s'est développée en allégeant cette hypothèse, soit en essayant de choisir le bon modèle (avec les travaux de HendryK par exemple) ou en proposant de faire des moyennes de modèles (comme développé récemment par Lili). Et plus généralement, alors que l'analyse économétrique (en particulier à des fins de politique économique) s'est développée plus récemment autour de l'inférence causale, les techniques d'apprentissage machine ont été vues, traditionnellement, autour de la prédiction (où la recherche de corrélations suffisamment fortes entre variables suffit) d'où leur popularité dans des usages plus industriels de classification, comme la reconnaissance de caractères, de signature, d'images, ou de traduction, comme le rappelle Bishop. En biologie, ces techniques ont été appliquées pour créer des classifications d'espèces animales en fonction d'analyse d'ADN, ou dans le domaine militaire et sécuritaire pour l'identification de cibles ou de terroristes (potentiels). Il faudra attendre les années 1990 pour voir des applications en finance avec Altman par exemple, ou Herbrich pour une revue de littérature sur les applications potentielles en économie. Si des applications sont aujourd'hui nombreuses, et si ces techniques concurrencent les modèles de micro-économétrie (on pourra penser au scoring bancaire, à la détection de fraude fiscale ou assurantielle, ou à l'identification de prospects en marketing), les algorithmes d'apprentissage sont devenus très populaires en reconnaissance de parole, puis d'images, et plus récemment avec les applications en ligne et les applications aux jeux (d'échec, et plus récemment de go). Si l'économétrie s'est développée au confluent des mathématiques et de l'économie, l'apprentissage machine (que l'on pourrait avoir tendance à rapprocher de l'intelligence artificelle) s'est développé à la frontière des mathématiques et de l'informatique (avec des résultats fondamentaux en optimisation - en particulier autour des méthodes de gradient stochastique - et sur les espaces « sparse » ou « parcimonieux »). §.§ De la grande dimension aux données massives Dans cet article, une variable sera un vecteur de ℝ^n, de telle sorte qu'en concaténant les variables ensemble, on puisse les stocker dans une matrice , de taille n× p, avec n et p potentiellement grands[Là encore, des extensions sont possibles, en particulier dans les données médicales avec des images de type IRM comme variables prédictives, ou des données climatiques avec des cartes en variables prédictives, ou plus généralement une variable tensorielle en dimension plus ou moins grande. Comme le montre Kolda il est toutefois possible de se ramener dans le cas usuel (de données sous formes de vecteurs) en utilisant la décomposition de Tucker.]. Le fait que n soit grand n'est, a priori, pas un problème en soi, au contraire. De nombreux théorèmes en économétrie et en statistique sont obtenus lorsque n→∞ (c'est la théorie asymptotique). En revanche, le fait que p soit grand est problématique, en particulier si p>n. Les deux dimensions sont à distinguer, car elles vont engendrer des problèmes relativement différents.Portnoy a montré que l'estimateur du maximum de vraisemblance conserve la propriété de normalité asymptotique si p reste petit devant n, ou plus précisément, si p^2/n→ 0 lorsque n,p→∞. Aussi, il n'est pas rare de parler de grande dimension dès lors que p>√(n). Un autre concept important est celui de sparcité, qui repose non pas sur la dimension p mais sur la dimension effective, autrement dit le nombre de variables effectivement significatives. Il est alors possible d'avoir p>n tout en ayant des estimateurs convergents.La grande dimension en terme de nombre de variables, p, peut faire peur à cause de la malédiction de la dimension, introduit par Bellman. L'explication de cette malédiction est que le volume de la sphère unité, en dimension p, tend vers 0 lorsque p→∞. On dit alors que l'espace est « parcimonieux » - c'est à dire que la probabilité de trouver un point proche d'un autre devient de plus en plus faible (on pourrait parler d'espace « clairsemé »). Ou de manière duale, pour reprendre la formulation de HastieEtal, le volume qu'il convient de considérer pour avoir une proportion donnée d'observations augmente avec p. L'idée de réduire la dimension en considérant une analyse en composante principale peut paraître séduisante, mais l'analyse souffre d'un certain nombre de défauts en grande dimension. La solution est alors souvent la sélection de variables, qui pose le problème des tests multiples, ou du temps de calcul, pour sélectionner k variables parmi p, lorsque p est grand.Pour reprendre la terminologie de Buhlmann, les problèmes que nous évoquons ici correspondent à ceux observés en grande dimension, qui est un problème essentiellement statistique. D'un point de vue informatique, on peut aller un peu plus loin, avec des données réellement massives (qui occupent énormément de place en mémoire). Dans ce qui précède, les données étaient stockées dans une matrice X, de taille n× p. Si cet objet formel est toujours bien défini, il peut y avoir des soucis à stocker une telle matrice, voire manipuler une matrice abondament utilisée en économétrie, X^ TX (matrice n× n). La condition du premier ordre (dans le modèle linéaire) est associée à la résolution de X^ T(Xβ-y)=0. En dimension raisonnable, on utilise la décomposition QR (c'est à dire la décomposition de Gram-Schmidt). En grande dimension, on peut utiliser des méthodes numériques de descente de gradient, où le gradient est approché sur un sous-échantillon de données (comme décrit par exemple dans Zinkevichetal) Cet aspect informatique est souvent oublié alors qu'il a été à la base de bon nombre d'avancées méthodologiques, en économétrie. Par exemple, HoerlKennard reviennent sur l'origine de l'utilisation de la régression Ridge: « Nous facturions90$ par jour pour notre temps, mais avons dû facturer 450$ par heure d'ordinateur sur un Univac I (⋯) Avec cette machine, il a fallu 75 minutes de traitement pour inverser une matrice40 × 40 en passant par une partition 4 × 4 de sous-matrices 10× 10, en utilisant des bandes magnétiques pour le stockage temporaire. Nous avons noté que les coefficients d'un régression linéaire calculés en utilisant les moindres carrés n'avaient pas toujours de sens. Les coefficients avaient tendance à être trop grands en valeur absolue, certains avaient même le mauvais signe, et ils pouvaient être instables avec de très petits changements dans les données (⋯) Comme la méthode que nous proposions attaquait l'une des vaches sacrées de la régression linéaire - les moindres carrés - nous avons fait face à une une résistance considérable». §.§ Statistique computationnelle et non-paramétrique L'objet de ce papier est d'expliquer les différences majeures entre l'économétrie et l'apprentissage statistique, correspondant aux deux cultures mentionnées par Breiman, lorsqu'il évoque en modélisation statistique la « data modeling culture » (reposant sur un modèle stochastique, comme la régression logistique ou le modèle de Cox) et la « algorithmic modeling culture » (reposant sur la mise en œuvre d'un algorithme, comme dans les forêts aléatoires ou les supports vecteurs machines, une liste exhaustive est présenté dans Shalev-Shwartz). Mais la frontière entre les deux est très poreuse. À l'intersection se retrouve, par exemple, l'économétrie non-paramétrique. Cette dernière repose sur un modèle probabiliste (comme l'économétrie), tout en insistant davantage sur les algorithmes (et leurs performances) plutôt que sur des théorèmes asymptotiques.L'économétrie non-paramétrique repose sur des décompositions dans des bases fonctionnelles. L'économétrie linéaire consiste à approcher la fonction m:↦[Y|=] par une fonction linéaire. Mais plus généralement, on peut considérer une décomposition dans une base fonctionnelle, et s'intéresser à une approximation obtenue sur un nombre fini de termes :m()=∑_j=0^∞ω_j g_j() et m()=∑_j=0^h^⋆ω_j g_j(),où les poids ω_j sont estimés, alors que le nombre de composantes h^⋆ est optimisé. On retrouvera ici les modèles additifs (dits gam), par exemple, étudiés dansTrevor. Une autre solution consiste à considérer un modèle simple, mais local. Par exemple un modèle constant, au voisinage de , obtenu en considérant seulement les observations proches de:g()=∑_i=1^n ω_ y_i par exemple g()=1/n_∑_i:‖_i-‖≤ h y_ioù n_ est le nombre d'observations au voisinage de . En mettant des poids fonctions de la distance à , on retrouve ici le modèle obtenu par Nadaraya et Watson, ou les méthodes de régression locale.Les différentes méthodes reposent sur des méta-paramètres - correspondant paramètres de lissage - c'est à dire h dans les exemples précédents. Pour un économètre, le paramètre « optimal » pour h est obtenu soit à l'aide de théorèmes asymptotiques, soit à l'aide de techniques de validation, comme en machine learning.On obtient alors une valeur numérique, mais on n'a pas d'interprétation en lien avec la taille de l'échantillon, ou les variances des différentes grandeurs. Si les économistes ont toujours la culture du tableau présentant la« sortie de régression », les méthodes non-paramétriques sont utiles pour détecter des mauvaises spécifications, des non-prises en compte de nonlinéarité, ou d'effets croisées (et les outils de « machine learning » que nous allons voir peuvent probablement jouer le même rôle). §.§ Plan de l'article Pour reprendre le titre de Varian, l'objet de cet article est de présenter les différences fondamentales entre l'économétrie et l'apprentissage machine, et surtout de voir comment ces deux techniques peuvent apprendre l'une de l'autre, dans un contexte où les bases de données deviennent massives. La Section <ref> reviendra sur la construction du modèle linéaire. Le modèle sera introduit ici à partir du modèle Gaussien « homoscédastique ». Ce modèle présente l'avantage d'avoir une élégante interprétation géométrique, en terme de projection sur le sous-espace des combinaisons linéaires des variables explicatives. La première extension que nous verrons est le passage du modèle linéaire à un modèle non-linéaire, tout en construisant un prédicteur linéaire. La seconde extension proposera de construire un modèle non-gaussien, pour modéliser une variable indicatrice ou un comptage Poissonnien, par exemple, donnant naissance aux modèles linéaires généralisés (construits pour des variables dans la famille exponentielle). Une fois rappelé l'origine des outils économétriques standards, dans la Section <ref> nous présenterons les outils et techniques développés dans le contexte de l'apprentissage machine. Si l'outil central des modèles économétriques est la distribution de la variable dépendante, Y, les techniques d'apprentissage reposent sur une fonction de perte, ℓ, représentant une « distance » entre la variable d'intérêt y, et le modèle m(·). Nous présenterons tout d'abord l'algorithme de boosting, reposant sur l'idée d'un apprentissage lent, en modélisant séquentiellement les résidus. Le danger des méthodes d'apprentissage est qu'il est aisé de construire un modèle « parfait », dont le pouvoir prédictif serait faible. Nous évoquerons alors les techniques de pénalisation, utilisées pour éviter le sur-apprentissage. Nous évoquerons en particulier les notions d'in-sample et out-of-sample, et les techniques de validation croisée. Pour conclure cette section, nous reviendrons sur les interprétations probabilistes des outils d'apprentissage, qui permettront de faire le lien entre les différentes approches, tout en restant sur une discussion générale sur la philosophie de ces deux cultures.Après cette section sur la philosophie des méthodes de machine learning, nous reviendrons dans la section <ref> sur quelques algorithmes importants : les réseaux de neurones, les supports vecteurs machine (SVM) et enfin les méthodes de type arbres et forêts.La Section <ref> proposera des exemples concrets de comparaison entre les différentes techniques, dans le cas de classifications (binaires) pour des variables y∈{0,1} (achat d'assurance, non-remboursement d'un crédit) et dans un contexte de régression (lorsque la variable d'intérêt n'est plus qualitative - ce que nous simplifierons en notant y∈ℝ). Nous reviendrons avant sur les courbes ROC, outils importants pour juger de la qualité d'un classifieur, malheureusement peu utilisés en économétrie. Nous verrons en particulier les méthodes de bagging, forêts aléatoires ou boosting. Nous reviendrons aussi sur les méthodes de choix de modèles et des méta-paramètres. À travers ces exemples d'application, nous verrons comment les modèles de type machine learning peuvent être utilisés pour mieux détecter la mauvaise spécification des modèles de régression paramétriques, à cause de non-linéarités, et/ou d'intéractions manquées.§ ÉCONOMÉTRIE ET MODÈLE PROBABILISTE L'importance des modèles probabilistes en économie trouve sa source dans les questionnements de Working et les tentatives de réponses apportées dans les deux tomes de Tinbergen. Ces derniers ont engendré par la suite énormément de travaux, comme le rappelle Duo dans son ouvrage sur les fondements de l'économétrie, et plus particulièrement dans le premier chapitre « The Probability Foundations of Econometrics ». Rappelons que Trygve Haavelmo a reçu le prix Nobel d’économie en 1989 pour sa « clarification des fondations de la théorie probabiliste de l'économétrie ». Car comme l'a montré Haavelmo (initiant un changement profond dans la théorie économétrique dans les années 1930, comme le rappelle le chapitre 8 de Morgan) l'économétrie repose fondamentalement sur un modèle probabiliste, et ceci pour deux raisons essentielles. Premièrement, l'utilisation de grandeurs (ou « mesures » ) statistiques telles que les moyennes, les erreurs-types et les coefficients de corrélation à des fins inférentielles ne peut se justifier que si le processus générant les données peut être exprimé en termes de modèle probabiliste. Deuxièmement, l'approche par les probabilitées est relativement générale, et se trouve être particulièrement adaptée à l'analyse des observations « dépendantes » et « non homogènes », telles qu'on les trouve souvent sur des données économiques. On va alors supposer qu'il existe un espace probabiliste (Ω,ℱ,ℙ) tel que les observations (y_i,_i) sont vues comme des réalisations de variables aléatoires (Y_i,_i). En pratique, la loi jointe du couple (Y,) nous intéresse toutefois peu : la loi deest inconnue, et c'est la loi de Y conditionnelle àqui nous intéressera. Dans la suite, nous noterons x une observation,un vecteur d'observations,X une variable aléatoire, etun vecteur aléatoire et, abusivement,pourra aussi désigner la matrice des observations individuelles (les _i), suivant le contexte. §.§ Fondements de la statistique mathématique Comme le rappelle l'introduction de Vapnik98, l'inférence en statistique paramétrique est basée sur la croyance suivante: le statisticien connaît bien le problème à analyser, en particulier, il connaît la loi physique qui génère les propriétés stochastiques des données, et la fonction à trouver s'écrit via un nombre fini de paramètres[On peut rapprocher cette approche de l'économétrie structurelle, telle que présentée par exemple dans Keen.]. Pour trouver ces paramètres, on adopte la méthode du maximum de vraisemblance. Le but de la théorie est de justifier cette approche (en découvrant et en décrivant ses propriétés favorables). On verra qu'en apprentissage, la philosophie est très différente, puisqu'on ne dispose pas d'informations a priori fiables sur la loi statistique sous-jacente au problème, ni-même sur la fonction que l'on voudrait approcher (on va alors proposer des méthodes pour construire une approximation à partir de données à notre disposition, pour reprendre Vapnik98). Un « âge d'or » de l'inférence paramétrique, de 1930 à 1960, a posé les bases de la statistique mathématique, que l'on retrouve dans tous les manuels de statistique, y compris aujourd'hui. Comme le dit Vapnik98, le paradigme paramétrique classique est basé sur les trois croyances suivantes:* Pour trouver une relation fonctionnelle à partir des données, le statisticien est capable de définir un ensemble de fonctions, linéaires dans leurs paramètres, qui contiennent une bonne approximation de la fonction souhaitée. Le nombre de paramètres décrivant cet ensemble est petit.* La loi statistique sous-jacente à la composante stochastique de la plupart des problèmes de la vie réelle est la loi normale. Cette croyance a été soutenue en se référant au théorème de limite centrale, qui stipule que dans de larges conditions la somme d'un grand nombre de variables aléatoires est approximée par la loi normale. * La méthode du maximum de vraisemblance est un bon outil pour estimer les paramètres.Nous reviendrons dans cette partie sur la construction du paradigme économétrique, directement inspiré de celui de la statistique inférentielle classique.§.§ Lois conditionnelles et vraisemblance L'économétrie linéaire a été construite sous l'hypothèse de données individuelles, ce qui revient à supposer les variables(Y_i,_i) indépendantes (s'il est possible d'imaginer des observations temporelles - on aurait alors un processus (Y_t,_t) - mais nous n'aborderons pas les séries temporelles dans cet article). Plus précisément, on va supposer que conditionnellement aux variables explicatives _i, les variables Y_i sont indépendantes. On va également supposer que ces lois conditionnelles restent dans la même famille paramétrique, mais que le paramètre est une fonction de . Dans le modèle linéaire Gaussien on suppose que :(Y|=)ℒ∼𝒩(μ(),σ^2) avec μ()=β_0+, et ∈ℝ^p.On parle de modèle linéaire car [Y|=]=β_0+ est une combinaison linéaire des variables explicatives. C'est un modèle homoscédastique si [Y|=]=σ^2, où σ^2 est une constante positive. Pour estimer les paramètres, l'approche classique consiste à utiliser l'estimateur du Maximum de Vraisemblance, comme l'avait suggéré initialement Ronald Fisher. Dans le cas du modèle linéaire Gaussien, la log-vraisemblance s'écrit :logℒ(β_0,,σ^2|,) = -n/2log[2πσ^2] - 1/2σ^2∑_i=1^n (y_i-β_0-_i)^2.Notons que le terme de droite, mesurant une distance entre les données et le modèle, va s'interpréter comme la déviance, dans les modèles linéaires généralisés. On va alors poser :(β_0,,σ^2)=argmax{logℒ(β_0,,σ^2|,)}.L'estimateur du maximum de vraisemblance est obtenu par minimisation de la somme des carrés des erreurs (estimateur dit des « moindres carrés » ) que nous retrouverons dans l'approche par machine learning.Les conditions du premier ordre permettent de retrouver les équations normales, dont l'écriture matricielle est [-]=0, que l'on peut aussi écrire ()=. Si la matriceest de plein rang colonne, alors on retrouve l'estimateur classique :=()= + ()en utilisant une écriture basée sur les résidus (comme souvent en économétrie), y=+ε. Le théorème de Gauss Markov assure que cette estimateur est l'estimateur linéaire sans biais de variance minimale. On peut alors montrer que ℒ∼𝒩(,σ^2 []), et en particulier :[]= et []=σ^2 []. En fait, l'hypothèse de normalité permet de faire un lien avec la statistique mathématique, mais il est possible de construire cet estimateur donné par l'équation (<ref>). Si on suppose que Y|=ℒ∼+ε, avec 𝔼[ε]=0, Var[ε]=σ^2 , Cov[ε,X_j]=0 pour tout j, alorsest un estimateur sans biais de([]=) et de variance minimale parmi les estimateurs sans biais linéaires, avec []=σ^2 []. De plus, cet estimateur est asymptotiquement normal√(n)(-)ℒ→𝒩(0,Σ)lorsquen→∞La condition d'avoirune matricede plein rang peut être (numériquement) forte en grande dimension. Si elle n'est pas vérifiée,=() n'existe pas. Si 𝕀 désigne la matrice identité, notons toutefois que(+λ𝕀) existe toujours, pour λ>0. Cet estimateur est appelé l'estimateur Ridge de niveau λ (introduit dans les années 60 par Hoerl, et associé à une régularisation étudiée par Tikhonov). Cette estimateur apparaît naturellement dans un contexte d'économétrie Bayesienne (nous le reverrons dans la section suivante présentant les techniques de machine learning).§.§ Les résidus Il n'est pas rare d'introduire le modèle linéaire à partir de la loi des résidus, comme nous l'avions mentionné auparavant. Aussi, l'équation (<ref>) s'écrit aussi souvent :y_i=β_0+_i+ε_ioù les ε_i sont des réalisations de variables aléatoires i.i.d., de loi 𝒩(0,σ^2). On notera parfois ℒ∼𝒩(0,σ^2𝕀), sous une forme vectorielle. Les résidus estimés sont définis par :ε_i=y_i - [β_0+_i]Ces résidus sont l'outil de base pour diagnostiquer la pertinence du modèle.Une extension du modèle décrit par l'équation (<ref>) a été proposé pour tenir compte d'un éventuel caractère hétéroscédastique :(Y|=)ℒ∼𝒩(μ(),σ^2())où σ^2() est une fonction positive des variables explicatives. On peut réécrire ce modèle en posant :y_i=β_0+_i+σ^2(_i)·ε_ioù les résidus sont toujours i.i.d., mais de variance unitaire, ε_i=y_i-[β_0+_i]/σ(_i).Si l'écriture à l'aide des résidus est populaire en économétrie linéaire (lorsque la variable dépendante est continue), elle ne l'est toutefois plus dans les modèles de comptage, ou la régression logistique.L'écriture à l'aide d'un terme d'erreur (comme dans l'équation (<ref>)) pose toutefois de nombreuses questions quant à la représentation d'une relation économique entre deux grandeurs. Par exemple, on peut supposer qu'il existe une relation (linéaire pour commencer) entre les quantités d'un bien échangé, q et son prix p. On peut ainsi imaginer une équation d'offreq_i=β_0+β_1 p_i+u_i(u_i désignant un terme d'erreur) où la quantité vendue dépend du prix, mais de manière tout aussi légitime, on peut imaginer que le prix dépend de la quantité produite (ce qu'on pourrait appeler une équation de demande),p_i=α_0+α_1 q_i+v_i(v_i désignant un autre terme d'erreur).Historiquement, le terme d'erreur dans l'équation (<ref>) a pu être inteprété comme une erreur idiosyncratique sur la variable y, les variables dites explicatives étant supposées fixées, mais cette interprétation rend souvent le lien entre une relation économique et un modèle économique compliqué, la théorie économique parlant de manière abstraite d'une relation entre grandeur, la modélisation économétrique imposant une forme spécifique (quelle grandeur est y et quelle grandeur est x) comme le montre plus en détails le chapitre 7 de Morgan.§.§ Géométrie du modèle linéaire gaussien Définissons le produit scalaire dans ℝ^n, ⟨,⟩=, et notons ‖·‖ la norme euclidienne associée, ‖‖ = √() (notée ·_ℓ_2 dans la suite). Notons ℰ_ l'espace engendré par l'ensemble des combinaisons linéaires des composantes(en rajoutant la constante). Si les variables explicatives sont linéairement indépendantes,est de plein rang colonne et ℰ_ est un sous-espace de dimension p+1 de ℝ^n. Supposons à partir de maintenant que les variableset la variable y sont ici centrées. Notons qu'aucune hypothèse de loi n'est faite dans cette section, les propriétés géométriques découlent des propriétés de l'espérance et de la variance dans l'espace des variables de variance finie.Avec cette notation, notons que le modèle linéaire s'écrit m()=⟨,β⟩. L'espace ℋ_z={∈ℝ^k:m()=z} est un hyperplan (affine) qui sépare l'espace en deux. Définissons l'opérateur de projection orthogonale sur ℰ_𝒳=ℋ_0, Π_𝒳=[]. Aussi, la prévision que l'on peut faire pourest :=[]_ Π_𝒳=Π_𝒳.Comme =-=(𝕀-Π_𝒳)=Π_𝒳^⊥, on note que ⊥, que l'on interprétera en disant que les résidus sont un terme d'innovation, imprévisible, au sens où Π_𝒳=0.Le théorème de Pythagore s'écrit ici :‖‖^2=‖Π_𝒳‖^2+‖Π_𝒳^⊥‖^2=‖Π_𝒳‖^2+‖-Π_𝒳‖^2=‖‖^2+‖ε‖^2qui se traduit classiquement en terme de somme de carrés :∑_i=1^n y_i^2_n×variance totale=∑_i=1^n y_i^2_n×variance expliquée + ∑_i=1^n (y_i-y_i)^2_n×variance résiduelleLe coefficient de détermination, R^2 (nous reviendrons sur ce coefficient dans la section <ref>) s'interprête alors comme le carré du cosinus de l'angle θ entreet Π_𝒳 :R^2=‖Π_𝒳‖^2/‖‖^2=1-‖Π_𝒳^⊥‖^2/‖‖^2=cos^2(θ).Une application importante a été obtenue par FrishWaugh, lorsque l'on partitionne les variables explicatives en deux groupes, =[ _1 |_2], de telle sorte que la régression devient :=β_0+ _1_1+_2_2+εFrishWaugh ont montré qu'on pouvait considérer deux projections successives. En effet, si _2^⋆=Π_𝒳_1^⊥ et _2^⋆=Π_𝒳_1^⊥_2, on peut montrer que_2=[_2^⋆_2^⋆]_2^⋆_2^⋆Autrement dit, l'estimation globale est équivalente à l'estimation indépendante des deux modèles si _2^⋆=_2, c'est à dire _2∈ℰ__1^⊥, que l'on peut noter _1⊥_2. On obtient ici le théorème de Frisch-Waugh qui garantie que si les variables explicatives entre les deux groupes sont orthogonales, alors l'estimation globale est équivalente à deux régressions indépendantes, sur chacun des jeux de variables explicatives. Ce qui est un théorème de double projection, sur des espaces orthogonaux. Beaucoup de résultats et d'interprétations sont obtenus par des interprétations géométriques (liées fondamentalement aux liens entre l'espérance conditionnelle et la projection orthogonale dans l'espace des variables de variance finie).Cette vision géométrique permet de mieux comprendre le problème de la sous-identification, c'est à dire le cas où le vrai modèle serait y_i=β_0+_1_1+_2_2+ε_i, mais le modèle estimé est y_i=β_0+_1b_1+η_i. L'estimateur du maximum de vraisemblance de b_1 est :b_1= (X_1^TX_1)^-1X_1^T= (X_1^TX_1)^-1X_1^T [X_1,iβ_1 + X_2,iβ_2 + ε] =(X_1^TX_1)^-1X_1^TX_1β_1 + (X_1^TX_1)^-1X_1^TX_2β_2 + (X_1^TX_1)^-1X_1^Tε= β_1 +(X_1'X_1)^-1X_1^TX_2β_2_β_12 +(X_1^TX_1)^-1X_1^Tε_ν_ide telle sorte que [b_1]=_1+_12, le biais étant nul uniquement dans le cas où X_1^TX_2=0 (c'est à dire X_1 ⊥X_2): on retrouve ici une conséquence du théorème de Frisch-Waugh.En revanche, la sur-identification correspond au cas où le vrai modèle serait y_i=β_0+_1_1+ε_i, mais le modèle estimé est y_i=β_0+_1b_1+_2b_2+η_i. Dans ce cas, l'estimation est sans biais, au sens où 𝔼(b_1)=β_1 mais l'estimateur n'est pas efficient. Et comme nous l'avons vu dans la section précédente, il n'est pas rare d'avoir des valeurs de b_2 qui sont considérées comme significativement non-nulles. Nous évoquerons dans la section suivante une méthode efficace de choix de variables (et éviter la sur-identification). §.§ Du paramétrique aunon-paramétrique La réécriture de l'équation (<ref>) sous la forme==[]_Π_𝒳 permet de voir la prévision directement comme une transformation linéaire des observations. De manière plus générale, on peut obtenir un prédicteur linéaire en considérant m()=_, où _ est un vecteur de poids, qui dépendent de , interprété comme un vecteur de lissage. En utilisant les vecteurs __i, calculés à partir des _i, on obtient une matriceS de taille n× n, et =S. Dans le cas de la régression linéaire décrite auparavant, _=[], et classiquement, trace( S) est le nombre de colonnes de la matrice(le nombre de variables explicatives). Dans ce contexte de prédicteurs linéaires,trace( S) est souvent vu comme un équivalent au nombre de paramètres (ou complexité, ou dimension, du modèle), et ν=n-trace( S) est alors le nombre de degrés de liberté (comme défini dans Ruppert et Simonoff). Le principe de parcimonie[« pluralitas non est ponenda sine necessitate » pour reprendre le principe énoncé par Guillaume d'Occam (les multiples ne doivent pas être utilisés sans nécessité).] consiste à minimiser cette dimension (la trace de la matrice S) autant que faire se peut. Mais dans le cas général, cette dimension est plus complexe à définir. Notons que l'estimateur introduit par Nadaraya et Watson, dans le cas d'une régression non-paramétrique simple, s'écrit également sous cette forme puisquem_h(x)=_x=∑_i=1^n s_x,iy_iavecs_x,i=K_h(x-x_i)/K_h(x-x_1)+⋯+K_h(x-x_n),où K(·) est une fonction noyau, qui attribue une valeur d'autant plus faible que x_i est proche de x, et h>0 est la fenêtre de lissage.L'introduction de ce méta-paramètre h pose un soucis, car il convient de le choisir judicieusement. En faisant des développement limités, on peut montrer que si X a pour densité f,biais[m_h(x)]=𝔼[m_h(x)]-m(x)∼h^2( C_1 /2m”(x)+C_2 m'(x)f'(x)/f(x))et Var[m_h(x)]∼C_3/nhσ(x)/f(x)pour des constantes que l'on peut estimer (voir Simonoff par exemple). Ces deux fonctions évoluent inversement en fonction de h, comme le rappelle la Figure <ref>. L'idée naturelle est alors de chercher à minimiser l'erreur quadratique moyenne, le MSE, biais[m_h(x)]^2+Var[m_h(x)], ce qui donne une valeur optimale pour h de la forme h^⋆=O(n^-1/5), ce qui rappelle la règle de Silverman. En plus grande dimension, pour des variablescontinues, on peut utiliser un noyau multivarié, de fenêtre matricielle H, 𝔼[m_H(x)]∼ m(x) +C_1/2trace(H^ Tm”(x)H)+C_2m'(x)^ THH^ T∇ f(x)/f(x) et Var[m_H(x)]∼C_3/n det(H)σ(x)/f(x).Si H est une matrice diagonale, avec le même terme h sur la diagonale, alors h^⋆=O(n^-1/(4+dim(x)))]. Cela dit, en pratique, on sera davantage intéressé par la version intégrée de l'erreur quadratique,MISE(m_h)=𝔼[MSE(m_h(X))]=∫ MSE(m_h(x))dF(x),dont on peut montrer queMISE[m_h] ∼ h^4/4(∫ x^2k(x)dx)^2∫[m”(x)+2m'(x)f'(x)/f(x)]^2dx^biais^2 +σ^2/nh∫ k^2(x)dx ·∫dx/f(x)^variance,lorsque n→∞ et nh→∞. On retrouve ici une relation asymptotique qui rappelle l'ordre de grandeur de Silverman,h^⋆ =n^-1/5(C_1∫dx/f(x)/C_2∫[m”(x)+2m'(x)f'(x)/f(x)]dx)^1/5,sauf que beaucoup de termes ici sont inconnus. On verra, le machine-learning propose des techniques computationnelles, lorsque l'économètre avait pris l'habitude de chercher des propriétés asymptotiques. §.§ Famille exponentielle et modèles linéaires Le modèle linéaire Gaussien est un cas particulier d'une vaste famille de modèles linéaires, obtenu lorsque la loi conditionnelle de Y appartient à la famille exponentiellef(y_i|θ_i,ϕ)=exp(y_iθ_i-b(θ_i)/a(ϕ)+c(y_i,ϕ)) avec θ_i=ψ(x_i^Tβ).Les fonctions a, b et c sont spécifiées en fonction du type de loi exponentielle (étudiée abondamment en statistique depuis les Darmois, comme le rappelle Brown), et ψ est une fonction bijective que se donne l'utilisateur.La log-vraisemblance a alors une expression relative simplelogℒ(θ,ϕ|y)=∏_i=1^n log f(y_i|θ_i,ϕ) =∑_i=1^ny_iθ_i-∑_i=1^nb(θ_i)/a(ϕ)+ ∑_i=1^n c(y_i,ϕ)et la condition du premier ordre s'écrit alors∂logℒ(θ,ϕ|y)/∂β = X^TW^-1[y-]=0pour reprendre les notations de Muller, où W est une matrice de poids (qui dépend de ). Compte tenu du lien entre θ et l'espérance de Y, au lieu de spécifier la fonction ψ(·), on aura plutôt tendance à specifier la fonction de lien g(·) définie par y=m()=[Y|=]=g().Pour la régression linéaire Gaussienne on prendra un lien Identité, alors que pour la régression de Poisson, le lien naturel (dit canonique) est le lien logarithmique. Ici, comme W dépend de(avecW=diag(∇ g()Var[])) il n'existe en général par de formule explicite pour l'estimateur du maximum de vraisemblance. Mais un algorithme itératif permet d'obtenir une approximation numérique. En posant=g()+(-)·∇ g()correspondant au terme d'erreur d'un développement de Taylor à l'ordre 1 de g, on obtient un algorithme de la formeβ_k+1 = [X^ TW_k^-1X]^-1X^ TW_k^-1z_kEn itérant, on notera β=β_∞, et on peut montrer que - moyennant quelques hypothèses techniques (cf Muller) - cet estimateur est asymptotiquement Gaussien, avec√(n)(β-β)ℒ→𝒩(0,I(β)^-1),où numériquement I(β)=ϕ·[X^ TW_∞^-1X].D'un point de vue numérique toujours, on résout la condition du premier ordre, et la loi de Y n'intervient pas réellement. Par exemple, on peut estimer une « régression de Poisson » même lorsque y∈ℝ_+, pas nécessairement y∈ℕ. Autrement dit, la loi de Y n'est qu'une interprétation donnée ici, et l'algorithme pourrait être introduit de manière différente (comme nous le verrons dans la section suivante), sans forcément avoir de modèle probabiliste sous-jacent.§.§ Régression logistique La régression logistique est le modèle linéaire généralisé obtenu avec une loi de Bernoulli, et une fonction de lien qui est la fonction quantile d'une loi logistique (ce qui correspond au lien canonique au sens de la famille exponentielle). Compte tenu de la forme de la loi de Bernoulli, l'économétrie propose un modèle pour y_i∈{0,1}, dans lequel le logarithme de la cote suit un modèle linéaire :log(ℙ[Y=1|=]/ℙ[Y≠ 1|=])=β_0+,ou encore :[Y|=]=ℙ[Y=1|=]=e^β_0+/1+e^β_0+=H(β_0+),oùH(·)=exp(·)/1+exp(·),correspondant à la fonction de répartition de la loi logistique. L'estimation de (β_0,) se fait par maximisation de la vraisemblance :ℒ=∏_i=1^n (e^x_i^ Tβ/1+e^x_i^ Tβ)^y_i(1/1+e^x_i^ Tβ)^1-y_iOn continuera à parler des modèles linéaires car les courbes d'isoprobabilités sont ici les hyperplans parallèles b_0+. À ce modèle, popularisé par Berkson1, certains préfèront le modèle probit (comme le raconte Berkson2), introduit par Bliss. Dans ce modèle :[Y|=]=ℙ[Y=1|=]=Φ(β_0+),où Φ désigne la fonction de répartition de la loi normale centrée réduite. Ce modèle présente l'avantage d'avoir un lien direct avec le modèle linéaire Gaussien, puisque y_i=1(y_i^⋆>0) avecy_i^⋆=β_0++ε_ioù les résidus sont Gaussiens, de loi 𝒩(0,σ^2). Un alternative est d'avoir des résidus centrés de variance unitaire, et de considérer une modélisation latente de la forme y_i=1(y_i^⋆>ξ) (où ξ sera à fixer). On le voit, ces techniques sont fondamentalement liées à un modèle stochastique sous-jacent. Mais dans la section <ref>, nous présenterons plusieurs techniques alternatives - tirées de la littérature en apprentissage - pour ce problème de classification (avec deux classes, ici 0 et 1). §.§ Régression en grande dimension Comme nous l'avions mentionné auparavant, la condition du premier ordre X(Xβ-y)=0 se résout numériquement en effectuant une décomposition QR, pour un coût en O(np^2) opérations (où p est le rang de ). Numériquement, ce calcul peut être long (soit parce que p est grand, soit parce que n est grand), et une stratégie plus simple peut être de faire du sous-échantillonnage. Soit n_s≪ n, et considérons un sous-échantillon de taille n_s de {1,⋯,n}. Alorsβ_s=(_s_s)^-1_sy_s est une bonne approximation de β comme le montre Dhilonetal2013. Cet algorithme est toutefois dangereux si certains points ont un pouvoir de levier important (i.e. L_i=_i()^-1_i). Tropp2011 propose de transformer les données (de manière linéaire), mais une approche plus populaire est de faire du sous-échantillonnage non uniforme, avec une probabilité liée à l'influence des observations (définie par I_i=ε_iL_i/(1-L_i)^2, et qui malheureusement ne peut être calculée qu'une fois le modèle estimé). De manière générale, on parlera de données massives lorsquela table de données de taille n × p ne tient pas en mémoire RAM de l'ordinateur. Cette situation est souvent rencontrée en apprentissage statistique de nos jours avec très souvent p ≪ n. C'est la raison pour laquelle, en pratique de nombreuses bibliothèques d'algorithmes assimilées à de l'apprentissage machine[comme, par exemple, celles du langage Python.] utilisent des méthodes itératives pour résoudre la condition du premier ordre. Lorsque le modèle paramétrique à calibrer est effectivement convexe et semi-différentiable, il est possible d'utiliser par exemple la méthode de descente de gradient stochastique comme le suggère bottou. Ce dernier permet de s'affranchir à chaque itération du calcul du gradient sur chaque observation de notre base d'apprentissage.Plutôt que d'effectuer une descente moyenne à chaque itération, on commence par tirer (sans remise) une observation X_i parmi les n disponibles. On corrige ensuite les paramètres du modèle de sorte à ce que la prédiction faite à partir de X_i soit la plus proche possible de la vraie valeur y_i. On réitère ensuite la méthode jusqu'à avoir parcourue l'ensemble des données. Dans cet algorithme il y a donc autant d'itération que d'observations. Contrairement à l'algorithme de descente de gradient (ou méthode de Newton) à chaque itération un seul vecteur de gradient est calculé (et non plus n). Il est néanmoins parfois nécessaire d'exécuter cette algorithme plusieurs fois pour augmenter la convergence des paramètres du modèle. Si l'objectif est par exemple de minimiser l'erreur quadratique ℓ entre l'estimateur f_β(X) et y l'algorithme peut se résumer ainsi : Etape 0: Mélange des donnéesEtape d'itérations: Pour t=1,...,n, on tire i ∈{ 1,⋯, n } sans remise, on poseβ^t+1 = β^t - γ_t∂l(y_i,f_β^t(X_i)) /∂βCet algorithme peut être réitérée plusieurs fois dans son ensemble selon le besoin de l'utilisateur. L'avantage de cette méthode est qu'à chaque itération, il n'est pas nécessaire de calculer le gradient sur toutes les observations (plus de somme). Elle est donc adaptée aux bases de données volumineuses. Cet algorithme s'appuie sur une convergence en probabilité vers un voisinage de l'optimum (et non pas l'optimum lui même). §.§ Qualité d'un ajustement et choix de modèle Dans le modèle linéaire Gaussien, le coefficient de détermination - noté R^2 - est souvent utilisé comme mesure de la qualité d'ajustement. Compte tenu de la formule de décomposition de la variance1/n∑_i=1^n (y_i-y̅)^2_variance totale=1/n∑_i=1^n (y_i-y_i)^2_variance résiduelle+1/n∑_i=1^n (y_i-y̅)^2_variance expliquéeon définit le R^2 comme le ratio de variance expliquée et de la variance totale, autre interpétation du coefficient que nous avions introduit à partir de la géométrie des moindres carrés.R^2 = ∑_i=1^n (y_i-y̅)^2-∑_i=1^n (y_i-y_i)^2/∑_i=1^n (y_i-y̅)^2Les sommes des carrés d'erreurs dans cette écriture peut se réécrire comme une log-vraisemblance. Or rappelons qu'à une constante près, dans les modèles linéaires généralisés, la déviance est définie parDeviance() = -2log[ℒ]que l'on peut aussi noterDeviance(). On peut définir une déviance nulle comme celle obtenue sans utiliser les variables explicatives , de telle sorte que y_i=y. On peut alors définir, dans un contexte plus généralR^2= Deviance(y)-Deviance(y)/Deviance(y)=1- Deviance(y)/Deviance(y). Toutefois, cette mesure ne peut être utilisée pour choisir un modèle, si on souhaite avoir au final un modèle relativement simple, car elle augmente artificiellement avec l'ajout de variables explicatives sans effet significatif. On aura alors tendance à préférer le R^2 ajustéR̅^2 = 1-(1-R^2)n-1n-p = R^2-(1-R^2)p-1n-p_pénalisation,où p est le nombre de paramètres du modèle (noté plus généralement ν dans la section <ref>). À la mesure de la qualité de l'ajustement, on va pénaliser les modèles trop complexes.Cette idée va se retrouver dans le critère d'Akaike, où AIC = Deviance + 2· p ou dans le critère de Schwarz, BIC = Deviance + log(n)· p. En grande dimension (typiquement p>√(n)), on aura tendance à utiliser un AIC corrigé, défini parAICc = Deviance + 2· p·n/n-p-1 Ces critères sont utilisés dans les méthodes dites « stepwise », introduisant les méthodes ensemblistes. Dans la méthode dite « forward », on commence par régresser sur la constante, puis on ajoute une variable à la fois, en retenant celle qui fait le plus baisser le critère AIC, jusqu'à ce que rajouter une variable augmente le critère AIC du modèle. Dans la méthode dite « backward », on commence par régresser sur toutes les variables, puis on enlève une variable à la fois, en retirant celle qui fait le plus baisser le critère AIC, jusqu'à ce que retirer une variable augmente le critère AIC du modèle. Une autre justification de cette notion de pénalisation (nous reviendrons sur cette idée en apprentissage) peut être la suivante. Considérons un estimateur dans la classe des prédicteurs linéaires,ℳ={ m: m()=s_h() oùS=(s(_1),⋯,s(_n) est la matrice de lissage}et supposons que y=m_0(x)+ε, avec 𝔼[ε]=0 et Var[ε]=σ^2𝕀, de telle sorte quem_0(x)=𝔼[Y|X=x]. D'un point de vue théorique, le risque quadratique, associé à un modèle estimé m, s'écrit ℛ(m)=𝔼[(Y-m(X))^2] = 𝔼[(Y-m_0(X))^2]_erreur+ 𝔼[(m_0(X)-𝔼[m()])^2]_biais+ 𝔼[(𝔼[m()]-m())^2]_variancesi m_0 désigne le vrai modèle. Le premier terme est parfois appelé « erreur de Bayes », et ne dépend pas de l'estimateur retenu, m.Le risque empirique quadratique, associé à un modèle m, est ici :ℛ_n(m)=1/n∑_i=1^n (y_i-m(x_i))^2 =1/ny-m(x)^2(par convention). On reconnaît ici l'erreur quadratique moyenne, mse, qui donnera plus généralement le « risque » du modèle m quand on utilise une autre fonction de perte (comme nous le discuterons dans la partie suivante). Notons que:𝔼[ℛ_n(m)]=1/nm_0(x)-m(x)^2+1/n𝔼(y-m_0(x)^2)On peut montrer que :n𝔼[ℛ_n(m)] =𝔼( y-m(x)^2) = (𝕀-S)m_0^2+σ^2 𝕀-S^2de telle sorte que le (vrai) risque de m est :ℛ_n(m)= 𝔼[ℛ_n(m)]+2σ^2/ntrace(S).Aussi, si trace(S)≥ 0, le risque empirique sous-estime le vrai risque de l'estimateur. On reconnaît ici le nombre de degrés de liberté du modèle, le terme de droite correspondant au C_p de Mallow, introduit dans Mallows utilisant non pas la déviance mais le R^2. §.§ Économétrie et tests statistiques Le test le plus classique en économétrie est probablement le test de significativité, correspondant à la nullité d'un coefficient dans un modèle de régression linéaire. Formellement, il s'agit du test de H_0:β_k=0 contre H_1:β_k≠ 0. Le test de Student, basésur la statistique t_k=β_k/se_β_k, permet a priori de trancher entre les deux alternatives, à l'aide de la p-value du test, définie par ℙ[|T|>|t_k|] avec Tℒ∼ Std_ν, où ν est le nombre de degrés de liberté du modèle (ν=p+1 pour le modèle linéaire standard). En grande dimension, cette statistique est néanmoins d'un intérêt très limité, compte tenu d'un FDR (False Discovery Ratio) important. Classiquement, avec un niveau de significativité α=0.05, 5% des variables sont faussement significatives. Supposons que nous disposions de p=100 variables explicatives, mais que 5 (seulement) sont réellement significatives. On peut espérer que ces 5 variables passeront le test de Student, mais on peut aussi s'attendre à ce que 5 variables supplémentaires (test faussement positif) ressortent. On aura alors 10 variables perçues comme significatives, alors que seulement la moitié le sont, soit un ratio FDR de 50%. Afin d'éviter cet écueil récurent dans les tests multiples, il est naturel d'utiliser la procédure de Benjamini. §.§ Quitter la corrélation pour quantifier un effet causal Les modèles économétriques sont utilisés pour mettre en oeuvre des politiques publiques. Il est alors fondamental de bien comprendre les mécanismes sous-jacents pour savoir quelles variables permettent effectivement d'agir sur une variable d'intérêt. Mais on passe alors dans une autre dimension importante de l'économétrie. C'est à Jerry Neyman que l'on doit les premiers travaux sur l'identification de mécanismes causaux, c'est Rubin qui a formalisé le test, appelé « modèle causal de Rubin » dans Holland. Les premières approches autour de la notion de causalité en économétrie se sont faites avec l'utilisation des variables instrumentales, des modèles avec discontinuité de régression, l'analyse de différences dans les différences, ainsi que des expériences naturelles ou pas. La causalité est généralement déduite en comparant l'effet d'une politique - ou plus généralement d'un traitement - avec son contrefactuel, idéalement donné par un groupe témoin, aléatoire.L'effet causal du traitement est alors défini comme Δ=y_1-y_0, c'est à dire la différence entre ce que serait la situation avec traitement (noté t=1) et sans traitement (noté t=0). Le souci est que seul y=t· y_1+(1-t)y_0 et t sont observés. Autrement dit l'effet causal de la variable t sur y n'est pas observé (puisque seule une des deux variables potentielles - y_0 ou y_1 est observée pour chaque individu), mais il est aussi individuel, et donc fonction de covariables . Généralement, en faisant des hypothèses sur la distribution du triplet (Y_0,Y_1,T), certains paramètres de la distribution de l'effet causal deviennent identifiables, à partir de la densitédes variables observables (Y, T). Classiquement, on sera intéressé par les moments de cette distribution, en particulier l’effet moyen du traitement dans la population, 𝔼[Δ], voire juste l'effet moyen du traitement en cas de traitement 𝔼[Δ|T=1]. Si le résultat (Y_0,Y_1) est indépendant de la variable d’accès au traitement T, on peut montrer que 𝔼[Δ]=𝔼[Y|T=1]-𝔼[Y|T=0]. Mais si cette hypothèse d'indépendance n'est pas vérifiée, on a un biais de sélection, souvent associé à 𝔼[Y_0|T=1]-𝔼[Y_0|T=0]. RosenbaumRubin proposent d'utiliser un score de propension à être traité, p()=ℙ[T=1|X=x], en notant que si la variable Y_0 est indépendante de l'accès au traitement T conditionnellement aux variables explicatives , alors elle est indépendante de T conditionnellement au score p() : il suffit de les apparier à l'aide de leur score de propension. Heckman2 propose ainsi un estimateur à noyau sur le score de propension, ce qui permet d'avoir simplement un estimateur de l'effet du traitement, conditionnellement au fait d’être traité. § PHILOSOPHIE DES MÉTHODES DE MACHINE LEARNING Parallèlement à ces outils développés par et pour des économistes, toute une littérature a été développée sur des questions similaires, centrées autour de la prévision. Pour Breiman, une première différence vient du fait que la statistique s'est développée autour du principe d'inférence (ou d'expliciter la relation liant y aux variables ) alors qu'une autre culture s'intéresse avant tout à la prédiction. Dans une discussion qui suit l'article, David Cox l'affirme très clairement « predictive success (...) is not the primary basis for model choice ».Nous allons présenter les fondements des techniques du machine learning (les exemples d'algorithmes étant présentés dans les sections suivantes). Le point important, comme nous allons le voir, est que la principale préoccupation de l'apprentissage machine est liée aux propriétés de généralisation d'un modèle, c'est-à-dire sa performance - selon un critère choisi a priori - sur des données nouvelles, et donc des tests hors échantillon. §.§ Apprentissage par une machine Aujourd'hui, on parle de « machine learning » pour décrire tout un ensemble de techniques, souvent computationnelles, alternatives à l'approche décrite auparavant (correspondant à l'économétrie classique). Avant de les caractériser autant que possible, notons juste qu'historiquement d'autres noms ont pu être donnés. Par exemple, Friedman97 propose de faire le lien entre la statistique (qui ressemble beaucoup aux techniques économétriques - test d'hypothèses, ANOVA, régression linéaire, logistique, GLM, etc) et ce qu'il appelait alors « data mining » (qui englobait alors les arbres de décisions, les méthodes des plus proches voisins, les réseaux de neurones, etc.). Le pont qu'il contribuera à construire correspond aux techniques d'apprentissages statistiques, décrites dans HastieEtal, mais l'apprentissage machine est un très vaste champ de recherche.L'apprentissage dit « naturel » (par opposition à celui d'une machine) est celui des enfants, qui apprennent à parler, à lire, à jouer. Apprendre à parler signifie segmenter et catégoriser des sons, et les associer à des significations. Un enfant apprend aussi simultanément la structure de sa langue maternelle et acquiert un ensemble de mots décrivant le monde qui l'entoure. Plusieurs techniques sont possible, allant d'un apprentissage par coeur, par généralisation, par découverte, apprentissage plus ou moins supervisé ou autonome, etc. L'idée en intelligence artificielle est de s'inspirer du fonctionnement du cerveau pour apprendre, pour permettre un apprentissage « artificiel » ou « automatique », par une machine. Une première application a été d'apprendre à une machine à jouer à un jeux (tic-tac-toe, échecs, go, etc). Une étape indispensable est d'expliquer l'objectif qu'il doit atteindre pour gagner. Une approche historique a été de lui apprendre les règles du jeu. Si cela permet de jouer, cela ne permettra pas à la machine de bienjouer. En supposant que la machine connaisse les règles du jeu, et qu'elle a le choix entre plusieurs dizaines de coups possible, lequel doit-elle choisir ? L'approche classique en intelligence artificielle utilise l'algorithme dit min-max utilisant une fonction d'évaluation : dans cet algorithme, la machine effectue une recherche en avant dans l'arbre des coups possibles, aussi loin que les ressources de calcul le lui permettent (une dizaine de coups aux échecs, par exemple). Ensuite, elle calcule différents critères (qui lui ont été indiqué au préalable) pour toutes les positions (nombre de pièces prises, ou perdues, occupation du centre, etc. dans notre exemple du jeu d'échec), et finalement, la machine joue le coup qui lui permet de maximiser son gain. Un autre exemple peut être celui de la classification et de la reconnaissance d'images ou de formes. Par exemple, la machine doit identifier un chiffre dans une écriture manuscrite (chèque, code postal). Il s'agit de prédire la valeur d'une variable y, en sachant qu'a priori y∈{0,1,2,⋯,8,9}. Un stratégie classique est de fournir à la machine des bases d'apprentissage, autrement dit ici des millions d'images labélisées (identifiées) de chiffres manuscrits. Une stratégie simple et naturelle est d'utiliser un critère de décision basé sur les plus proches voisins dont on connaît l'étiquette (à l'aide d'une métrique prédéfinie).La méthode des plus proches voisins (« k-nearest neighbors ) peut être décrit de la manière suivante : on considère (comme dans la partie précédante) un ensemble de n observations, c'est à dire des paires (y_i,x_i) avec x_i∈ℝ^p. Considérons une distance Δ sur ℝ^p (la distance Euclienne ou la distance de Mahalanobis, par exemple). Étant donnée une nouvelle observation x∈ℝ^p, supposons les observations ordonnées en fonction de la distance entre les x_i et x, au sens oùΔ(x_1,x)≤Δ(x_2,x)≤⋯≤Δ(x_n,x)alors on peut considérer comme prédiction pour y la moyenne des k plus proches voisins,m_k(x)=1/k∑_i=1^k y_i.L'apprentissage fonctionne ici par induction, à partir d'un échantillon (appelé base d'apprentissage).Le Machine Learning englobe ces algorithmes qui donnent aux ordinateurs la capacité d'apprendre sans être explicitement programmé (comme l'avait défini Arthur Samuel en 1959). La machine va alors explorer les données avec un objectif précis (comme chercher les plus proches voisins dans l'exemple que nous venons de décrire). Tom Mitchell a proposé une définition plus précise en 1998 : on dit qu'un programme d'ordinateur apprend de l'expérience E par rapport à une tâche T et une mesure de performance P, si sa performance sur T, mesurée par P, s'améliore avec l'expérience E. La tâche T peut être un score de défaut par exemple, et la performance P peut être le pourcentage d'erreurs commise. Le système apprend si le pourcentage de défauts prédit augmente avec l'expérience.On le voit, l'apprentissage machine est fondamentalement un problème d'optimisation d'un critère à partir de données (dites d'apprentissage). Nombreux sont les ouvrages de programmation qui proposent des algorithmes, sans jamais faire mention d'un quelconque modèle probabiliste. Dans Watt par exemple, il n'est fait mention du mot « probabilité » qu'une seule fois, avec cette note de bas de page qui surprendra et fera sourire les économètres, « logistic regression can also be interpreted from a probabilistic perspective » (page 86). Mais beaucoup d'ouvrages récents proposent une relecture des approches d'apprentissage machine à l'aide de théories probabilistes, suite aux travaux de Vaillant et Vapnik. En proposant le paradigme de l'apprentissage « probablement à peu près correct » (PAC), une saveur probabiliste a été rajouté à l'approche jusqu'alors très computationnelle, en quantifiant l'erreur de l'algorithme d'apprentissage (dans un problème de classification). §.§ Le tournant des années 80/90 et le formalisme probabiliste On dispose d'un échantillon d'apprentissage, avec des observations (x_i,y_i) où les variables y sont dans un ensemble 𝒴. Dans le cas de la classification, 𝒴={-1,+1}, mais on peut imaginer un ensemble relativement général. Un prédicteur est une fonction m à valeurs dans 𝒴, permettant d'étiqueter (ou de classer) les nouvelles observations à venir. On suppose que les étiquettes sont produites par un classifieur f appelé cible. Pour un statisticien, cette fonction serait le vrai modèle. Naturellement, on veut construire m le plus proche possible de f. Soit ℙ une distribution (inconnue) sur 𝒳. L'erreur de m relativement à la cible f est définie parℛ_ℙ,f(m)=ℙ[m(X)≠ f(X)] oùX∼ℙ,ou écrit de manière équivalente,ℛ_ℙ,f(m)=ℙ[{x∈𝒳:m(x)≠ f(x)}].Pour trouver notre classifieur, il devient nécessaire de supposer qu'il existe un lien entre les données de notre échantillon et le couple (ℙ,f), c'est à dire un modèle de génération des données. On va alors supposer que les x_i sont obtenus par des tirages indépendants suivant ℙ, et qu'ensuite y_i=f(x_i) .On peut ici définir le risque empirique d'un modèle m,ℛ(m)=1/n∑_i=1^n 1(m(x_i)≠ y_i). Il est alors important d'admettre qu'on ne peut pas trouver un modèle parfait, au sens où ℛ_ℙ,f(m)=0. En effet, si on considère le cas le plus simple qui soit, avec 𝒳={_1,_2} et que ℙ soit telle que ℙ({_1})=p et ℙ({_2})=1-p. Il est aussi possible d'observer x_1 et x_2, et malgré tout, de se tromper sur les étiquettes. Aussi, au lieu de chercher un modèle parfait, on peut tenter d'avoir un modèle approximativement correct. On va alors chercher à trouver m tel que ℛ_ℙ,f(m)≤ϵ, où ϵ est un seuil spécifié a priori.Une fois admis ce premier écueil, qui fait penser à l'errreur de modèle, notons aussi un second problème. Sur notre exemple à deux valeurs, la probabilité de ne jamais observer x_2 parmi n tirages suivant ℙ est p^n. Il sera alors impossible de trouver m(x_2) car cette valeur n'aura jamais été observée. Autrement dit, aucun algorithme ne peut nous assurer d'avoir avec certitude, avec n observations, ℛ_ℙ,f(m)≤ϵ. On va alors chercher à être probablement approximativement correct (PAC). Pour se faire, on autorise l'algorithme à se tromper avec une probabilité δ, là aussi fixée a priori.Aussi, quand on construit un classifieur, on ne connaît ni ℙ, ni f, mais on se donne un critère de précision ϵ, et un paramètre de confiance δ, et on dispose de n observations. Notons que n, ϵ et δ peuvent être liés. On cherche alors un modèle m tel que ℛ_ℙ,f(m)≤ϵ avec probabilité (au moins) 1-δ, de manière à être probablement approximativement correct.Wolpert96 a montré (détaillé dans Wolpert97) qu'il n'existe pas d'algorithme d'apprentissage universel. En particulier, on peut montrer qu'il existe ℙ telle que ℛ_ℙ,f(m) soit relativement grande, avec une probabilité (elle aussi) relativement grande.L'interprétation qui en est faite est qu'il est nécessaire d'avoir un biais pour apprendre. Comme on ne peut pas apprendre (au sens PAC) sur l'ensemble des fonctions m, on va alors contraindre m à appartenir une classe particulière, notée ℳ. Supposons pour commencer que ℳ contienne un nombre fini de modèles possibles. On peut alors montrer que pour tout ϵ et δ, que pour tout ℙ et f, si on dispose d'assez d'observations (plus précisément n≥ϵ^-1log[δ^-1|ℳ|], alors avec une probabilité plus grande que 1-δ, ℛ_ℙ,f(m^⋆)≤ϵ oùm^⋆∈m∈ℳargmin{1/n∑_i=1^n 1(m(x_i)≠ y_i)}autrement dit m^⋆ est un modèle dans ℳ qui minimise le risque empirique.Un peut aller un peu plus loin, en restant dans le cas où 𝒴={-1,+1}. Une classe ℳ de classifieurs sera dite PAC-apprenable s'il existe n_ℳ:[0,1]^2→ℕ tel que, pour tout ϵ, δ, ℙ et si on suppose que la cible f appartient à ℳ, alors en utilisant n>n_ℳ(ϵ,δ) tirages d'observations x_i suivant ℙ, étiquetés y_i par f, alors il existe m∈ℳ tel que, avec probabilité 1-δ, ℛ_ℙ,f(m)≤ϵ. La fonction n_ℳ est alors appelée complexité d'échantillon pour apprendre. En particulier, nous avons vu que si ℳ contient un nombre fini de classifieurs, alors ℳ est PAC-apprenable avec la complexité n_ℳ(ϵ,δ)=ϵ^-1log[δ^-1|ℳ|].Naturellement, on souhaiterait avoir un résultat plus général, en particulier si ℳ n'est pas fini. Pour cela, il faut utiliser la dimension VC de Vapnik-Chervonenkis, qui repose sur l'idée de pulvérisation de nuages de points (pour une classification binaire). Considérons k points {x_1,⋯x_k}, et considérons l'ensemble ℰ_k={(m(x_1),⋯,m(x_k)) pour m∈ℳ)}.Notons que les éléments de ℰ_k appartiennent à {-1,+1}^k. Autrement dit |ℰ_k|≤ 2^k. On dira que ℳ pulvérise l'ensemble des points si toutes les combinaisons sont possibles, c'est à dire |ℰ_k|= 2^k. Intuitivement, les étiquettes de l'ensemble de points ne procurent pas assez d'information sur la cible f, car tout est possible. La dimension VC de ℳ est alors VC(ℳ)=sup{ k tel que ℳ pulvérise {x_1,⋯x_k}}.Par exemple si 𝒳=ℝ et que l'on considère l'ensemble des modèles (simples) de la forme m_a,b=1_±(x∈[a,b]). Aucun ensemble de points {x_1,x_2,x_3} ordonnés ne peut être pulvérisé car il suffit d'assigner respectivement +1, -1 et +1 à x_1, x_2 et x_3 respectivement, donc VC<3. En revanche {0,1} est pulvérisé donc VC≥ 2. La dimension de cet ensemble de prédicteur est 2.Si on augmente d'une dimension, 𝒳=ℝ^2 et que l'on considère l'ensemble des modèles (simples) de la forme m_a,b=1_±(x∈[a,b]) (où [a,b] désigne le rectangle), alors la dimension de ℳ est ici 4.Pour introduire les SVM, plaçons nous dans le cas où 𝒳=ℝ^k, et considérons des séparations par des hyperplans passant par l'origine (on dira homogènes), au sens où m_w(x)=1_±(w^ Tx≥ 0). On peut montrer qu'aucun ensemble de k+1 points ne peut être pulvérisé par ces deux espaces homogènes dans ℝ^k, et donc VC(ℳ)=k. Si on rajoute une constante, au sens où m_w,b(x)=1_±(w^ Tx+b≥ 0), on peut montrer qu'aucun ensemble de k+2 points ne peut être pulvérisé par ces deux espaces (non homogènes) dans ℝ^k, et donc VC(ℳ)=k+1.De cette dimension VC, on déduit le théorème dit fondamental de l'apprentisssage : si ℳ est une classe de dimension d=VC(ℳ), alors il existe des constante positives C et C telles que la complexité d'échantillon pour que ℳ soit PAC-apprenable vérifieCϵ^-1(d+log[δ^-1]) ≤ n_ℳ(ϵ,δ) ≤Cϵ^-1(dlog[ϵ^-1]+log[δ^-1]).§.§ Le choix de l'objectif et la fonction de perte Ces choix sont essentiels, et dépendent du problème considéré. Commençons par décrire un modèle historiquement important, le « perceptron » de Rosenblatt, introduit dans des problèmes de classification, où y∈{-1,+1}, inspiré par McCulloch. On dispose de données {(y_i,_i)}, et on va construire de manière itérative un ensemble de modèles m_k(·), où à chaque étape, on va apprendre des erreurs du modèle précédent. Dans le perceptron, on considère un modèle linéaire de telle sorte que :m()=1_±(β_0+β≥ 0)={[ +1 si β_0+β≥ 0; -1 si β_0+β< 0 ].,où les coefficientssont souvent interprétés comme des « poids » attribués à chacune des variables explicatives. On se donne des poids initiaux (β_0^(0),β^(0)), que l'on va mettre à jour en tenant compte de l'erreur de prédiction commise, entre y_i et la prédiction y_i^(k) :y_i^(k)=m^(k)(_i)=1_±(β_0^(k)+β^(k)≥ 0), avec, dans le cas du perceptron :β_j^(k+1)=β_j^(k)+η(y-y^(k))^ T_=ℓ(y,y^(k))_joù ici ℓ(y,y')=1(y ≠ y') est une fonction de perte, qui permettra de donner un prix à une erreur commise, en prédisant y'=m() et en observant y. Pour un problème de régression, on peut considérer une erreur quadratique ℓ_2, telle que ℓ(y,m())=(y - m())^2 ou en valeur absolue ℓ_1, avec ℓ(y,m())=| y - m()|. Ici, pour notre problème de classification, nous utilisions une indicatrice de mauvaise qualification (on pourrait discuter le caractère symétrique de cette fonction de perte, laissant croire qu'un faux positif coûte autant qu'un faux négatif). Une fois spécifiée cette fonction de perte, on reconnaît dans le problème décrit aurapavant une descente de gradient, et on voit que l'on cherche à résoudre :m^⋆()=m∈ℳargmin{∑_i=1^n ℓ(y_i,m(_i)) }pour un ensemble de prédicteurs ℳ prédéfini.Tout problème d'apprentissage machine est mathématiquement formulé comme un problème d'optimisation, dont la solution détermine un ensemble de paramètres de modèle (si la famille ℳ est décrite par un ensemble de paramètres - qui peuvent être des coordonnées dans une base fonctionnelle). On pourra noter ℳ_0 l'espace des hyperplans de ℝ^p au sens oùm∈ℳ_0signifie m()=β_0+ avec ∈ℝ^p,engendrant la classe des prédicteurs linéaires. On aura alors l'estimateur qui minimise le risque empirique. Une partie des travaux récents en apprentissage statistique vise à étudier les propriétés de l'estimateur m^⋆, dit « oracle », dans une famille d'estimateurs ℳ,m^⋆ = m∈ℳargmin{ℛ(m,m) }.Cet estimateur est, bien entendu, impossible à définir car il dépend de m, le vrai modèle, inconnu.Mais revenons un peu davantage sur ces fonctions de perte. Une fonction de perte ℓ est une fonction ℝ^d×ℝ^d→ℝ_+, symmétrique, qui vérifie l'inégalité triangulaire, et telle que ℓ(,)=0 si et seulement si =. La norme associée est ·, telle que ℓ(,)=-=ℓ(-,0) (en utilisant le fait que ℓ(,+z)=ℓ(-,z) - nous reverrons cette propriété fondamentale par la suite).Pour une fonction de perte quadratique, on notera que l'on peut avoir une interprétation particulière de ce problème, puisque :y = m∈ℝargmin{∑_i=1^n 1/n [y_i-m]^2 } = m∈ℝargmin{∑_i=1^nℓ_2(y_i,m) },où ℓ_2 est la distance quadratique usuelle Si l'on suppose - comme on le faisait en économétrie - qu'il existe un modèle probabiliste sous-jacent, et en notant que :𝔼(Y) = m∈ℝargmin{‖ Y-m‖^2_ℓ_2} =m∈ℝargmin{𝔼( [Y-m]^2 )}= m∈ℝargmin{𝔼[ℓ_2(Y,m)] }on notera que ce que l'on essaye d'obtenir ici, en résolvant le problème (<ref>) en prenant pour ℓ la norme ℓ_2, est une approximation (dans un espace fonctionnel donné, ℳ) de l'espérance conditionnelle ↦[Y|=]. Une autre fonction de perte particulièrement intéressante est la perte ℓ_1, ℓ_1(y,m)=| y-m|. Rappelons quemédiane(y) =m∈ℝargmin{∑_i=1^nℓ_1(y_i,m) }.Le problème d'optimisation :m^⋆=m∈ℳ_0argmin{∑_i=1^n |y_i-m(_i)|}est obtenu en économétrie si on suppose que la loi conditionnelle de Y suit une loi de Laplace centrée sur m(), et en maximisant la (log) vraisemblance (la somme des valeurs absolues de erreurs correspond à la log-vraimenblance d'une loi de Laplace). On pourra noter d'ailleurs que si la loi conditionnelle de Y est summétrique par rapport à 0, la médiane et la moyenne conïncident Si on réécrit cette fonction de perte ℓ_1(y,m)=| (y-m)(1/2-1_y≤ m)|, on peut obtenir une généralisation pour τ∈(0,1) :m^⋆_τ=m∈ℳ_0argmin{∑_i=1^n ℓ_τ^ q (y_i,m(_i))} avec ℓ_τ^ q(x,y)= (x-y)(τ-1_x≤ y)est alors la régression quantile de niveau τ (voir Koenker et dHaultefoeuille). Une autre fonction de perte, introduite par Aigneretal et analysée dans Walltrup, est la fonction associée à la notion d'expectiles :ℓ^ e_τ(x,y)= (x-y)^2·|τ-1_x≤ y|avec τ∈[0,1]. On voit le parallèle avec la fonction quantile :ℓ^ q_τ(x,y)= | x-y|·|τ-1_x≤ y|.KoenkerMachado et YuMoyeed ont d'ailleurs noté un lien entre cette condition et la recherche du maximum de vraisemblance lorsque la loi conditionnelle de Y suit une loi de Laplace assymétrique.En lien avec cette approche, Gneiting a introduit la notion de « statistique ellicitable » - ou de « mesureellicitable » dans sa version probabiliste (ou distributionnelle) : T sera dite « ellicitable » s'il existe une fonction de perte ℓ: ℝ×ℝ→ℝ_+ telle que :T(Y)=x∈ℝargmin{∫_ℝℓ(x,y)dF(y)} =x∈ℝargmin{𝔼[ ℓ(x,Y)] oùYℒ∼ F }.La moyenne (espérance mathématique) est ainsi ellicitable par la distance quadratique, ℓ_2, alors que la médiane est ellicitable par la distance ℓ_1. Selon Gneiting, cette propriété est essentielle pour construire des prédictions. Il peut alors exister un lien fort entre des mesures associées à des modèles probabilistes et les fonctions de perte. Enfin, la statistique Bayésienne propose un lien direct entre la forme de la loi a priori et la fonction de perte, comme l'ont étudié Berger et Bernardo. Nous reviendrons sur l'utilisation de ces différentes normes dans la section sur la pénalisation.§.§ Boosting et apprentissage séquentiel Nous l'avons vu auparavant: la modélisation repose ici sur la résolution d'un problème d'optimisation, et résoudre le problème décrit par l'équation (<ref>) est d'autant plus complexe que l'espace fonctionnel ℳ est volumineux. L'idée du Boosting,tel qu'introduit par ShapireFreund, est d'apprendre, lentement, à partir des erreurs du modèle, de manière itérative. À la première étape, on estime un modèle m_1 pour , à partir de , qui donnera une erreur _1. À la seconde étape, on estime un modèle m_2 pour _1, à partir de , qui donnera une erreur _2, etc. On va alors retenir comme modèle, au bout de k itération :m^(k)(·)=m_1(·)_∼+m_2(·)_∼_1+m_3(·)_∼_2+⋯+ m_k(·)_∼_k-1=m^(k-1)(·)+m_k(·).Ici, l'erreur ε est vue comme la différence entre y et le modèle m(), mais elle peut aussi être vue comme le gradient associé à la fonction de perte quadratique. Formellement,peut être vu comme un ∇ℓ dans un contexte plus géneral (on retrouve ici une interprétation qui fait penser aux résidus dans les modèles linéaires généralisés).L'équation (<ref>) peut se voir comme une descente du gradient, mais écrit de manière duale. En effet, la descente de gradient permettant d'obtenir le minimum d'une fonction f repose sur une équation de la formef(x_k)_⟨ f,x_k ⟩∼f(x_k-1)_⟨ f,x_k-1⟩+ (x_k-x_k-1) _α_k∇f(x_k-1)_⟨∇ f,x_k-1⟩Le problème (<ref>) est dual dans le sens où c'est la fonction f qui doit être optimisée. On pourrait alors écrire une descente de gradient de la forme :f_k(x)_⟨ f_k,x⟩∼f_k-1(x)_⟨ f_k-1,x⟩+ (f_k-f_k-1)_β_k⋆_⟨f_k-1,∇x⟩où le terme ⋆ peut être interprété comme un gradient, mais dans un espace fonctionnel, et non plus dans ℝ^p. Le problème (<ref>) va alors se réécrire comme un problème d'optimisation :m^(k)=m^(k-1)+h∈ℋargmin{∑_i=1^n ℓ(y_i-m^(k-1)(x_i)_ε_k,i,h(x_i))}où l'astuce consiste à considérer un espace ℋ relativement simple (on parlera de « weak learner » ). Classiquement, les fonctions ℋ sont des fonctions en escalier (que l'on retrouvera dans les arbres de classification et de régression) appelés stumps. Afin de s'assurer que l'apprentissage est effectivement lent, il n'est pas rare d'utiliser un paramètre de « shrinkage », et au lieu de poser, par exemple, ε_1=y-m_1(), on posera ε_1=y-α· m_1() avec α∈[0,1]. On notera que c'est parce qu'on utilise pour ℋ un espace non-linéaire, et que l'apprentissage est lent, que cet algorithme fonctionne bien. Dans le cas du modèle linéaire Gaussien, rappelons en effet que les résidus =- sont orthogonaux aux variables explicatives, , et il est alors impossible d'apprendre de nos erreurs. La principale difficulté est de s'arrêter à temps, car après trop d'itérations, ce n'est plus la fonction m que l'on approxime, mais le bruit. Ce problème est appelé sur-apprentissage.Cette présentation a l'avantage d'avoir une heuristique faisant penser à un modèle économétrique, en modélisant de manière intérative les résidus par un modèle (très) simple. Mais ce n'est souvent pas la présentation retenue dans la littérature en apprentissage, qui insiste davantage sur une heuristique d'algorithme d'optimisation (et d'approximation du gradient). La fonction est apprise de manière itérative, en partant d'une valeur constante,m^(0)=m∈ℝargmin{∑_i=1^n ℓ(y_i,m) }.puis on considère l'apprentissage suivantm^(k)=m^(k-1)+h∈ℋargmin∑ _i=1^nℓ(y_i,m^(k-1)(_i)+h(_i)),qui peut s'écrire, si ℋ est un ensemble de fonctions différentiables,m^(k)=m^(k-1)-γ _k∑ _i=1^n∇ _m^(k-1)ℓ(y_i,m^(k-1)(_i)),oùγ _k=γargmin ∑ _i=1^nℓ(y_i,m^(k-1)(_i)-γ∇ _m^(k-1)ℓ(y_i,m^(k-1)(_i))).Pour mieux comprendre le lien avec l'approche décrite auparavant, à l'étape k, on définit des pseudo-résidus en posantr_i,k=-.∂ℓ(y_i,m(_i))/∂ m(_i)|_m()=m^(k-1)() pour i=1,⋯,n.On cherche alors un modèle simple pour expliquer ces pseudo-résidus en fonction des variables explicatives _i, i.e. r_i,k=h^⋆(_i), où h^⋆∈ℋ. Dans un second temps, on cherche un multiplicateur optimal en résolvantγ_k = γ∈ℝargmin{∑_i=1^n ℓ(y_i,m^(k-1)(_i)+γ h^⋆(_i)) }puis on met à jour le modèle en posant m_k()=m_k-1()+γ_k h^⋆(). Plus formellement, on passe de l'équation (<ref>) - qui montre clairement qu'on construit un modèle sur les résidus - à l'équation (<ref>) - qui sera ensuite retraduit comme une problème de calcul de gradient - en notant que ℓ(y,m+h)=ℓ(y-m,h). Classiquement, les fonctions ℋ sont construites avec des arbres de régression. Il est aussi possible d'utiliser une forme de pénalisation en posant m_k()=m_k-1()+νγ_k h^⋆(), avec ν∈(0,1). Mais revenons un peu plus longuement sur l'importance de la pénalisation avant de discuter les aspects numériques de l'optimisation. §.§ Pénalisation et choix de variables Dans la section <ref>, nous avions évoqué le principe de parcimonie, populaire en économétrie. Le critère d'Akaike était basé sur une pénalisation de la vraisemblance en tenant compte de la complexité du modèle (le nombre de variables explicatives retenues). Si en économétrie, il est d'usage de maximiser la vraisemblance (pour construire un estimateur asymptotiquement sans biais), et de juger de la qualité du modèle ex-post en pénalisant la vraisemblance, la stratégie ici sera de pénaliser ex-ante dans la fonction objectif, quitte à construire un estimateur biaisé. Typiquement, on va construire :(β_0,λ,_λ)=argmin{∑_i=1^n ℓ(y_i,β_0+)+λ pénalisation( β) },où la fonction de pénalisation sera souvent une norme ‖·‖ choisie a priori, et un paramètre de pénalisation λ (on retrouve en quelque sorte la distinction entre AIC et BIC si la fonction de pénalisation est la complexité du modèle - le nombre de variables explicatives retenues). Dans le cas de la norme ℓ_2, on retrouve l'estimateur Ridge, et pour la norme ℓ_1, on retrouve l'estimateur lasso (« Least Absolute Shrinkage and Selection Operator »). La pénalisation utilisée auparavant faisait intervenir le nombre de degrés de liberté du modèle, il peut alors paraître surprenant de faire intervenir ‖β‖_ℓ_2 comme dans la régression Ridge. On peut toutefois envisager une vision Bayésienne de cette pénalisation. Rappelons que dans un modèle Bayésien :ℙ[θ|y]_a posteriori∝ℙ[y|θ]_vraisemblance·ℙ[θ]_a priori soit logℙ[θ|y]= logℙ[y|θ]_log vraisemblance + logℙ[θ]_pénalisation.Dans un modèle linéaire Gaussien, si on suppose que la loi a priori de θ suit une loi normale centrée, on retrouve une pénalisation basée sur une forme quadratique des composantes de θ.Avant de revenir en détails sur ces deux estimateurs, obtenus en utilisant la norme ℓ_1 ou la norme ℓ_2, revenons un instant sur un problème très proche : celui du meilleur choix de variables explicatives. Classiquement (et ça sera encore plus vrai engrande dimension), on peut disposer d'un grand nombre de variables explicatives, p, mais beaucoup sont juste du bruit, au sens où β_j=0 pour un grand nombre de j. Soit s le nombre de covariables (réellement) pertinentes, s=#𝒮 avec 𝒮()={ j=1,⋯,p; β_j≠ 0}. Si on note _𝒮 la matrice constituée des variables pertinentes (en colonnes), alors on suppose que le vrai modèle est de la forme y=_𝒮_𝒮+ε. Intuitivement, un estimateur intéressant serait alors β_𝒮=[_𝒮_𝒮]^-1_𝒮, mais cet estimateur n'est que théorique car 𝒮 est ici inconnue. Cet estimateur est l'estimateur oracle évoqué auparavant. On peut alors être tenté de résoudre (β_0,s,_s)=argmin{∑_i=1^n ℓ(y_i,β_0+)}, sous la contrainte #𝒮()=s. Ce problème a été introduit par FosterGeorge94 en introduisant la norme ℓ_0. Plus précisément, définissons ici les trois normes suivantes‖a‖_ℓ_0= ∑_i=1^d 1(a_i≠ 0), ‖a‖_ℓ_1= ∑_i=1^d |a_i| et ‖a‖_ℓ_2=( ∑_i=1^d a_i^2)^1/2, pour a∈ℝ^d. Considérons les problèmes d'optimisation de la Table <ref>. Si on considère le problème classique où ℓ est la norme quadratique, les deux problèmes de l'équation (ℓ1) de la Table <ref> sont équivalents, au sens où, pour toute solution (β^⋆,s^⋆) au problème de gauche, il existe λ^⋆ tel que (β^⋆,λ^⋆) soit solution du problème de droite; et inversement. Le résultat est également vrai pour les problèmes (ℓ2)[ Pour (ℓ1), s'il y a équivalence au niveau théorique, il peut exister des soucis numériques car il n'y a pas forcément unicité de la solution.]. Il s'agit en effet de problèmes convexes. En revanche, les deux problèmes (ℓ0) ne sont pas équivalents : si pour (β^⋆,λ^⋆) solution du problème de droite, il existe s^⋆ tel que β^⋆ soit solution du problème de gauche, la récriproque n'est pas vraie. Plus généralement, si on veut utiliser une norme ℓ_p, la sparsitée est obtenue si p≤ 1 alors qu'il faut avoir p≥ 1 pour avoir la convexité du programme d'optimisation.On peut être tenté de résoudre le programme pénalisé (ℓ0) directement, comme le suggère FosterGeorge94. Numériquement, c'est un problème combinatoire complexe en grande dimension (Natarajan note que c'est un problème NP-difficile), mais il est possible de montrer que si λ∼σ^2 log(p), alors𝔼([-_0]^2) ≤𝔼(_𝒮_𝒮-_0]^2)_=σ^2 #𝒮·(4log p+2+o(1)).Notons quand dans ce cas_λ,j^ sub = {[ 0sij∉𝒮_λ(); _j^ ols sij∈𝒮_λ(), ].où 𝒮_λ() désigne l'ensemble des coordonnées non nulle lors de la résolution de (ℓ0).Le problème(ℓ2) est strictement convexe si ℓ est la norme quadratique, autrement dit, l'estimateur Ridge est toujours bien défini, avec en plus une forme explicite pour l'estimateur, β_λ^ ridge=(+λ𝕀)^-1 =(+λ𝕀)^-1()β^ ols.Aussi, on peut en déduire quebiais[β_λ^ ridge]=-λ[+λ𝕀]^-1 β^ ols et Var[β_λ^ ridge]= σ^2[+λ𝕀]^-1[+λ𝕀]^-1.Avec une matrice de variables explicatives orthonormées (i.e. =𝕀), les expressions se simplifientbiais[β_λ^ ridge]=λ/1+λ β^ ols et Var[β_λ^ ridge]=σ^2/(1+λ)^2𝕀.Notons que Var[β_λ^ ridge]<Var[β^ ols]. En notant quemse[β_λ^ ridge]=kσ^2/(1+λ)^2+λ^2/(1+λ)^2,on obtient une valeur optimale pour λ: λ^⋆=kσ^2 /. En revanche, si ℓ n'est plus la norme quadratique mais la norme ℓ_1, le problème(ℓ1) n'est pas toujours strictement convexe, et en particulier, l'optimum n'est pas toujours unique (par exemple siest singulière). Mais le fait que ℓ soit strictement convexe β sera unique. Notons de plus que deux solutions sont forcément cohérentes en terme de signe des coefficients : il n'est pas possible d'avoir β_j<0 pour une solution et β_j>0 pour une autre. D'un point de vue heuristique, le programme (ℓ1) est intéressant car il permet d'obtenir dans bon nombre de cas une solution en coin, qui correspond à une résolution de problème de type (ℓ0) - comme le montre de manière visuelle la Figure <ref>. Considérons un modèle très simple: y_i=x_iβ+ε, avec une pénalité ℓ_1 et une fonction de perte ℓ_2. Le problème (ℓ2) sécrit alorsmin{-2β+ββ+2λ|β|}La condition du premier ordre est alors-2 + 2β± 2λ=0.le signe du dernier terme dépend du signe de β. Supposons que l'estimateur par moindre carrés (obtenu en posant λ=0) soit (strictement positif), autrement dit >0. Si λ n'est pas trop grand, on peut imaginer que β soit du même signe que β^ mco, et donc la condition devient-2 + 2β+ 2λ=0.et la solution est β_λ^ lasso=-λ/.En augmentant λ, on va arriver à un moment où β_λ=0. Si on augmente encore un peu β_λ ne devient pas négatif car dans ce cas le dernier terme de la condition du premier ordre change, et dans ce cas on cherche à résoudre-2 + 2β- 2λ=0.dont la solution est alorsβ_λ^ lasso=+λ/.Mais cette solution est positive (nous avions supposé >0), et donc il est possible d'avoir en même temps β_λ<0. Aussi, au bout d'un moment, β_λ=0, qui est alors une solution de coin. Les choses sont bien entendu plus compliquées en dimension plus grande (TibWasserman revient longuement sur la géométrie des solutions) mais comme le note Candes, sous des hypothèses minimales garantissant que les prédicteurs ne sont pas fortement corrélées, le Lasso obtient une erreur quadratique presque aussi bonne que si l'on dispose d'un oracle fournissant des informations parfaites sur quels β_j sont non nulles. Moyennant quelques hypothèses techniques supplémentaires, on peut montrer que cet estimateur est « sparsistent »au sens où le support de β_λ^ lasso est celui de , autement dit Lasso a permis de faire de la sélection de variables (plus de discussions sur ce point peuvent être obtenues dans HTT). De manière plus générale, on peut montrer que β_λ^ lasso est un estimateur biaisé, mais qui peut être de variance suffisamment faible pour que l'erreur quadratique moyenne soit plus faible qu'en utilisant des moindres carrés. Pour comparer les trois techniques, par rapport à l'estimateur par moindre carrées (obtenu quand λ=0), si on suppose que les variables explicatives sont orthonormées, alorsβ_λ,j^ sub=β_j^ ols1_|β_λ,j^ sub|>b, β_λ,j^ ridge=β_j^ ols/1+λ et β_λ,j^ lasso=signe[β_j^ ols]·(|β_j^ ols|-λ)_+.§.§ Optimisation et aspects algorithmiques En économétrie, l'optimisation (numérique) est devenu omniprésente dès que l'on a quitté le modèle Gaussien. Nous l'avions rapidement évoqué dans la section sur la famille exponentielle, et l'utilisation du score de Fisher (descente de gradient) pour résoudre la condition du premier ordre X^TW(β)^-1[y-]=0. En apprentissage, l'optimisation est l'outil central. Et il est nécessaire d'avoir des algorithmes d'optimisation efficaces, pour résoudre des problèmes de la forme :∈∈ℝ^pargmin{∑_i=1^n ℓ(y_i,β_0+)+λ‖β‖}. Dans certains cas, au lieu de faire de l'optimisation globale, il suffit de considérer de l'optimisation par coordonnées (largement étudiée dans Daubechies). Si f:ℝ^d→ℝ est convexe et différentiable, alors sivérifie f(+he_i)≥ f() pour tout h>0 et i∈{1,⋯, d}, alorsf()=min{f},où e=(e_i) est la base canonique deℝ^d. Cette propriété n'est toutefois pas vraie dans le cas non-différentiable. Mais si on suppose que la partie non-différentiable est séparable (additivement), elle redevient vraie. Plus précisément, si f()=g()+∑_i=1^d h_i(x_i) avec {[ g: ℝ^d→ℝ convexe-différentiable; h_i: ℝ→ℝ convexe. ].C'est le cas pour la régression Lasso, f()=-X_ℓ_2+λ_ℓ_1, comme le montre Tsen. On peut alors utiliser un algorithme de descente par coordonnées: à partir d'une valeur initiale ^(0), on considère (en itérant)x_j^(k)∈argmin{ f(x_1^(k),⋯,x_k-1^(k),x_k,x_k+1^(k-1), ⋯,x_n^(k-1)) } pour j=1,2,⋯,n.Ces problèmes algorithmiques peuvent paraître secondaires à des économètres. Ils sont pourtant essentiels en apprentissage machine: une technique est intéressante s'il existe un algorithme stable et rapide, qui permet d'obtenir une solution. Ces techniques d'optimisation sont d'ailleurs transposables : par exemple, on pourra utiliser cette technique de descente par coordonnées dans le cas des méthodes svm (dit « à support vecteur») lorsque l'espace n'est pas linéairement séparable, et qu'il convient de pénaliser l'erreur de classification (nous reviendrons sur cette technique dans la prochaine section).§.§ In-sample, out-of-sample et validation croisée Ces techniques semblent intellectuellement intéressantes, mais nous n'avons pas encore abordé le choix du paramètre de pénalisation λ. Mais ce problème est en fait plus général, car comparer deux paramètres _λ_1 et _λ_2 revient en fait à comparer deux modèles. En particulier, si on utilise une méthode de type Lasso, avec des seuils λ différents, on compare des modèles qui n'ont pas la même dimension. Dans la section <ref>, nous avions abordé le problème de la comparaison de modèles sous l'angle économétrique (en pénalisant les modèles trop complexes). Dans la littérature en apprentissage, juger de la qualité d'un modèle sur les données qui ont servi à le construire ne permet en rien de savoir comment le modèle se comportera sur des nouvelles données. Il s'agit du problème dit de « généralisation ». L'approche classique consiste alors à séparer l'échantillon (de taille n) en deux : une partie qui servira à entraîner le modèle (la base d'apprentissage, in-sample, de taille m) et une partie qui servira à tester le modèle (la base de test, out-of-sample, de taille n-m). Cette dernière permet alors de mesure un vrai risque prédictif. Supposons que les données soient générées par un modèle linéaire y_i=_i_0 + ε_i où les ε_i sont des réalisations de lois indépendantes et centrées. Le risque quadratique empirique in-sample est ici1/m∑_i=1^m𝔼([_i-_i_0]^2)=𝔼([_i-_i_0]^2),pour n'importe quelle observation i. En supposant les résidus ε Gaussiens, alors on peut montrer que ce risque vaut σ^2trace(Π_𝒳)/m soit σ^2p/m. En revanche le risque quadratique empirique out-of-sample est ici𝔼([-_0]^2)oùest une nouvelle observation, indépendante des autres. On peut noter que 𝔼([-_0]^2|)=σ^2()^-1,et en intégrant par rapport à ,𝔼([-_0]^2)= 𝔼(𝔼([-_0]^2|))= σ^2trace(𝔼[]𝔼[()^-1]).L'expression est alors différente de celle obtenue in-sample, et en utilisation la majoration de GrovesRothenberg69, on peut montrer que𝔼([-_0]^2) ≥σ^2p/m,ce qui est assez intuitif, finalement. Hormis certains cas simple, il n'y a pas de formule simple. Notons toutefois que si ∼𝒩(0,σ^2𝕀), alorssuit une loi de Wishart, et on peut montrer que𝔼([-_0]^2)=σ^2p/m-p-1.Si on regarde maintenant la version empirique: si β est estimé sur les m premières observations,ℛ^ IS=∑_i=1^m [y_i-x_iβ]^2et ℛ^ OS=∑_i=m+1^n [y_i-x_iβ]^2,et comme l'a noté Leeb, ℛ^ IS -ℛ^ OS≈ 2 ·ν où ν représente le nombre de degrés de libertés, qui n'est pas sans rappeler la pénalisation utilisée dans le critère d'Akaike.La Figure <ref> montre l'évolution respective de ℛ^ IS et ℛ^ OS en fonction de la complexité du modèle (nombre de degrés dans une régression polynomiale, nombre de noeuds dans des splines, etc). Plus le modèle est complexe, plus ℛ^ IS va diminuer (c'est la courbe rouge). Mais ce n'est pas ce qui nous intéresse ici : on veut un modèle qui prédise bien sur de nouvelles données (autrement dit out-of-sample). Comme le montre la Figure <ref>, si le modèle est trop simple, il prédit mal (tout comme sur les données in-sample). Mais ce que l'on peut voir, c'est que si le modèle est trop complexe, on est dans une situation de « sur-apprentissage » : le modèle va commencer à modéliser le bruit. Au lieu de séparer la base en deux, avec une partie des données qui vont servir à calibrer le modèle et une autre à étudier sa performance, il est aussi possible d'utiliser la validation croisée. Pour présenter l'idée générale, on peut revenir au « jackknife », introduit par Quenouille1 (et formalisé par Quenouille2 et Tukey) et utilisé en statistique pour réduire le biais. En effet, si on suppose que { y_1,⋯,y_n} est un échantillon tiré suivant une loi F_θ, et que l'on dispose d'un estimateur T_n()=T_n(y_1,⋯,y_n), mais que cet estimateur est biaisé, avec [T_n()]=θ+O(n^-1), il est possible de réduire le biais en considérant :T_n()=1/n∑_i=1^n T_n-1(_(i)) avec _(i)=(y_1,⋯,y_i-1,y_i+1,⋯,y_n).On peut alors montrer que [T_n()]=θ+O(n^-2).L'idée de la validation croisée repose sur l'idée de construire un estimateur en enlevant une observation. Comme on souhaite construire un modèle prédictif, on va comparer la prévision obtenue avec le modèle estimé, et l'observation manquante :ℛ^ CV=1/n∑_i=1^n ℓ(y_i,m_(i)(_i))On parlera ici de méthode « leave-one-out » (loocv).On utilise classiquement cette technique pour trouver le paramètre optimal dans les méthodes de lissage exponentiel, pour des séries chronologiques. Dans le lissage simple, on va construire une prédiction de la forme _ty_t+1=α·_t-1y_t+(1-α)· y_t, avec α∈[0,1], et on va considérer :α^⋆ = α∈[0,1]argmin{∑_t=2^T ℓ(_t-1y_t,y_t) },comme le décrit Hyndman.Le principal problème de la méthode « leave-one-out » est qu'elle nécessite de calibrer n modèles, ce qui peut être problématique en grande dimension. Une méthode alternative est la validation croisée par k-blocs (dit « k-fold cross validation ») qui consiste à utiliser une partition de {1,⋯,n} en k groupes (ou blocs) de même taille, ℐ_1,⋯,ℐ_k, et notons ℐ_j={ 1,⋯,n}\ℐ_j. En notant m_(j) construit sur l'échantillon ℐ_j, on pose alors :ℛ^k- CV=1/k∑_j=1^k ℛ_j oùℛ_j=k/n∑_i∈ℐ_jℓ(y_i,m_(j)(_i)).La validation croisée standard, où une seule observation est enlevée à chaque fois (loocv), est un cas particulier, aveck=n. Utiliser k=5,10 a un double avantage par rapport à k=n : (1) le nombre d'estimations à effectuer est beaucoup plus faible, 5 ou 10 plutôt que n ; (2) les échantillons utilisés pour l'estimation sont moins similaires et donc, moins corrélés les uns aux autres, ce qui tend à éviter les excès de variance, commme le rappelle James.Une autre alternative consiste à utiliser des échantillons boostrappés. Soit ℐ_b un échantillon de taille n obtenu en tirant avec remise dans {1,⋯,n} pour savoir quelles observations (y_i,_i) seront gardées dans la population d'apprentissage (à chaque tirage). Notons ℐ_b={ 1,⋯,n}\ℐ_b. En notant m_(b) construit sur l'échantillon ℐ_b, on pose alors :ℛ^ B=1/B∑_b=1^B ℛ_b oùℛ_b=n_b/n∑_i∈ℐ_bℓ(y_i,m_(b)(_i)),où n_b est le nombre d'observations qui n'ont pas été conservées dans ℐ_b. On notera qu'avec cette technique, en moyenne e^-1∼ 36.7% des observations ne figurent pas dans l'échantillon boostrappé, et on retrouve un ordre de grandeur des proportions utilisées en créant un échantillon de calibration, et un échantillon de test.En fait, comme l'avait montré Stone, la minimization du AIC est à rapprocher du critère de validation croisée, et Shao97 a montré que la minimisation du BIC correspond à de la validation croisée de type k-fold, avec k=n/log n.§ QUELQUES OUTILS DE MACHINE LEARNING §.§ Réseaux de Neurones Les réseaux de neurones sont des modèles semi-paramétriques. Néanmoins, cette famille de modèles peut être appréhendée de la même manière que les modèles non-paramétriques: la structure des réseaux de neurones (présentée par la suite) peut être modifiée afin d'étendre la classe des fonctions utilisées pour approcher une variable d'intérêt. Plus précisément, Cybenko a démontré que l'ensemble des fonctions neuronales est dense dans l'espace des fonctions continues sur un compact. En d'autres termes, on a un cadre théorique permettant de garantir une forme d'approximation universelle. Il impose en outre une définition d'un neurone et met en avant l'existence d'un nombre de neurones suffisant pour approcher toute fonction continue sur un compact. Ainsi, un phénomène continue peut être approché par une suite de neurones: on appellera cette suite « réseau de neurones à une couche ». Si ce théorème d'approximation universelle est démontré en 1989, le premier neurone artificiel fonctionnel fut introduit par Franck Rosenblatt au milieu du XXième siècle, dans Rosenblatt. Ce neurone, qualifié de nos jours de « neurone élémentaire », porte le nom de « Perceptron ». Il a permis dans ses premières utilisations de déterminer le sexe d'un individu présenté aux travers d'une photo. Si ce premier neurone est important, c'est qu'il introduit le premier formalisme mathématique d'un neurone biologique. On peut décrire un neurone artificiel par analogie avec une cellule nerveuse : -les synapses apportant l'information à la cellule sont formalisés par un vecteur réel. La dimension du vecteur d'entrée du neurone (qui n'est d'autre qu'une fonction) correspond biologiquement au nombre de connections synaptiques;-chaque signal apporté par un synapse est ensuite analysé par la cellule. Mathématiquement, ce schéma est transcrit par la pondération des différents éléments constitutifs du vecteur d'entrée;- en fonction de l'information acquise, le neurone décide de retransmettre ou non un signal. Ce phénomène est répliqué par l'introduction d'une fonction d'activation. Le signal de sortie est modélisé par un nombre réel calculé comme image par la fonction d'activation du vecteur d'entrée pondéré.Ainsi, un neurone artificiel est un modèle semi-paramétrique. Le choix de la fonction d'activation est en effet laissé à l'utilisateur. Nous introduisons dans le paragraphe qui suit une formalisation rigoureuse qui nous permettra de poser le modèle, et de faire le lien avec les notations économétriques usuelles. On peut alors définir un neurone élémentaire formellement par :-un espace d'entrée 𝒳, généralement ℝ^k avec k ∈ℕ^*;-un espace de sortie 𝒴, généralement ℝ ou un ensemble fini (classiquement {0,1}, mais on préférera ici {-1,+1});-un vecteur de paramètres w∈ℝ^p -une fonction d'activation ϕ : ℝ→ℝ.Cette fonction doit être dans l'idéal monotone, dérivable et bornée (on dira ici « saturante») afin de s'assurer de certaines propriétés de convergence. Cette dernière fonction ϕ fait penser aux transformations logistique ou probit, populaire en économétrie (qui sont des fonctions de répartition, à valeur dans [0,1], idéal quand 𝒴 est l'ensemble {0,1}). Pour les réseaux de neurones, on utilisera plutôt la tangente hyperbolique, la fonction arctangente ou les fonctions sigmoïdes pour des problèmes de classification (𝒴={-1,+1}. On appellera neurone toute application f_w de 𝒳 dans 𝒴 définie par : y=f_w() = ϕ (wx), ∀x∈𝒳. Pour le perceptron introduit par Rosenblatt, on assimile un neurone élémentaire à la fonction :y=f_w(x)=signe(wx) ∀x∈𝒳On remarque que selon cette formalisation, beaucoup de modèles statistiques comme par exemple les régressions logistiques pourraient être vus comme des neurones. En effet si l'on regarde d'un peu plus près, tout modèle glm (« Generalized Linear Model ») pourrait s'interpréter comme un neurone formel où la fonction d'activation ϕ n'est d'autre que l'inverse de la fonction de lien canonique (par exemple). Si g désigne la fonction de lien du glm , w le vecteur de paramètres, y la variable à expliquer et x le vecteur des variables explicatives de même dimension que w : g(𝔼[Y|X=x])=wx On retrouve la modélisation neuronale en prenant ϕ = g^-1. Cependant, là où réside la différence majeure entre les glm et le modèle neuronale est que ce dernier n'introduit aucune hypothèse de distribution sur Y|X (on n'a d'ailleurs pas besoin d'introduire ici de modèle probabiliste). D'autre part, lorsque le nombre de neurones par couche augmente, la convergence n'est pas nécessairement garantie si la fonction d'activation ne vérifie pas certaines propriétés (qu'on ne retrouve pas dans la majorité des fonctions de liens canoniques des glm).Cependant, comme énoncé précédemment, la théorie des réseaux de neurones introduit des contraintes mathématiques supplémentaires sur la fonction g (détaillé dans Cybenko). Ainsi par exemple, une régression logistique peut être perçue comme un neurone alors que les régressions linéraires généralisées ne vérifient pas toutes les hypothèses nécessaires.Toujours par analogie avec le fonctionnement du système nerveux, il est alors possible de connecter différents neurones entre eux. On parlera de structure de réseaux de neurones par couche. Chaque couche de neurones recevant à chaque fois le même vecteur d'observation. Pour revenir à une analogie plus économétrique, on peut imaginer passer par une étape intermédiaire (on reviendra sur cette construction dans la Figure <ref>), par exemple en ne faisant pas une régression sur les variables brutesmais un ensemble plus faible de variables orthogonales, obtenues par exemple suite à une analyse en composantes principales. Soit A la matrice associée à cette transformation linéaire, avec A de taille k× p si on souhaite utiliser les p premières composantes. Notons z la transformation de x, au sens où z = A, ou encore z_j=A_j. Un généralisation du modèle précédant peut être de posery=f(x) = ϕ (wz)= ϕ (wA)=, ∀x∈𝒳,où cette fois w∈ℝ^p. On a ici une transformation linéaire (en considérant une analyse en composante principale) mais on peut imaginer une généralisation avec des transformée non-linéaire, avec une fonction de la formey=f(x) = ϕ (w F_A()=, ∀x∈𝒳,où F est ici une fonction ℝ^k→ℝ^p. C'est le réseau de neurone à deux couches. Plus généralement, pour formaliser la construction, on introduit les notations suivantes : - K ∈ℕ^* : nombre de couches;- ∀ k ∈{ 1 ,⋯ K } , p_k représente le nombre de neurones dans la couche k;- ∀ k ∈{ 1 ,⋯ K } ,W_k désigne la matrice des paramètres associés à la couche k. Plus précisément, W_k est une matrice p_k× p_k-1 et pour tout l ∈{ 1 ,⋯ p_k } , w_k,l∈ℝ^ p_k-1 désigne le vecteur de poids associé au neurone élémentaire l de la couche k;- on appellera W = {W_1,..,W_K}, l'ensemble des paramètres associés au réseau de neurones.- F^k_W_k : ℝ^p_k-1→ℝ^p_k désigne la fonction de transfert associé à la couche k. Pour des raisons de simplification, on pourra également écrire F^k;- ŷ_k ∈ℝ^p_k représentera le vecteur image de la couche k ∈{ 1 ,⋯, K };- on appellera F= F_W = F^1 ∘⋯∘ F^K la fonction de transfert associée au réseau global. A ce titre, si x∈𝒳, on pourra noter y= F_W(x).La Figure <ref> permet d'illustrer les notations présentées ici[Source: http://intelligenceartificielle.org.]. Chaque cercle représente un neurone élémentaire. Chaque rectangle englobant plusieurs cercles représente une couche. On parle de couche d'entrée pour la première couche prenant en « input » les observation x∈𝒳, de couche de sortie pour la couche fournissant en « output » la prédiction ŷ∈𝒴. Les autres couches sont couramment appelées couches cachées. Un réseau de neurones multicouches est donc également un modèle semi-paramétrique dont les paramètres sont l'ensemble des composantes des matrices W_k pour tout entier k de { 1,⋯, K }. Chaque fonction d'activation associée à chaque neurone (chaque cercle de la Figure <ref>) est à déterminer par l'utilisateur.Une fois que les paramètres à calibrer du modèle sont identifiés (ici les réels constituant les matrices W_k pour chaque couche k ∈{1,⋯,K }) , il est nécessaire de fixer une fonction de perte ℓ. En effet, on rappelle que l'objectif de l'apprentissage supervisé sur une base d'apprentissage de n ∈ℕ^* couples (y_i,x_i) ∈𝒴×𝒳 est de minimiser le risque empirique : ℛ_n(F_W) = 1/n∑_i=1^n ℓ(y_i,F_W(x_i)) Afin d'illustrer les propos, intéressons nous à l'exemple suivant qui illustrera également la démarche opérée. Supposons que nous observons un phénomène y aux travers de n observations y_i ∈ [-1,1]. On souhaiterait expliquer ce phénomène à partir des variables explicativesque l'on suppose à valeurs réelles. La « théorie de l'approximation universelle » nous indique qu'un réseau à une couche de neurones devrait permettre de modéliser le phénomène (sous hypothèse qu'il soit continue). On note toutefois que ce théorème ne donne pas de vitesse de convergence. Il est alors laissé à l'utilisateur le choix de la structure. Ainsi par exemple une première structure pourrait être un simple neurone dont la fonction d'activation serait la fonction tangente hyperbolique.On aurait ainsi comme premier modèle:y_1 = tanh(w_0 + w_1 x) où les paramètres w_0, w_1sont les paramètres à optimiser de sorte que sur les données d'apprentissage, le risque empirique soit minimal.Si l'on suit toujours la philosophie du théorème d'approximation universelle, en ajoutant plusieurs neurones, l'erreur est censée diminuer. Cependant,ne connaissant pas la fonction à estimer, on ne peut l'observer qu'aux travers de l'échantillon. Ainsi, mécaniquement, on s'attendà ce que plus on ajoute de paramètres, plus l'erreur sur la base d'apprentissage diminue. L'analyse de l'erreur sur la base de test permet alors d'évaluer notre capacité à généraliser (cf partie précédente). On peut ainsi s'intéresser à un second modèle qui cette fois utilise plusieurs neurones. Par exemple, considérons le modèley_2 = w_a tanh(w_0 + w_1 x) + w_b tanh(w_2 + w_3 x) + w_c tanh(w_4 + w_5 x) où les paramètres w_0,..,w_5 ainsi que w_a, w_b, w_csont les paramètres à optimiser.Calibrer un réseaux de neurones revient alors à réitérer ces étapes de modification de la structure jusqu'à minimisation du risque sur la base de test. Pour une structure de réseau de neurones fixée (c'est-à-dire nombre de couches, nombre de neurones par couches et fonctions d'activation fixés), le programme revient donc à déterminer l'ensemble de paramètres W^*=(W_1,...,W_K) de sorte que :W^*∈ W=(W_1,...,W_K)argmin{1/n∑_i=1^n ℓ(y_i,F_W(_i)) }.De cette formule apparaît l'importance du choix de la fonction ℓ. Cette fonction de perte, elle quantifie l'erreur moyenne commise par notre modèle F_W sur la base d'apprentissage. A priori ℓ peut être choisie arbitrairement. Cependant, dans l'optique de résoudre un programme d'optimisation, on préféra des fonctions de coût sous-différentiables et convexes afin de garantir la convergence des algorithmes d'optimisation.Parmi les fonctions de perte classiques, en plus de la fonction de perte quadratique ℓ_2 on retiendra la fonction dite « Hinge » - ℓ(y,ŷ)=max(0, 1-yŷ) - ou la fonction dite logistique - ℓ(y,ŷ)=log(1-e^-yŷ).En définitive les réseaux de neurones sont des modèles semi-paramétriques dont le nombre de paramètres est croissant avec le nombre de couches et de neurones par couche. Il est laissé à l'utilisateur de choisir les fonctions d'activation et la structure du réseau. Ceci explique l'analogie avec la philosophie des modèles non-paramétriques faite auparavant.Les réseaux de neurones ont été utilisés très tôt en économie et en finance, en particulier sur les défauts d'entreprises - Tam ou Altman - ou plus récemment la notation de crédit - Blanco ou Khashman. Cependant les structures telles que présentées précédemment sont généralement limitées. L'apprentissage profond (ou « deep learning ») caractérise plus particulièrement des réseaux de neurones plus complexes (parfois plus d'une dizaine de couches avec parfois des centaines de neurones par couche). Si aujourd'hui ces structures sont très populaires en analyse du signal (image, texte, son) c'est qu'elles sont capables à partir d'une quantité d'observations très importante d'extraire des informations que l'humain ne peut percevoir et de faire face à des problèmes non linéaires, comme le rappelle LeCun. L'extraction d'informations peut, par exemple, se faire grâce à la convolution. Procédé non supervisé, il a permis notamment d'obtenir d'excellente performance dans l'analyse d'image. Techniquement, cela peut s'apparenter à une transformation à noyaux (comme utilisé dans les techniques SVM). Si une image peut être perçue comme une matrice dont chaque coordonnée représente un pixel, une convolution reviendrait à appliquer une transformation sur un point (ou une zone) de cette matrice générant ainsi une nouvelle donnée. Le procédé peut ainsi être répété en appliquant des transformations différentes (d'où la notion de couches convolutives). Le vecteur final obtenu peut alors enfin alimenter un modèle neuronal comme introduit dans le paragraphe précédant. En fait, plus généralement, une couche de convolution peut être perçue comme un filtre qui permet de transformer la donnée initiale. Une explication intuitive pour laquelle l'apprentissage approfondi, en particulier les réseaux nerveux profonds, est si puissant pour décrire des relations complexes dans les données, c'est leur construction autour de l'approximation fonctionnelle simple et l'exploitation d'une forme de hiérarchie, comme le note Lin. Néanmoins les modèles de type « deep learning » sont plus difficiles à appréhender car ils nécessitent beaucoup de jugement empirique. En effet, si aujourd'hui les bibliothèques open sources (keras, torch, etc.) permettent de paralléliser plus facilement les calculs en utilisant par exemple les GPU (Graphical Processor Units), il reste néanmoins à l'utilisateur de déterminer la structure du réseau de neurones le plus approprié.§.§ Support Vecteurs Machine Comme nous l'avions noté auparavant, dans les problèmes de classification en apprentissage machine (comme en traitement du signal) on préférera avoir des observations dans l'ensemble {-1,+1} (plutôt que {0,1}, comme en économétrie). Avec cette notation, Cortes ont posé les bases théorique des modèles dit svm, proposant une alternative aux réseaux de neurones alors très populaires comme algorithme de classification dans la communauté de l'apprentissage machine. L'idée initiale des méthodes de « Support Vectors Machine » (svm) consiste à trouver un hyperplan séparateur divisant l'espace en deux ensembles de points le plus homogène possible (i.e. contenant des labels identiques). En dimension deux, l'algorithme consiste à déterminer une droite séparant l'espace en deux zones les plus homogènes possibles. La résolution de ce problème possédant parfois une infinité de solution (il peut en effet exister une infinité de droites qui séparent l'espace en deux zones distinctes et homogènes), on rajoute généralement une contrainte supplémentaire. L'hyperplan séparateur doit se trouver le plus éloigné possible des deux sous-ensembles homogènes qu'il engendre. On parlera ainsi de marge. L'algorithme ainsi décrit est alors un svm linéaire à marge.Si un plan peut être caractérisé entièrement par un vecteur directeur w orthogonal à ce dernier et une constante b, appliquer un algorithme SVM à un ensemble de n∈ℕ^* points _i de ℝ^p labellisés par y_i ∈{-1,1} revient alors à résoudre un programme d'optimisation sous contrainte similaire à celui d'un lasso (distance quadratique sous contrainte linéaire). Plus particulièrement, on sera amené à résoudre :(w^⋆,b^⋆) = w,bargmin{w^2}=w,bargmin{ww}, sous contrainte ∀ i ∈{ 1, ⋯, n }, {[ ω^ Tx_i+b ≥ +1 lorsque y_i=+1; ω^ Tx_i+b ≤ -1 lorsque y_i=-1; ].La contrainte peut être relâchée en autorisant que dans un sous-ensemble, un point puisse ne pas être du même label que la majeure partie des points de ce sous-ensemble à condition de ne pas être trop loin de la frontière. C'est ce qu'on appelle les SVM linéaire à marge légère (soft margin). De manière heuristique, comme en pratique, bien souvent, on ne peut pas avoir y_i(wx_i +b)-1≥ 0 pour tout i∈{ 1, ⋯, n }, on relâche en introduisant des variables positives ξ telle que {[ ω^ Tx_i+b ≥ +1-ξ_i lorsque y_i=+1; ω^ Tx_i+b ≤ -1+ξ_i lorsque y_i=-1; ].avec ξ_i≥ 0. On a une erreur de classification si ξ_i>1, et on va alors introduire une pénalité, un coût à payer pour chaque erreur commise. On cherche alors à résoudre un problème quadratiquemin{1/2ω^ Tω+C1^T1_ξ>1}sous la contrainte (<ref>), qui pourra être résolu de manière numérique très efficacement par descente de coordonnées (décrit auparavant).S'il n'est pas possible de séparer les points, une autre astuce possible consiste à les transformer dans une dimension supérieure, de sorte que les données deviennent alors linéairement séparables. Trouver la bonne transformation qui sépare les données est toutefois très difficile. Cependant, il existe une astuce mathématique pour résoudre ce problème avec élégance, en définisant les transformations T (·) et les produits scalaires via un noyau K (x _1, x _2) = ⟨ T (x _1),T (x _2)⟩. L'un des choix les plus courants pour une fonction de noyau est la fonction de base radiale (noyau gaussien) K (x _1, x _2) = exp (-‖x _1- x _2) ‖ ^ 2 ). Il n'existe néanmoins pas de règles à ce jour permettant de choisir le « meilleur » noyau. Comme mentionné au début de la section précédante, cette technique est basé sur de la minimisation de distance, et il n'a aucune prévision de la probabilité d'être positif ou négatif (mais une interprétation probabiliste est néanmoins possible, comme le montre Grandvalet, par exemple). §.§ Arbres, Bagging et Forêts Aléatoires Les arbres de classification ont été introduits dans Quinlan mais c'est surtout Breiman qui a assuré la popularité de l'algorithme. On parle de modèle CART pour « Classification And Regression Tree ».L'idée est de diviser consécutivement (par une notion de branchement) les données d'entrée jusqu'à ce qu'un critère d'affectation (par rapport à la variable cible) soit atteint, selon une règle prédéfinie. L'intuition de la construction des arbres de classification est la suivante. L'entropie H (x) est associée à la quantité de désordre dans les données x par rapport aux modalités prises par la variable de classification y, et chaque partition vise à réduire ce désordre. L'interprétation probabiliste est de créer les groupes les plus homogènes possible, en réduisant la variance par groupe (variance intra), ou de manière équivalente en créant deux groupes aussi différents que possible, en augmentant la variance entre les groupe (variance inter). À chaque étape, nous choisissons la partition qui donne la plus forte réduction de désordre (ou de variance). L'arbre de décision complet se développe en répétant cette procédure sur tous les sous-groupes, où chaque étape k aboutit à une nouvelle partition en 2 branches, qui subdivise notre ensemble de données en 2. Enfin, on décide quand mettre fin à cette constitution de nouvelles branches, en procédant à des affectations finales (nœuds dits foliaires). Il existe plusieurs options pour mettre fin à cette croissance. L'une est de construire un arbre jusqu'à ce que toutes les feuilles soient pures, c'est à dire composées d'une seule observation. Une autre option est de définir une règle d'arrêt liée à la taille, ou à la décomposition, des feuilles. Les exemples de règles d'arrêt peuvent être d'une taille minimale (au moins 5 éléments par feuille), ou une entropie minimale. On parlera alors d'élagage de l'arbre: on laisse l'arbre grossir, puis on coupe certaines branches a posteriori (ce qui est différent de l'introduction d'un critère d'arrêt a priori au processus de croissance de l'arbre - par exemple en imposant une taille minimale aux feuilles, ou d'autres critères discutés dans Breiman).À un nœud donné, constitué de n_0 observations (x_i,y_i) avec i∈ℐ_0, on va couper en deux branches (une à gauche et une à droite), partitionnant ainsi ℐ_0 en ℐ_g et ℐ_d. Soit I le critère d'intérêt, comme l'entropie du nœud (ou plutôt du nœud vu en tant que feuille):I(y_0)=-n_0 p_0log p_0oùp_0=1/n_0∑_i∈ℐ_0y_i,ou la variance du nœud:I(y_0)=n_0 p_0(1- p_0)oùp_0=1/n_0∑_i∈ℐ_0y_i,ce dernier étant également l'indice d'impureté de Gini. On partitionnera entre la branche gauche et la branche droite si le gain I(y_0)-[I(y_g)+I(y_d)]est suffisamment important. Lors de la construction des arbres, on va chercher la partition qui donne le gain le plus important possible. Ce problème combinatoire étant complexe, le critère suggéré par Breiman est de considérer un découpage suivant une des variables, avec ℐ_g={ i∈ℐ_0:x_k,i<s} et ℐ_d={ i∈ℐ_0:x_k,i>s}, pour une variable k et un seuil s (si la variable est continue, sinon on considère des regroupements de modalités pour des variables qualitatives). Les arbres de décision ainsi décrits sont simples à obtenir et faciles à interpréter (comme le montre la Figure <ref> sur les données du Titanic[Ce jeu de données, contenant des informations sur tous les passagers (et membres d'équipage) du Titanic, dont la variable y indiquant si la personne a survécu a été abondamment utilisé pour illustrer les techniques de classification, voir https://www.kaggle.com/c/titanic/data.]), mais ils sont peu robustes, et leur pouvoir prédictif est souvent très faible, en particulier si l'arbre est très profond. Une idée naturelle est de développer un ensemble de modèles d'arbres à peu près indépendants, qui prédisent conjointement mieux qu'un modèle d'arbre unique. On va utiliser le bootstrap, en tirant (avec remise) n observations parmi {(_i,y_i)}. Ces ensembles d'arbres - naturellement appelés « forêts » - une fois agrégés donnent souvent de bien meilleurs résultats que les arbres isolés, mais elles sont difficiles à interpréter. Ces techniques ressemblent toutefois beaucoup à ce qui est fait lorsque l'on utilise les techniques de bootstrap en régression (par exemple pour construire des tubes de confiance dans une régression fonctionnelle).Le principe du « bagging », pour « bootstrap aggregating »,consiste à générer des échantillons aléatoires, en tirant avec remise dans l'échantillon d'origine, comme pour le bootstrap. Chaque échantillon ainsi générépermet d'estimer un nouvel arbre de classification, formant ainsi une forêt d'arbres. C'est l'aggrégation de tous ces arbres qui conduit à la prévision. Le résultat global est moins sensible à l'échantillon initial et donne souvent de meilleurs résultats de prévision.Les forêts aléatoires, ou « random forests » reposent sur le même principe que le « bagging », mais en plus, lors de la construction d'un arbre de classification, à chaque branche, unsous-ensemble de m covariables est tiré aléatoirement. Autrement dit, chaque branche d'un arbre ne s'appuie pas sur le même ensemble de covariables. Cela permet d'amplifier la variabilité entre les différents arbres et d'obtenir, au final, une forêt composée d'arbres moins corrélés les uns aux autres.§.§ Sélection de modèle de classification Étant donné un modèle m(·) approchant [Y|=], et un seuil s∈[0,1], posonsy^(s)=1[m()>s] = {[1si m()>s; 0si m()≤ s ].La matrice de confusion est alors le tableau de contingence associé aux comptages N=[N_u,v] avecN_u,v^(s)=∑_i=1^n 1(y^(s)_i=u,y_j=v)pour (u,v)∈{0,1}. La Table <ref> présente un tel tableau, avec le nom de chacun des éléments : TP (true positive) sont les vrais positifs, correspondant aux 1 prédit en 1,TN (true negative) sont les vrais négatifs, correspondant aux 0 prédit en 0, FP (false positive) sont les faux positifs, correspondant aux 0 prédit en 1, et enfin FN (false negative) sont les faux négatifs, correspondant aux 1 prédit en 0).Plusieurs quantités sont dérivées de ce tableau. La sensibilité correspond à la probabilité de prédire 1 dans la population des 1, ou taux de vrais positifs. La spécificité estla probabilité de prédire 0 dans la population des 0 ou taux de vrais négatifs. On s'intéressera toutefois davantage au taux de faux négatifs,c'est à dire la probabilité de prédire 1 dans la population des 0. La représentation de ces deux valeurs lorsque s varie donne la courbe ROC (« receiver operating characteristic ») :ROC_s=(FP_s/FP_s+TN_s,TP_s/TP_s+FN_s)=( sensibility_s,1- specificity_s)pour s∈[0,1].Une telle courbe est présentée dans la partie suivante, sur des données réelles.Les deux grandeurs intensivement utilisées en machine learning sont l'indice κ, qui compare la précision observée avec celle espérée, avec un modèle aléatoire (tel que décrit dans LandisKoch) et l'AUC correspondant à l'aire sous la courbe ROC. Pour le premier indice, une fois choisi s, notons N^⊥ le tableau de contingence correspond aux cas indépendants (défini à partir de N dans le test d'indépendance du chi-deux). On pose alorsprécision totale=TP+TN/nalors queprécision aléatoire=[TN+FP]·[TP+FN]+[TP+FP]·[TN+FN]/n^2On peut alors définirκ=précision totale-précision aléatoire/1-précision aléatoireClassiquement s sera fixé égal à 0.5, comme dans une classification bayésienne naïve, mais d'autres valeurs peuvent être retenues, en particulier si les deux erreurs ne sont pas symmétriques (nous reviendrons sur ce point dans un exemple par la suite).Il existe des compromis entre des modèles simples et complexes mesurés par leur nombre de paramètres (ou plus généralement les degrés de liberté) en matière de performance et de coût. Les modèles simples sont généralement plus faciles à calculer, mais peuvent conduire à des ajustements plus mauvais (avec un biais élevé par exemple). Au contraire, les modèles complexes peuvent fournir des ajustements plus précis, mais risquent d'être coûteux en termes de calcul. En outre, ils peuvent surpasser les données ou avoir une grande variance et, tout autant que des modèles trop simples, ont de grandes erreurs de test. Comme nous l'avons rappelé auparavant, dans l'apprentissage machine, la complexité optimale du modèle est déterminée en utilisant le compromis de biais-variance. §.§ De la classification à la régression Comme nous l'avons rappelé en introduction, historiquement, les méthodes de machine learning se sont orientées autour des problèmes de classification (avec éventuellement plus de 2 modalités[Par exemple dans le cas dereconnaissance de lettres ou de chiffres]), et assez peu dans le cas où la variable d'intérêt y est continue. Néanmoins, il est possible d'adapter quelques techniques, comme les arbres et les forêts aléatoires, le boosting, ou les réseaux de neurones.Pour les arbres de régression, Morganetal ont proposé la méthode AID, basée sur la formule de décomposition de la variance de l'équation (<ref>), avec un algorithme proche de celui de la méthode CART décrite auparavant. Dans le contexte de la classification, on calculait, à chaque nœud (dans le cas de l'indice d'impureté de Gini) en sommant sur la feuille de gauche {x_k,i<s} et celle de droite {x_k,i>s}I=∑_i:x_k,i<sy_g(1-y_g) +∑_i:x_k,i>sy_d(1-y_d)où y_g et y_d désignent les fréquences de 1 dans la feuille de gauche et de droite, respectivement. Dans le cas d'un arbre de régression, on utiliseraI=∑_i:x_k,i<s(y_i-y_g)^2 +∑_i:x_k,i>s(y_i-y_d)^2qui va correspondre à la somme (pondérée) des variances intra. Le partage optimal sera celui qui aura le plus de variance intra (on veut les feuilles les plus homogènes possibles) ou de manière équivalente, on veut maximiser la variance intra.Dans le contexte des forêts aléatoires, on utilise souvent un critère majoritaire en classification (la classe prédite sera la classe majoritaire dans une feuille), alors que pour la régression, on utilise la moyenne des prédictions, sur tous les arbres. Dans la partie précédente, nous avons présenté la dimension « apprentissage » du machine learning en présentant le boosting. Dans un contexte de régression (variable y continue), l'idée est de créer une succession de modèles en écrivant l'équation (<ref>) sous la forme :m^(k)(x)=m^(k-1)(x)+α_k h∈ℋargmin{∑_i=1^n (y_i,m^(k-1)(x)+h(x))^2}où α_k est un paramètre de « shrinkage », où le second terme correspond à un arbre de régression, sur les résidus, y_i-m^(k-1)(x_i).Mais il existe d'autres techniques permettant d'apprendre de manière séquentielle. Dans un modèle additif (gam) on va chercher une écriture de la formem(x)=∑_j=1^p m_j(x_j)=m_1(x_1)+⋯+m_p(x_p)L'idée de la poursuite de projection repose sur une décomposition non pas sur les variables explicatives, mais sur des combinaisons linéaires. On va ainsi considérer un modèlem(x)=∑_j=1^k g_j(ω_jx)=g_1(ω_1x)+⋯+g_k(ω_kx).Tout comme les modèles additifs, les fonctions g_1,⋯,g_k sont à estimer, tout comme les directions ω_1,⋯,ω_k. Cette écriture est relativement générale, et permet de tenir compte d'intéractions et d'effets croisés (ce que nous ne pouvions pas faire avec les modèles additifs qui ne tiennent compte que de non-linéarités). Par exemple en dimension 2, un effet multiplicatif m(x_1,x_2)=x_1· x_2 s'écritm(x_1,x_2)=x_1· x_2=(x_1+x_2)^2/4-(x_1-x_2)^2/4autrement dit g_1(x)=x^2/4, g_1(x)=-x^2/4, ω_1=(1,1) et ω_1=(1,-1). Dans la version simple, avec k=1, avec une fonction de perte quadratique, on peut utiliser un développement de Taylor pour approcher [y_i-g(ωx_i)]^2, et construire classiquement un algorithme itératif. Si on dispose d'une valeur initiale ω_0, notons que∑_i=1^n[y_i-g(ωx_i)]^2 ≈∑_i=1^n g'(ω_0x_i)^2[ωx_i+y_i-g(ω_0x_i)/g'(ω_0x_i)- ωx_i]^2qui correspondrait à l'approximation dans les modèles linéaires généralisés sur la fonction g(·) qui était la fonction de lien (supposée connue). On reconnaitun problème de moindres carrés pondérés. La difficulté ici est que les fonctions g_j(·) sont inconnues.§ APPLICATIONS Les données massives ont rendu nécessaire le développement de techniques d'estimation permettant de pallier les limites des modèles paramétriques, jugés trop restrictifs, etdes modèles non-paramétriques classiques, dont l'estimation peut être difficile en présence d'un nombre élevé de variables. L'apprentissage statistique, ou apprentissage machine, propose de nouvelles méthodes d'estimation non-paramétriques, performantes dans un cadre général et en présence d'un grand nombre de variables.[Entre autres, voir HastieEtal et James.] Toutefois, l'obtention d'une plus grande flexibilité s'obtient au prix d'un manque d'interprétation qui peut être important.En pratique, une question importante est de savoir quel est le meilleur modèle ? La réponse à cette question dépend du problème sous-jacent. Si la relation entre les variables est correctement approximée par un modèle linéaire, un modèle paramétrique correctement spécifié devrait être performant. Par contre, si le modèle paramétrique n'est pas correctement spécifié, car la relation est fortement non-linéaire et/ou fait intervenir des effets croisés non-négligeables, alors les méthodes statistiques issues du machine learning devraient être plus performantes.La bonne spécification d'un modèle de régression est une hypothèse souvent posée, elle est rarement vérifiée et justifiée. Dans les applications qui suivent, nous montrons comment les méthodes statistiques issues du machine learning peuvent être utilisées pour justifier la bonne spécification d'un modèle de régression paramétrique, ou pour détecter une mauvaise spécification. Des applications en classification sont présentées dans un premier temps, sections <ref>, <ref> et <ref>. D'autres applications sont ensuite présentées dans le contexte de régression classique, sections <ref> et <ref>. §.§ Les ventes de sièges auto pour enfants (classification)Nous reprenons ici un exemple utilisé dans James. Le jeu de données contient les ventes de sièges auto pour enfants dans 400 magasins (Sales), ainsi que plusieurs variables, dont la qualité de présentation en rayonnage (Shelveloc, égal à « mauvais », « moyen », « bon ») et le prix (Price).[C'est le jeu de données Carseats de la bibliothèque ISLR.] Une variable dépendante binaire est artificiellement crée, pour qualifier une forte vente ou non (High=« oui » si >8 et à « non » sinon). Dans cette application, on cherche à évaluer les déterminants d'un bon niveau de vente.Dans un premier temps, on considère un modèle de régression linéaire latent:y^⋆ = γ ++ ε, ε∼ G(0,1),oùest composé de kvariables explicatives,est un vecteur de k paramètres inconnus et ε est un terme d'erreur i.i.d. avec une fonction de répartition G d'espérance nulle et de variance égale à un. La variable dépendante y^⋆ n'est pas observé, mais seulement y, avec:y=1siy^⋆ > ξ, 0siy^⋆≤ξ.On peut alors exprimer la probabilité d'avoir y égal à 1, comme suit :ℙ(Y=1)=G(β_0+)où β_0=γ-ξ.[ℙ[Y=1]=ℙ[Y^⋆>ξ]=ℙ[γ ++ ε>ξ]=ℙ[ ε>ξ-γ - ]=ℙ[ε<γ-ξ + ]. En posant γ-ξ=β_0, on obtient ℙ[Y=1]=G(β_0+). En général, on suppose que le terme d'erreur est de variance σ^2, auquel cas les paramètres du modèle (<ref>) deviennent β_0/σ et /σ, ce qui veut dire que les paramètres du modèle latent (<ref>) ne sont pas identifiables, ils sont estimés à un paramètre d'échelle près.] L'estimation de ce modèle se fait par maximum de vraisemblance en sélectionnant a priori une loi paramétrique G. Si on suppose que G est la loi Normale, c'est un modèle probit, si on suppose que G est la loi logistique, c'est un modèle logit. Dans unmodèlelogit/probit, il y a deux sources potentielles de mauvaise spécification : (i) la relation linéaire β_0+ est mal spécifiée(ii) la loi paramétrique utilisée G n'est pas la bonneEn cas de mauvaise spécification, de l'une ou l'autre sorte, l'estimation n'est plus valide. Le modèle le plus flexible est le suivant :ℙ[Y=1|X=x]=G(h(x))où h est une fonction inconnue et G une fonction de répartition inconnue. Les modèles de bagging, de forêt aléatoire et de boosting permettent d'estimer ce modèlegénéral sans faire de choix à priori sur la fonction het sur la distribution G. L'estimation du modèle logit/probit est néanmoins plus performante si h et G sont correctement spécifiés. Nous estimons le modèle (<ref>) avec la loi logistique pour G, et le modèle (<ref>) avec les méthodes de bagging, de forêt aléatoire et de boosting. Nous faisons une analyse de validation croisée par 10 blocs. Les probabilités individuelles des données out-of-sample, c'est à-dire de chacun des blocs non-utilisée pour l'estimation, sont utilisées pour évaluer la qualité de la classification. La figure <ref> présente la courbe ROC, ainsi que l'aire sous la courbe (AUC), pour les estimations logit, bagging, random forest et boosting. La courbe ROC est un graphique qui représente simultanément la qualité de la prévision dans les deux classes, pour des valeurs différentes du seuil utilisé pour classer les individus (on parle de « cutoff »). Une manière naturelle de classer les individus consiste à les attribuer dans la classe pour laquelle ils ont la plus grande probabilité estimée. Dans le cas d'une variable binaire, cela revient à prédire la classe d'appartenance pour laquelle la probabilité estimée est supérieure à 0.5. Mais un autre seuil pourrait être utilisé. Par exemple, dans la figure <ref>, un point de la courbe ROC du modèle logit indique qu'enprenant un seuil égal à 0.5, la réponse "non" est correctement prédite à 90.7% (specificity), et la réponse « oui »à86% (sensitivity). Un autrepoint indique qu'en prenant un seuil égal à 0.285, la réponse « non » est correctement prédite à 86% (specificity), et la réponse « oui »à92.7% (sensitivity). Comme décrit auparavant, un modèle de classification idéal aurait une courbe ROC de la forme Γ. Autrement dit, le meilleur modèle est celui dont la courbe est au-dessus des autres. Un critère souvent utilisé pour sélectionner le meilleur modèle est celui dont l'aire sous la courbe ROC est la plus grande (AUC). L'avantage d'un tel critère est qu'il est simple à comparer et qu'il ne dépend pas du choix du seuil de classification.Dans notre exemple, la courbe ROC du modèle logit domine les autres courbes, et son aire sous la courbe est la plus grande (AUC=0.9544). Ces résultats indiquent que ce modèlefournit les meilleures prévisions de classification. N'étant dominé par aucun autre modèle, ce constat suggère que le modèle linéaire logit est correctement spécifié et qu'il n'est pas utile d'utiliser un modèle plus général et plus complexe.§.§ L'achat d'une assurance caravane (classification)Nous reprenons à nouveau un exemple utilisé dans James. Le jeu de données contient 85 variables sur les charactéristiques démographiques de 5822 individus.[C'est le jeu de données Caravan de la bibliothèque ISLR sous R.]La variable dépendante (Purchase) indique si l'individu a acheté une assurance caravane, c'est une variable binaire, égale à « oui » ou « non ». Dans le jeu de données, seulement 6% des individus ont pris une telle assurance. Les classes sont donc fortement désequilibrées. Nous estimons le modèle (<ref>) avec la loi logistique et le modèle (<ref>) avec les méthodes bagging, forêt aléatoire et boosting (les paramètres de « tuning » sont ceux de James, n.trees=1000 et shrinkage=0.01).Nous faisons une analyse de validation croisée par 10 blocs. Les probabilités individuelles des données out-of-sample, c'est à-dire de chacun des blocs non-utilisée pour l'estimation, sont utilisées pour évaluer la qualité de la classification.La figure <ref> présente la courbe ROC, ainsi que l'aire sous la courbe (AUC), pour les estimations logit, bagging, random forest et boosting. La courbe du modèle boosting domine les autres courbes, son aire sous la courbe est la plus grande (AUC=0.7691). Ces résultats indiquent que le boostingfournit les meilleures prévisions de classification. Notons que, comparées à l'exemple précédent,les courbes sont assez éloignées de la forme en coude,ce qui suggère que la classification ne sera pas aussi bonne.Il faut faire attention aux résultats d'une classification standard, c'est-à-dire avec un seuil de classification égal à 0.5, qui est souvent pris par défaut dans les logiciels (la prédiction de la réponse de l'individu i est « non » 1 si la probabilité estimée qu'il réponde « non » est supérieure à 0.5, sinon c'est « oui »). La partie gauche du tableau <ref> présente les taux de classifications correctes avec ce seuil (0.5 cutoff), pour les différentes méthodes. Avec le meilleur modèle et le seuil standard (boosting et seuil à 0.5), les réponses « non » sont correctes à 99.87% (spécificité, specificity) et les réponses « oui » sont toutes fausses (sensitivité, sensitivity). Autrement dit, cela équivaut à utiliser un modèle qui prédit que personne n'achète d'assurance caravane. Sélectionner un tel modèle est absurde pour l'analyste, qui est surtout intéressé par les 6% des individus qui en ont pris une. Ce résultat s'explique par la présence de classes fortement déséquilibrées. En effet, dans notre exemple, en prévoyant que personne n'achète d'assurance, onfait « seulement » 6% d'erreur. Mais ce sont des erreurs qui conduisent à ne rien expliquer. Plusieurs méthodes peuvent être utiles pour pallier à ce problème, lié aux classes fortement désequilibrées (pour plus d'informations, voir KuhnJohnson, chapitre 16). Une solution simple consiste à utiliser un seuil de classification différent. La courbe ROC présente les résultats en fonction de plusieurs seuils de classification, où la classification parfaite est illustrée par le couple (specificity, sensitivity)=(1,1), c'est à-dire par le coin supérieur gauche dans le graphique. Aussi, on choisit comme seuil de classification optimal celui qui correspond au point de la courbe ROC qui est le plus proche du point (1,1), ou du coin supérieur gauche. La partie droite du tableau <ref> présente les taux de classifications correctes avec les seuils optimaux (optimal cutoff), pour les différentes méthodes (les seuils optimaux des méthodes logit, bagging, forêt aléatoire et boosting sont, respectivement, égaux à 0.0655 , 0.0365 , 0.0395, 0.0596). Avec le boosting et un seuil optimal, les réponses "non" sont correctes à 68.6% (specificity) et les réponses «oui » à 73.85%(sensitivity). L'objet de l'analyse étant de prévoir correctement les individus susceptibles d'acheter une assurance caravane (classe "oui"), et les distinguer suffisamment des autres (classe «non »), le choix du seuil optimal est beaucoup plus performant que le seuil standard 0.5. Notons qu'avec un modèle logit et un seuil optimal, le taux de classifications correctes de la classe "non" est de 72.78%, celui de la classe "oui" est de 63.51%. Par rapport au boosting, le logit prédit un peu mieux la classe "non", mais nettement moins bien la classe «oui ». §.§ Les défauts de remboursement de crédits particuliers (classification) Considérons la base allemande de crédits particuliers, utilisée dans Nisbet et Tuffery, avec 1000 observations, et 19 variables explicatives, dont 12 qualitatives c'est à dire, en les disjonctant (en créant une variable indicatrice pour chaque modalité), 48 variables explicatives potentielles.Une question récurrente en modélisation est de savoir quelles sont les variables qui mériteraient d'être utilisées. La réponse la plus naturelle pour un économètre pourrait être une méthode de type stepwise (parcourir toutes les combinaisons possibles de variables étant a priori un problème trop complexe en grande dimension). La suite des variables dans une approche forward est présentée dans la première colonne du tableau <ref>. Une approche mentionnée avant qui peut être utile est le Lasso, en pénalisant convenablement la norme ℓ_1 du vecteur de paramètres . On peut ainsi, séquentiellement, trouver les valeurs du paramètre de pénalisation λ, qui permet d'avoir une variable explicative supplémentaire, non nulle. Ces variables sont présentées dans la dernière colonne. On note que les deux premières variables considérées comme non nulles (pour un λ assez grand) sont les deux premières à ressortir lors d'une procédure forward. Enfin, une dernière méthode a été proposée parBreiman2001, en utilisant tous les arbres créé lors de la construction d'une forêt aléatoire : l'importance de la variable x_k dans une forêt de T arbres est donnée par:Importance(x_k)=1/T∑_t=1^n ∑_j∈ N_t,k p_t(j)Δℐ(j)où N_t,k désigne l'ensemble des nœuds de l'arbre t utilisant la variable x_k comme variable de séparation, p_t(j) désigne la proportion des observations au nœud j, et Δ(j) est la variation d'indice au nœud j (entre le nœud précédant, la feuille de gauche et celle de droite). Dans la colonne centrale du tableau <ref> sont présentées les variables par ordre d'importance décroissante, lorsque l'indice utilisé est l'indice d'impureté de Gini.Avec l'approche stepwise et l'approche lasso, on reste sur des modèles logistiques linéaires. Dans le cas des forêtes aléatoires (et des arbres), des intéractions entre variables peuvent être prises en compte, lorsque 2 variables sont présentes. Par exemple la variable residence_since est présente très haut parmi les variables prédictives (troisième variable la plus importante).§.§ Les déterminants des salaires (régression) Afin d'expliquer les salaires (individuels) en fonction du niveau d'étude, de l'expérience de la personne, et son genre, il est classique d'utiliser l'équation de salaire de Mincer - décrite dans Mincer - tel que le rappelle Lemieux:log () = β_0+β_1+β_2+β_3^2+β_4+εoù ed est le niveau d'études, ex l'expérience professionnelle et fe une variable indicatrice, égale à 1 si l'individu est une femme et à 0 sinon. D'après la théorie du capital humain, le salaire espéré augmente avec l'expérience, de moins en moins vite, pour atteindre un maximum avant de diminuer. L'introduction du carré de exp permet de prendre en compte une telle relation. La présence de la variablefe permet quand à elle de mesurer une éventuelle discrimination salariale entre les hommes et les femmes.Le modèle (<ref>) impose une relation linéaire entre le salaireet le niveau d'étude, et une relation quadratique entre le salaire et l'expérience professionnelle. Ces relations peuvent paraître trop restrictives. Plusieurs études montrent notamment que le salaire ne diminue pas après un certain age, et qu'une relation quadratique ou un polynôme de degré plus élevé est plus adapté (comme décrit dans MurphyWelch etBazen).Le modèle (<ref>) impose également que la différence salariale entre les hommes et les femmes est indépendente du niveau d'étude et de l'expérience. Il est trop restrictif si, par exemple, on suspecte que l'écart de salaire moyen entre les hommes et les femmes est faible pour les postes non-qualifiés et fort pour les postes qualifiés, ou faible en début de carrière et fort en fin de carrière (effets d'intéractions). Le modèle le plus flexible est le modèle entièrement non-paramétrique :log () = m() +εoù m(·) est une fonction quelconque. Il a l'avantage de pouvoir tenir compte de relations non-linéaires quelconques et d'intéractions complexes entre les variables. Mais, sa grande flexibilité se fait au détriment d'une interprétation plus difficile du modèle. En effet, il faudrait un graphique en 4-dimensions pour représenter la fonction m. Une solution consiste à représenter la fonction m en 3 dimensions, en fixant la valeur de l'une des variables, mais la fonction représentée peutêtre très différente avec une valeur fixée différente.Nous utilisons les données d'une enquête de l'US Census Bureau daté de mai 1985, issues de l'ouvrage de Berndt et disponibles sour R.[C'est le jeu de données CPS1985 de la bibliothèque AER.] Nous estimons les deux modèles et utilisons une analyse de validation croisées par 10 blocs pour sélectionner la meilleure approche. Le modèle paramétrique (<ref>) est estimé par Moindres Carrés Ordinaires (ols). Le modèle entièrement non-paramétrique (<ref>) est estimé par la méthode des splines,car il en comprend peu de variables, ainsi que par les méthodes bagging, random forest et boosting.Le tableau <ref> présente les résultats de lavalidation croisée en 10 blocs (10-fold cross-validation). Le meilleur modèle est celui qui minimise le critèreℛ^10- CV. Les résultats montrent que le modèle (<ref>) est au moins aussi performant que le modèle (<ref>), ce qui suggère que le modèle paramétrique (<ref>) est correctement spécifié. §.§ Les déterminants des prix des logements à Boston (régression) Nous reprenons ici l'un des exemples utilisé dans James, dont les données sont disponibles sous R. Le jeu de données contient les valeurs médianes des prix des maisons (medv) dans n=506 quartiers autour de Boston, ainsi que 13 autres variables, dont le nombre moyen de pièces par maison (rm), l'age moyen des maisons (age) et le pourcentage de ménages dont la catégorie socio-professionnelle est peu élevée (lstat).[C'est le jeu de donnéesdela librairie . Pour une description complète des données, voir: https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/Boston.html.] Considérons le modèle de régression linéaire suivant := α ++εoù =[] est un vecteur en dimension 13 etest un vecteur de 13 paramètres. Ce modèle spécifie une relation linéaire entre la valeur des maisons et chacune des variable explicatives.Le modèle le plus flexible est le modèle entièrement non-paramétrique := m( ) +ε.L'estimation de ce modèle avec les méthodes du noyau ou les splines peut être problématique, car le nombre de variables est relativement élevé (il y a ici 13 variables), ou au moins trop élevé pour envisager estimer une surface en dimension 13.Nous estimons les deux modèles et utilisons une analyse de validation croisée par 10-blocs pour sélectionner la meilleure approche. Le modèle paramétrique (<ref>) est estimé par Moindres Carrés Ordinaires (ols) et le modèle entièrement non-paramétrique (<ref>) est estimé par trois méthodes différentes: bagging, forêt aléatoire et boosting (nous utilisons ici les valeurs par défaut utilisées dans James, pp. 328-331). Le tableau <ref> présente les résultats de lavalidation croisée en 10 blocs (10-fold cross-validation). La première ligne (in-sample) présente la qualité de l'ajustement des modèles en utilisant seulement les données d'apprentissage, c'est-à-dire celles qui ont servi à estimer le modèle, pour calculer le mse. La deuxième ligne (out-of-sample) présente la qualité de l'ajustement en utilisant d'autres données que celles ayant servies à estimer le modèle, pour calculer l'erreur quadratique.À partirdes résultats in-sample, les méthodes de bagging et de random forest paraissent incroyablement plus performantes que l'estimation ols du modèle linéaire (<ref>), le critèreℛ^10- CV passant de 21.782 à 1.867 et 1.849. Les résultats out-of-sample vont dans le même sens, mais la différence estmoins importante, le critèreℛ^10- CV passant de 24.082 à 9.59 et 9.407. Ces résultats illustrent un phénomène classique des méthodes non-linéaires, comme le bagging et la forêt aléatoire, qui peuvent être très performantes pour prédire les données utilisées pour l'estimation, maismoins performantes pour prédire des données hors-échantillon. C'est pourquoi la sélection de la meilleure estimation est habituellement basée sur une analyse out-of-sample, telle que présentée dans la deuxième ligne.La différence entre l'estimation du modèle linéaire (<ref>) et du modèle entièrement non-paramétrique (<ref>) est importante (24.082 vs 9.590, 9.407 et 11.789). Un tel écart suggère que le modèle linéaire est mal spécifié, et que des relations non-linéaire et/ou des effets d'intéractions sont présentes dans la relation entre le prix des logements, medv, et les variables explicatives . Ce résultat nous conduit àchercher une meilleure spécification paramétrique. À partir du modèle paramétrique(<ref>), et afin de prendre en compte d'éventuelles non-linéarités, le modèle additif généralisé (gam) suivant peut être considéré := m_1 (x_1)+m_2 (x_2)+…+m_13 (x_13) +ε,où m_1,m_2, … m_13 sont des fonctions inconnues. L'avantage de ce modèle est qu'il permet de considérer n'importe quelle relation non-linéaire entre la variable dépendante et chacune des variables explicatives. De plus, il ne souffre pas du problème du fléau de la dimension, car chacune des fonction est de dimension 1, et il est facilement interprétable. Toutefois, il ne prend pas en compte d'éventuels effets d'intéractions. L'estimation du modèle additif généralisé (<ref>) par la méthode des splines, dans le cadre d'une analyse de validation croisée par 10-blocs, donne une valeurℛ^10- CV=13.643. Par rapport au modèle paramétrique (<ref>), il y a un gain important (13.643 vs. 24.082). Mais la différence avec le modèle entièrement non-paramétrique (<ref>) reste conséquente (13.643 vs 9.590, 9.407, 11.789). Une telle différence suggère que la prise en compte de relations individuelles pouvant être fortement non-linéaires n'est pas suffisante, et que des effets d'intéractions entre les variables sont présents. Nous pourrions inclure dans le modèle les variables d'intéractions les plus simples entre toutes les paires de variables (x_i× x_j), mais cela impliquerait de rajouter un très grand nombre de variables au modèle initial (78 dans notre cas), qui ne serait pas sans conséquence sur la qualité de l'estimation du modèle. Quoi qu'il en soit, nous pouvons dire pour le moment que le modèle linéaire est mal spécifié et qu'il existe des effets d'intéractions pouvant être forts dans la relation entre medv et X, l'identification de tels effets restant délicat. Afin d'aller plus loin, les outils développés en apprentissage statistique peuvent être à nouveau d'un grand recours. Par exemple, l'estimation random forest s'accompagne de mesures de l'importance de chacune des variables dans l'estimation du modèle (décrit dans la section précédante). Le tableau <ref> présente ces mesures dans le cadre du modèle (<ref>), estimé surl'échantillon complet. Les résultats suggèrent que les variables rm et lstat sont les variables les plus importantes pour expliquer les variations des prix des logements medv. Ce constat nous conduit à enrichir la relation initiale, en rajoutant les intéractions liées à ces deux variables seulement, qui sont les plus importantes.Nous estimons le modèle additif généralisé incluant les variables d'intéractions, sur l'échantillon complet:= m_1(x_1)+ ⋯ + m_13(x_13)+ (:x)γ + (:x)δ +ε,où (:x) représente les variables d'intéractions de rm avec toutes les autres variables de x et (:x) représente les variables d'intéractions de lstat avec toutes les autres variables de x.[On a(:x)=[rm×chas, rm×nox,rm×age, rm×tax, rm×indus,rm×rad, rm×dis, rm×lstat, rm×crim, rm×black, rm×zn, rm×ptratio] et(:x)=[lstat×chas, lstat×nox, lstat×age, lstat×tax, lstat×indus,lstat×rad, lstat×dis, lstat×crim, lstat×black, lstat×zn, lstat×ptratio]. ] L'analyse des résultats de cette estimation suggère que les fonctions m̂_i sont linéaires pour toutes les variables, sauf pour la variable dis, dont la relation estimée est présentée dans la figure <ref>.Cette variable mesure la distance moyenne à cinq centres d'emplois de la région. L'effet semble diminuer plus rapidement avec la distance, lorsque celle-ci n'est pas très élevée. Au delà d'une certaine distance (au delà de 2, en log), l'effet est réduit, il continue à diminuer mais plus doucement. Cette relation non-linéaire peutêtre approchée par une régression linéaire par morceaux en considérant un nœud.Finalement, l'analyse précédente nous conduit à considérerle modèle linéaire suivant:= α + + (-2)_+θ + (: x)γ + (:x) δ +εoù (-2)_+ est égal àla valeur de son argument si ce dernier est positif, et à 0 sinon. Par rapport au modèle linéaire initial, ce modèle inclut une relation linéaire par morceaux avec la variable dis, ainsi que des effets d'intéractions entre rm, lstat et chacune des autres variables de . Le tableau <ref> présente les résultats de lavalidation croisée en 10 blocs (10-fold cross-validation) de l'estimation des modèles paramétriques (<ref>) et (<ref>),estimés par Moindres Carrés Ordinaires (ols), et du modèle additif généralisé (<ref>) estimé par les splines. Il montre que l'ajout des variables d'intéractions et de la relation linéaire par morceaux dans le modèle (<ref>) donne des résultat beaucoup plus performants que le modèle initial (<ref>): le critèreℛ^10- CV est divisé par plus de deux, il passe de 24.082 à 11.759. En comparant ces résultats avec ceux du tableau<ref>, on constate également que le modèle paramétrique (<ref>), estimé par ols, est aussi performant que le modèle général (<ref>) estimé par boosting ( ℛ^10- CV=11.789). La différence avec les méthodes bagging et forêt aléatoire n'est quant à elle pas très importante ( ℛ^10- CV=9.59, 9.407)Finalement, les méthodes bagging, forêt aléatoire et boosting ont permis de mettre en évidence une mauvaise spécification du modèle paramétrique initial, puis de trouver un modèle paramétrique beaucoup plus performant, en prenant compte des effets de non-linéarités et d'intéractions appropriées.§ CONCLUSION Si les « deux cultures » (ou les deux communautés) de l'économétrie et du machine learning se sont développées en parallèle, le nombre de passerelles entre les deux ne cesse d'augmenter. Alors que Varian présentait les apports importants de l'économétrie à la communauté du machine learning, nous avons tenté ici de présenter des concepts et des outils développés au fil du temps par ces derniers, qui pourraient être utiles aux économètres, dans un contexte d'explosion du volume de données. Si nous avons commencé par opposer ces deux mondes, c'est aussi pour mieux comprendre leurs forces et leurs faiblesses. Les fondements probabilistes de l'économétrie sont incontestablement sa force, avec non seulement une interprétabilité des modèles, mais aussi une quantification de l'incertitude. Néanmoins, nous l'avons vu à plusieurs reprises sur des données réelles, les performances prédictives des modèles de machine learning sont intéressantes, car elles permettent de mettre en avant une mauvaise spécification d'un modèle économétrique. De la même manière que les techniques non-paramétriques permettent d'avoir un point de référence pour juger de la pertinence d'un modèle paramétrique, les outils de machine learning permettent d'améliorer un modèle économétrique, en détectant un effet non-linéaire ou un effet croisé oublié.Une illustration des interactions possibles entre les deux communautés se trouve par exemple dans BelloniBelloni2, dans un contexte de choix d'instrument dans une régression. Reprenant les données deAngristKrueger sur un problème de réussite scolaire, ils montrent comment mettre en œuvre efficacement les techniques d'économétrie instrumentale quand on peut choisir parmi 1530 instruments disponibles (problème qui deviendra récurrent avec l'augmentation du volume de données). 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"authors": [
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"Antoine Ly"
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Statistics of shared components in complex component systems Matteo Osella[To whom correspondence should be addressed.Email: [email protected]] May 29, 2018 ========================================================================================= § INTRODUCTIONIn the last decade, the dynamics of active particles has attracted much attention from various viewpoints of nonequilibrium science <cit.>.This is a broad field concerning spontaneous dynamical motion accompanied by symmetry breaking, examples of which are found in both biological and artificial systems <cit.>.Some active particles are rigid with a prescribed fixed shape such as self-propelled colloids <cit.> and microswimmers such as bacteria <cit.> and Chlamydomonas <cit.>.Deformable active particles also exist, the shape of which changes in time.In particular, shape deformation is of great importance for biological organisms such as eukaryotic cells <cit.>.Deformable active particles are also realized by using liquid droplets and vesicles that undergo chemical reactions on the interface <cit.>. Depending on the environment, active particles are also classified into two different groups.One is swimming-type active particles, which self-propel in a fluid or on the interface between liquid and gas.Such swimmers include active colloidal particles and microorganisms in a solvent and camphor solids on the interface of an aqueous phase <cit.>.The other is crawling active particles, which migrate on solid or soft substrates, such as self-propelled liquid droplets <cit.> and eukaryotic cells <cit.>.In the latter case, the adhesion to the substrate plays an important role <cit.>.In order to elucidate the dynamics of these particles from a theoretical viewpoint, one needs to conduct studies of nonlinear dynamics and nonequilibrium statistical physics.A number of elaborate models have been studied for each specific system, both for swimming particles <cit.> and for crawling ones <cit.>, by taking into account the details of internal mechanisms.However, since the example of active particles includes both biological and synthetic systems, a basic theoretical description of active particles is also needed.For the case of deformable active particles, we have developed a general description <cit.>.By focusing on shape deformation, we derived nonlinear time-evolution equations from symmetry considerations. In contrast to active motion, the classical passive motion of a particle is a mechanical reaction to external forces.While the external forces acting on a particle do not vanish when integrated over the whole particle, the force relevant to active motion vanishes, which is known as the force-free and torque-free property of active particles.Still, force-free and torque-free active particles undergo a variety of dynamical motion even without external forces <cit.>.Here, symmetry breaking plays an important role so that active particles achieve spontaneous motion <cit.>. Then, a naturally arising question is what dynamics appear if the active particles move under external stimuli.In fact, in most realistic situations, there are various external effects caused by the environment.One is chemotaxis <cit.>, where the particles sense a chemical concentration gradient.This is particularly relevant to biological cells <cit.>, which migrate either towards the higher chemical concentration or away from it.Phototaxis is another important external factor <cit.> especially for some biological cells <cit.>.External stimuli can also be mechanical such as gravity <cit.> and confinement <cit.>. An important particular external stimulus for swimming active particles is a solvent flow field.The simplest external flow profile is a linear shear flow, which has been studied theoretically for rigid circular active particles <cit.> and for deformable active particles <cit.>, as well as experimentally by using bacteria <cit.>.Another flow profile that is more practical is a Poiseuille flow through tubes <cit.>.A realistic situation is swirl flows, which occur naturally, including in turbulent systems.Interestingly, such a flow is also induced by active particles themselves <cit.>.The dynamics of active particles in a swirl flow was investigated theoretically including particle shape deformation <cit.>, which was followed by a study of an artificial model swimmer <cit.>. It has also been studied experimentally using bacterial suspensions <cit.>. In this paper, we review our recent studies of active deformable particles in an external flow field.The organization of this review is as follows.In the next section (Sect. <ref>), we introduce a set of model equations that describes the dynamics of active deformable particles in a solvent flow field.Sections. <ref> and <ref> are devoted to the dynamics in a linear shear flow <cit.> and in a swirl flow <cit.>, respectively.In addition to the steady-state solutions, the collision dynamics of active particles with the swirl flow are presented in Sect. <ref>.§ MODELHere, we introduce the equations of motion that describe the dynamics of an active deformable particle under the influence of an external flow field.In order to make the analysis general, a set of coupled nonlinear dynamical equations is derived from symmetry considerations.These equations are proposed for the general case of three spatial dimensions and an unspecified flow profile.Afterwards, we confine ourselves to two dimensions and consider a linear shear flow and a swirl flow. First, we consider the flow field without the presence of the active particles.For simplicity, we assume that the flow profile is externally imposed and thus prescribed.We denote the external flow velocity by u, which varies as a function of space.The spatial dependence is characterized by the elongational part 𝖠 and the rotational part 𝖶, which are defined by <cit.>A_ij = ( ∂_i u_j +∂_j u_i ) /2,W_ij = ( ∂_i u_j -∂_j u_i ) /2,where the indices i and j label the Cartesian coordinates. An active particle moves spontaneously in the external flow field.That is, on the one hand, it is passively advected by the external flow and, on the other hand, it can actively self-propel with respect to the surrounding fluid.The active velocity measured with respect to the surrounding fluid flow is denoted by v.Here, we consider a small particle so that the Reynolds number is sufficiently small.Then, the time evolution of the center-of-mass position of the particle x is given byd x_i/dt = u_i + v_i. In the same manner, the rotation of the particle is divided into two parts.One is the passive rotation due to the external flow field, which is denoted by 𝖶,and the other is the active rotation that the particle exhibits spontaneously, which is denoted by Ω.The total rotation of the particle is represented by𝖶 +Ω.From Eq. (<ref>), 𝖶 is an antisymmetric tensor, and so is Ω.The antisymmetric tensor of the active rotation is related to the angular velocity vector ω asΩ_ij = ϵ_ijkω_k,where ϵ_ijk denotes the (i,j,k) component of the Levi-Civita tensor. Summation over repeated indices is implied, as throughout the remainder of this review. Now, we introduce the description of the shape deformation of the particle.For simplicity, we first consider two spatial dimensions and then explain the three-dimensional case.The shape of the particle is determined by the position of the interface, which is represented by the local radius with respect to the particle center of mass:R(θ̃,t) = R_0 +δ R(θ̃,t),where the angle θ̃ measures the direction around the center of mass.R_0 in Eq. (<ref>) stands for the equilibrium shape without deformation, for which we assume a circular shape.Thus, R_0 is a positive constant.The deformation is then described by the deviation from the equilibrium circular shape, δ R(θ̃,t), which may depend on time. Here, we assume that the shape deformation is not very large so that the local radius is a single-valued function with respect to the angle θ̃.Generally, the deformation is expanded in a Fourier series asδ R (θ̃,t) = ∑_m=2^∞( z_m(t) e^imθ̃ + z_-m(t) e^-imθ̃).Here, the zeroth mode is excluded by assuming that the original circular shape is sufficiently stable. The first Fourier mode represents the translation of the center of mass, which is therefore included in the center-of-mass velocity v.The lowest-mode deformation is thus given by the second Fourier mode z_±2, which represents an elliptical deformation.In two dimensions, we can define the second-rank traceless symmetric tensor as𝖲 =( [ S_11 S_12; S_21 S_22 ]) = ( [s cos 2θs sin 2θ;s sin 2θ -s cos 2θ ]),where we define z_±2 = (s/2) exp(∓ 2i θ).Note that the symmetric tensor can also be defined for each of the higher-order deformation modes <cit.>.The symmetric-tensor description of the deformation is general in the sense that the same form is applicable for both two and three dimensions.In the case of three spatial dimensions, the deviation δ R must be expanded into spherical harmonics Y_ℓ m with coefficients c_ℓ m, to which the symmetric deformation tensor is related likewise <cit.>.The lowest mode of the deformation is given by ℓ=2, representing an ellipsoidal deformation.Hereafter, we take into account only the lowest-mode deformation 𝖲. In total, we have introduced three central dynamical variables to characterize the state of a deformable active particle:the active propulsion velocity v, the active rotation Ω, and the elliptical deformation 𝖲.For the sake of generality, the time evolution equations for these variables are derived on the basis of symmetry arguments.We consider the following coupled nonlinear equations <cit.>: d v_i/d t = α v_i -(v_k v_k) v_i -a_1 S_ik v_k -a_2 ( W_ik +Ω_ik ) v_k, d Ω_ij/d t = ζΩ_ij -μ_0 ( Ω_kℓΩ_kℓ ) Ω_ij +μ_1 ( Ω_ik S_kj - Ω_jk S_ki ) +μ_2 S_ikΩ_kℓ S_ℓ j,d S_ij/d t = -κ S_ij +b_1 [ v_i v_j -1/d ( v_k v_k ) δ_ij]-b_2 [ (W_ik +Ω_ik) S_kj + (W_jk +Ω_jk) S_ki ]+b_3 [ Ω_ik S_kℓΩ_ℓ j -1/d ( Ω_mk S_kℓΩ_ℓ m ) δ_ij] +b_4 ( Ω_kℓΩ_kℓ ) S_ij +ν_1 ( A_ij -1/d A_kkδ_ij) +ν_2 [ A_ik S_kj +A_jk S_ki -2/d ( A_kℓ S_ℓ k ) δ_ij].Here, δ_ij denotes the Kronecker delta and d is the dimension of the space.All the coefficients are phenomenological parameters.The meaning of each term in Eqs. (<ref>)–(<ref>) is explained below.In principle, more terms and higher-order couplings can be included, but the current model already covers the main physical aspects that we intend to describe.Note that we consider the effect of the external flow on the particle dynamics but the inverse effect is not included here.Such an effect is important for, for example, hydrodynamic interactions between particles <cit.>.In this paper, we devote ourselves to the single-particle dynamics. We start with the first two terms on the right-hand side of Eq. (<ref>).They can be rewritten as-∂ℱ/∂ v_i with ℱ = -α/2 (v_k v_k) +1/4 (v_k v_k)^2,where ℱ is a Lyapunov functional controlling the spontaneous self-propulsion.By increasing α, the particle exhibits a supercritical pitchfork bifurcation from v=0 to v≠0 at α=0, corresponding to the onset of self-propulsion.Such a structure is also considered in continuum descriptions of flocks of active particles <cit.>.The drift bifurcation formula, Eq. (<ref>), together with the time derivative term on the left-hand side of Eq. (<ref>), was derived for the isolated domain solution of reaction-diffusion equations in both two <cit.> and three dimensions <cit.>.It was also derived from the Stokes equation for a droplet catalyzing a chemical reaction on its inside, which changes the interfacial tension <cit.>.The Marangoni flow arising from the nonuniform distribution of the interfacial tension causes the droplet to move.The equation for the migration velocity is derived in the limit of an infinitesimally thin interface.The time derivative term of the center-of-mass velocity appears as a consequence of the time delay, that is, the relaxation of the concentration is much slower than that of the fluid velocity. The same bifurcation structure is taken into account for the active rotation of the particle Ω in the first line of Eq. (<ref>).Indeed, the spontaneous spinning motion of active particles is found in experiments <cit.>.The parameter ζ characterizes the active rotation strength; the particle does not exhibit a spontaneous rotation if ζ≤0; otherwise, it spins with angular velocity √(ζ/2μ_0).Here, μ_0 is a positive constant. In contrast, a spontaneous deformation of the particle is not considered here.The linear damping term with coefficient κ>0 in Eq. (<ref>) describes the relaxation of the deformation back to the spherical (circular) shape.The deformation is then caused through the coupling terms to v and Ω and by stretching due to the external flow. The elongational part of the externally imposed flow field 𝖠 deforms the particle shape through the terms with coefficients ν_1 and ν_2.These terms are identical to those in a previous study on the dynamics of a nonactive droplet in a fluid flow, which also considered elliptical shape deformation <cit.>.Note that the term with coefficient ν_2 vanishes for a two-dimensional incompressible flow.The deformation is also induced by the active velocity through the second term on the right-hand side of Eq. (<ref>).The coefficient b_1 determines the magnitude of the effect and also the tendency of the elongation direction.When b_1 is positive (negative), the particle tends to elongate parallel (perpendicular) to the self-propulsion direction.The third term on the right-hand side of Eq. (<ref>) is the counter term of the b_1 term if a_1 b_1 <0.That is, both terms are derived from the functional derivative of b_1 v_k S_kℓ v_ℓ <cit.>.However, here we take the same sign for a_1 and b_1 considering the nonvariational case. Then, because of the a_1 term, the self-propulsion speed changes and its direction reorients depending on the particle deformation.These terms are the leading-order coupling terms between the velocity v and the deformation 𝖲. The effect of these coupling terms between the active velocity and second-order deformation was studied previously by using Eqs. (<ref>) and (<ref>) without the active rotation (Ω=0) <cit.>.After undergoing the bifurcation from the motionless state v=0 without any deformation 𝖲=0 to the self-propulsion regime v≠0 at α=0, the particle migrates spontaneously in a straight trajectory with its shape elliptically deformed 𝖲≠0. The straight solution is stable as long as 0<α<α^*, while it becomes unstable and the particle starts to migrate in a circular trajectory for α>α^*.Here the bifurcation threshold from the straight motion to the circular motion is given by <cit.>α^* = κ^2/a_1 b_1 +κ/2. The active rotation Ω also influences the deformation 𝖲 through the coupling terms with coefficients b_3 and b_4 on the right-hand side of Eq. (<ref>).For b_3>0 and b_4>0, the spontaneous rotation of the particle enhances the degree of the deformation, while it reduces it for b_3<0 and b_4<0.In two dimensions, these terms are equivalent for b_3=2b_4.In a three-dimensional space, the term with coefficient b_3 includes an additional effect that rotates the deformed particle, in contrast to the term with b_4.Note that the b_3 term is corrected in Eq. (<ref>), which was not traceless in three dimensions in the previous formula in Ref. .This correction, however, does not change any results discussed in Refs. andsince we considered only two-dimensional dynamics.The third and fourth terms on the right-hand side of Eq. (<ref>) have a similar effect on the rotation Ω induced by the deformation 𝖲.Note that the term with coefficient μ_1 vanishes in two dimensions. Finally, both the active rotation of the particle Ω and the passive rotation due to the external flow field 𝖶 reorient the particle active velocity and the particle configuration.This effect is included by the term with coefficient a_2 in Eq. (<ref>) and by the term with coefficient b_2 in Eq. (<ref>).Since an active particle can follow a prescribed rule on how to react to an external rotational flow, the numerical value of the coefficients cannot be fixed at this point.In principle, the contribution with a_2>0 describes so-called Magnus effect, a force acting on the particle in the direction perpendicular to the velocity and angular velocity. Generally, the model equations of Eqs. (<ref>) and (<ref>)–(<ref>) apply to a three-dimensional setup. For simplicity, however, we confine ourselves to two spatial dimensions for the remainder of this review. § LINEAR SHEAR FLOWAs the simplest example of a flow profile, a linear steady shear flow is considered in two spatial dimensions.The flow velocity is given byu = ( γ̇ y, 0 ),where γ̇ is the shear rate. For the analytical investigation below, we parametrize the position, the velocity, and the angular velocity by x=(x,y),v = ( vcosϕ, vsinϕ ), Ω = ( [0ω; -ω0 ]).Then, Eqs. (<ref>) and (<ref>)–(<ref>) are rewritten asd x/dt = vcosϕ +γ̇ y, d y/dt = vsinϕ,dv/dt = α v -v^3 -a_1 v s cos 2(θ-ϕ), dϕ/dt = -a_1 s sin 2(θ-ϕ) +a_2 ( -γ̇/2 +ω),dω/dt = ζω - 2μ_0 ω^3 -μ_2 s^2 ω,ds/dt = -κ s +b_1 v^2/2cos 2(θ-ϕ) +b̃ s ω^2 +ν_1 γ̇/2sin2θ,dθ/dt = -b_1 v^2/4 ssin 2(θ-ϕ) +b_2 ( -γ̇/2 +ω) +ν_1 γ̇/4 scos 2θ,where we define b̃ = b_3 +2 b_4. From these expressions, it is obvious that both x and y do not affect the dynamics of the other variables, and so we can solve Eqs. (<ref>)–(<ref>) separately from Eq. (<ref>).The dynamics governed by Eqs. (<ref>)–(<ref>), and thus, by Eqs. (<ref>)–(<ref>) with Eq. (<ref>), are invariant under the simultaneous transformations (a_1,b_1)→(-a_1,-b_1) and ϕ→ϕ +π/2.This invariance ensures that the choice of signs for a_1 and b_1 does not change the dynamical structure.As we have explained in Sect. <ref>, the choice of signs determines the characteristic direction of the self-propulsion with respect to that of the elliptical deformation; the particle tends to self-propel in the longitudinal (lateral) direction of the elliptic shape if a_1>0 and b_1> 0 (if a_1<0 and b_1<0).The dynamics of the particle position, Eq. (<ref>), is given by a superposition of the dynamics governed by Eqs. (<ref>)–(<ref>) and the simple advection due to the external flow u.Therefore, although the particle trajectory is shifted slightly, the dynamical structures such as the transition from one dynamical solution to another are not affected by the choice of the signs of a_1 and b_1. In the following, we first consider the limited case of a circular shape without deformation to see the role of the particle activity and also to clarify the connection to a model of a rigid active particle.Then, we describe the dynamics of a self-propelled deformable particle but without active rotation.§.§ Rigid active particleHere, we consider the special case of a rigid active particle of circular shape.We neglect the deformation, and thus we set s=0 in Eqs. (<ref>)–(<ref>), and drop Eqs. (<ref>) and (<ref>) from our model equations. We now assume that the magnitude of the velocity v and that of the relative rotation ω relax quickly so that they are given by the steady-state solutions of Eqs. (<ref>) and (<ref>), i.e., v=v_0 and ω=±ω_0, where v_0 = √(α),ω_0 = √(ζ /2μ_0).The positive (negative) sign in front of ω_0 corresponds to the counterclockwise (clockwise) rotation. For these solutions, Eqs. (<ref>) and (<ref>) are solved asx(t) = v_0 ( ω̃ -γ̇ / ω̃^2 ) [ sinϕ(t) -sinϕ_0 ]+γ̇ [ (v_0 / ω̃ ) cosϕ_0 +y_0 ] t +x_0,y(t) = - (v_0 /ω̃) [ cosϕ(t) -cosϕ_0 ] +y_0,ϕ(t) = ω̃ t +ϕ_0,where (x_0,y_0) and ϕ_0 are the position of the center of mass and the direction of the velocity vector at t=0, respectively.This set of solutions represents a cycloidal trajectory as long as ω̃≠ 0.Here, we have defined the effective angular velocityω̃ = a_2 [ -(γ̇/2) ±ω_0 ]. In the special case that the effective angular velocity vanishes, ω̃=0, the passive rotation due to the external flow is balanced by the particle active rotation.Then, Eq. (<ref>) gives ϕ(t) = ϕ_0 and the solutions of Eq. (<ref>) becomex(t) = ( γ̇ v_0 /2) ( sinϕ_0 ) t^2 +( v_0 cosϕ_0 +γ̇ y_0 ) t +x_0,y(t) = ( v_0 sinϕ_0) t + y_0.The meanings of (x_0,y_0) and ϕ_0 are the same as above.This set of solutions causes the particle to move in a parabolic trajectory instead of a cycloid. Similar equations of motion and results were also obtained for an active Brownian particle under a linear shear flow <cit.>.In this case, the equations of motion are in the overdamped limit.The particle possesses a polarity, along which it tends to self-propel.Its self-propulsion speed as well as its active rotation, which rotates the polarity, fluctuate around constant values.In the limit of no fluctuation, the same trajectories as above are obtained.§.§ Active deformable particleNow, we consider the dynamics of a self-propelled deformable particle in a linear shear flow. For simplicity, we include only the active velocity and eliminate the active rotation by setting ζ<0. We numerically integrate Eqs. (<ref>) and (<ref>)–(<ref>) with ζ=-0.1.In the following simulation results, we numerically confirmed that the active rotation always vanishes (Ω=0).Here, we discuss the perpendicular case with a_1=b_1=-1.The other parameters are chosen as a_2=b_2=μ_2=ν_1=1 and b̃=1.The results of the numerical simulation are summarized in Fig. <ref>.Panel (a) shows the dynamical phase diagram, where the shear rate γ̇ of the imposed linear shear flow and the magnitude of the particle self-propulsion α are varied for the intermediate damping rate of deformation κ_2=0.5.Panels (b)–(f) display the trajectories in real space (left and right columns) and the attractors in ϕ-θ space for each dynamical solution.The figures in the right column are obtained for a_1=b_1=1 and the others are for a_1=b_1=-1. As mentioned in Sect. <ref>, there are three solutions in the absence of an external flow: the motionless solution for α<0, the straight solution for 0<α<α^*, and the circular solution for α^*<α.From Eq. (<ref>), α^*=0.5 for our present parameters.These solutions are indicated by “motionless”, “straight”, and “circular” in Fig. <ref>(a), respectively. If the external shear flow is switched on, the particle in the motionless state α<0 then exhibits a trivial solution, where it is slightly elongated and simply advected parallel to the flow for the parameters indicated by the cyan filled pentagons in Fig. <ref>(a).For positive α, the particle self-propels on top of the passive advection.If 0<α<α^*, it undergoes active straight motion, as denoted by the green open pentagons in Fig. <ref>(a).For this solution, the active velocity is finite and time-independent, and thus the trajectory becomes straight if seen from the frame comoving with the flow, as shown in Fig. <ref>(b).On the other hand, the particle describes a cycloidal trajectory for α>α^*, where the shape deformation rotates with its center-of-mass trajectory.The cycloidal trajectories with counterclockwise and clockwise rotations are depicted in Figs. <ref>(c) and <ref>(d), respectively.We refer to these solutions as cycloidal I motion to distinguish them from the other cycloidal solution that we explain shortly.The parameter region where the cycloidal I motion was obtained is plotted by the red open squares in Fig. <ref>(a).As the shear rate increases, the cycloidal I motion rotating in the opposite direction to the external flow, i.e., in the counterclockwise direction, first becomes unstable, and then that with the clockwise rotation loses its stability and starts to undergo winding motion, as shown by the purple filled triangles in Fig. <ref>(a).Unlike the cycloidal I motion, the shape deformation of the particle for the winding solution does not rotate but oscillates as displayed in Fig. <ref>(e).The magnitude of the oscillation decreases with increasing γ̇ and finally vanishes so that the particle undergoes the active straight motion. Note that there is a wide range of γ̇ where the active straight solution was obtained for α>0 in Fig. <ref>(a).For a much larger shear rate, the particle with α>0 exhibits the cycloidal II motion, where its shape is always elongated almost horizontally and does not rotate, as displayed in Fig. <ref>(f).This solution is found for the shear rates denoted by the gray filled squares in Fig. <ref>(a). As noted at the beginning of Sect. <ref>, Eqs. (<ref>)–(<ref>) can be solved independently of Eq. (<ref>) and they are invariant with respect to the simultaneous transformations (a_1,b_1)→(-a_1,-b_1) and ϕ→ϕ+π/2.Indeed, the numerical simulation of Eqs. (<ref>) and (<ref>)–(<ref>) with a_1=b_1=1 results in the same dynamical phase diagram as Fig. <ref>(a), which was obtained for a_1=b_1=-1.However, the real-space trajectories are slightly modified as shown in the plots in the right column of Figs. <ref>(b)–<ref>(f), which should be compared with those in the left column of Figs. <ref>(b)–<ref>(f).Here, all the other parameters are kept the same. On the whole, for small shear rates, the particle undergoes the motion that is obtained as the superposition of the passive advection due to the flow and the active motion, which it exhibits in the absence of the external flow.As the shear rate increases, the effect of the external shear flow increases and the dynamics becomes complicated.This feature is the same even if the active rotation exists Ω≠0 <cit.>.§ SWIRL FLOWNext, we consider the dynamics of an active deformable particle under a rotational flow (swirl).Swirl flows occur naturally in many situations including turbulence.The flow velocity of the swirl that we consider here is of the formu = ( - σ y /(x^2 +y^2), σ x /(x^2 +y^2) ),where σ sets the strength of the vortex.Since this flow profile possesses rotational symmetry, we measure the particle center-of-mass position byx = ( r cosη, r sinη )with distance r and direction η with respect to the center of the vortex flow, whereas the center-of-mass velocity and deformation are parametrized by Eq. (<ref>).Note that the vortex flow given by Eq. (<ref>) has a flow potential such that u = - ∇ U, where U = μarctan ( x/y ).Consequently, ∇×u = 0, which implies that there is no local rotational contribution, i.e., 𝖶 = 0.In contrast, the stretching contribution does not vanish and is calculated as𝖠 = ( [σ r^-2sin 2η -σ r^-2cos 2η; -σ r^-2cos 2η -σ r^-2sin 2η ]).From this, one can see that the equations of the active velocity [Eq. (<ref>)] and the shape deformation [Eq. (<ref>)] depend on the position of the particle in the case of the swirl, unlike the case of the linear shear flow in the previous section.This implies that the dynamics of the parallel configuration (a_1,b_1>0) and the perpendicular configuration (a_1,b_1<0) likely differ. We first investigate the steady-state solutions of Eqs. (<ref>) and (<ref>)–(<ref>) with Eq. (<ref>) by numerically integrating them.Afterwards, we consider the scattering dynamics of active deformable particles by the swirl flow.The setup of a swirl has a geometrical similarity to that of a collision and scattering experiment and therefore possesses a resemblance to the classical Kepler and Rutherford problem.We distinguish the two cases of self-propulsion in the parallel and perpendicular directions with respect to the elongation of the particle shape.§.§ Steady-state solutionsFirst, we discuss the steady-state solutions of an active deformable particle in the swirl flow.The model equations are too complicated to solve analytically, and therefore, we numerically integrate Eqs. (<ref>) and (<ref>)–(<ref>) with Eq. (<ref>).Since the flow profile [Eq. (<ref>)] possesses rotational symmetry, we only vary the initial distance from the flow center. Here, we distinguish the particles that tend to align the active velocity parallel to the elongation of the shape deformation and those that tend to self-propel perpendicularly.The parallel particles are realized by setting a_1=b_1=1, and the perpendicular ones by a_1=b_1=-1.In both cases, the other parameters are fixed as κ=0.5, ν_1=1, and σ=1.Note that there are no contributions from the terms with coefficients a_2, b_2, and b̃ since we omit the active rotation, i.e., Ω=0, and the swirl flow [Eq. (<ref>)] does not possess any rotational contribution 𝖶=0. Before discussing the case of active particles, we consider the motion of a passive particle in the swirl, i.e. α<0.In this case, the particle is always passively advected by the circular flow, following a circular trajectory around the vortex center.The circular trajectory is marginally stable in the radial direction, that is, its radius can take any value depending on the initial conditions.In Fig. <ref>, the red pluses indicate the radius of the marginally stable passive circular motion, which are obtained starting from different initial distances.However, if α becomes close to the bifurcation threshold of self-propulsion located at α=0, a region appears where the particles cannot stay and where they are repelled from, as shown by the horizontal gray arrows in Fig. <ref>.The theoretical analysis <cit.> reveals that the radius of the passive rotation r_0 should satisfy the stability condition r_0≤ r_ min or α +2σ^2 (r_ min^-4 -r_0^-4 )^1/2 (κ^2 r_0^4 +4 σ^2)^-1/2 < 0for r>r_ min.Here, we have definedr_ min = ( 2 |σ| )^1/2 [ (a_1 ν_1)^2 -κ^2 ]^-1/4.In this passive case, there is no difference between the parallel case a_1,b_1>0 and the perpendicular case a_1,b_1<0.However, a difference appears if the particle possesses an active velocity, i.e., α>0, as we will see next. First, we discuss the perpendicular case (a_1=b_1=-1), where the active velocity tends to align perpendicular to the elongation direction of the shape deformation.When 0<α<α^*, where α^*=0.5 for the current parameters, the particle undergoes circular motion around the vortex center, as displayed in Fig. <ref>(a).We refer to this as active circular motion because, in contrast to the passive circular motion that appears for α<0, only one radius of the circular trajectory r_0 is selected for each self-propulsion strength α, as shown by the cyan diamonds in Fig. <ref>.Depending on the initial conditions, the swimmer either asymptotically approaches this orbit or it manages to escape from the swirl to an infinite distance, as shown in Fig. <ref>(b).When α> α^*, the situation becomes markedly different.Starting sufficiently close to the radius r_0, we still observe the steady-state active circular motion as indicated in Fig. <ref>.However, another type of motion occurs depending on the initial conditions.We refer to it as lunar-type motion, the typical trajectory of which is depicted in Fig. <ref>(c) for α=1. This trajectory is understood as the circular motion that already occurs in the absence of the swirl for α>α^* <cit.> superimposed onto the circular convection due to the vortex flow.In this case, both rotation directions, the one of the smaller revolution and the one of the larger revolution, are the same as that of the fluid flow.The radius of the larger revolution depends on the initial conditions. On the other hand, the active circular motion is not found in the parallel case (a_1=b_1=1), where the particle tends to self-propel in the elongation direction of the shape deformation.Instead, all particles escape far from the vortex center when 0<α<α^*=0.5, as shown in Fig. <ref>(a).In contrast, for α>α^*, a particle again undergoes lunar-type motion as displayed in Fig. <ref>(b).However, the smaller revolution and the fluid flow have opposite directions of rotation, whereas the rotation directions of the larger revolution and the fluid flow are identical.Again, the radius of the larger revolution depends on the initial conditions. When α>0.7, the situation becomes more complex in the parallel case.Depending on the initial conditions, multicircular motion can emerge as illustrated in Fig. <ref>(c), where the lighter gray, cyan, and red lines show trajectories of different time intervals.To obtain the multicircular motion, the swimmer was initially placed relatively close to the vortex center. In summary, in a swirl flow, the difference between the parallel (a_1=b_1=1) and perpendicular (a_1=b_1=-1) configurations has a strong effect on the steady-state solutions, unlike the case of the linear shear flow shown in Sect. <ref>.The active deformable particles with the perpendicular configuration either escape from the swirl flow or are captured depending on the initial conditions.In the latter case, they exhibit the active circular motion around the vortex center for 0<α<α^* and the lunar-type motion for α>α^*.Active deformable particles with the parallel configuration always escape from the swirl for 0<α<α^*, while for α>α^*, they undergo the lunar-type motion or, if they are initially placed very close to the vortex center, the multicircular motion.§.§ Scattering dynamicsNow, we study the collision dynamics of the active deformable particles with the swirl flow.This is performed in analogy to a classical scattering experiment.As will be explained shortly, the particles are either scattered or captured by the swirl.If the particles are scattered and manage to escape from the vortex, we measure the scattering angle of the event.For this purpose, we determine the scattering angle η_ scat between the initial velocity orientation and the final velocity orientation when the particle has reached a certain distance r_ scat from the vortex center.Owning to the swirl geometry, the event of passing the vortex center on one side differs from that of passing it on the other side.Therefore, in the following numerical simulations, we measure the scattering angles η_ scat by integrating the changes in the particle velocity during the course of scattering. To make the setup meaningful in the sense of a scattering experiment, we set the propulsion strength to values 0<α<α^*.For these values, the particle undergoes straight motion in the absence of the flow field <cit.>.We provide this solution as an initial condition and place the particle at a comparatively large distance r_ init = 1.5 × 10^4, with its active velocity heading towards the vortex center.If the particle was not affected by the flow field of the swirl, it would propel exactly in the direction of its initial velocity orientation.The distance d_ imp by which it would then miss the vortex center is called the impact parameter.A swimmer of d_ imp=0 would hit the center of the vortex if it were not affected by the swirl flow.We define d_ imp >0 when the particle velocity is initially oriented towards the side of the oppositely directed fluid flow.In contrast, we set d_ imp < 0 when the particle initially propels towards the side of the identically directed fluid flow.See Figs. <ref>(b) and <ref>(b) for an illustration of the definition of the sign of d_ imp. After numerically integrating Eqs. (<ref>), (<ref>), and (<ref>) with Eq. (<ref>), we measure the scattering angle at the distance r_ scat = 10^4 if a scattering event occurs.We varied the values of the propulsion strength α and the impact parameter d_ imp, while the other parameters were chosen as before.Our results are summarized in Figs. <ref> and <ref>, which display the scattering angles as functions of the impact parameter in panel (a) and the real-space trajectories for α=0.3 on a large scale in panel (b) and on a small scale in the vicinity around the swirl center in panels (c) and (d).Again, we distinguish between the perpendicular configuration (a_1 = b_1 = -1) and the parallel configuration (a_1 = b_1 = 1).§.§.§ Perpendicular configurationsFirst, we show the collision of the active deformable particles with the perpendicular configuration with the swirl.Generally, the vortex causes the particle trajectory to deviate from the original straight one, as in Fig. <ref>(b).For negative impact parameters d_ imp, the particle is only weakly deformed and stays far from the vortex center so that it finally leaves the vortex with basically the same velocity orientation as the incident velocity.Thus, the scattering angle η_ scat becomes almost zero, as plotted in Fig. <ref>(a).This is true even for slightly positive impact parameters, as illustrated in Fig. <ref>(a) andFigs. <ref>(b) and <ref>(c) for d_ imp = 60.With increasing impact parameter, the scattering angle increases, as shown in Fig. <ref>(a), where the particle approaches even closer to the vortex center and its trajectory becomes significantly bent as displayed in Fig. <ref>(c) for d_ imp = 70 and 85.The particle circles around the vortex center before it escapes from the swirl as depicted in Fig. <ref>(c) for d_ imp = 89.This strong effect of the swirl on the particle dynamics is indicated by the scattering angles η_ scat>π in Fig. <ref>(a). With further increasing d_ imp, the scattering angle seems to diverge in Fig. <ref>(a).Indeed, at even higher impact parameters, the particle is eventually caught by the swirl and cannot escape from it, as shown in Fig. <ref>(d) for d_ imp = 90, 100, and 103.Interestingly, in all these cases, the trajectories end in the same circle around the vortex center.This attractive trajectory corresponds to the active circular motion discussed in Sect. <ref>.Finally, when the impact parameter is very large, the particle no longer moves sufficiently close to the swirl center to be effectively captured.Instead, it is scattered again; however, it passes the vortex center on the other side, where it propels against the flow velocity as shown in Fig. <ref>(c) for d_ imp = 104 and 110.This scattering event where the particle propels in the opposite direction to the flow velocity is indicated by the scattering angle η_ scat approximately -2π in Fig. <ref>(a).The stability of the active circular motion and the robustness of the capturing event were studied numerically by introducing a stochastic noise term to the equation of the active velocity [Eq. (<ref>)] <cit.>.The latter was also investigated by changing the initial distance, which reveals that the active deformable particles are captured in qualitatively the same manner independent of the initial distance.Although the active circular motion is stable for large fluctuation intensities, the capturing event turns out to be much more fragile.This is because the trajectory can be shifted considerably owning to the fluctuation on the path towards the swirl center so that the particle does not even reach the close vicinity of the vortex center to be captured. §.§.§ Parallel configurationsNext, we consider the collision of the particles with the parallel configuration a_1=b_1=1.In this case, no permanent capturing by the swirl was observed, in contrast to the case of the perpendicular configuration.While they are heading towards the vortex, the situation for active particles with the parallel configuration is much different from those with the perpendicular configuration; the trajectory becomes curved in the opposite direction [compare Fig. <ref>(b) with Fig. <ref>(b)].Therefore, significant scattering now takes place for negative impact parameters d_ imp, as demonstrated in Figs. <ref>(b)–<ref>(d).For negative impact parameters d_ imp of large magnitude, the particle trajectory is only slightly affected by the swirl.The propulsion direction suffers a slight change, i.e., η_ scat is almost zero, as displayed in Fig. <ref>(c) for d_ imp=-75.With increasing impact parameter, the particle appraoches closer to the vortex center and the scattering angle η_ scat increases; see the trajectory for d_ imp=-71 in Fig. <ref>(c).Interestingly, as the impact parameter is increased, the scattering angle jumps discontinuously as in Fig. <ref>(a).The trajectory for d_ imp=-70 in Fig. <ref>(d) shows the drastic event that occurs in this case, which explains the jump in the scattering angle.The particle becomes close to the vortex center and is temporarily caught by the swirl, describing a loop around the vortex center before it escapes.As indicated by the trajectories for the other d_ imp in Fig. <ref>(d), the same behavior is obtained upon further increasing the impact parameter, although the scattering angle decreases continuously.As highlighted by the inset of Fig. <ref>(d), in all these cases, the particle performs a loop around the center in the same direction as the fluid flow. Finally, another discontinuous jump of the scattering angle occurs in Fig. <ref>(a) at even higher impact parameters.After the jump, the particle no longer performs a narrow loop around the vortex center.Instead, its trajectory features a simple bend around the swirl, as depicted in Fig. <ref>(c) for d_ imp=-35 and -25.The complete trajectory of the scattering event for d_ imp=-25 is depicted in Fig. <ref>(b).Again, the scattering events where the particle propels against the flow velocity are indicated by the shift of the scattering angle η_ scat by -2π in Fig. <ref>(a).To sum up, the collision dynamics of active deformable particles with the swirl is divided into two: weak collisions, where the particles are slightly affected but are finally scattered by the swirl, and strong collisions, where the particles reach the close vicinity of the vortex center.The weak collisions occur for small and large impact parameters, whereas the strong collisions are observed for intermediate impact parameters.In the case of the perpendicular configuration (a_1=b_1=-1), the particles with small and large impact parameters are simply advected in addition to the self-propulsion before being scattered from the vortex center as in Figs. <ref>(b) and <ref>(c).For positive intermediate impact parameters, strong collisions are obtained, where the particles are captured by the swirl as in Fig. <ref>(d).In contrast, for the parallel configuration (a_1=b_1=1), the particles self-propel even against the flow velocity as shown in Figs. <ref>(b) and <ref>(c) so that a strong collision is obtained for negative impact parameters.In this case, the particles describes a small loop around the vortex center in the same direction as the flow velocity and are finally scattered far away by the swirl as depicted in Fig. <ref>(d).§ SUMMARYWe have reviewed the dynamics of active deformable particles under an external flow field.By focusing on the shape deformation, we have introduced a general model, which includes the effect of an external flow, based on symmetry considerations.Our model does not depend on any details of the specific system such as a mechanical origin of the activity.Since the variables are described by tensors, the obtained formulae are applicable to both two and three spatial dimensions. We have investigated the dynamics in two dimensions under two different flow profiles, a linear shear flow as the simplest case and a swirl flow.In both cases, the time-evolution equations are solved numerically.In addition, the dynamics of the collision between an active particle and the swirl flow was investigated numerically, revealing a capturing event and a complicated scattering trajectory depending on the impact parameters.An important flow that has not yet been studied for the deformable active particles is a Poiseuille flow.This type of flow profile often appears in tubes; therefore, the interaction with the boundary wall plays a major role <cit.>.Additionally, when the density of the suspension is high, the interaction between the active deformable particles needs to be taken into account.Moreover, hydrodynamic interactions can be included either by solving the full fluid dynamic equations <cit.> or by employing the Green's function method <cit.>. 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"authors": [
"Mitsusuke Tarama"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170726092134",
"title": "Dynamics of Deformable Active Particles under External Flow Field"
} |
Ekaterina Vylomova et al.Ekaterina Vylomova, Andrei Shcherbakov, Yuriy Philippovich, Galina Cherkasova The University of Melbourne, Melbourne, Australia,[email protected] & [email protected], Moscow Polytech, Moscow, Russia,[email protected] Institute of the Science of Language, Moscow, [email protected] Men Are from Mars, Women Are from Venus: Evaluation and Modelling of Verbal Associations Ekaterina Vylomova1 Andrei Shcherbakov1 Yuriy Philippovich2 Galina Cherkasova3 December 30, 2023 ======================================================================================== We present a quantitative analysis of human word association pairs and study the types of relations presented in the associations. We put our main focus on the correlation between response types and respondent characteristics such as occupation and gender by contrastingsyntagmatic and paradigmatic associations. Finally, we propose a personalised distributed word association model and show the importance of incorporating demographic factors into the models commonly used in natural language processing. § INTRODUCTION Most of contemporary approaches in natural language processing (NLP) mainly rely on well-annotated and clean textual corpora. For instance, language as well as translation models are typically trained over Europarl<cit.> or the Wall Street Journal corpora. As Eisenstein<cit.> noted, most of such corpora present language used by a very specific social group. For example, Hovy<cit.> showed that the models trained over the Wall Street Journal perform better for old language users. And it becomes extremely troublesome to adapt the models trained over these corpora to new domains such as Twitter. In most cases researchers either normalize the data (for instance, by using string and distributional similarity as in Han<cit.>) or apply various techniques of domain adaptation and knowledge transfer. Recently several studies in sociolinguistics demonstrated how the NLP models could be improved by considering social factors (see Volkova<cit.>, Stoop<cit.>). This inspired us to exploit associative experiments approach to demonstrate how specialization and gender might affect associations. In this paper we first propose the dataset for associative pairs of Russian native speakers[The dataset is available at <http://github.com/ivri/RusAssoc>] and then show how association types vary across gender and occupation. We also present a simple PPMI-based model of associations and demonstrate the difference in the model's predictions depending on the social characteristics.The paper is structured as follows. We first discuss previously organized associative experiments, then we introduce the dataset for Russian speakers associations. In Section 4 we analyse how the associations depend on demographic factors and, finally, we present a personalised associative vector model.§ RELATED WORK Introduced by Sir Francis Halton in 1870s, associative experiments became a common approach to study human cognition. Nowadays various researchers organized the experiments on many languages. Most of the experiments present English (American and British) native speakers (see Deese<cit.>, Cramer<cit.>, Kiss<cit.>, Nelson<cit.>). De Groot<cit.> and De Deyne<cit.> conducted them for Dutch, and Shaps<cit.> for Swedish. There are also some for Eastern languages, such as Japanese (see Okamoto & Ishizaki<cit.> and Joyce<cit.>), Korean (Jung <cit.>); and Hebrew (Rubinstein<cit.>) for Semitic group. Lots of research had been done on Slavic languages as well. Novak<cit.> organized the experiment for Czech, Ufimtseva<cit.> presented Slavic Associative Thesaurus comprising of Russian, Belarusian, Bulgarian, and Ukrainian. Finally, Russian thesauri were developed by Leontiev<cit.> and Karaulov<cit.>. The latter one has been conducted in three stages during 1986-1997 and is one of the largest experiments. In addition to associations the dataset also contains demographic information such as age, gender, specialization, and location. Most of the previous research had been focused on the study of reactions: their distribution and cross-lingual commonalities. Some of the researchers (e.g. Steyvers & Tenenbaum<cit.>) also studied the structure of human associative networks. They represented a network as a directed graph in which stimuli and reactions correspond to nodes whereas associations are edges connecting them. They showed that the node's degree (the number of different reactions given for a stimulus) follows a power law distribution[i.e. associative networks are scale-free.]. In other words, there are several “hub” nodes with many connections and many “weak” nodes with small degree.But very little had been done in terms of quantitative evaluation of demographic factors in associations. Current research fills up this gap. We investigate the reaction types distribution in regards to gender and speciality.§ DATASETThe experiment conducted by Karaulov's group, although being one of the most lasting ones, very quickly becomes outdated. Moreover, it has only been focused on the regions of Central Russia. To address these issues as well as to analyse the change of the associations over time, we additionally organized the associative experiments in various Russian regions, including Siberia and the Urals. The age of participants ranged from 16 to 26[People in psycholinguistics typically assume that the core of the verbal associations becomes stable and does not significantly change after the age of 18.], most of them were either undergraduate or postgraduate university students of ≈50 specialities. The experiments were organised as follows. A respondent received a questionnaire of 100 single-word stimuli. For each stimulus the respondent had to provide a reaction. There were no constraints on the reaction types, but the total time was limited to 10-15 minutes, i.e. the participants had 6-9 seconds for each stimulus. Most of the reactions appeared to be also single-word. Several association pairs are presented on Table 1.In total, the dataset contains 4,997 questionnaires. The list of stimuli comprises of 1,213 various lemmas partially taken from Leontiev's list as well as most common reactions of previous Russian associative experiments from Karaulov<cit.>. The total number of different reactions received from the respondents is 50,359 (37,895 lemmas). Table <ref> shows the top-10 most frequently used reactions. Surprisingly, we see a large overlap between current study and the experiment conducted in 1986-1997. Besides that, there is also an overlap with the most frequently used Russian words from Sharoff's list<cit.>. We also did not observe a significant cross-gender difference in the top reactions. § EXPERIMENTS §.§ Association types analysisAiming to observe possible differences in distribution of association patterns among categories of respondents, we match associations against two major patterns as follows. First, we check whether a stimulus - response pair matches an ngram observed in text corpora. We measure a smoothed sum of matched ngram frequencies: S = ∑_a ∈ A{[ log f(a),f > 0,;0, f = 0 ]}where a is an association pair, A is a set of all association pairs, f(a) is a corpus frequency of an ngram produced of a association.We consider bigrams for single-word responses (the vast majority of cases). If we have two-word response, we match it against trigrams. Responses containing more than two words are treated as non-matching any known ngrams. We tried to match each association both in forward (stimulus→response) and backward (response→stimulus) direction, and each side (stimulus and response) was supplied both as is and in a lemmatized form.[We used mystem<cit.> to extract lemmas.] By doing that, we actually match each association against eight candidate ngrams, and we pick the maximum frequency observed over those ngrams. We used National Corpus of Russian Language[http://www.ruscorpora.ru/corpora-freq.html] as source for ngram frequencies. Second, we extract associations that correspond to basic thesaurus relations such as synonyms, antonyms, hypernyms/hyponyms, meronyms/holonyms, and cause/effect. We use Russian WordNet[http://wordnet.ru] <cit.> as our initial data source, where we count the number of matching associations for every relation type. Table 1 presents a list of associations and corresponding extracted ngram frequencies and relations.We measure the values listed above as percentages to the total number of responses. We have done it for the full dataset and also for slices selected by respondent's gender or specialization.Figures <ref> and <ref> show that men are more biased towards using semantically inspired associations (paradigmatic) whereas women are more likely to produce ngrams (syntagmatic)[With p-value<0.001]. We observe a similar pattern (i.e. syntagmatic inversely related to paradigmatic) by looking at the specializations. Figure <ref> presents S values for the top-20 most popular specializations. For instance, “chemistry” presents the highest scores in semantic relations whereas lower than average for ngrams. On the other hand, in the case of “sales” it is completely opposite. Note that most of the technical specializations and natural sciences demonstrate high scores for paradigmatic association types. We supposed that this is due to correlation between gender and occupation and the fact that gender still plays a significant role in the process of choosing the future career. In order to test that hypothesis, we calculated ngram usage figures normalized over gender (Fig. <ref>). In this context, a normalized value is a half-sum of two corresponding average values, each one being computed over its respective respondent gender. The normalization didn't seem to smooth differences in ngram usage over various specializations. Therefore, one may conclude that the specialization standalone plays as a significant factor influencing word association patterns. §.§ Personalization of vector modelsNow we turn to our experiments with associative vector models. We propose gender and specialization specific associative models. In order to create the models, we first slice the data by the proposed attribute. For instance, for “gender” we take two subsets corresponding to “male” and “female”, respectively.[The dataset is quite balanced and we have roughly the same number of questionnaires for both male and female participants.] As we described earlier, their association frequency distributions present some differences which we would like our model to capture. Unlike traditional language modelling task, here we only rely on stimulus-reaction pair frequencies. Therefore, we consider SVD-PPMI<cit.> approach to get the distributed vector representations. The method had been shown to perform on par with neural models<cit.>, such as word2vec<cit.>. It is also less expensive in terms of the time complexity and better fits our task setting.[We used the model implementation from <https://bitbucket.org/omerlevy/hyperwords>. We set the size of the context window to 1 (left and right words), embedding size to 100, context distribution smoothing of 0.75, token threshold value of 5, all the other parameters were left with their default values.]We additionally train a baseline model on the full dataset to compare it to the personalised models. We would like to emphasize that usage of distributed vector models allows us to go beyond the scope of direct associations and generalize better.Table <ref> presents several examples for the top 10 nearest neighbours of “male”, “female” and baseline models. In this case, we observe mainly semantic differences. Notice a substantial variation in the predictions of the models if we provide them with “I” stimulus. Table <ref> illustrates the models work for “sales” and “publishing” occupation types. In general, we find the ability to provide gender- and specialization-sensitive information useful for addressing the issues related to social language variation. Table <ref> additionally provides the difference in the model's predictions for twolocations in Chelyabinsk oblast: a small town of Asha and an industrial city of Magnitogorsk.There is no consensus in the research community on how to evaluate the associative models. Most of the methods are based on direct comparison of statistical characteristics of the distributions of reactions each of the models generates. Moreover, to our knowledge, no theoretical framework or quality assessment or measures had been proposed for that so far. Therefore, we consider this part of research for our future studies.§ CONCLUSION We presented a new dataset for Russian verbal associations. We also showed that social factors such as gender and specialization provide a significant amount of information on the type of association. Finally, we also proposed a gender-sensitive associative model and demonstrated the significance of incorporating of social factors into the traditional NLP models. § ACKNOWLEDGMENTSWe would like to thank all reviewers for their valuable comments and suggestions for future research directions. The first author was supported by the Melbourne International Research Scholarship (MIRS). | http://arxiv.org/abs/1707.08458v1 | {
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"Yuriy Philippovich",
"Galina Cherkasova"
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"published": "20170726142012",
"title": "Men Are from Mars, Women Are from Venus: Evaluation and Modelling of Verbal Associations"
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1 2014 2014 0 999 0 asna.201400000Near-UV transit photometry of HAT-P-32 b M. Mallonn & H.R. WakefordLeibniz Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany Planetary Systems Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA XXXX XXXX XXXXBroad-band exoplanet transit photometry can characterize the planetary atmosphere when observed at multiple selected filters. This observing technique can reveal gradients in the spectra of extrasolar planets, for example the slope of decreasing opacity from short to long optical wavelengths caused by aerosol scattering. In this work we observed a transit of the hot Jupiter HAT-P-32 b in the shortest wavelength possible from the ground using the Large Binocular Telescope (LBT). The data comprise the best-quality ground-based U-band taken so far of an exoplanet transit. Compared to broad-band observations of intermediate and long optical wavelength published previously, a clear scattering slope in the planetary transmission spectrum is revealed. Most likely, the scattering particles are magnesium silicate aerosols larger than 0.1 μm. We define a spectral index to compare this scattering feature of HAT-P-32 b to published results of other exoplanets. It turns out to be very typical in amplitude within the comparison sample. Furthermore, we searched for correlation in this sample of the spectral index with planetary equilibrium temperature, surface acceleration and stellar activity indicator, but could not reveal any. Near-UV transit photometry of HAT-P-32 b with the LBTBased on observations made with the Large Binocular Telescope (LBT). The LBT is an international collaboration among institutions in the United States, Italy and Germany. LBT Corporation partners are: LBT Beteiligungsgesellschaft, Germany, representing The Leibniz Institute for Astrophysics Potsdam, the Max-Planck Society, and Heidelberg University; The University of Arizona on behalf of the Arizona Board of Regents; Istituto Nazionale di Astrofisica, Italy; The Ohio State University, and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia.: Silicate aerosols in the planetary atmosphere Matthias Mallonn1Correspondence: [email protected] Hannah R. Wakeford2 31 July 2017 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONThe majority of known host stars of transiting extrasolar planets are solar-type stars, i. e. main sequence stars of spectral type late F, G, and K. These stars emit only little flux at ultraviolet (UV) wavelengths, hence UV observations of transit events often suffer from photon noise and do not reach the same very high quality as we are used to nowadays for optical transit photometry <cit.>. This is an unfortunate situation since the UV wavelengths are key for at least two open questions in exoplanet science. One concerns the interaction of close-in gas giants with the stellar corona and the potential formation of a bow shock in front of the planet, which might be detectable in absorption in the UV <cit.>. The second open question is on the presence of a scattering signature in the transmission spectra of extrasolar planets, which would be most pronounced at the bluest wavelengths <cit.>.WASP-12 b was observed by the Hubble Space Telescope (HST) to show an asymmetric transit light curve at near-UV wavelengths <cit.>. The transit started significantly earlier at these short wavelengths than its counterparts at optical wavelengths. This early ingress was confirmed by follow-up observations of WASP-12 b at 278.9–282.9 nm <cit.>. Two main models for explaining the origin of an early ingress were given, both including the formation of a bow-shock, one of magnetic origin <cit.>, the other of non-magnetic origin <cit.>. Recently, <cit.> observed a large sample of hot Jupiter transit events in the Johnson U band to investigate whether the phenomenon of an early transit ingress is common among close-in gas giants. The precision of their measurements were sufficient to rule out similar transit asymmetries as in the HST measurements of WASP-12 b. Thus, the observation of bow-shock phenomena might be constrained to wavelengths shortward of 300 nm.Cloud- and haze-free atmospheres of extrasolar planets are expected to show an increased opacity towards blue optical wavelengths and the near-UV because of light scattering by H_2 molecules <cit.>. Also small-sized haze particles (aerosols) cause scattering with a dominant slope in the near-UV and optical transmission spectrum <cit.>. This increased opacity by scattering is measurable as an increased effective planetary radius and therefore increased transit depth during transit compared to longer wavelength. In a large and homogeneous study, <cit.> analyzed ten close-in gas giants and found in all cases an enlarged planetary radius at bluest wavelength. The amplitude can be compared among different planets in terms of their atmospheric pressure scale height. All system showed an increase in effective planetary radius, however with diverse amplitudes <cit.>. Thus, is the blue scattering signature a spectral signature common to all transmission spectra of close-in gas giants? This question can be answered by high-quality transit light curves taken at very short wavelengths like the Johnson U band at 360 nm. There are numerous examples of ground-based U band transit observations in the literature <cit.>, however the derived value of planetary size rarely reaches a precision of two scale heights or below. Whether a variation in planetary size of only a few scale heights is measurable as a variation in transit depth depends on the planet-star radius ratio and the size of the scale height. Additionally, also the brightness of the host star matters to lower the photon noise in the transit photometry. One very favorable exoplanet is HAT-P-32 b with a fairly bright F-type host star of V = 11.4 mag, an inflated planetary radius of ∼1.7 R_J and a scale height of ∼1000 km <cit.>. Several publications exist on the optical transmission spectrum of HAT-P-32 b <cit.>, and they all agree on the absence of pressure-broadened absorption features predicted by cloud-free atmosphere models. However, the current data bluer than about 450 nm could not definitively clarify about a scattering signature because of a decrease in precision towards these short wavelengths. In what follows, we present the analysis of a near-UV broad-band observation of a transit of HAT-P-32 b observed with the LBT. In Section 2 we describe the observation and data reduction, while in Section 3 the analysis and results are presented. In Section 4 we discuss the broad-band spectrum of HAT-P-32 b in regard to literature spectra of other close-in exoplanets. Section 5 contains the conclusion.§ OBSERVATION AND DATA REDUCTION We observed one transit with the Large Binocular Camera <cit.> at the Large Binocular Telescope (LBT) on November 29, 2016. The LBC consists of two prime-focus, wide-field imagers mounted on the left and right arm of LBT, and is optimized for blue and red optical wavelengths, respectively. The observation was performed with the LBT in binocular mode using the blue optimized LBC on one side, and the Multi-Object Double Spectrograph (MODS) on the other side. The analysis of the MODS data is work in progress and is not included in this paper. The LBC already proved its capabilities for exoplanet transit and secondary eclipse observations <cit.>. We employed the U_spec filter, which has a response curve very similar to Sloan u' but increased efficiency. The exposure time was 15 s, and we reduced the overheads by a read-out window to 29 s. The transit lasted from November 30, 07:03 to 10:09 UT, and the observation was executed from 05:25 to 10:25 UT in which we gathered 375 science images. The object was setting over the course of the observation, thus the airmass increased towards the end to 2.25. The telescope was heavily defocused to avoid detector saturation.The data reduction was done as described in our previous work on HAT-P-32 b <cit.>. Bias and flat-field correction was done in the standard way, with the bias value extracted from the overscan regions. We performed aperture photometry with the publicly available software SExtractor using the option of a fixed circular aperture MAG_APER and the automatically adjusted aperture MAG_AUTO. The set of comparison stars (flux sum) was chosen to minimize the root mean square (rms) of the light curve residuals after subtraction of a second order polynomial over time plus transit model using literature transit parameter. Using the same criterion of a minimized rms, we also determined and applied the best aperture width to be MAG_AUTO with a Kron parameter of 2.75 (see the SExtractor manual[http://www.astromatic.net/software/sextractor] for details). The best circular aperture with a diameter of 72 pixel gave a slightly larger point-to-point scatter. § LIGHT CURVE ANALYSIS AND RESULTS The transit light curve analysis was performed homogeneously to our previous studies on the same target <cit.> to allow for a direct comparison. We model the transit light curve with the publicly available software JKTEBOP <cit.> in version 34. The transit fit parameters consist of the sum of the fractional planetary and stellar radius, r_⋆ + r_p, and their ratio k=r_p/r_⋆, the orbital inclination i, the transit midtime T_0, the host-star limb-darkening coefficients (LDC) u and v of the quadratic limb darkening law, and the coefficients c_0,1,2 of a polynomial over time. The index “⋆” refers to the host star and “p” refers to the planet. The dimensionless fractional radius is the absolute radius in units of the orbital semi-major axis a, r_⋆ = R_⋆/a, and r_p = R_p/a. The planetary eccentricity is fixed to zero as determined by <cit.> and the orbital period P_orb is fixed to 2.15000825 days according to <cit.>.The stellar limb darkening was approximated by the quadratic law <cit.>. Theoretical values of the LDC were taken from <cit.>, we used the stellar parameters of the host star derived by <cit.> for the circular orbit solution. Homogeneously to <cit.>, we fitted the linear LDC u and kept v fixed to its theoretical value.After an initial run of transit modeling, we noticed an increase in point-to-point scatter with time for the light curve residuals. This increase is not reflected in the SExtractor calculation of the photometric uncertainties that takes into account the photon noise of the target and the measured noise in the local background. We quadratically combined the uncertainties of the target and the comparison stars. In the first part of our time-series, the point-to-point scatter is approximately 1.3 above the theoretical SExtractor uncertainty, while towards the end it is larger by more than a factor of two. <cit.> noticed that scintillation noise caused by the Earth atmosphere strengthens towards shorter wavelengths, thus we calculated the scintillation noise with the approximating Equation 10 of <cit.> (Fig. <ref>). And indeed, during the second half of our observations the scintillation noise is much larger than the photon noise. In an attempt to use the most realistic photometric uncertainties in our light curve modeling, we derived the standard deviation per data point in the light curve residuals with a sliding window of 60 data points width (a “running standard deviation” in analogy to a running mean) and applied this as uncertainty. As shown in Figure <ref>, this photometric uncertainty varies from ∼ 0.5 to ∼ 1.0 mmag (time sampling of 44 seconds) with a mean of 0.78 mmag, which is to our knowledge the best ground-based near-UV transit light published so far.Additionally, we calculated the so-called β factor, a concept introduced by <cit.> and <cit.> to include the contribution of correlated noise in the light curve analysis. It describes the evolution of the standard deviation σ of the light curve residuals when they become binned in comparison to Poisson noise. In the presence of correlated noise, σ of the binned residuals is larger by the factor β than with pure uncorrelated (white) noise. The value of β = 1.15 derived for our light curve is the average of the values for a binning from 10 to 30 minutes in two minute steps. We enlarged the photometric uncertainty by this factor.A smooth trend in the out-of-transit baseline is obvious in Fig. <ref>. We tried to model this trend with a linear combination of linear functions of airmass, detector position, width of the point-spread-function, and sky background <cit.>, but in our case none of the terms was verified by the Bayesian Information Criterion <cit.>. We found a simple second-order polynomial over time to approximate the trend very well and to minimize the BIC.As a consistency check to previous studies on HAT-P-32 b we fitted our near-UV transit light curve with all transit and detrending parameters free. Our derived values of i = 88.7±0.7 and a/R_⋆ = 6.09±0.27 agree to within 1 σ, while the k = 0.1548±0.0011 is larger than literature values by 2 to 3 σ. The transit midtime T_0 = 2 457 722.859898±0.00012 BJD_TDB is in 1 σ agreement to the ephemeris of <cit.>. The estimation of the transit parameter uncertainties was done with“task 8” in<cit.>, which is a Monte Carlo simulation, and with “task 9” <cit.>, which is a residual-permutation algorithm that takes correlated noise into account. We run the Monte Carlo simulation with 5000 steps. As final parameter uncertainties we adopted the larger value of both methods.We want to investigate the wavelength dependence of the effective planetary radius. We assume a/R_⋆ and i to be wavelength independent and fix them to the values used in <cit.> and <cit.> for comparability. We assume their uncertainties to be a common source of noise to all bandpasses, negligible in the search for relative variations of k over wavelength. Therefore, the derived error on k is a relative uncertainty. The free parameters of the light curve fit are r_p, u, and c_0,1,2. The LDC v is fixed to its theoretical value as done in <cit.> and <cit.>. The result is shown in Fig. <ref> in comparison to <cit.>, who derived a U band data point from a single ULTRACAM transit light curve. We performed a joint fit of both U band light curves with JKTEBOP, allowing for an individual detrending of both light curves, but a common value for r_p and u. The results are summarized in Tab. <ref>.§ DISCUSSION§.§ No early ingress – a symmetric transit light curve Our data yield no indication of an asymmetry in the transit light curve or an early ingress, potentially caused by a bow shock in the stellar corona as for the transit observations of WASP-12 b shortward of 300 nm <cit.>. The transit depth of HAT-P-32 b in the U band is only slightly larger than at optical wavelengths without indications of absorption in a planetary exosphere extending its Roche lobe. With this result we are in agreement to the observational result of <cit.>, who did not find an asymmetry or early ingress for 15 different hot Jupiters using ground-based U band observations. The study of the origin of these transit asymmetries appears to be possible only at UV wavelengths from space. §.§ Aerosol scattering in the planetary atmosphere The analysis of the U band light curve of HAT-P-32 b resulted in an increased planet-star radius ratio k compared to longer wavelength measurements of <cit.>. Under the assumption of a stellar radius independent of wavelength, it translates into an increased near-UV effective radius of the planet. The parameter k could in principle also be modified by third light from another stellar body falling in the photometric aperture. And indeed, <cit.> found a near-by M dwarf companion only 2.9 ” to HAT-P-32. However, the third light contribution of this M dwarf towards the total system light at near-UV wavelengths is smaller than 0.1 %, and therefore the modification in k is smaller by orders of magnitude than its uncertainty <cit.>. In principle, also brightness inhomogeneities on the stellar surface (star spots) could cause variations in k <cit.>. However, <cit.> found the host to be photometrically stable, therefore star spots are very unlikely to cause a significant increase in k towards blue wavelengths. The most plausible explanation for a larger effective radius of the planet at near-UV wavelengths is scattering in the planetary atmosphere. The scattering particles could be hydrogen molecules in gas phase <cit.>. However, this scenario is unlikely since it would require a cloud/haze-free atmosphere, which is disfavored by all previous spectroscopic studies of this planet <cit.>. Certainly, the scattering takes place at aerosols, which are thought to be a widespread phenomenon in the atmospheres of hot Jupiter exoplanets <cit.>. In a recent work, <cit.> found all of the 10 planets of the HST spectral survey <cit.> to contain aerosols at the probed latitudes, however at varying pressure levels.We show the temperature-pressure (T-P) profile of HAT-P-32b in Fig. <ref> with the pressures probed by transmission spectral measurements constrained by vertical lines to indicate the region of the atmosphere measured through observations. It is expected that the U band measurements presented in this paper are probing the top regions of the atmosphere at higher altitudes than would be observed in the near-IR, however, further constraints cannot be placed on the T-P profile. We use a series of condensation curves () to approximate the different cloud forming species expected in the upper atmosphere of HAT-P-32b. These condensation curves indicate that the most likely cloud forming species at the pressures probed in transmission would be composed of magnesium silicates, which have been hypothesized as sources of enhanced optical scattering in a number of hot Jupiters (e.g. ) and are expected to be abundant in giant exoplanet atmospheres (e.g. ).In an attempt to put our new HAT-P-32 b optical transmission spectrum in the context of the existent observations of other exoplanets, we defined the spectral index Δ k_u-z. It measures the difference in k between U band wavelengths (here defined as 300<λ<420 nm) and z' band wavelengths (here defined as 880<λ<1000 nm) and expresses it in terms of the atmospheric pressure scale height H for reasonable comparability among different planets. The scale height H is defined byH = k_B T_eq/μ_m gwith T_eq as the planetary equilibrium temperature, k_B the Boltzmann constant, μ_m the mean molecular mass equaling 2.2, and g as the local gravitational acceleration. For the calculation of H we used published values of T_eq and g (see Tab. <ref>). The index Δ k_u-z can only be calculated for planets for which the U and z' band planet-star radius ratios have been derived consistently, i.e. with a common value for i and a/R_⋆, and the use of the same limb darkening law. Together with the values for HAT-P-32 b of this work, we could therefore include nine planets investigated by <cit.>, the results of GJ3470 b from <cit.>, and an independent measurement of HAT-P-12 b by <cit.>, a planet also included in the sample of <cit.>. If the wavelength interval used here for U and z' band are spectroscopically resolved by multiple data points, we use a weighted average.Under the assumption of the equilibrium temperature being a correct description of the average temperature at the terminator, the value Δ k_u-z can be used as an indicator for the particle size of the scattering aerosols. Particles of the size 0.1 μm and smaller cause a steep Rayleigh scattering slope, while larger particles cause a shallower slope. The transition to a flat spectrum happens for particle sizes larger than ∼ 0.5 to 1 μm with a slight dependence on atmospheric temperature <cit.>. The optical spectral slope Δ k_u-z of HAT-P-32 b of 2.18 ± 0.95 H is slightly smaller than the expectation of ∼ 4 H for a pure Rayleigh-scattering slope. In comparison to the other investigated systems in Tab. <ref>, the value for HAT-P-32 b appears to be about average. A majority of targets show a Δ k_u-z of between one and three H. On the contrary, the results of Δ k_u-z > 5 H of HD189733b and GJ3470b are unusually large. This favors a usually larger atmospheric particle size for most of the hot Jupiters than for these two planets, i. e. larger than 0.1 μm, but smaller than ∼ 0.5 μm. Another valid explanation for an optical slope shallower than expected for Rayleigh-scattering could be a difference in the scale height of the aerosols to that of the surrounding gas. The value Δ k_u-z ∼ 5 H of HD189733 b is indicative for Rayleigh scattering by aerosols with a similar scale height as the gas, while the lower value of Δ k_u-z ∼ 2 H for, e. g., WASP-12 b could potentially be explained by Rayleigh scattering of aerosols with a lower scale height than the gas <cit.>. One way to measure the aerosol scale height might be the detection of vibrational mode absorption features of the aerosols at mid-infrared wavelengths with JWST <cit.> and a comparison of their amplitudes with theoretical models.The temperature of hot Jupiters ranges from below 1000 K to 3000 K or above. The elemental composition of a cloud is expected to vary with temperature because of different condensation temperature for the different potential condensation species. We searched for a correlation of the spectral index Δ k_u-z, potentially being related to the particle size, with T_eq, but none was found (upper panel of Fig. <ref>). The coolest planet in the sample, GJ3470b, has the strongest optical spectral slope. However, all other planets follow a roughly constant value of Δ k_u-z over T_eq. We also searched for a relation of Δ k_u-z and the planetary surface gravity g, because for larger g values, larger particles might have been settled below observable altitudes. Thus, we might expect a positive correlation of Δ k_u-z and g. We find HD189733b with its largest g value among our sample to show a large Δ k_u-z value. However, all other planets do not show a significant correlation (middle panel of Fig. <ref>). Another test, which we performed, looked for a correlation among Δ k_u-z and the stellar activity indicator log(R'_HK). The aerosols in the planetary atmospheres might be formed by condensation or photochemistry. The latter is expected to be important for cooler hot Jupiter atmospheres, while the former might dominate the hotter atmospheres <cit.>. Photochemistry is linked to the incident UV flux that the planet receives. To quantify this, we use the indicator log(R'_HK), since stellar activity correlates with enhanced UV flux <cit.>. Hydrocarbon species formed by photochemistry are expected to be small in particle size <cit.>, therefore we would intuitively expect a positive correlation between Δ k_u-z and log(R'_HK). However, the planets of our sample form a wide range in log(R'_HK), and except HD189733b having an extreme value of log(R'_HK) and a large value of Δ k_u-z, there is no indication for a dependence. § CONCLUSION We observed one transit of the inflated hot Jupiter HAT-P-32 b with the LBT in the U band. With an average point-to-point scatter (rms) of 0.78 mmag and a time sampling of 44 seconds, these data form the best-quality ground-based near-UV transit light curve so far. We do not find indication of an asymmetry or an early ingress in the transit light curve, indicating that studies on bow shock phenomena in the stellar corona or on the exosphere of hot Jupiters are bound to wavelengths shortward of 300 nm. Such observations cannot be obtained from the ground. We reveal an increase in planet-star radius ratio in the U band compared to optical wavelengths previously published. A plausible explanation is scattering in the planetary atmosphere caused by aerosols. By a comparison of the planetary T-P profile with condensation curves of potential cloud forming species we find the aerosols to be most likely formed by magnesium silicates. The scattering signature in the planetary broad-band transmission spectrum is smaller than expected for Rayleigh-scattering, which is indicative for an aerosol particle size larger than 0.1 up to 0.5 μm. We defined the spectral index Δ k_u-z, which measures the difference in the planet-star radius ratio between the U band and the z' band in units of the planetary atmospheric pressure scale height H. This index allowed a comparison of the scattering feature derived for HAT-P-32 b in this work with published results of other exoplanets. We find that the scattering feature of HAT-P-32 b is rather typical for hot Jupiter systems. More pronounced scattering features like in the atmosphere of HD189733 b are rather an exception. We searched for correlation of Δ k_u-z with the planetary equilibrium temperature, the planetary surface gravity and the stellar magnetic activity indicator, but did not find any. Our sample of 11 targets with measurements of Δ k_u-z might still be too small. We also urge the community to observe targets more widespread in the T_eq–g parameter space to reveal dependencies of the optical spectral scattering slope. We thank Jonathan J. Fortney for providing the planetary atmosphere models and the planetary T-P profile. We also like to thank Jesper Storm and the LBT Science Operation Team for performing the observations at the LBT and the helpful discussions regarding their preparation. This research has made use of the SIMBAD data base and VizieR catalog access tool, operated at CDS, Strasbourg, France, and of the NASA Astrophysics Data System (ADS).an | http://arxiv.org/abs/1707.08328v2 | {
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"primary_category": "astro-ph.EP",
"published": "20170726091251",
"title": "Near-UV transit photometry of HAT-P-32 b with the LBT: Silicate aerosols in the planetary atmosphere"
} |
Phase-matching-free parametric oscillators based on two dimensional semiconductors C. Conti^4,5 December 30, 2023 ================================================================================== According to implicit ligand theory, the standard binding free energy is an exponential average of the binding potential of mean force (BPMF), an exponential average of the interaction energy between the ligand apo ensemble and a rigid receptor. Here, we use the Fast Fourier Transform (FFT) to efficiently estimate BPMFs by calculating interaction energies as rigid ligand configurations from the apo ensemble are discretely translated across rigid receptor conformations. Results for standard binding free energies between T4 lysozyme and 141 small organic molecules are in good agreement with previous alchemical calculations based on (1) a flexible complex (R ≈ 0.9 for 24 systems) and (2) flexible ligand with multiple rigid receptor configurations (R ≈ 0.8 for 141 systems). While the FFT is routinely used for molecular docking, to our knowledge this is the first time that the algorithm has been used for rigorous binding free energy calculations.Noncovalent Binding Free Energy, Implicit Ligand Theory, Fast Fourier Transform, Protein-Ligand, T4 Lysozymeiblabel[1]#1. apsrev § INTRODUCTIONTo accelerate absolute binding free energy calculations between proteins and small molecules, we developed a method based on the fast Fourier transform and then tested it, finding reasonable agreement with more expensive calculations.The binding of small molecules to biological macromolecules plays a critical role in cellular processes including enzymatic reactions, signal transduction, and gene regulation. Moreover, noncovalent associations with specific targets are essential to the mechanism of most drugs. Molecular simulations to estimate standard binding free energies between small molecules and biological macromolecules, therefore, have become increasingly important for understanding biological mechanisms <cit.> and for computer-aided drug design <cit.>.The most rigorous methods for estimating standard binding free energies are alchemical pathway methods <cit.>,in which conformations of both the ligand and receptor are sampled from a series of possibly nonphysical thermodynamic states along a pathway connecting two end states of interest. These methods can be categorized into two flavors:relative and absolute binding free energy methods. The former calculations attempt to calculate the relative binding free energy of one ligand compared to another.In these calculations, the pathway involves transforming one ligand into another. The latter attempt to calculate binding free energies without comparison to a reference ligand. In absolute binding free energy calculations, the pathway may involve turning off interactions between the ligand and receptor or physically separating them <cit.>. Simulations may be done in explicit or implicit <cit.> solvent, with the latter increasing speed at the expense of accuracy <cit.>. In an increasing number of publications, alchemical pathway methods have successfully calculated accurate protein-ligand binding free energies <cit.>.If the binding free energy of many ligands is desired, however,sampling from a series of thermodynamic states for each receptor-ligand pair can be computationally expensive. The ability to perform large-scale free energy calculations efficiently is desirable for the purposes of computer-aided drug discovery. Absolute binding free energy calculations for a large library of diverse compounds may be used in lead discovery. Relative binding free energy calculations between a congeneric series may be used in lead optimization.A number of strategies are emerging to facilitate large-scale free energy calculations. In λ-dynamics <cit.>, the alchemical parameter specifying the identity of a substituent is treated as a dynamical variable. Its extension to multiple substituents at multiple locations on a scaffold, multi-site λ-dynamics <cit.>, enables many relative binding free energy estimates based on a single simulation. Another strategy involves pre-computing ensembles of the protein complexed with a reference molecule <cit.>, with a reference ligand <cit.>, or of the protein by itself, the apo ensemble <cit.>. This approach is advantageous because it allows for receptor conformations to be exhaustively sampled once and used for a large set of ligands. For a series of molecules that are similar to the reference molecule, a one-step perturbation is often sufficient to estimate their relative binding free energies.Implicit ligand theory (ILT) <cit.> is a formalism that enables large-scale absolute binding free energy calculations based a pre-computed apo ensemble of the receptor. The theoretical basis for ILT is similar to the basis for estimating standard binding free energies based on implicit solvent <cit.>. While the implicit solvent formalism involves an integral over all solvent degrees of freedom to obtain a potential of mean force for solvation,ILT invokes an integral over all ligand degrees of freedom to obtain a binding potential of mean force (BPMF) - the binding free energy between a flexible ligand and rigid receptor. The solvent potential of mean force is typically estimated by a continuum dielectric electrostatics model.In contrast, the BPMF has hitherto been estimated by an alchemical pathway method <cit.>.Recently, <cit.> demonstrated the feasibility of using ILT <cit.> (ILT) to estimate absolute binding free energies based on multiple rigid conformations of a protein. Receptor conformations were first drawn from alchemical binding free energy calculations between T4 lysozyme and 6 different ligands. The BPMF was then calculated between 141 ligands and each of the receptor conformations. In accordance with ILT, the standard binding free energy was calculated as an exponential average of the BPMFs. ILT-based calculations closely reproduced (with Pearson's R = 0.9 and RMSE = 1.59 kcal/mol) results obtained by YANK <cit.>, a program based on alchemical pathway calculations with a flexible receptor.Here, we perform similar calculations using an alternate way to estimate BPMFs.<cit.> computed BPMFs using the program Alchemical Grid Dock <cit.> (AlGDock), which implements an alchemical pathway method in which the receptor is held rigid. While these calculations were much faster than simulations with a flexible receptor, they still required sampling from multiple thermodynamic states.To understand the alternate approach, it is helpful to consider the definition of the BPMF between a flexible ligand and a rigid receptor <cit.>,B(r_R) = - β ^ -1ln< I(ζ_L) exp[ - βΨ(r_RL) ]> _ζ_L, r_L,where β = (k_B T)^-1. Internal coordinates of the complex, r_RL, include internal coordinates of the receptor and ligand, r_R and r_L, respectively, and external degrees of freedom of the ligand, ζ_L. I(ζ_L) is an indicator function between 0 and 1 that specifies whether the receptor and ligand are bound. Ψ(r_RL) is the effective interaction energy between the receptor and ligand, the difference in energy between the solvated complex and solvated receptor and solvated ligand. The bracket < ·>_ζ_L, r_L denotes the ensemble average over the external and conformational degrees of freedom of the ligand in the apo ensemble. In this paper, the apo ensemble of a ligand (or receptor) refers to the probability density of the molecule in the unbound state, alone in implicit solvent. Equation <ref>, therefore, specifies that BPMFs can be calculated by sampling ligand internal coordinates r_L from the apo ensemble, sampling external degrees of freedom from the distribution proportional to I(ζ), calculating interaction energies, and estimating B(r_R) with a sample mean,B̂(r_R) = -β^-1ln1/N∑_n=1^Nexp[ -βΨ(r_RL,n) ]. The fast Fourier transform (FFT) facilitates the use of Equation <ref> providing an efficient way to compute interaction energies as the ligand is translated across the receptor. If interaction energies are calculated at M translational positions per dimension,the complexity of direct calculations is O(M^6). <cit.> pioneered an alternate procedure based on the FFTin which interaction energies are calculated from the cross-correlation between three-dimensional grids that represent each binding partner. Their grids were very simple - points were 0 outside the protein, 1 on its surface, and a constant value for the interior - but more recent studies have incorporated elements of a molecular mechanics nonbonded interaction energy, including electrostatics <cit.>, van der Waals interactions <cit.>, or both <cit.>. After mapping binding partners onto the grids, a discrete Fourier transform is performed, the two grids are convoluted, and an inverse Fourier transform is performed, yielding interaction energies for relative translations specified by grid point positions. Using the FFT, this algorithm is of order M^3 ln(M^3) or less! Due to this speedup, docking algorithms based on the FFT are routinely used for protein-protein docking <cit.> and have been applied to docking fragments to proteins <cit.>. Qin and Zhou <cit.> have implemented an FFT-based method for calculating an exponential average of the intermolecular interaction energy (similar to Equation <ref>) and determining the chemical potential of a tracer protein in a crowded macromolecular solution.In this study, we develop a similar FFT-based method for estimating BPMFs and standard binding free energies between T4 lysozyme L99A and the same set of ligands as in <cit.>. We will refer to our approach as FFTΔG. To our knowledge, this is the first time that the FFT has been used for rigorous binding free energy calculations.§ METHODOLOGY Our calculations were performed on the same systems as in the prior study <cit.>: T4 lysozyme L99A complexed with 141 ligands with experimentally measured activities. In a thermal denaturation shift assay, 69 of the ligands were determined to be active and 71 inactive <cit.>. Another of the ligands, iodobenzene, was determined to be active by isothermal titration calorimetry. We used the same AMBER <cit.> topology and coordinate files for the ligands and receptor as before <cit.>. BPMFs were calculated between these ligands and the same T4 lysozyme conformations, extracted from alchemical pathway simulations for six ligands: 1-methylpyrrole, benzene, p-xylene, phenol, N-hexylbenzene, and (±)-camphor.§ LIGAND SAMPLING To sample ligand conformations from the apo ensemble, we performed parallel tempering <cit.> in the gas phase with a script based on OpenMM 6.3.1 <cit.>. Langevin dynamics simulations were performed at eight geometrically spaced temperatures between 300 and 600 K for 1 ns at each temperature using a time step of 2 fs. Exchanges between nearest neighbors were attempted every 1 ps. Ligand snapshots from the simulation at 300 K were saved every 1 ps,resulting in the total of 1000 snapshots per ligand.In the apo state, the energy is isotropic with respect to translation and rotation. To sample from this state, therefore, each snapshot was randomly rotated about its centroid. Translational sampling was based on the FFT.§ FFT-BASED BPMF ESTIMATIONBPMFs between each ligand and receptor snapshot were estimated based on the thermodynamic cycle shown in Figure <ref>. Solvent contributions to the BPMF are (1) the desolvation of the rigid receptor and flexible ligand from state (A) to (B), and (2) the solvation of the complex from state (C) to (D). The gas phase BPMF is the free energy difference between states (B) and (C),estimated by using the FFT to calculate interaction energies for a set of discrete translations of the ligand relative to the receptor.Interaction energies were calculated using grid-based terms for electrostatic and repulsive and attractive van der Waals interactions. The general form for these terms is <cit.>, Ψ_gen = ∑_i ∈ Lγ_i∑_j ∈ Rγ_j V(r_ij), where the inner sum is over all receptor atoms and the outer one over all the ligand atoms. V(r_ij) is a function of the distance r_ij between atoms i and j. For the electrostatic energy,Ψ_ELE = ∑_i ∈ L q_i∑_j ∈ R q_j 1/r_ij,where q_i and q_j are atomic partial charges. As partial charges were specified in electronic charge units and r_ij in Å, Ψ_ELE was multiplied by 332.05 such that the energy is in units of kcal/mol. The van der Waals interaction energy is based on the Lennard-Jones potential using the geometric (opposed to arithmetic) mean to combine the well depth of two atoms as ϵ_ij = (ϵ_iiϵ_jj)^1/2 and radii as σ _ij = (σ_iiσ_jj)^1/2. It includes a repulsive term,Ψ_LJr = ∑ _ i ∈ Lϵ ^ 1/2 _ iiσ ^ 6 _ ii∑ _ j ∈ Rϵ ^ 1/2 _ jjσ ^6 _jj/ r ^ 12 _ ijand an attractive term,Ψ_LJa = - ∑ _ i ∈ L 2 ϵ ^ 1/2 _ iiσ ^ 3 _ ii∑ _ j ∈ Rϵ ^ 1/2 _ jjσ ^3 _jj/ r ^ 6 _ ij.Combining all three terms, the total interaction energy is Ψ_ELE + Ψ_LJr + Ψ_LJa.Each of these terms was calculated by discretizing Equation <ref> onto a pair of three-dimensional grids: one for the receptor and one for the ligand. Values on the receptor grid are based on the inner sum of Equation <ref>,G_R (n) = ∑_j ∈ Rγ _ j V ( r_j - n),where n is the position vector of a grid point. To discretize the ligand grid G_L (n),first, the ligand was placed near the corner (i, j, k) = (0,0,0) such that its lowest x, y and z coordinates are 2 grid spacings away from the planes i=0, j=0 and k=0, respectively. Then the generalized “charges” γ _ i of the ligand are distributed into its ten nearest grid points as described by <cit.>. With these discretizations, the interaction energy between the protein, in its original position, and the ligand, translated by the vector m, is approximated by the cross-correlation function,Ψ_gen(m) ≈ C(m) ≡∑ _ n G_R (n) G_L (n + m) .where the approximation is due to discretization. According to the cross-correlation theorem, it is possible to evaluate the cross-correlation function C for every position on the grid by,C = FT^-1[ FT(G_R) ·FT(G_L)^* ],where FT and FT^-1 denotes the forward and inverse discrete Fourier transforms, respectively, and the dot symbol · denotes the element-wise product of matrices. While direct evaluation of grid-based interaction energies for all points has a complexity of O(M^6), the complexity of FFT-based estimates scales as M^3 ln(M^3) or better, where M is number of translations per dimension.FFT-based interaction energy calculations were performed for two grid sizes. In one set of calculations, we used a smaller cubic grid of 16 Å along each dimension, centered around the binding site (as defined in <cit.>). For a subset of 21 ligands in which experimental binding free energies are available,we also performed calculations with a larger cubic grid of 62 Å along each dimension, which encompasses the whole receptor surface.Interaction energies computed directly and by the FFT-based method were compared for three grid spacings: 0.125 Å, 0.25 Å, and 0.5 Å. Direct calculations based on equations <ref>, <ref>, and <ref> were performed using a custom python script. Based on the comparison, grid spacings of 0.25 Å and 0.125 Å were used for the small grid and a spacing of 0.25 Å was used for the larger grid.While the FFT evaluates interaction energies at all grid points, not all of the points are useful for BPMF estimation.First, because the FFT is periodic, an interaction energy can be evaluated for an unphysical situation where a ligand is split between opposite sides of a receptor grid. To filter out configurations corresponding to these unphysical translations,a ligand box was defined as a rectangular region containing all ligand coordinates and two additional grid spacings. Interaction energies corresponding to translations where the ligand box is split across the system were discarded.Second, the receptor grid given by Equation <ref> implies that when an atom is very close to a grid point n, the repulsive van der Waals term will cause G_R (n) to be an extremely large positive number. This can lead to a floating point overflow in the FFT calculation.To avoid extremely large grid values, we set V(r) in Equation <ref> to zero for r values less than the van der Waals parameter 2^-1/6σ _ ii. (Note that setting V(r) to zero for translations that result in steric clash is arbitrary and does not affects the final result because we did not use the interaction energy values given by FFT calculation for these translations. Instead, we assumed that they are infinite.) To keep track of translational positions that give rise to a steric clash, we used occupancy grids, defined the same way for both the receptor and ligand as,G_occ (n) = 1if ∃ i: r_i - n < 2^-1/6σ_ii, 0otherwise.The FFT correlation function of the ligand and receptor occupancy grids gives positive values whenever there is at least one pair of atoms that sterically clash. It gives zero otherwise.Interaction energies for translations that have steric clashes were assumed to be infinite.Interaction energies (including infinite values) were collected for all sampled translations, random rotations, and conformations of the ligand to estimate the BPMF using Equation <ref>.Solvent contributions to the BPMF were estimated based on a single-step perturbation,B̂_j,k(r_R) = -β^-1ln1/N∑_n=1^Nexp[ -βΔ U_j,k(r_RL,n) ],where B̂_j,k(r_R) is the free energy difference between the sampled state j and target state k and Δ U_j,k(r_RL,n) is the difference in the potential energy of the complex in state k versus state j. In the target state, the receptor, ligand, or complex, was solvated in* Onufriev-Bashford-Case (OBC2) generalized Born / surface area implicit solvent <cit.>, implemented in OpenMM 6.3.1 <cit.>; or* Poisson-Boltzmann / surface area (PBSA) implicit solvent, implemented in the sander program from AmberTools <cit.>. For desolvation of the rigid receptor and flexible ligand, the sampled state was (B) and target state was (A). Since the FFT approach does not directly sample from either state (C) or (D), we performed sampling importance resampling from the conformations in state (B).That is, 100 conformations were drawn from state (B) with weight proportional to e^-βΨ, where Ψ is the total interaction energy. The weight, e^-βΨ, is the ratio of probabilities for observing the conformation in the bound state versus the apo state. After drawing from state (C) using sampling importance resampling, the free energy difference is estimated using Equation <ref> with state (C) as the sampled state and (D) as the target state.§ STANDARD BINDING FREE ENERGY ESTIMATION According to implicit ligand theory <cit.>, the standard binding free energy Δ G^∘ is given by an exponential average of BPMFs over the apo ensemble of receptor conformations, Δ G^∘ = - β ^-1ln< exp[ - β B(r_R) ] > _ r_R - βln( Ω C^∘/8 π^2) , where Ω = ∫ I(ζ_L) dζ_L = 8π^2 V_site.The 8π^2 comes from an integral over the rotational degrees of freedom on the ligand. Because there are no restraints on this rotation, the integral is over the full range of Euler angles. V_site is the volume of the binding site, defined as the rectangular region where the ligand box does not go outside of the receptor grid (see above for the definition of the “ligand box”). Since each FFT translation yields a slightly different box size, the average volume of all rectangular regions is used to calculate V_site. C^∘ is the standard state concentration.Standard binding free energies were estimated using a weighted sample mean of the BPMFs of different receptor snapshots, ΔĜ^∘ = -β^-1ln∑_r_R W(r_R) exp[ -β B(r_R) ] + β^-1ln( Ω C^∘/8 π^2), where W(r_R) is the normalized weight (∑_r_R W(r_R) = 1) associated with each receptor configuration r_R in the apo ensemble. <cit.> explained in detail why we used the weighted mean and how to obtain the weights. Briefly, receptor conformations were not sampled from the apo state but from a series of alchemical states in the presence of a ligand using the program YANK <cit.>. Therefore we had to reweigh these conformations to the apo ensemble. To this end we used the multistate Bennett Acceptance Ratio <cit.> to obtain weights for all the receptor snapshots in the apo ensemble. However, the set of receptor conformations selected for binding free energy calculations is just a small subset of all the snapshots from YANK simulations. Therefore, there is no obvious way to assign weights for snapshots in the selected subset given the weights of all snapshots. As in <cit.>, we combined MBAR <cit.> weights using six weighting schemes:* Each snapshot is assigned its own MBAR weight; each YANK simulation has equal weight.* Each snapshot is assigned its own MBAR weight; each apo state has equal weight.* Each snapshot is assigned the cumulative MBAR weight of the thermodynamic state it represents; each YANK simulation has equal weight.* Each snapshot is assigned the cumulative MBAR weight of the thermodynamic state it represents; each apo state has equal weight.* Each snapshot is assigned the MBAR weight of its neighbors; each YANK simulation has equal weight.* Each snapshot is assigned the MBAR weight of its neighbors; each apo state has equal weight. § CORRELATION AND ERROR STATISTICS To quantify the agreement between different data sets, we used the Pearson's R, the root mean square error (RMSE), and adjusted RMSE (aRMSE). The RMSE between two series of data points {x_1, x_2, ..., x_N} and {y_1, y_2, ..., y_N} is,ϵ = √(1/N∑_n=1^N[ x_n - y_n ]^2 ).The aRMSE is <cit.>,ϵ = √(1/N∑_n=1^N[ x_n - y_n - (x̅ - y̅) ]^2 ),where the x̅ and y̅ are the sample mean of x and y, respectively. The aRMSE accounts for systematic deviation between the series and is useful for assessing whether relative binding free energies are accurate.§ RESULTS AND DISCUSSION § FFT-BASED INTERACTION ENERGY ACCURACY DEPENDS ON GRID SPACING Distributing generalized ligand charges onto a grid results in a discretization error that is more pronounced for a larger grid spacing (Figure <ref>). For a grid spacing of 0.125 Å, the FFT interaction energies are very close to the direct computation values and discretization error is essentially nonexistent. Increasing grid spacing to 0.25, 0.5, and 0.8 Å leads to a gradual reduction of the Pearson's R to 0.89 and increase of the root mean square error to 0.63 kcal/mol (Figure S1 in the supplementary material).While the accuracy at larger spacing may still be acceptable, increased spacing also reduces the number of samples.Therefore, for BPMF calculations with the smaller grid with 16 Å edges, we used grid spacings of 0.25 Å and 0.125 Å. For the larger grid with 62 Å edges, we reduced computational expense by only using a grid spacing of 0.25 Å.Our free energy calculations employ a finer grid spacing than Qin and Zhou <cit.> used to estimate the excess chemical potential of proteins in a crowded solution. In these studies, grid spacings between 0.15 Å and 0.75 Å were all sufficient to calculate the fraction of proteins that do not clash with a crowding molecule <cit.>. However, chemical potential estimates were found to suffer from a systematic bias that increases with the grid spacing, and is approximately 0.75 kcal/mol at 0.6 Å<cit.>. To improve accuracy while retaining the speed of a larger spacing, <cit.> corrected for these discretization errors by adjusting atomic radii and partial charges; this strategy may be pursued in future work on FFT-based protein-ligand interaction energies.§ FFT-BASED INTERACTION ENERGIES FAVOR THE BINDING SITE AND PROTEIN SURFACE A cross-section of the interaction energy grid between benzene and the whole protein (Figure <ref>) shows the lowest values in the binding site and near the protein surface and many infinite values within the protein. Low values in the binding site are expected, but low values on the surface may indicate either alternative binding sites or an artifact of the force field. The existence of alternative sites was suggested by <cit.>, who predicted that several ligands including benzene bind to multiple sites on T4 lysozyme and that each site contributes to the binding affinity. Interaction energies for surface sites may be comparable to the main binding site because the gas phase interaction energy does not account for solvation. To elaborate, placing the small hydrophobic molecule benzene in the main binding site would be favorable because it would be almost entirely desolvated, but this effect is not captured in our treatment of the interaction energy. The infinite interaction energies result from steric clashes between the ligand and receptor. The existence of many translational vectors with steric clashes is common in FFT-based docking, evident from even the first paper in the area <cit.>. § BPMF CALCULATIONS APPARENTLY CONVERGE Ligand conformational and binding pose sampling appears to be thorough. Parallel tempering and random rotation leads to ligands being uniformly oriented in three-dimensional space (Figure <ref>a).When ligand conformations are translated across the receptor binding pocket,a large number of translations result in steric clashes and infinite interaction energies. However, due to the diversity of sampled ligand conformations and orientations, a substantial number of poses have finite interaction energies (Figure <ref>b).Ligand sampling is sufficient for BPMF estimates to apparently converge (Figure <ref>). As the number of ligand conformations increases, the standard deviation of the BPMF decreases and, for the ten arbitrarily chosen systems in the figure, starts to level off at around 1000 configurations. At this point, the standard deviations of the BPMFs range from 0.04 to 0.3 kcal/mol. The main caveat of this analysis is that exponential averages such as Equation <ref> are known to suffer from convergence issues due to conformational sampling limitations and finite sample bias <cit.>. As ligand conformations important to the holo ensemble may differ from those in the apo ensemble,many ligand conformations may be required to sample from the energetic minima in the holo ensemble. Moreover, a large number of samples may need to be sampled from within each minima in order to observe rare events that have the largest contribution to the exponential average. The established approach to address these issues of configuration space overlap and finite sample bias is to compute free energy differences between similar thermodynamic ensembles in an alchemical pathway. Comparison to the alchemical pathway method in AlGDock allows us to better evaluate the accuracy of the FFT-based single-step perturbation.FFT-based and AlGDock BPMF estimates are highly correlated but there is an error of 3.31 ± 0.21 kcal/mol(Figure <ref>). As seen in Figure <ref>b, much of the deviation from AlGDock stems from using a single-step perturbation instead of an alchemical pathway to estimate the gas-phase BPMF. Some error also results from the complex solvation free energy, but the ligand solvation free energy estimates are very consistent.§ FFTΔG IS CONSISTENT WITH YANK For 24 ligands binding to T4 lysozyme, using the FFT-based minimum interaction energy as an approximation to the BPMF yields results that are highly correlated with the binding free energy from YANK, but with large error (Figure <ref>). In the context of ILT, using the minimum interaction energy instead of the exponential average is referred to as the dominant state approximation <cit.>. When the minimum interaction energy from all receptor structures is used as a free energy estimate, the correlation with free energies is high, with a Pearson's R of 0.90, but the slope is much greater than one and RMSE is large, at 12.70 kcal/mol. Using an exponential average of BPMFs estimated by the dominant state approximation retains the high correlation with a reduction in RMSE.The full FFTΔG procedure of using the Equation <ref> yields binding free energy estimates that are highly correlated with YANK and have much smaller error (Figure <ref> and Table <ref>).The RMSE for the full set of ligands ranges between 2.07 and 2.31 kcal/mol and Pearson's R between 0.87 and 0.90 (with the exact value depending on the weighting scheme). The slope of the linear regression line is close to 1 (≈ 1.2). When considering only the 11 active ligands, the RMSE is only slightly different from the full data set but the Pearson's R much lower and has greater uncertainty. This large uncertainty in the Pearson's R likely results from the relatively small spread of free energies among active ligands. On the other hand, due to a large spread in YANK free energies of inactive ligands, the subset of 13 inactive ligands achieves even higher correlation (Pearson's R ≈ 0.96) than the full set of 24 ligands. A notable outlier is N-hexylbenzene, whose free energy is estimated to be around -2.5 and -8 kcal/mol by FFTΔG and YANK, respectively. N-hexylbenzene is the largest active ligand and possesses a long and flexible hydrocarbon chain. As it is difficult to sample the bound conformation of this chain from a simulation of the ligand by itself, its binding free energy is overestimated.The performance of different weighting schemes are fairly similar to one another. Overall, weighting scheme (d) gives the lowest RMSE for the full set of ligands and sets containing only active or inactive ligands (see Tab. <ref>).However, the differences among weighting schemes in the present work are not significant due to the sizable error bars of the RMSE and Pearson's R. To be consistent with <cit.>,who found that scheme (c) led to the greatest consistency between AlGDock and YANK results,we will hitherto use weighting scheme (c).In recapitulating YANK, FFTΔG is less accurate than AlGDock. AlGDock yields RMSEs in the range 1.49 to 1.76 kcal/mol <cit.>.This reduction in accuracy may be linked to ligand being sampled only from the apo ensemble in the FFT approach as opposed to a thorough sampling from a series of alchemical states in AlGDock method <cit.>.With both methods, the slope is slightly greater than one, likely because we used the same receptor conformations and these conformations did not include those relevant to the weakest-binding ligands.This makes the weakest-binding ligands appear even weaker, leading to an increased slope.§ CONSISTENCY WITH YANK REQUIRES A DIVERSE SET OF RECEPTOR SNAPSHOTS The consistency between FFTΔG and YANK is sensitive to the choice of receptor snapshots. If only snapshots from a single alchemical simulation are used,there is generally a weaker correlation and larger RMSE (and uncertainty in the RMSE) compared to YANK (Table <ref> and Figure S3). However, there are exceptions. When only snapshots from alchemical simulations of T4 lysozyme in complex with p-xylene or (±)-camphor are used, FFTΔG gives rather strong correlation and relatively low RMSE compared to YANK.These two ligands are the second largest active and the largest inactive ligands, respectively.The quality of these calculations can be attributed to the fact that larger ligands tend to open up the binding pocket more widely and hence improve the FFT translational sampling of ligands.However, N-hexylbenzene, the largest active ligand with a long carbon chain, gives the worst FFT-based free energy estimates.YANK simulations with this ligand may have induced a binding pocket shape that is unfavorable for binding to most of other ligands. These trends are consistent with our previous results <cit.>, although N-hexylbenzene snapshots did not perform as poorly when using AlGDock.In contrast to <cit.>, using only snapshots from alchemical simulations of T4 lysozyme complexes with four active ligands (Tab. <ref>) results in worse consistency with respect to YANK than using all snapshots. This suggests that FFTΔG may require a larger set of receptor snapshots to obtain converged binding free energy estimates, likely due to the large number of steric clashes.§ FFTΔG CORRELATES WITH ALGDOCK For 141 ligands, binding free energy estimates based on FFTΔG and AlGDock are highly correlated (Figure <ref> and Figure S3 in the supplementary material). The RMSE for 140 ligands (excluding one ligand with high free energies) is 2.24 (0.13) kcal/mol and Pearson's R is 0.82 (0.04).The subsets of 69 active and 71 inactive ligands maintain essentially the same level of consistency with AlGDock results with RMSE/Pearson's R at 0.70 (0.08) / 2.23 (0.18) kcal/mol for active ligands and 0.86 (0.04) / 2.24 (0.19) kcal/mol for inactive ligands. For the full set of 141 ligands, the performance is similar (Figure S3 in the supplementary material). In the majority of deviations between the two estimates, FFTΔG has a higher value. The likely cause of these deviations is that the FFT-based procedure does not sample poses with sufficiently low interaction energies.Using a larger grid spacing of 0.25 Å does not appear to have a significant effect on the accuracy of FFTΔG with respect to YANK, but reduces accuracy with respect to AlGDock (Figures S2 and S4 in the supplementary material). With respect to YANK, the Pearson's R is not much lower and RMSE not much higher. However, accuracy is more sensitive to the weighting scheme. On the other hand, for the larger data set of 141 ligands, the Pearson's R and RMSE with respect to AlGDock is 0.73 and 3.28 kcal/mol, respectively; the RMSE is slightly higher than with 0.125 Å grid spacing. Because interaction energies are fairly accurate at 0.25 Å grid spacing, the likely cause of the deviation is reduced sampling of translational positions.The overall consistency with respect to YANK and AlGDock suggests that it is feasible to estimate protein-ligand binding free energies using FFTΔG.§ FFTΔG CONVERGENCE REQUIRES MORE RECEPTOR SAMPLING THAN ALGDOCK Compared to AlGDock, FFTΔG requires a greater number of receptor snapshots in order for Δ G^∘ estimates to converge. If receptor snapshots are randomly selected, AlGDock requires about 200 snapshots in order for the free energy to converge <cit.>. In contrast, the Pearson's R and RMSE values do not level off until about 400 snapshots processed by FFTΔG (red curves in Figure <ref>). When receptor snapshots are selected in the order of increasing the DOCK 6 scores, the convergence of Pearson's R and RMSE is improved significantly (green curves in Figure <ref>). They all level off after about 100 snapshots. Nevertheless, the convergence of FFT method is still much slower that AlGDock <cit.>, which needs only a few snapshots to recover the best possible correlation and RMSE when snapshots were selected from lowest to highest docking scores. The best convergence can be achieved by selecting receptor snapshots in the order of increasing BPMF as shown by blue curves in Figure <ref>. These curves serve as an ideal unobtainable reference point because a priori, we do not know which snapshots in the set give lowest BPMFs unless we carry out the calculation for all of them. The convergence analysis done here and in <cit.> suggest that it is better to use docking scores to sort out receptor snapshots before performing BPMF calculations. Convergence is likely worse with FFTΔG than with AlGDock because the ligand has no flexibility and must truly fit as a lock-and-key into the selected receptor conformations to obtain accurate free energies.§ FFTΔG IS CONSISTENT WITH EXPERIMENT Binding free energy estimates based on FFTΔG are moderately consistent with experimental results (Figure <ref>), with similar but slightly worse performance than AlGDock.Isothermal titration calorimetry has been used to measure binding free energies between T4 lysozyme L99A and 21 active ligands <cit.>. Calculated binding free energies for iodine-containing ligands, as in <cit.>, were found to be highly overestimated. Therefore, iodobenzene was excluded from the present analysis (but is included in the supplementary material). The correlation of FFTΔG estimates and measured values is dependent on the solvent model. Comparing the results from the OBC2 model and experiment, the RMSE is 1.29 ± 0.14 kcal/mol, aRMSE is 1.08 ± 0.08 kcal/mol, and Pearson's R is 0.49 ± 0.17.When the PBSA implicit solvent model is used, the RMSE is 3.42 ± 0.42 kcal/mol, aRMSE is 1.99 ± 0.28 kcal/mol, and Pearson's R is 0.52 ± 0.19, which has slightly worse error but comparable correlation.For a point of comparison, <cit.> used the PBSA implicit solvent model in AlGDock to attain a RMSE of 2.81 ± 0.32, aRMSE of 1.35 ± 0.27, and Pearson's R of 0.65 ± 0.05. In both the present results and in <cit.>, the reason for the poor RMSE is that the PBSA model causes a positive shift in the estimated binding free energies.Using alternate estimators for the binding free energy increases the error but does not significantly reduce the correlation with experiment.The error is largest using the dominant state approximation, the minimum interaction energy, as the BPMF. Error is reduced by taking the exponential average of the minimum interaction energy, and is least with FFTΔG. These results are consistent with comparisons to YANK, where we found that these approximations also increase the error with respect to YANK calculations. They are also consistent with <cit.>, who found that using the exponential average is no better than the minimum energy for T4 lysozyme. However, they found that in other, more complex systems, an exponential average was superior to the minimum interaction energy for reproducing experimental free energies.We hypothesized that agreement with experiment could be improved by using a larger grid (62 Å on each side) to account for binding to alternative sites. This grid size encompasses the whole surface of the receptor. However, in spite of being more computationally demanding, the FFT calculations with large grid size do not lead to improved agreement with experiment (Figure <ref>).When the OBC2 solvent model is used, the Pearson's R, RMSE, and aRMSE with respect to experiment are 0.43 ± 0.18), 2.30 ± 0.19 kcal/mol, and 1.04 ± 0.12 kcal/mol, respectively. When PBSA solvent model is used, the Pearson's R slightly increases to 0.53 ± 0.13 but the RMSE becomes slightly larger, 3.50 ± 0.24 kcal/mol, and the aRMSE is 1.08 ± 0.16 kcal/mol. The lack of improvement in agreement with experiment suggests that weak binding of these ligands outside of the L99A cavity may not make a significant contribution to their binding free energy.§ FFTΔG IS A SLIGHTLY BETTER BINARY CLASSIFIER THAN MOLECULAR DOCKING Binding free energy methods can also be used to predict whether a molecule is active or inactive against a target. Here we use receiver operating characteristic (ROC) <cit.> curves, the area under the ROC curve (AUC), and the area under the semi-log ROC curve (AUlC) to assess the ability of FFT free energy estimates to discern active from inactive molecules.A ROC curve illustrates the fraction of true positives versus the fraction of false positives as the threshold separating two categories is changed.An ideal ROC consists of a vertical line from (0,0) to (0,1), and then a horizontal line from (0,1) to (1,1), meaning that all active molecules are more highly ranked than any inactive molecules.The AUC ranges from 0 for completely incorrect to 0.5 for random to 1 for completely correct classification. The intent of the AUlC <cit.> metric is to emphasize top-ranked molecules, which are more likely to be pursued in subsequent experiments and calculations. For a random classifier, the AUlC is 0.14.These different metrics show that DOCK 6 is essentially random, FFTΔG is slightly better, and AlGDock is the best binary classifier (Figure <ref> and Table <ref>). With the OBC2 model, ROC curves of AlGDock and FFT methods (based on the minimum interaction energy or BPMF) are about the same and slightly better than DOCK 6. With the PBSA implicit solvent model, AlGDock has better, but FFTΔG has worse ROC curves. The relatively poor binary classification performance of FFTΔG with the PBSA implicit solvent model is consistent with its relatively poor correlation with experimental Δ G^∘ measurements.As observed in <cit.>, computing a binding free energy opposed to the minimum or mean interaction energy has no benefit to binary classification of active and inactive ligands to T4 lysozyme L99A. As the ligands for T4 lysozyme are small and relatively rigid, the entropic contribution to binding may not change much from ligand to ligand. § FFTΔG IS FASTER THAN ALGDOCK FOR COARSE AND SMALL GRIDS While FFTΔG is less accurate than AlGDock, its main potential benefit is speed. To compare the computational cost of FFTΔG and AlGDock, we carried out BPMF calculations for a small and a large ligand. Both methods basically consist of two steps: sampling and post-processing with implicit solvent. The post-processing step happens in the same way for both methods and expected to consume more or less the same amount of CPU time. Therefore we only compare timing for the sampling step. For the purposes of this comparison, we sampled the binding pose using AlGDock <cit.> based on Hamiltonian replica exchange as previously described <cit.> andin FFTΔG using random rotation and the cross-correlation.Our benchmark calculations (Table <ref>) show that FFTΔG is faster than AlGDock for coarse (0.25 Å spacing) and small grids (16 Å on each edge) and is less sensitive to the ligand size. For a finer grid with 0.125 Å spacing, the speed of FFTΔG is comparable or slower than AlGDock. It is considerably slower for a large grid, but the comparison is unfair because the AlGDock calculation is restricted to smaller binding site opposed to the entire protein surface. One apparent advantage of the FFT approach is that it is less sensitive to the size of the ligand. That is, the larger ligand, N-hexylbenzene, requires approximately twice the amount of computer time with AlGDock, but takes around the same amount of time with FFTΔG. The reason that the calculations take about the same amount of time is that they depend on the number of ligand grid points, not the number of ligand atoms.A comparison between FFTΔG and YANK is less meaningful because YANK calculations were performed on graphical processing units. As mentioned in <cit.>, each YANK calculation took about one week on a single graphical processing unit.§ CONCLUSIONS We have demonstrated that the FFT may be used to estimate standard binding free energies based on implicit ligand theory. To our knowledge, this is the first time that the FFT has been used to calculate noncovalent binding free energies. For the binding of T4 lysozyme to 141 small molecules, FFTΔG is less accurate than AlGDock at reproducing YANK, less correlated with experiment, and less capable of classifying active and inactive molecules. With small grids, however, FFT-based sampling is considerably faster than AlGDock and less sensitive to ligand size.Given the benefits and limitations of FFTΔG, the approach is mostly likely to be useful for larger chemical libraries than AlGDock. Virtual screening can be performed by first using FFTΔG on a large library.Subsequently, AlGDock may be used for a refined library with the greatest predicted affinity.Finally, a flexible-receptor technique such as YANK may be used on yet a smaller library prior to experimental validation.Reflecting a niche of FFT-based docking, another possible use of FFTΔG may be for computing binding free energies between relatively inflexible fragments and a protein. The FFT is an efficient approach for translating the ligand across an entire protein surface. ILT provides a rigorous formalism for combining interaction energies from multiple sites into a single binding free energy.Due to its relative insensitivity to the number of ligand atoms, FFTΔG is likely to be useful for larger ligands than AlGDock. However, FFTΔG may not be efficient if the large ligands are too flexible. Ultimately, another niche of FFTΔG may be the same as the present niche of FFT-based molecular docking: docking of folded protein domains to each other. In these cases, the ligands are large but relatively ordered; it may be feasible to recapitulate their flexibility with molecular dynamics simulations.As a closing comment, we would like to point out a possible way to improve the scoring function of FFT-based docking methods.The current paradigm in FFT-based docking, as in other docking strategies, is to search for the configuration with the lowest interaction energy between binding partners.Once this position is obtained, all other interaction energies are ignored and the binding strength is scored solely based on the lowest value. We suggest an alternative approach that requires essentially no additional computational expense: following the approach of our present study, one may calculate an exponential average of the interaction energy and use the BPMF to score the strength of binding.§ ACKNOWLEDGMENTS We thank Paula Bianca Viana Pinheiro for participating in this project as an undergraduate summer intern through the Brazil Scientific Mobility Program. We thank OpenEye scientific software for providing a free academic license to their software. Computer resources were provided by the Open Science Grid <cit.>. This research was supported by the National Institutes of Health (R15GM114781 to DDLM and R35GM118091 to HXZ).Additional Supporting Information may be found in the online version of this article. 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Ser. volume78, pages012057 (year2007). | http://arxiv.org/abs/1708.07045v1 | {
"authors": [
"Trung Hai Nguyen",
"Huan-Xiang Zhou",
"David D. L. Minh"
],
"categories": [
"q-bio.BM",
"cond-mat.stat-mech"
],
"primary_category": "q-bio.BM",
"published": "20170727123954",
"title": "Using the Fast Fourier Transform in Binding Free Energy Calculations"
} |
Reconstruction via shape optimization in EIT]Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions.Primary 35R30, 65N21; Secondary 49Q10 [ M. I. Krivoruchenko^1,2,3 December 30, 2023 =============================§ INTRODUCTION Electrical Impedance Tomography (EIT) is a noninvasive technique, which aims to detect the conductivity inside a body from voltage and current boundary measurements. The mathematical problem arising from EIT, known as the inverse conductivity problem, was introduced for the first time by A.-P. Calderón in the early 80's <cit.>. Even though it was first motivated by an application in geophysical prospecting <cit.>, EIT has been having big impact also in medical imaging and nondestructive testing of materials <cit.>.The conductivity problem can be stated mathematically as follows. Consider a bounded domain Ω⊂^d, with d=2,3, equipped with an electrical conductivity σ∈ L^∞(Ω) such that σ(x) ≥λ >0.The corresponding Neumann-to-Dirichlet (ND) or current-to-voltage map is the operator _σ: H_0^-1/2(∂Ω) → H_0^1/2(∂Ω), defined by_σ(g) = u|_∂Ω,where H^s_0(∂Ω) = { f ∈ H^s(∂Ω) : ∫_∂Ω fds = 0}, g ∈ H_0^-1/2(∂Ω) and u is the unique H^1(Ω)-weak solution of the Neumann problem for the conductivity equation{[ -∇· (σ∇ u) = 0, in Ω,;σ∂ u/∂ν =g, on ∂Ω, ].where ν is the unit outward normal to ∂Ω, satisfying the normalization condition∫_∂Ω uds = 0. The following inverse boundary value problem arises from this framework. Inverse conductivity problem. Given _σ, find σ in Ω. Since the seminal paper by A.-P. Calderón, much interesting mathematics has been developed in order to address the issues of uniqueness, stability and reconstruction for this problem. Concerning uniqueness and reconstruction, we mention the breakthrough results in <cit.>.The conductivity problem is severely ill-posed as was noted by G. Alessandrini in <cit.>. Despite a-priori smoothness assumptions on the unknown conductivity, a logarithmic-type continuous dependence of the conductivity on the data is the best possible one <cit.>. This fact makes crucial the analysis of the instability and of suitable regularization strategies in order toobtain successful computational reconstructions. Several recovery methods and procedures have been developed in the last decades. Without being exhaustive, the possible approaches to reconstruction can be divided into two main streams: * iterative methods, based on ad-hoc regularization strategies; * direct methods, where an explicit reconstruction formula of the solution is used.The first group includes variational-type methods, which reduce the inverse problem to a minimization problem for a least-squares constrained type functional with a suitable regularization. A pioneering paper in this direction is represented by <cit.>, which applies one step of a Newton method with a constant conductivity as an initial guess. In <cit.> the authors introduce a Mumford-Shah type functional, in <cit.> alevel set representation and a total variation regularization is introduced, while in <cit.> an augmented Lagrangian method is proposed. All these methodsare particularly suited to recover piecewise constant conductivities.Concerning direct methods, we would like to mention the factorization method <cit.>, the D-bar method <cit.>, the enclosure method <cit.>, and the monotonicity method <cit.>.Finally, statistical inversion has shed interesting insights into EIT reconstruction as well <cit.>. Despite the impressive progress, there remains a big interest in developing new algorithms that take advantage of a-priori information arising from applications.An a-priori assumption physically relevant in many applications is to assume the conductivity to be of the form σ=∑_j=1^Nσ_jχ_P_j,where 𝒫={P_j}_j=1^N is a polygonal (polyhedral) partition of the background body Ω,χ_P denoting the characteristic function associated with the generic region P⊂Ω. Such an assumption arises, for example, in geophysics, medical imaging, and nondestructive testing of materials, where the body under investigation contains regions, represented by the subdomains {P_j}_j=1^N, with different electrical properties.Additionally, this kind of a-priori information restores the well-posedness of the inverse problem; in particular, in the case of a given known partition, Lipschitz dependence estimates of the coefficients from the data can be shown <cit.>. From the results obtained for the Helmholtz equation in <cit.> and <cit.>, where Lipschitz stability holds, we expect that a similar result should be true also when the partition is unknown. As shown in <cit.>, a crucial role to prove Lipschitz stabilityis played by the differentiability properties of the Neumann-to-Dirichlet map with respect to motions of the partition and by the derivation of an explicit formula for the derivative. This is also a crucial step towards reconstruction if we use an optimization approach to solve the inverse conductivity problem.In this paper, we consider a cost functional, J(σ), representing the L^2-norm of the difference between the potential due to the applied current and the measured potential on the boundary. This functional will be minimized in the class of conductivities σ of the form (<ref>).To solve this minimization problem, we introduce an iterative gradient type method which requires the computation of the shape derivative of the functional J(σ). This derivative has been obtained rigorously by several authors in the case of a single sufficiently smooth inclusion ω⊂Ω, i.e. for σ = σ_1 χ_ω+σ_2χ_Ω\ω (see, for example,<cit.>).In our case, to implement the optimization procedure, we need to differentiate J(σ) with respect to variations of a partition.In <cit.>, the authors derived, for the first time, a rigorous formula for the shape derivative of the functional J(σ) in the planar case and for conductivities of the formσ=σ_1χ_P+σ_2χ_Ω\ Pwhere P is a polygon strictly contained in Ω. The shape derivative is expressed in terms of an integral over the boundary of P and has, surprisingly, exactly the same form as the one derived in <cit.> in the case of a smooth interface, despite the presence of singularities of the gradient of the solutions to the conductivity equation at the vertices of P. The extension of this boundary formula seems not to be possible in the case of an arbitrary partition since the singularity of the gradient might become too strong at the vertices <cit.>.In this paper, we follow the idea suggested in <cit.> and in <cit.> of using a more general distributed shape derivative of the functional expressed in terms ofan integralover Ω. The advantage of this formula is twofold: on the one hand, it allows to consider piecewise constant conductivities on very general partitions; on the other hand, it is numerically more accurate (we refer to <cit.> for a thorough comparison of the two formulas from a numerical point of view). In <cit.>, the authors establish the distributed shape derivative in the case of a single measurable conductivity inclusion strictly contained in Ω by using a Lagrangian approach. Here, we establish the formula computing directly and rigorously the derivative of J in terms of the material derivative of the solution to a certain boundary value problem (see Lemma 2.1). The formula is valid for any partition 𝒫 in dimension d = 2,3. Successively, we take advantage from this result to implement our reconstruction procedure in the two dimensional case, based on a gradient type method.The reconstruction algorithm we present is very similar to the one introduced in <cit.>. Nevertheless, there are two major differences in the implementation, which greatly affect the numerical results. The first one is a regularization step, applied at each iteration, where the number of the sides of each polygon in the partition 𝒫 changes in order to preserve a uniform length. This considerably reduces the artifacts that typically appear in EIT reconstructions. The second major difference lies in the choice of the descent direction for the shape of the partition. While this was done by solving an additional variational problem, we propose a more direct computation which exploits the assumptions made on the conductivity.The plan of the paper is the following one. In Section 2, we state the problem with the main assumptions, derive the shape derivative of the cost functional and show the equivalence with the boundary shape derivative established in <cit.> for suitable two-dimensional partitions. In Section 3, we describe the reconstruction algorithm. In Section 4 we present some numerical examples which corroborate the reliability and the accuracy of the proposed approach.§ MATHEMATICAL FRAMEWORK §.§ Main AssumptionsLet Ω⊂^d, d = 2,3, be a bounded domain with Lipschitz boundary. Let = {P_j}_j=1^N be a polytopal partition of Ω, i.e., P_j are open bounded polytopes (i.e., polygons in 2D, polyhedra in 3D) such that:⋃_j = 1^ NP_j = Ω,P_j ∩ P_k = ∅ forj ≠ k. Let N ∈ be an integer with N>1, and λ >0 be a positive real number. We define the space L^∞(Ω,N,λ) as the collection of conductivities σ∈ L^∞(Ω) such that σ(x) ≥λ > 0 for all x ∈Ω, and such that there exists a polytopal partition = {P_j}_j=1^N' with N' ≤ N, such that σ can be written asσ = ∑_j=1^N'σ_j χ_P_j, with σ_l ≠σ_mifP_lisadjacent toP_m. Now let σ̂, σ∈ L^∞(Ω,N,λ). We denote by σ̂ the unknown conductivity and by σ a (generally) different one, which will be used in the reconstruction scheme.Let M be the number of measurements. For 1 ≤ j ≤ M, let g_j ∈ H_0^-1/2(∂Ω) be a given function representing the applied current density on ∂Ω, and f_j ∈ H^1/2(∂Ω) the corresponding measurement of the voltage on ∂Ω.More precisely, f_j =û_j|_∂Ω, where û_j is a solution of{[ -∇· (σ̂∇û_j) = 0, in Ω,;σ̂∂û_j/∂ν =g_j, on ∂Ω. ].In order to recover σ̂, we minimize the following Dirichlet least-squares fitting cost functional:J(σ) = 1/2∑_j=1^M∫_∂Ω|u_j -f_j|^2ds,for σ∈ L^∞(Ω,N,λ), where the state function u_j solves{[ -∇· (σ∇ u_j) = 0, in Ω,;σ∂ u_j/∂ν =g_j, on ∂Ω, ].with the normalization condition∫_∂Ω u_j ds= ∫_∂Ω f_jds. Notice that the functional J(σ) depends on the values {σ_j}_j=1^N and the partition = {P_j}_j=1^N, where σ = ∑_j=1^Nσ_j χ_P_j. The next subsections are devoted to the computation of the gradient of J(σ) with respect to the above variables. §.§ Gradient of J with respect to– the shape derivativeThe gradient of J with respect to the partitionis actually a shape derivative. We thus want to compute the shape derivative ⟨∇_ J, U ⟩ of the functional J at the partitionin the direction of a vector field U = (U_1,…,U_d). We assume that U ∈ W^1,∞(^d) and U = 0 in a neighborhood of ∂Ω. This derivation has already been carried out in <cit.> in the case of smooth inclusions, and in <cit.> for a single polygonal inclusion. We present here a more general formula that is valid for any finite partition of a domain.In order to compute the derivative, consider the transformation Φ_t( x) = x + t U( x): ^d →^d, as smooth as U. We assume that t ≤ 1/(2U_W^1,∞) so that Φ_t^-1 exists globally. For instance, Φ_t is a piecewise affine function that moves the nodes of the partition(i.e., the vertices of the P_j's). Note that Φ_t |_∂Ω = Id, Id denoting the identity mapping. This assumption is not a real restriction since in EIT the boundary ∂Ω is always assumed to be known and fixed. Nevertheless, it is crucial in the derivation of the main results of this section. The following matrix-valued functions will be useful in the following:A(t)= (DΦ_t^-1) (DΦ_t^-1)^T(DΦ_t),= . d A/dt|_t=0 = (U)I - (DU+DU^T),where D Φ_t^-1 and DU are the Jacobian matrices of Φ_t^-1 and U, respectively.Consider the deformed partition _t = Φ_t() and σ_t = σ∘Φ_t^-1 the corresponding conductivity. Let G(t) be defined asG(t) = J(σ_t). Let now u be any of the solutions u_j, with Neumann data g, where g is any of the currents g_j, j=1,…,M, and let f be a boundary measurement corresponding to g. First, we need to study the material derivative of the solution u. The solution u to problem (<ref>) has a material derivative u̇∈ H^1(Ω) that solves∫_Ωσ∇u̇·∇ w dx = - ∫_Ωσ∇ u ·∇ w dx, ∀ w ∈ H^1(Ω),with the normalization condition ∫_∂Ωu̇ ds = 0. We follow Step 1 and 2 of the proof of <cit.>. Let u_t be the solution of{[ -∇· (σ_t ∇ u_t)= 0, in Ω,;σ_t ∂ u_t/∂ν = g, on ∂Ω, ].with the normalization condition ∫_∂Ω u_t ds = ∫_∂Ω f ds. Then the transported solution ũ_t = u_t ∘Φ_t solves the variational equation:∫_Ωσ A(t) ∇ũ_t ·∇ w dx -∫_∂Ωg w ds= 0, ∀ w ∈ H^1(Ω).Subtracting to (<ref>) the variational equation solved by u and dividing by t, we find∫_Ωσ A(t) ∇ũ_t -∇ u/t·∇ w dx = ∫_ΩσI- A(t)/t∇ u ·∇ w dx, ∀ w ∈ H^1(Ω).Using ũ_t- u as test function we obtain1/2min_x ∈Ωσ(x) ∇ũ_t - ∇ u/t_L^2(Ω)≤A(t) - I/t_∞∇ u _L^2(Ω).Thus, we have found that (ũ_t - u)/t is bounded in H^1(Ω). Therefore, the sequence is weakly convergent in H^1(Ω) and its weak limit is the material derivative u̇ of u. Passing to the limit in (<ref>), we find that u̇ solves the desired variational formulation (<ref>). By the Lax-Milgram lemma, since the right-hand side is an H^-1(Ω) function (because of our assumptions on U), every solution to the variational problem (<ref>) lies in H^1(Ω). In particular, the trace on ∂Ω is well defined.Actually, we have strong convergence. Plugging w = (ũ_t - u)/t into (<ref>), we obtain∫_Ωσ A(t) ∇ w ·∇ wdx= ∫_ΩσI- A(t)/t∇u ·∇ wdx = B_1,t+B_2,t,where B_1,t = ∫_Ωσ ( A(t)-I) ∇w ·∇ wdx and B_2,t= ∫_ΩσI- A(t)/t∇ũ_t ·∇ wdx.Thanks to the weak convergence of (ũ_t - u)/t, we obtainB_1,t→ 0 and B_2,t→ - ∫_Ωσ∇ u ·∇u̇ dx ast → 0.Now, the variational formulation (<ref>) yields B_2,t→∫_Ωσ∇u̇·∇u̇ dx. So we have found that ∇ũ_t -∇ u/t converges strongly to ∇u̇ in L^2(Ω). Using the normalization conditions for u̇, u and ũ_t (which coincides with u_t on ∂Ω), we get the strong convergence of ũ_t - u/t to u̇ in H^1(Ω), via the Poincaré inequality.Now we can derive the formula for the shape derivative of the functional J.We have⟨∇_ J , U ⟩ = ∑_j=1^M ∫_Ωσ∇ u_j ·∇ z_j dx,where z_j solves {[-∇· (σ∇ z_j)= 0, in Ω,; σ∂ z_j/∂ν = f_j-u_j, on ∂Ω, ].with the normalization ∫_∂Ωz_j ds = ∫_∂Ω f_j ds. We need to compute the derivative d G/dt|_t=0, where G is defined in (<ref>). Let u_j,t, ũ_j,t be defined as u_t, ũ_t in the proof of Lemma <ref>, with Neumann data g_j. By the assumption that Φ_t|_∂Ω = Id, we haveG(t) = 1/2∑_j=1^M ∫_∂Ω(u_j,t-f_j)^2 ds = 1/2∑_j=1^M ∫_∂Ω(ũ_j,t -f_j)^2 ds.Note thatG(t)-G(0)/t=∑_j=1^M ∫_∂Ω( ũ_j,t-u_j/t)( ũ_j,t+u_j/2 -f_j) ds.Using the strong convergence ũ_j,t-u_j/t→u̇_j in H^1(Ω) and the convergence ũ_j,t→ u_j in L^2(∂Ω) (which follows from the fact that ũ_j,t = u_j,t on ∂Ω and u_j,t→ u_j in H^1(Ω)), we can pass to the limit as t → 0 in (<ref>), thus obtaining ⟨∇_ J , U ⟩= dG/dt|_t=0 = ∑_j=1^M ∫_∂Ω (u_j-f_j)u̇_j ds,where u̇_j is the material derivative of u_j. Using the variational formulation of the adjoint z_j, ∫_Ωσ∇ z_j ·∇ w dx + ∫_∂Ω (u_j-f_j) w dσ = 0, ∀ w ∈ H^1(Ω),and of the material derivative, (<ref>), we find⟨∇_ J , U ⟩= - ∑_j=1^M ∫_Ωσ∇ z_j ·∇u̇_j dx = ∑_j=1^M ∫_Ωσ∇ u_j ·∇ z_j dx,which is the desired formula. We now recall also the gradient of J with respect to {σ_j }_j=1^N, that will be used in the reconstruction algorithm: d J/d σ_j = ∑_k=1^M ∫_P_j∇ u_k ·∇ z_k dx,j = 1,…,N,where z_k solves (<ref>) with a normalization (e.g., ∫_∂Ωz_k ds = ∫_∂Ω f_k ds). §.§ Equivalence of the distributed and of the boundary integral formulas in the case of a single polygonal inclusionWe emphasize that, in the case of a single polygonal inclusion, the shape derivative of the functional J(σ) has been computed rigorously in <cit.>, and expressed as an integral on the boundary of such an inclusion.It is unclear if this boundary representation of the shape derivative is still valid in the case of an arbitrary partition.Let P be a polygon strictly contained in Ω and letu^+=u|_Ω\P̅, z^+=z|_Ω\P̅.and u^-=u|_ P, z^-=z|_P.Assume Ω=P∪(Ω\ P), with P a polygon strictly contained in Ω, and let σ|_P = k and σ = 1 outside P.Then, we have the following equivalent formulas for the shape derivative of the functional J at P in the direction given by U:.d G/dt|_t=0 =∑_j=1^M ∫_Ωσ( x) ∇ u_j( x) ·∇ z_j(x) dx= (k-1) ∑_j=1^M∫_∂ P(1/k∂ u_j^+/∂ν∂ z^+_j/∂ν+∇_τu_j ·∇_τz_j)U_ν ds,where ν is the unit outward normal vector to ∂ P, ∇_τ is the tangential gradient, U_ν = U·ν, functions u_j,z_j satisfy (<ref>) and (<ref>), respectively.We can assume, without loss of generality, M=1 and denote by u and zthe solutions corresponding to the datum f. Let B_ε be the union of balls of radius ε centered at each vertex of P, and let us consider the splitting ∫_Ωσ∇ u ·∇ z dx=∫_Ω\ B_εσ∇ u ·∇ z dx+∫_B_εσ∇ u ·∇ z dx.Denote by Ω_ε^+=Ω\P∪ B_ε, Ω_ε^-=P\ B_ε,and observe that, by standard regularity results (see, for example, <cit.>), it can be shown that u^+,z^+∈ H^2(Ω_ε^+), andu^-,z^-∈ H^2(Ω_ε^-). Then,applying Green's formula in Ω_ε^±, observing that U has compact support in Ω, and using the following identity∇ u ·∇ z= -(b) + (U·∇ u) Δ z +(U·∇ z) Δ u=-(b) in Ω_ε^+∪Ω_ε^- =Ω\B̅_ε,where b = (U·∇ u) ∇ z + (U·∇ z) ∇ u- (∇ u ·∇ z) U, we easily derive∫_Ω\B̅_εσ∇ u ·∇ z dx=∫_∂ P\B̅_ε[σ b]·ν ds+∫_∂ B_εσ b·ν ds,where we use notation [f] = f^+-f^- to denote the jump off across ∂ P. Using the transmission conditions satisfied by u and z across ∂ P, we end up with the following relation∫_Ω\B̅_ε σ∇ u ·∇ z dx = (k-1) ∫_∂ P(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds-(k-1) ∫_∂ P∩B̅_ε(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds+∫_∂ B_εσ b·ν ds,where U_ν=U·ν. Hence,∫_Ω σ∇ u ·∇ z dx = (k-1) ∫_∂ P(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds -(k-1) ∫_∂ P∩B̅_ε(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds_R_1+∫_∂ B_εσ b·ν ds_R_2+∫_B_εσ∇ u ·∇ z dx_R_3 =(k-1) ∫_∂ P(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds+R_1+R_2+R_3. From <cit.>, we have the following upper bounds of the gradients of u and z in a neighbourhood of the vertices:|∇ u (x)|≤ C |x-x̃|^α-1,|∇ z (x)| ≤ C |x-x̃|^α-1,for some constant C>0, α > 1/2 and x sufficiently close to a vertex x̃, with x≠x̃.This, jointly with the regularity assumptions on the vector field U, implies that ∫_Ω σ∇ u ·∇ z dx =(k-1) ∫_∂ P(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds+O(ϵ^2α-1),and, since α>1/2, letting ϵ→ 0 in the last equation, we finally obtain∫_Ωσ∇ u ·∇ z dx=(k-1) ∫_∂ P(1/k∂ u^+/∂ν∂ z^+/∂ν+∇_τu ·∇_τz)U_ν ds,which ends the proof.It is straightforward to check that the boundary formula derived in <cit.> (and hence also the last proposition) extends to the case of a finite number of well separated, polygonal inclusions at a positive distance to the boundary of Ω.The formula for the shape derivative of the functional J can also be obtained following the Lagrangian approach as in <cit.>.In fact, defining the following Lagrangian Ł̃(u,z,t,U)= 1/2∫_∂Ω|u -f|^2 dσ+ ∫_Ωσ A(t)∇ u ·∇ z dx - ∫_∂Ω g z dσ,where A(t) is defined as in (<ref>), while functions u and z satisfy (<ref>) and (<ref>), respectively corresponding to a given Neumann datum g,it is possible to show that dG/dt|_t=0= ∂/∂ tŁ̃(u,z,t,U)|_t=0= ∫_Ωσ(x) ∇ u(x) ·∇ z(x) dx.§ RECONSTRUCTION ALGORITHMWe use the descent gradient method to solve the minimization problem involving the functional J in (<ref>). Although the algorithm holds in dimension 2 and 3, we focus on the planar case, since computational experiments are in 2D. The algorithm adopted in this paper differs from reconstruction algorithms available in the literature (<cit.>) because of two major features, which are separately addressed in the next sections. §.§ RegularizationThe first novelty adopted in the reconstruction algorithm consists inincreasing or decreasing the number of vertices of the polygons in the partition in order to impose some regularity on the reconstruction. This simple trick considerably improves the reconstruction quality, smoothing out artifacts and irregularities that often characterize EIT reconstructions (see Section 4 below for practical examples). §.§ Construction of the descent directionThe second distinguishing feature of the adopted reconstruction algorithmconcerns the computation of the descent direction associated with the partition vertices. A standard approach consists in solving a discretized version of the equation⟨θ^k, δθ⟩ + ⟨∇_ J(σ^k), δθ⟩ = 0 for every δθ∈ X,where θ^k ∈ X is the descent direction at iteration k, σ^k is the conductivity obtained at the iteration k (see Section <ref> for a precise definition), while X ⊂ W^1,∞(Ω,^2). There are several approaches to discretize and solve equation (<ref>). We refer to <cit.> for a thorough discussion on this subject. Here we propose a different, more straightforward, derivation. Since the partition is defined by its vertices, we compute the descent direction for each vertex individually, and then we update the partition. The formula we use at iteration k to define the descent direction, θ_l^k ∈^2, for a given vertex, V_l^k, l=1,…,N^k_V, with N^k_V the number of the partition vertices at the iteration k, is:θ_l^k = -(⟨∇_ J(σ^k), U^k_l,1⟩, ⟨∇_ J(σ^k), U^k_l,2⟩),where the vector fields U^k_l,1,U^k_l,2 are chosen as follows: (1) U^k_l,1,U^k_l,2∈ W^1,∞(Ω,^2) withsupport strictly contained in Ω(2) U^k_l,1,U^k_l,2 piecewise linear on the edges of the partition and such thatU^k_l,1(V_j^k) = (δ_jl,0),U^k_l,2(V_j^k) = (0,δ_jl), where δ_jl indicates the Kronecker delta.With this choice, the vector (⟨∇_ J(σ^k), U^k_l,1⟩, ⟨∇_ J(σ^k), U^k_l,2⟩) represents, at the iteration k, the gradient of the functional J with respect to the position of a single vertex, V_l^k.In practice, in view of a finite element discretization, we choose the vector fieldsU^k_l,1 and U^k_l,2 as U^k_l,1 = (φ^k_l,0),U^k_l,2 = (0,φ^k_l),where φ^k_l is the hat function associated with the node V^k_l of a coarse mesh that contains both the vertices and the edges of the partitionbut with no additional nodes on the sides of the polygons. More precisely, ϕ^k_l (V^k_j) = δ_jl, and it is piecewise linear. Examples of coarse meshes adapted to a partition can be seen in every plot in Section 4. Thus, any other hat function corresponding to such a coarse mesh would produce the same results, since the shape derivative is supported on the edges of the partition.We remark that we need to assume the knowledge of the number N of polygons in the partition, since the algorithm is unable to make changes in the topology. A possible approach to overcome this limitation, using topological derivatives and a level set formulation, has been studied in <cit.>. §.§ The reconstruction algorithmLet N^k_V be the number of vertices of the partition ^k = { P^k_j}_j=1^N at iteration k, and { V^k_l }_l =1^N^k_V be the corresponding set of vertices. Let N^k_j be the number of vertices of the polygon P^k_j, for j=1,…,N. At each iteration, we consider the conductivity σ^k = ∑_j=1^N σ_j^k χ_P^k_j. In the algorithm we introduce the parameters δ_1, δ_2 >0 as threshold for the distances between consecutive vertices in the regularization step and a tolerance tol >0 is chosen to control the size of the shape gradient as a stopping criterion. Finally, the step sizes α^j_k > 0 and β_k >0 can be fixed or obtained by line search. However, in all numerical experimentsbelow, we keep them fixed.The stopping criteria for the algorithm may be improved following the error analysis in <cit.>. For the sake of simplicity, in the present work, we decided to focus on the new shape updates in the algorithm, neglecting other issues already investigated in earlier works. Theinclusion of an advanced stopping criterion as well as of a robust line search algorithm for the step sizes constitute possible topics for a future investigation.§ NUMERICAL TESTSIn this section, we show the effectiveness of the proposed reconstruction scheme. A series of numerical experiments have been carried out in order to assess numerically that the algorithm enable us to recover simultaneously the partition and the values of the conductivity for some significant configurations.For the sake of simplicity, we consider Ω = (0,1)^2. For each test, we create a mesh adapted to the unknown conductivity that is used to provide the boundary data. For the reconstruction, a coarse mesh adapted to the approximate partition is generated at each iteration. A refined mesh is also constructed in order to compute the state and the adjoint problem.The boundary data are generated in the following manner. We set σ∂ u/∂ν = 1 on one of the four sides of Ω, -1 on another side and 0 elsewhere. In this way, we obtain 6 independent current patterns, corresponding to 4 electrodes, one for each side. Then, we divide each side of Ω in half. We set σ∂ u/∂ν = 1 on one half, -1 on another half side and 0 elsewhere. This gives 28 independent current patterns, corresponding to 8 electrodes. We iterate this procedure one more time. In this way, we construct sets of 6, 28 and 120 boundary data, corresponding to 4, 8 and 16 electrodes, respectively. See Figure <ref> for a scheme of the electrode position corresponding to the data sets.We also add a uniform noise to the data. More precisely, given a noiseless boundary measurement f_j ∈ H^1/2(∂Ω), j=1, …,M, the noisy data f̃_j is obtained by adding to f_j a uniform noise in the following way:f̃_j (x) = f_j (x) +εf_L^2(∂Ω),where x ∈∂Ω is a boundary vertex of the mesh that generated f_j and ε is a uniform random real in (-γ,γ), where γ> 0 is chosen according to the noise level. To measure the noise level, we use the relative error on the boundary in the L^2-norm, that is the following quantity:√(∑_j=1^M f̃_j - f_j^2_L^2(∂Ω))/√(∑_j=1^Mf_j^2_L^2(∂Ω)). The regularization parameters, δ_1 and δ_2, are chosen experimentally. Since the initial guess is always a regular polygon (or a collection of regular polygons), we choose δ_1 = α_1 δ, δ_2 = α_2 δ, where δ is the length of the side of the initial guess, with α_1 < 1, α_2 > 1.5. This is done in order to have, at each iteration, a partition with edges of similar length.All the computations are performed using FreeFem++ <cit.>. §.§ Shape reconstruction In this first set of examples, we assume to know the values of the conductivity, and we only reconstruct the shape of the partition. In Figure <ref>, we consider a piecewise conductivity which is equal to 10 on a convex pentagon and to 1 in the background. The reconstruction is carried out using 6 boundary measurements. The initial guess is a regular polygon of 14 sides (top-left). We present a noiseless reconstruction, with (top-right) and skipping (bottom-left) the regularization step, and a reconstruction with regularization when 3% of noise is added to the data (bottom-right). The regularization parameters are set to δ_1 = 0.7 δ and δ_2 = 1.8 δ, where δ is the length of the side of the initial guess. Figure <ref> shows that the three reconstructions well identify the shape with a comparable precision.A non-convex polygon is considered in Figure <ref>, where the target conductivity is 10 inside and 1 in the background. Here, we employ 28 boundary measurements. The initial guess is a regular polygon of 24 sides (top-left). For this phantom, we present a noiseless reconstruction with (top-right) and without (bottom-left) the regularization step, and a reconstruction post regularization in the presence of a 3% of noise added to the data. The regularization parameters are chosen as δ_1 = 0.85 δ and δ_2 = 1.8 δ, where δ is the length of the side of the initial guess. We notice here that the regularization step helps to reconstruct more precisely the non-convex part of the unknown. It is also interesting to observe that the shape is well identified also in the noisy case.In Figure <ref>, we consider the so-called heart and lung phantom <cit.>, with background conductivity 1, two ellipses with conductivity 0.5 and a disk with conductivity 2 (both ellipses and the disk are approximated with 16-sided polygons). The initial guess coincides with three identical regular polygons of 16 sides each (top-left). The regularization parameters are δ_1 = 0.9 δ and δ_2 = 1.8 δ, where δ is the length of the side of one of the initial guess polygons. The reconstruction is done using 6 boundary measurements. Here, the difference between the regularized (top-right) and the non-regularized (bottom-left) reconstruction is significative. Moreover, the reconstruction is very robust to the 3% noise added to the data (bottom-right). The three phantoms just presented have different contrast. It is therefore natural to study the dependence of the reconstruction on the contrast. We observed that the algorithm converges faster in case of higher contrast, yet the reconstruction quality is the same. For this reason we decided to not include reconstructions of the same phantom with different contrast values.§.§ Reconstruction from a misplaced initial guess With the example in Figure <ref>, we show the robustness of the algorithm in recovering an unknown shape from a misplaced initial guess. The target is a piecewise conductivity with value 10 inside a square and 1 in the background. The initial guess is an octagon, far from the exact square (top-left). The regularization parameters are set to δ_1 = 0.8 δ and δ_2 = 1.7 δ, being δ the length of the side of the octagon. The algorithm is able to recover the square (bottom-right). The striking result strongly depends on the choice of the regularization parameters. Indeed, after the first few iterations, the octagon becomes a triangle (top-right). This reduction in the degrees of freedom avoids potential degeneracy of the shape. The triangle eventually changes and becomes a square when approaching the target (bottom-left). §.§ Simultaneous reconstructionIn this section, we present reconstructions obtained with the full algorithm, i.e., the conductivity values are also unknown and updated at each iteration.We focus on the heart and lung phantom, this configuration being characterized by more interesting features compared with the other ones. The background is assumed to be known and is equal to 1, while the coefficients to be recovered are the lungs, with value 0.5, and the heart, with value 2. The reconstructions presented in Figure <ref> are obtained from 28 boundary data. The initial guess (top-left) identifies the shapes the algorithm acts on, while three different values are adopted for the initial conductivity during the reconstruction procedure. The regularization parameters are set to δ_1 = 0.9 δ and δ_2 = 1.8 δ independently of the run, with δ the length of the side of one of the initial guess polygons.The first reconstruction deals with noiseless data. The values of the initial guess are 0.55 in the lungs and 2.05 in the heart. The reconstructed values are 0.49 and 2.05, respectively, and the shape is well reconstructed for the three inclusions (top-right). The second reconstruction starts from the same initial guess, by adding a 5% noise to the data. The reconstructed conductivity values are 0.37 in the lungs and 2.06 in the heart. The shape of the lungs is better recovered than the heart (bottom-left).The last reconstruction is essentially blind, where we assume that the initial guess is provided by a constant conductivity equal to 1 (the background). Moreover, data contains 1% of additive noise. Despite the challenging setting, some features are correctly recovered. The reconstructed values are 0.44 in the lungs and 0.94 in the heart. The shape and the values of the lungs are well reconstructed, whereas this is not the case for the heart(bottom-right), due to the lack of a priori information and the central position of the middle polygon in the initial guess. §.§ Sensitivity to the number of measurementsUsing the heart and lung phantom (always with value 0.5 in the lungs, 2 in the heart, and 1 in the background), we check now how the reconstructions change using a different set of measurements. The regularization parameters are always chosen as δ_1 = 0.9 δ and δ_2 = 1.8 δ, with δ the length of the side of one of the initial guess polygons.In Figure <ref>, we present shape reconstructions (conductivity values are known) starting from noiseless data sets of 6, 28 and 120 boundary measurements. While there is an evident improvement passing from 6 to 28 data, the reconstruction quality obtained from 28 and 120 measurements looks very similar in this example. This is due to the ill-posedness of the problem that limits the resolution. §.§ Sensitivity to the noise levelIn this section, we show how the algorithm is stable to noise in the measurements. In Figure <ref>, we present reconstructions using the full algorithm (unknown shape and values) for the heart and lung phantom. The regularization parameters are always chosen as δ_1 = 0.9 δ and δ_2 = 1.8 δ, being δ the length of the side of one of the initial guess polygons.The initial guess has values 0.7 for the lungs and 1.5 for the heart, far from the exact values 0.5 and 2, respectively. Data contain 0.5% (top-right), 5% (bottom-left) and 20% (bottom-right) of additive noise, respectively. The tolerance in the stopping criterion is chosen experimentally, according to the noise level: 0.004 for both 0.5% and 5% of noise, 0.02 for 20% of noise.We observe that the shape of the lungs is well recovered even with very noisy data, whereas the heart is poorly recovered, due to the values of the initial guess and the central position of the middle disk. The reconstructed values exhibit a low sensitivity to the noise level. They are 0.4 in the lungs and 1.51 in the heart for both 0.5% and 5% of noise, while we obtain 0.37 in the lungs and 1.49 in the heart when considering 20% of noise.§ CONCLUSIONS We have presented a new shape optimization approach that is able to well identify a piecewise constant conductivity on a polytopal partition in electrical impedance tomography. Despite the ill-posedness of EIT, the a-priori assumption on the conductivity to be piecewise constant on a polygonal partition regularizes the problem. This fact, coupled with the effectiveness of the algorithm, allows us to recover an unknown partition (or at least a subset of it) even in the case of noisy data and a wrong initial guess. This is possible due to the use of a distributed shape derivative, to a new regularization step and a new computation of the descent direction.The algorithm performs better when a good approximation of the values of the conductivity is available. When the coefficients are not known, we noticed that it takes more iterations to approximate their values than to identify the shapes. These values could be obtained, for instance, from a one-step reconstruction <cit.> and used as a first guess to recover the interfaces more accurately (see also <cit.> for a hybrid one-step method). The algorithm can be accelerated using more advanced minimization methods and more efficient forward solvers. 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"authors": [
"Elena Beretta",
"Stefano Micheletti",
"Simona Perotto",
"Matteo Santacesaria"
],
"categories": [
"math.AP",
"math.NA",
"35R30, 65N21 (Primary), 49Q10 (Secondary)"
],
"primary_category": "math.AP",
"published": "20170726125444",
"title": "Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT"
} |
^1Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Malaysia.[Also at ]Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Skudai, Johor Bahru, Malaysia. [email protected] ^2Research Center for Nuclear Physics, Osaka University, Osaka, Japan. ^3Department of Physics, Osaka University,Osaka, Japan. ^4Material Life Science Facility, J-PARC,Tokai, Japan. The negative-muon capture reaction (MCR) on the enriched ^100Mo isotope was studied for the first time to investigate neutrino nuclear responses for neutrino-less double beta decays and supernova neutrino nuclear interactions. MCR on ^100Mo proceeds mainly as ^100Mo(μ,xn)^100-xNb with x being the number of neutrons emitted from MCR. The Nb isotope mass distribution was obtained by measuring delayed γ-rays from radioactive ^100-xNb. By using the neutron emission model after MCR, the neutrino response (the strength distribution) for MCR was derived. Giant resonance (GR)-like distribution at the peak energy around 11-14 MeV, suggests concentration of the MCR strength at the muon capture GR region.DOIPACS number(s)23.40.-s, 13.35.Hb, 14.60.St.Valid PACS appear here Muon capture reaction on ^100Mo to study nuclear responses for double beta decays and astro-neutrinos I.H.Hashim^1,2,Ejiri^2, T.Shima^2,A.Sato^3,Y.Kuno^3,N.Kawamura^4, S.Miyake^4, K.Ninomiya^4 December 30, 2023 =====================================================================================================§ INTRODUCTION Neutrino nuclear responses (square of nuclear matrix element, NME) are crucial for neutrino studies in nuclei. Neutrino properties such as the Majorana nature, the absolute mass scale and others beyond the standard electro-weak model are studied by investigating neutrino-less double beta decays (DBD) <cit.>. Here the ν responses for DBD (β^- and β^+ responses) are necessary to get the neutrino properties beyond the standard model. The DBD matrix element is expressed in terms of the product of β^- and β^+ matrix elements of M(β^-) and M(β^+). Astro-neutrino nucleo-syntheses and astro-neutrino nuclear reactionsare studied by investigating astro neutrino nuclear interactions. Then one needs nuclear responses for astro neutrinos and astro anti-neutrinos <cit.>. So far, the single β ^- matrix element M(β ^-) and the neutrino response have been extensively studied by charge exchange reactions <cit.>.The present report aims to show that negative muon (μ) capture reaction (MCR) is used to study the neutrino nuclear responses relevant to the β^+ response involved in DBD and anti-neutrino response associated with astro anti-neutrino reactions. The MCR response for ^100Mo shows a giant resonance (GR) distribution at the peak energy around 11-14 MeV.MCR is a kind of muon charge exchange reaction (MCER) via the weak boson W^±, where muon becomes muon neutrino andproton in the target nucleus transform into neutron. The response represents the square of the nuclear matrix element M(μ). The reaction and the NME are expressed as μ + ^A_ZX → ^A_Z-1Y + ν_μ, M(μ),where ^A_ZX with A and Z being the mass number and the atomic number of the target nucleus and^A_Z-1Y is the nucleus after MCR. The corresponding astro anti-neutrino reaction and the NME are given as ν̅_e + ^A_ZX → ^A_Z-1Y + e^+, M(ν̅).DBD via the light Majorana ν exchange process with the β^+ and β^- responses is written as the neutrino emission and re-absorption process as^A_Z X→ ^A_Z-1Y + ν_e + e^+, M(β ^+). ^A_Z-2X→ ^A_Z-1Y + ν̅_e+ e^-, M(β ^-),where ^A_Z-1Y is the intermediate nucleus. Note that the μ, ν̅ and β ^+ NMEs are associatedwith τ ^+ p→n transition, while ν and β ^- NMEs are with τ ^-n→ p one.These reaction and decay schemes are shown in Fig. <ref>. Here the DBD NME M(ββ) is given by the sum of NMEs M_i(ββ ) over all relevant states in the intermediate nucleus ^A_Z-1Y. The NMEs M_i(ββ) are associated indirectly with the single β NMEs of M_i(β^+) and M_i(β ^-) via the intermediate state i. Thus information of M(β ^±) is used to help evaluateM(ββ ).Theoretical calculations of NMEs for neutrino nuclear responses are very hard since they are very sensitive to nuclear correlations, nuclear medium effects and nuclear models, as discussed in review articles <cit.>. Thus experimental studies of them are very valuable. Experimental studies of the neutrino nuclear responses for DBDs and astro neutrinos are described in review articles <cit.>. The NMEs M(ν) for astro neutrino and the M(β ^-) for DBD (the left hand side of Fig. <ref>) have been studied extensively by using high energy-resolution (^3He,t) experiments at RCNP <cit.>. On the other hand, there are no appropriate high energy-resolution nuclear probes for the astro anti-neutrino M(ν̅) and DBD M(β^+) (the right hand side of Fig.<ref>).The present MCR provides useful information relevant to M(ν̅) and M(β^+). Unique features of MCR is the large energy and momentum regions of E=0-50 MeV and p= 20-100 MeV/c, which are the regions involved in neutrino-less DBD neutrinos and medium energy supernova neutrinos. In MCR, a well-bound proton in the target nucleus ^A_ZX is shifted up to a vacant neutron shell, and one getsmostly the excited nucleus ^A_Z-1Y^* with the excitation energy E. If particle bound, it decays by emitting γ-rays to the ground state of ^A_Z-1Y. On the other hand, if ^A_Z-1Y^* is unbound, it de-excites by emitting a number (x) of neutrons in case of medium heavy nucleus since proton emission is suppressed much by the Coulomb barrier. Finally one residual nucleus^A-x_Z-1Y is obtained. If it is radioactive, we identify it by measuring characteristic γ-rays of the residual nucleus. The number of theemitted neutrons reflects the excitation energy E, larger x corresponds to the higher E region. Thus one can derive the MCR response, i.e. the strength distribution, as a functionof E, from the mass (A-x) distribution of the residual isotopes, as suggested in 1972 <cit.>, and also in 2001's <cit.>. The MCER and the neutron emission schemes are illustrated in Fig.<ref>.So far nuclear MCR γ-rays were measured to study the nuclear reaction mechanisms <cit.>. Prompt γ-rays from bound states excited by μ capture ^AX(μ,γ)^AY reactions were investigated tostudy β^+ responses for low lying bound states <cit.>. Here it is hard to extract the β^+strengths to individual states because the states are populated not only directly by MCR but also indirectly from so many higher states populated by MCR via γ transitions. In the present work, we focus on gross structure of the MCER strength distribution by measuring the mass distribution of residual isotope (RI) ^A-xY. The RI yield is well obtainedby measuring the yield of γ-rays from ^A-xY and the known γ-branching ratio. A neutron emission code wasdeveloped to link the mass number A-x of the residual nucleus to the initial excitation energy E of ^A_Z-1Y^*.§ MUON IRRADIATION EXPERIMENT The present MCR experiment is made on ^100_42Mo as a typical medium heavy nucleus for astro-physics and DBD interests <cit.>. The target used is a thick ^100Mo with 40 mg/cm^2. Low energy μ^- beams with p = 28 MeV/c from the D-beam line at J-PARC MLF were used to irradiate the target for 7 hours. Here most muons were stopped and captured into the target nucleus to form excited states in ^100_41Nb. Then γ-rays from short-lived Nb isotopes with A=100 and 99 were measured on-line using planar and coaxial-end type HPGe detectors as shown in the left hand side of Fig.<ref>, while γ-rays from long-lived Nb isotopes and other isomers were measured off-line at a separate room by using 2 coaxial Ge detectors as shown in the right hand side of Fig.<ref>. § GAMMA RAYS FROM MCR ON ENRICHED MO AND NB MASS DISTRIBUTION The prominent γ-ray peaks from short-lived and long-lived Nb isotopes were clearly observed as shown in Fig.<ref>. The typical γ-rays from the MCR products are listed in Table<ref>. The 535.0 keV γ-ray from the short-lived ^100Nb and 137 keV γ-ray from theshort-lived^99Nb were measured by the on-line Ge detector setting, while others from the long-lived^99Mo, ^99Tc^m, ^98Nb, ^97Nb, ^96Nb, and ^95Nbwere measured by the off-line Ge detector setting. Among them, the 66 hr ^99Mo is the β ^- decay product from ^99Nb and the 6 hr ^99Tc^m is the isomeric state produced by the β ^- decay from ^99Nb. The number of the Nb RIs ^100-xNb produced by MCR on ^100Mo was evaluated from the observed γ-ray yields corrected for the Ge detector efficiency, the γ-ray branching ratio, and their decays during the muon irradiation and the γ-ray measurement. The obtained RI mass (A-x) distribution is shown in Fig.<ref>. The ^100Nb yield at x=0 is quite small, but jumps up drastically at x=1, and decreases gradually as x increases down to the mass A=95 and x=5. This is similar to the distributions in other target nuclei <cit.>.Let us evaluate the RI mass distribution on the basis of the MCR strength distribution and the statistical neutron emission model. MCR excitations are expressed in terms of the vectorexcitations with the spin transfers of Δ J^π = 0^+, 1^-, 2^+ and the axial-vector ones with Δ J^π = 1^+, 2^-. Among them the 0^+ Fermi and the 1^+ GT excitationsare reduced much since the 0ħω Fermi and GT excitations for the β ^+ and the anti-neutrino responses are blocked by the neutron excess in medium heavy nuclei of thepresent interest. The 1^- excitation with the 1ħω jump may show the giant resonance (GR) like the E1 GR in case of the photon capture reaction (PCR). The vector 2^+ and the axial-vector 2^- excitationsmay show a broad GR-like distributions as the 2ħω and spin dipole GRs. Accordingly, we assume MCR strength distribution of B(μ,E) given by the sum of the two GR strengths of B_1(μ,E) and B_2(μ,E), B(μ,E)=B_1(μ,E) + B_2(μ,E), B_i(μ,E)=B_i(μ)/(E-E_Gi)^2 + (Γ_i /2)^2,where E_Gi and Γ_i with i=1 and 2 are the GR energy and the width for the ith GR, and the constant B_i(μ) is expressed as B_i(μ)=σ_i Γ _i/(2π)withσ_i being the total strength integrated over the excitation energy. Excited states ^A_Z-1Y^* populated by MCR de-excite mostly by emitting neutrons at the pre-equilibrium (PEQ) and equilibrium (EQ) stages <cit.>. If the states are neutronunbound, and do by emitting γ-rays to the ground state if they are bound. Here we ignore proton emission which isprohibited by the Coulomb barrier in case of the medium and heavy nuclei. The energy spectrum of the first neutron E_n^1 is given by <cit.>.S(E_n^1) = k [E_n^1exp (-E_n^1/T_EQ(E) + p E_n^1exp (-E_n^1/T_PEQ(E)) ],where E_n^1 is the first neutron kinetic energy, T_EQ(E) and T_PEQ(E) are the EQ and PEQ nuclear temperatures and p is the fraction of the PEQ neutron emission. The neutron emission from the EQ stage is a kind of neutron evaporation from thermal equilibrium phase. The EQ temperature is expressed as T_EQ(E)=√(E/a) with a being the level density parameter <cit.>. The parameter a is expressed as a=A/8 MeV for the nucleus withmass number A. T_PEQ(E) is given by b× T_EQ(E) with b≈3 for MCR with low-momentum (≈ 50-90 MeV/c) and low-excitation (E≈ 10-50 MeV). The PEQ contribution for the first neutron emission depends on the nuclear size, getting smaller as the nuclear size becomes larger. It is estimated to be around p≈ 0.6 A^-1/3 for the present MCR case by referring to the observed neutron energy spectra <cit.>.The residual nucleus ^A-1_Z-1Y after the first neutron emission de-excites by emitting the second neutron or γ-rays depending on the excitation energy above or below the neutron threshold energy. The ground state of ^A-1_Z-1Y is populated after the γ emission. The2nd neutron n_2 is the EQ evaporation neutron, and then the 3rd neutron is emitted if the residual nucleus after the 2nd neutron emission is neutron-unbound, and so on. Then, one getsfinally the residual isotopes of ^A-x_Z-1Y with x = 0,1,2,3,... depending on the excitation energy E and the number x of the emitted neutrons. Some of them are β-unstable RIs.The neutron number x and the mass number A-x distributions reflect the strength distribution B(μ,E) of the nucleus ^A_Z-1Y^* after MCR, the highly excited states around 30-40 MeV emits 3-4 neutrons while the low excited states around 11-14 MeV emit one neutron as illustrated in Fig.<ref>. In other words, the GR-like strength around 11-14 MeV leads preferentially to population of ^A-1_Z-1Y after 1 neutron emission, and the population of ^A-x_Z-1y decrease as x increases. These features are just what have been observed <cit.>.We compare the observed RI mass distribution for MCR on^100Mo with the calculation based on the strength distribution and the EQ/PEQ neutron emission model. Theobtained RI mass distribution is compared with the observed one in Fig.<ref>. The agreement with the observeddata is quite good where χ ^2 is 0.06. The parameters used for the calculation are E_G1=12 MeV with Γ_1=8 MeV, E_G2=30 MeV with Γ_2=8 MeV, and the cross sectionratio is σ_1/σ_2=1/6. The first GR corresponds to the large population of the mass A-1 with x=1 neutron emission, while the second GR match with the population of the RIs with the mass around A-3 and A-4 with x=3-4 neutron emission.The MCR GR may be compared with the photon capture reaction giant resonance (PCR GR). The energy of 12 MeV is a bit smaller than the PCR GR energy of 14 MeV, but the width of 8 MeV is much larger than the width of 5 MeV for PCR GR. MCR GR consists ofmixed components of J^π=1^-, 1^+, 2^-,... while PCR GR is only one component of J^π=1^-.MCRs in other nuclei have been studies as discussed in the review paper <cit.>. The one neutron emission is dominant in most MCRs, being consistent with the present observation on ^100Mo, and with the GR strength corresponding to the one neutron emission. Neutron energy spectra were observed for MCRs on ^32S, ^40Ca, ^207Pb and ^209Bi. They reported low-energy EQ and medium energy PEQ components from neutron time-of-flight (TOF) measurement. They are reproduced by the EQ/PEQ neutron emission model with the GR1 and GR2 strength distribution given in equation (6). § CONCLUDING REMARKS AND PERSPECTIVES MCR, as the lepton-sector charge exchange reaction (CER) via the weak boson, is shown to be used to study neutrino nuclear responses relevant to DBD andastro-neutrino reactions. Itprovides unique information on β^+ side DBD NMEs and astro anti-neutrino NMEs in the energy and momentum regions of E≈5-50 MeV and p≈95-50 MeV/c. These are just the regions associated with the neutrino-less DBD and supernova neutrinos. Nb RIs produced by MCR on ^100Mo are identified by measuring delayed γ-rays characteristic of the RIs. The Nb mass distribution shows large yield at A=99 after x=1 neutron emission, and decreases gradually as A decreases till A=95 (x=5). The neutron emission is analyzed in terms of the EQ/PEQ neutron emission model. The observed RI distribution reflects the MCR strength distribution with a broad peak at E≈ 11-14 MeV and the small bump at E≈ 30-40 MeV. The broad peak is a kind of μ-capture GR analogous to the E1 photon-capture GR. It is noted that similar GRs are observed for the DBD β^--side responses and the astro neutrino responses <cit.>.The RCNP MuSIC DC muon beamand the J-PARC MLF pulsed muon beam are promising for further studies of neutrino nuclear responses. The lifetime measurement is under progress to study the absolute strength (square of absolute NME). The absolute response, together with the strength distribution, help theories to evaluate the DBD NMEs and astro-neutrino synthesis/interaction NMEs. In other words, nuclear models to be used for DBD NME calculations should reproduce the MCR strength distribution as observed. Experimental studies of MCRs on all DBD nuclei are under progress <cit.>.The EQ/PEQ neutron emission code was developed for the neutron emission following MCR on the present ^100Mowith the large neutron excess and Z=42. One has to include proton emissions as well in medium heavy nuclei with less neutron excess since the proton binding energy gets lower relative to the neutron one, and also in light nuclei where the Coulomb barrier gets lower. The EQ/PEQ code with both neutrons and protons are being developed at UTM and RCNP.The present experiment is made on delayed γ-rays from RIs produced by MCR to get the yields of the RIs since theγ-ray branching ratios are well known. Then we get the gross structure of the strength distribution up toE≈ 60 MeV. The strength (response) to individual states below the neutron threshold energy can be studied by measuring prompt γ-rays from MCR.The population of the ith state is in principle obtained from the difference between the summed yield of all the γ-rays from the ith state and that of all the γ-rays feeding the ith state from higher states. However, in medium heavy nuclei with rather high level density, it is a challenge to measure accurately all the γ-rays from and to the individual states.10 § REFERENCESref1 Ejiri, H. Phys.Rep.C 338 2000 265 and refs. therein ref2 Ejiri, H. Czechoslovak J.Phys. 56 5 2006 459. ref3 Ejiri, H. Progress in Particle and Nuclear Physics 64 2010 249. ref4 Ejiri, H. J.Phys.Soc.Jap. 74 8 2005 2101. ref5 Vergados, J., Ejiri, H. and Simkovic, F. Rep. Prog. Phys. 75 106301 2012 52pp. ref6 Ejiri, H. Proc. EM interactions, Sendai 1972. ref7 Measday, D.F. Phys. Reports 354 2001 243. ref8 Ejiri, H., Hashim, I.H., et al. J. Phys. Soc. of Japan 82 2013 044202.ref9 Evans, H.J. Nuclear Physics A, 207 1972 379-400. ref10 Lucas, G.R., Jr., Martin, P.,Miller, G. H., Welsh, R. E., Jenkins, D. A., Powers, R.J. and Kunselman, A.R. Physical Review C, 74 1973 1678-1686. ref11 Egorov, V. et al. Czech. J. Phys. 56 2006 453. ref12 Ejiri, H. et al. Phys. Rev. Lett. 85 2000 2917 . ref13 Ejiri, H., Engel, J., and Kudomi, N. Phys. Lett. B 530 2002 27 . ref14 Ejiri, H. and de Voigt, M.J.A. Gamma-Ray and Electron Spectroscopy in Nuclear Physics Oxford, NY 1989. ref15 Hashim, I.H. PhD Osaka University 2015. ref16 Capote, R. et al. Nuclear Data Sheets 110 2009 3107-3214. ref17 Raphael,R. and Uberall,H. et al. Physics Letters B, 24 1 1966 15-18. ref18 Hashim, I.H., Ejiri, H. et. al. Proposal for WSS-MuSIC beamtime Osaka University 2016. | http://arxiv.org/abs/1707.08363v2 | {
"authors": [
"I. H. Hashim",
"H. Ejiri",
"T. Shima",
"A. Sato",
"Y. Kuno",
"N. Kawamura",
"S. Miyake",
"K. Ninomiya"
],
"categories": [
"nucl-ex",
"nucl-th"
],
"primary_category": "nucl-ex",
"published": "20170726103905",
"title": "Muon capture reaction on $^{100}Mo$ to study nuclear responses for double beta decays and astro-neutrinos"
} |
Caching Policy for Cache-enabled D2D Communications by Learning User PreferenceBinqiang Chen and Chenyang Yang Binqiang Chen and Chenyang Yang are with the School of Electronics and Information Engineering, Beihang University, Beijing, China, Emails: {chenbq,cyyang}@buaa.edu.cn. Part of this work has been presented in IEEE VTC Spring 2017 <cit.>. January 15, 2018 ======================================================================================================================================================================================================================================================================================================Prior works in designing caching policy do not distinguish content popularity with user preference. In this paper, we illustrate the caching gain by exploiting individual user behavior in sending requests. After showing the connection between the two concepts, we provide a model for synthesizing user preference from content popularity. We then optimize the caching policy with the knowledge of user preference and active level to maximize the offloading probabilityfor cache-enabled device-to-device communications, and develop a low-complexity algorithm to find the solution. In order to learn user preference, we model the user request behavior resorting to probabilistic latent semantic analysis, and learn the model parameters by expectation maximization algorithm. By analyzing a Movielens dataset, we find that the user preferences are less similar, and the active level and topic preference of each user change slowly over time. Based on this observation, we introduce a prior knowledge based learning algorithm for user preference, which can shorten the learning time. Simulation results show remarkable performance gain of the caching policy with user preference over existing policy with content popularity, both with realistic dataset and synthetic data validated by the real dataset. User preference, Content popularity, Caching policy, D2D, Machine learning, data analysis. § INTRODUCTION Caching at the wireless edge has become a trend for content delivery <cit.>, which can improve network throughput and energy efficiency as well as user experience dramatically <cit.>.Owing to the small storage size of each node at the wireless edge, saybase station (BS) or user device, caching in a proactive manner is critical to achieve the performance gain, where future user demand statistics is exploited <cit.>. In wireless networks, the contents can be precached at each BS<cit.> or even directly pushed to user device <cit.>. By caching at BS, backhaul traffic can be offloaded and backhaul cost can be reduced. By caching at user, wireless traffic can be further offloaded from peak time to off-peak time <cit.>. To boost the cache hit rate by precaching contents at each user that has very limited cache size, cache-enabled device-to-device (D2D) communications and coded-multicast strategy are proposed <cit.>.To heap the proactive caching gain, various caching policies have been optimized with diverse objectives for different networks. Most existing works assume known content popularity, defined as the probability distribution that every content in a catalog is requested by all users. For example, the policies for caching at BSs were optimized to minimize average download delay in <cit.> and to maximize coverage probability in <cit.>. The policies for caching at users were optimized to maximize the offloading gain of cache-enabled D2D networks in <cit.>. Coded caching policy was optimized to maximize the average fractional offloaded traffic and average ergodic rate of small-cell networks in <cit.>. In these works, every user is assumed to request files according to content popularity. However, in practice a user actually sends requests according to its own preference, which may not be identical to content popularity. Noticing such fact, caching policies at the user groups with different group popularity were optimizedin <cit.> to minimize the average delay of cache-enabled D2D communications.To implement above-mentioned proactive caching policies, content popularity needs to be predicted <cit.>. Popularity prediction has been investigated for diverse applications such as advertisement, where content popularity is defined as the accumulated number of requests every content in a catalog received or the request arrival rate for every content. Numerous methods have been proposed<cit.>. By using these prediction methods, the content popularity defined with probability in<cit.> can be obtained as a ratio of the number of requests for each file to the number of all requests.In cellular networks, the number of users in a cell is much less than that in a region covered by a content server, and a mobile user may send requests to more than one BS. Since popularity depends on the group of users who send requests, the local popularity in a cell may differ from the global popularity observed at a server. Designing proactive caching policy at wireless edge with predicted global popularity leads to low cache hit rate<cit.>.Optimizing caching policy for a BS should be based on the local popularity, which is formed by the users in a cell. In <cit.>, local popularity was predicted as the number of requests received for each file at a small BS divided by the observation time, i.e., the request arrival rate. Then, the popularity is adjusted with a perturbation term, which is learned by applying multi-arm bandit algorithm, and finally the predicted popularity with perturbation term is used for caching policy optimization. The prediction is based on the cumulative growth method<cit.> and under the assumption that only the requests for already cached files can be observed, hence the learning algorithm is slow.In <cit.>, the content popularity was predicted with a real dataset measured from cellular network. By converting the number of requests received at each BS into rating, the local popularity was predicted by a widely-used collaborative filtering technique, matrix factorization <cit.>, with which the files with largest ratings are cached at the BS.§.§ Motivation and Contributions Since caching at wireless edge is motivated by the observation that the majority of requests are initiated for a minority of popular contents, a large body of priori works assume that all users send their requests according to content popularity.The following facts, which are widely recognized in the communities studying recommendation problem and analyzing user behavior with real data, are largely overlooked in the community of studying caching at wireless edge: (i) as a demand statistic of multiple users, content popularity can not reflect the personal preference of each individual user <cit.>, (ii) only a small portion of users are active in creating traffic <cit.>. In fact, existing works do not differentiate content popularity from user preference. In practice, user preferences are heterogeneous although they may exhibit similarity to a certain extent. The caching policy designed under unrealistic assumptions inevitably yields performance loss. In this paper, we investigate the gain of optimizing caching policy by learning user preference and active level over content popularity. To this end, we take cache-enabled D2D communications as an example system and offloading probability as an example objective. Because there are different definitions in the domains of computer science and wireless communications, we first define user preference and active level as well as content popularity to be used in this paper, and provide a probabilistic model to synthesize user preference from content popularity by introducing similarity among user preferences. We then formulate an optimization problem with known user preference and active level to maximize offloading probability. Since the problem is NP-hard, a local optimal algorithm is proposed to reduce the complexity, which achieves at least 1/2 optimality. In order to learn user preference and active level, we model user request behavior resorting to probabilistic latent semantic analysis (pLSA) originally proposed for natural language processing <cit.>, whose model parameters can be learned using approximate inference methods such as expectation maximization (EM) <cit.>. With the help of pLSA model to decompose the user behavior into different components and based on the observation from analyzing a real dataset that active level and topic preference change slowly over time, we provide a prior knowledge based algorithm, whichcan quickly learn user preference.The major contributions of this paper are summarized as follows: * We illustrate the caching gain of exploiting user preference and active level by optimizing an caching policy for D2D communications, and predict the behavior of each individual user by estimating model parameters of pLSA. We introduce a prior knowledge based algorithm to learn user preference, which shows the potential of transfer learning. * We characterize the connection between content popularity and user preference, provide a probabilistic model for synthesizing user preference from content popularity, and validate the method by the MovieLens 1M dataset <cit.>. We analyze the relation between file catalog size and the number of users, the statistics of active level and topic preference of each user, and the user preference by the real dataset, which are critical for the caching gain. * Simulation results with both synthesized data and MovieLens dataset show remarkable performance gain of the caching policy with user preference over that with local popularity, no matter the user demands are assumed known or learned from historical requests.§.§ Related WorksConsidering that the tastes of different users are not identical, caching policies were optimized to minimize the average delay of cache-enabled D2D communications in <cit.> and to maximize the cache hit rate of mobile social networks in <cit.>, by assuming user preferences as Zipf distributions with different ranks. However, both works assume that all users have the same active level, do not validate the assumption for user preference, and do not show the gain over caching with popularity. Until now, there exists no method to synthesize user preference validated by real dataset, and the gain of caching with user preference is unknown.There are few works that consider the relation between content popularity and user preference. In <cit.>, local popularity is computed as a weighted average of preferences for the users associated with each BS, where the weight is the number of requests sent by each user and the user preference was assumed as uniform distribution. Then, the most popular files at each BS were cached. Differing from this early work, we characterize the connection between the collective and the individual user request behavior in a probabilistic framework, and illustrate the gain of exploiting user preference over popularity by optimizing a caching policy. These priori works assume known user preference <cit.>. To facilitate proactive caching, user preference needs to be predicted, which is a key task in recommendation problem. Collaborative filtering is the most commonly used technique to predict user preference<cit.>, which can be mainly classified into memory based method including user-based and item-based approaches, and model based method that is based on models with parameters estimated from historical records<cit.>. Typical model based methods employ matrix factorization, latent Dirichlet allocation <cit.>, and pLSA <cit.> as models <cit.>. For recommendation problem, user preference is defined as the rating that a user gives for a file, such as 0 ∼ 5 or simply “like” and “dislike”. Most collaborative filtering methods predict the ratings for unrated contents of each user, which however cannot be used in optimization for wireless caching. To optimize caching policy in wireless edge, where various metrics are in statistical sense <cit.>, user preference needs to be defined in probabilistic form, but there is no widely-accepted approach to map the rating into probability. In this paper, we resort to pLSA to model and predict user preference, which is originally developed for classification in automatic indexing <cit.> and then is applied to predict ratings in <cit.>.The rest of the paper is organized as follows. Section <ref> provides the relation between content popularity and user preference, and a model to synthesize user preference. Section <ref> optimizes the caching policy with known user preference. Section <ref> presents the learning algorithms. Section <ref> analyzes the statistics of user demands from and validate the synthetic model by a MovieLens dataset. Section <ref> provides simulation results. Section <ref> concludes the paper. § CONTENT POPULARITY AND USER PREFERENCE In this section, we first define content popularity, user preference and active level to be used in the following, show their connection, and then provide a probabilistic model with a free parameter to synthesize user preference.§.§ Definition and RelationshipConsider a content library ℱ = { f_1, f_2,..., f_F} consisting of F files that K users in an areamay request, where f_f denotes the fth file. Content popularity is defined as the probability distribution that each file in the library is requested by all users, denoted asp = [p_1,p_2,...,p_F ], where p_f≜ P( f_f) is the probability that f_f is requested, ∑_f=1^F p_f = 1, p_f∈ [0,1], and 1 ≤ f ≤ F. If the area only consists of a single cellular cell, then p is called local popularity. User preference is defined as the conditional probability distribution that a user requests a file given that the user sends a request, denoted as q_k= [q_1|k,q_2|k,...,q_F|k ] for the kth user (denoted as u_k)), where q_f|k≜ P( f_f| u_k) is the conditional probability that the kth user requests the fth file when the user sends a file request, ∑_f=1^F q_f|k= 1, q_f|k∈ [0,1], 1 ≤ f ≤ F and 1 ≤ k ≤ K. We use matrix Q= (q_f|k)^K × F to denote the preferences of all users, where (q_f|k)^K × F represents a matrix with K rows and F columns and q_f|k as the element in the kth row and fth column.Active level is defined as the probability that a request is sent by a user, denoted as w_k ≜ P( u_k) for the kth user, which reflects how active the user is, where ∑_k=1^K w_k = 1 and w_k ∈ [0,1]. Then, the vector w = [w_1,w_2,...,w_K] denotes the active level distribution of the K users.Content popularity p reflects the collective request behavior of a group of users, while q_k and w_k characterize the individual request behavior of the kth user. To show their connection, we consider a K × F user-content matrix <cit.>, where each of its element n_k,f represents the number of requests sent by u_k for f_f. Denote N = ∑_k=1^K∑_f=1^F n_k,f as the overall number of requests sent by all the K users for all the F files, n_f = ∑_k=1^K n_k,f as the total number of requests sent by all users for f_f (i.e., the sum of all elements in the fth column), and n_k = ∑_f=1^F n_k,f as the total number of requests sent by u_k for all files (i.e., the sum of all elements in the kth row). Considering that n_f/N, n_k/N and n_k,f/n_k are respectively multinomial distributions with F, K, and F parameters, it is not hard to show that they are respectively the maximum likelihood estimate of p_f, w_k and q_f|k. From their definitions, we have∑_k=1^Kn_k/N_w_kn_k,f/n_k_q_f|k= ∑_k=1^Kn_k,f/N = n_f/N_p_f,and hence each element of p can be expressed as the average of user preferences weighted by their active levels,p_f = ∑_k=1^K w_k q_f|k,1 ≤ f ≤ F.In practice, users have different tastes, i.e., q_k ≠ q_m, and hence q_f|k≠ p_f, despite that users may have similar preferences, say for popular contents. Besides, not all users send requests with equal probability. We can use cosine similarity to reflect the similarity of preferences between two users, which is frequently used in collaborative filtering<cit.> and defined assim( q_k, q_m) = ∑_f=1^F q_f|m q_f|k/√(∑_f=1^F q_f|m^2 ∑_f=1^F q_f|k^2).To show the similarity among K users, we can define average cosine similarity as𝔼_k,m [ sim( q_k, q_m) ]= 2/K(K-1)∑_k,m∑_f=1^F q_f|k q_f|m/√(∑_f=1^F q_f|k^2 ∑_f=1^F q_f|m^2). §.§ Modeling and Synthesizing User Preference Content popularity can be modeled as a Zipf distribution according to the analyses for many real datasets <cit.>. The probability that the fth file is requested by all users isp_f = f^-β/∑_j=1^F j^-β, 1 ≤ f ≤ F,where the files in the library are indexed in descending order of popularity, and the content popularity is more skewed with larger value of β. User preference model obtained from real datasets is unavailable in the literature so far. Inspired by the method in <cit.> to synthesize local popularity of a cell from that of a core network, we represent users and files in a shared one-dimensional latent space, which bears the same spirit as latent factor model widely applied in collaborative filtering <cit.>. To connect withcontent popularity, we model user preference from the following generative process: * u_k is associated with a feature value X_k, which is randomly selected from [0,1]. * f_f is associated with a feature value Y_f, which is again chosen uniformly from [0,1]. * The joint probability that the fth file is requested by the kth user is given byP( u_k,f_f) = w_kq_f|k = p_f g(X_k,Y_f)/∑_k'=1^K g(X_k',Y_f),and then u_k's active level is w_k = ∑_ f_f ∈ℱP( u_k ,f_f ) and its preference is q_f|k = P( u_k ,f_f )/w_k, where g(X_k,Y_f) ∈ [0,1] is a kernel function used to control the correlation between the kth user and the fth file. When g(X_k,Y_f) = 0, the fth file will never be requested by the kth user. When g(X_k,Y_f) = 1, the file is a preferred file of the user. Intuitively, the value of X_k can be interpreted as the likelihood that the kth user prefers a topic, and the value of Y_f can be interpreted as the likelihood that the fth file belongs to a topic.Various kernel function can be applied, e.g., Gaussian, logarithmic and power kernels. To control the average similarity among the user preferences by introducing a parameter α in kernel function, we choose power kernel with expression g(X_k,Y_f) = (1-|X_k-Y_f|)^(1/α^3-1)∈ [0,1] (0<α≤ 1), which exhibits a linear relation between 𝔼_k,m [ sim( q_k, q_m) ] and α in a wide range.Remark 1: When α = 1, g(X_k,Y_f) = 1 for ∀ k,f. Then, we can see from (<ref>) that all user preferences are identical and equal to the content popularity, and then from(<ref>) we can obtain 𝔼_k,m[sim ( q_k, q_m) ]=1. When α→ 0, g(X_k,Y_f) → 0 for X_k ≠ Y_f, and g(X_k,Y_f) = 1 only for X_k = Y_f. Because X_k and Y_f are uniformly chosen from [0,1], i.e., ℙ(X_k = Y_f) → 0, we have g(X_k,Y_f)g(X_k',Y_f) → 0, k ≠ k'. Then, according to (<ref>) and (<ref>), 𝔼_k,m[ sim ( q_k, q_m) ] → 0, i.e., no user has the same preference. When α is small, g(X_k,Y_f) is low, which means that the number of interested files of u_k is small. Because each user randomly chooses feature values, the interested file sets among different users are less overlapped, and hence the average similarity among the users is low.This probabilistic model is appropriate for generating synthetic data of user preference and active level, which differs from pLSA that can be used for predicting the individual behavior. This model will be validated later by a real dataset and the parameter will be fitted. § CACHING POLICY OPTIMIZATION: AN ILLUSTRATION In this section, we illustrate how to optimize the caching policy with known request behavior of each individual user. For comparison, we also provide the corresponding caching policy optimization problem with known content popularity, whose solution reflects the existing policy in literature. To focus on the performance gain brought by distinguishing user preference from content popularity, we consider a simple objective: the offloading gain of D2D communications. To reduce the time complexity in finding the solution, we provide a local optimal algorithm. §.§ System Model Multiple BSs in the area are connected to core network via backhaul to serve the K uniformly distributed users, which constitutes a set of users 𝒰 = { u_1, u_2,..., u_K} that request the files in content library ℱ. Assume that each file is with same file size, but the results are applicable for general case with different sizes <cit.> by dividing each file into chunks of approximately equal size.[We can also formulate another optimization problem with different file sizes, which can be shown as a knapsack problem.] Each single-antenna user has a local cache to store M files, and can act as a helper to share files via D2D link. To provide high ratetransmission with low energy cost at each user device, we consider a user-centric D2D communication protocol as in <cit.>. A helper can serve as a D2D transmitter and send its cached files to a user only if their distance is smaller than a collaboration distancer_ c, which reflects the coverage of the helper. Each BS is aware of the cached files at and the locations of the users, and coordinates the D2D communications. Proactive caching consists of content delivery phase and content placement phase.In content delivery phase, each user requests files according to its own preference. If a user can find its requested file in local cache (i.e., fetching locally), it directly retrieves the file with zero delay. If not, the user sends the request to a BS. If the BS finds that the file is cached in the local caches of helpers adjacent to the user, it informs the request to the closest helper, and then a D2D link is established between the user and the closest helper (i.e., fetching via D2D link). Otherwise, the BS fetches the file via backhaul to serve the user. For simplicity, both fetching locally and via D2D link are called fetching via D2D links in the sequel.Denote the file requests matrix after a period as N=(n_k,f)^K × F (i.e., the user-content matrix), where n_k,f≥ 0 is the number of requests from u_k ∈𝒰 to f_f ∈ℱ.Assume that a central processor (CP) can record the requests history of users. To predict user preference, the CP needs to be deployed in the mobile core, such that most requests of users can be recorded. To determine where to deploy CP, one needs to consider both coverage area and computational cost. In content placement phase, the CP learns the user preferences Q and active levels w from the requests history N, and then optimizes the caching policy for users and informs the cached files of the users to the BSs.We consider deterministic caching policy,[We do not consider probabilistic caching policy, which is designed under the assumption that a group of nodes share the same caching distribution <cit.>, and hence is not appropriate for a system with heterogeneous user preferences. ] denoted as a vector c_k = [c_k,1,c_k,2,...,c_k,F] for the kth user, where c_k,f=1 iff_f is cached at u_k, c_k,f=0 otherwise, and ∑_f=1^F c_k,f≤ M.Denote the caching policy for all users as C= (c_k,f)^K × F. After being informed about the files to be cached at the users in its cell, a BS fetches the files from the core network and refreshes the caches of the users during the off-peak time, say every several hours, noticing that traffic load varies on the timescale of hours as measured by real cellular data <cit.>. Due to user preference similarity, some users may have common contents to precacheas shown later, which can be pre-downloaded by the BS via multicast to reduce wireless traffic. §.§ Caching Policy Optimization with Individual Request BehaviorWe use offloading probability to reflect the offloading gain introduced by cache-enabled D2D communications, defined as the probability that a user can fetch the requested file via D2D links, which represent the average ratio of requests able to be offloaded.When optimizing caching policy in the content placement phase, it is hard to know where a mobile user will be located in the content delivery phase. Therefore, it is hard to know when and how long the users will contact. Fortunately, data analysis shows that users always periodically reappear at the same location with high probability <cit.>. Consequently, it is reasonable to assume that the contact probability is known a priori <cit.>. Let A= (a_i,j)^K × K represent the contact probability among users, where a_i,j∈ [0,1] is the probability that the distance between the ith user and the jth user is less than r_c. When all users do not move, a_i,j=0 or 1.In D2D communications, adjacent helpers may have overlapped coverage. Since different helpers need to serve different groups of users, which depend not only on r_c but also on the cached files at the adjacent helpers, the “local popularity” observed at a helper differs from that observed at another helper and relies on the caching policy. As a result, the caching policy can not be designed based on the “local popularity”.Denote p^ d_k,f( A,C) as the probability that the kth user can fetch the fth file via D2D links given contact probability A and caching policy C. The complementary probability of p^ d_k,f( A,C) is the probability that the fth file is not cached at any users in proximity to the kth user, which can be derived as ∏_m=1^K (1- a_k,mc_m,f). Then, we can obtain the offloading probability asp_ off ( Q,w,A,C) = ∑_k=1^K w_k∑_f=1^F q_f|k p^ d_k,f( A,C) =∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_m=1^K (1- a_k,mc_m,f) ).With known user preference and active level, the caching policy can be optimized to maximize the offloading probability by solving the following problem, P1:max_c_m,fp_ off (Q,w,A,C)s.t.∑_f=1^F c_m,f≤ M, c_m,f∈{0,1}, 1 ≤ m ≤ K, 1 ≤ f≤ F. Remark 2: If all users are with equal active level and equal preference, then (<ref>)becomesp_ off ( Q,w,A,C)= 1/K∑_f=1^F p_f∑_k=1^K p^ d_k,f( A,C) ≜ p^ pop_ off ( p,A,C). Remark 3: If the collaboration distance r_c →∞, then a_k,m=1, and (<ref>) becomesp_ off ( Q,w,A,C) = ∑_f=1^F(1-∏_m=1^K (1- c_m,f) ) ∑_k=1^K w_k q_f|k = ∑_f=1^F(1-∏_m=1^K (1- c_m,f) ) p_f.It is easy to show that p_ off ( Q,w,A,C)=p^ pop_ off ( p,A,C) in this extreme case where D2D links can be established between any two users in the area even with heterogeneous user preferences.If the assumptions in the two remarks hold, then the “local popularity” observed at every helper will be identical, which is equal to the content popularity of the area with the K users. In practice, the assumptions are not true, hence the caching gain from exploiting user preference differs from exploiting content popularity.With known content popularity, the caching policy is optimized by maximizing p^ pop_ off ( p,A,C) in (<ref>) under constraint (<ref>), called problem P2,[The solution of P2 slightly differs from <cit.>, wherethe future user location is exactly known in <cit.> and completely unknown in<cit.> when optimizing caching policy, where the contact probability is knownin P2.] which is actually a special case of P1. By setting the contact probability a_i,j as 1 or 0 (i.e., all users are static), we can obtain a special case of problem P2, which has the same structure as aNP-hard problem formulated in <cit.>. Since P2 is a special case of P1, problem P1 is NP-hard. As a consequence, it is impossible to find its global optimal solution within polynomial time. By using similar way of proofin <cit.>, it is not hard to prove that P1 is equivalent to maximizing a submodular function over matroid constraints. Thus, we can resort to greedy algorithm, which is commonly used to provide a solution achieving at least 1/2 of the optimal value for such type of problem <cit.>.[The best algorithm with polynomial time complexity for such problem can achieve (1-1/e) optimality guarantee, which is based on continuous greedy process and pipage rounding techniques <cit.>. However, when K=100 and F=3000 in the considered setting as detailed later, its complexity is O((FK)^8) = O(6.5×10^43), which is too complex for our problem.] The greedy algorithm starts with zero elements for the caching matrix, i.e., C= (0)^K × F. In each step, the value of one element in C is changed from zero to one with the maximal incremental caching gain defined asv_ C(m,f) = p_ off (Q,w,A,C|_c_m,f=1) - p_ off (Q,w,A,C)(a)=∑_k=1^K w_k q_f|k( p^ d_k,f( A, C|_c_m,f=1)-p^ d_k,f( A,C)),where (a) follows by substituting (<ref>), C is the caching matrix at previous step, and C|_c_m,f=1 is the matrix by letting c_m,f=1 in C. The algorithm is summarized in Algorithm <ref>.The loops in step <ref> of Algorithm <ref> takeKM iterations, because there are totally KM files that are possible to be cached at all users. The step <ref> for finding the element in C that introduces the highest incremental caching gain takes at most KF iterations. For each time of computing v_ C(m,f) in (<ref>), the time complexity is O(K^2), and thus computing all v_ C(m,f) is O(K^3F). Hence the total time complexity for Algorithm <ref> is O(KM(KF+K^3F)) = O(K^2FM(K^2+1)), which is high especially when the numbers of users K and files F are large. §.§ A Low Complexity Algorithm with 1/2 Optimality GuaranteeSince the greedy algorithm is with high time complexity, finding a low-complexity algorithm is worthwhile for practice use. In what follows, wepropose an alternating optimization algorithm, which improves the offloading gain at every iteration and converges to a local optimal solution.To be specific, we fix the caching policy at users c_m(m≠ k', 1 ≤ m ≤ K) and optimize c_k'. Then, from problem P1 we obtain the optimization problem with respect to c_k' asP1':max_c_k',f f_ off( c_k') = ∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_m=1,m≠ k'^K (1- a_k,mc_m,f) (1- a_k,k'c_k',f) )s.t.∑_f=1^F c_k',f≤ M, c_k',f∈{0,1}, 1 ≤ f ≤ F. P1' can be solved with polynomial time complexity O(F(K^2+M)). See Appendix <ref>. Based on the proof of Proposition <ref>, we propose an algorithm to iteratively solve problem P1' by changing k' from 1 to K until convergence. The algorithm starts with a given initial value of C.In every iteration, by fixing c_m(m ≠ k', 1 ≤ m ≤ K), it respectively computes the offloading gain introduced by caching the fth file at the k'th userb_k',f = ∑_k=1^K w_kq_f|k a_k,k'∏_m=1,m≠ k'^K (1- a_k,mc_m,f), 1 ≤ f≤ F, 1 ≤ k' ≤ K.Then, the algorithm finds the file indices with the maximal M values of b_k',f to constitute a set ℐ_k', and obtain c^*_k' asc^*_k',f ={ 1 , f ∈ℐ_k'0 , f ∉ℐ_k'. .The detailed algorithm is presented in Algorithm <ref>. The loops in step <ref> takeK iterations. Step <ref> is with complexity O(F(K^2+M)) according to Proposition <ref>. Hence the total time complexity of Algorithm <ref> is O(t_A2KF(K^2+M)), where t_A2 is the number of iterations for step <ref>. Algorithm <ref> monotonically increases the objective function of problem P1 and finally converges to achieve at least 1/2 optimality. See Appendix <ref>. It is noteworthy that Algorithm <ref> and Algorithm <ref> can also solve P2 by lettingq_f|k = p_f, ∀ k,f in Q. Solutions for P1 and P2 obtained with Algorithm <ref> are called S1-A1 and S2-A1, and solutions using Algorithm <ref> for P1 and P2 are called S1-A2 and S2-A2, respectively.§ LEARNING USER PREFERENCE AND ACTIVE LEVEL In this section, we first use pLSA to model content request behavior of an individual user. We then learn the model parameters by maximizing likelihood function, either without the pLSA model for comparison or with the model using the EM algorithm, which is efficient for ML parameter estimation with latent variables <cit.>. Finally, we present a prior knowledge based algorithm to learn user preference. §.§ Modeling Individual User Behavior in Requesting Contents To characterize the request behavior of a user, pLSA associates each request with a topic, which may be unobservable but can be intuitively interpreted as comedy, adventure, etc.By introducing latent topic set 𝒵 = { z_1, z_2,..., z_Z} with cardinality|𝒵|=Z, pLSA associates each topic z_j ∈𝒵 with each possible user request, i.e., u_k ∈𝒰 requestsf_f ∈ℱ.Specifically, the request of each user can be modeled as the following steps with three model parameters: * A request is sent by u_k with probability P( u_k) (i.e., active level), * u_k chooses a topic z_j with probability P( z_j| u_k) (i.e., topic preference, ∑_j=1^Z P( z_j| u_k) = 1), * u_k requestsf_f in topic z_j with probability P( f_f| z_j), ∑_f=1^F P( f_f | z_j) = 1, where conditional independence assumption is used. In particular, conditioned on a request being sent byu_k who chooses topic z_j, u_k chooses f_f with probability P( f_f| z_j, u_k ) = P( f_f| z_j), i.e., P( f_f | u_k ) = ∑_ z_j ∈𝒵P( f_f | z_j) P( z_j| u_k). In other words, no matter which user sends a request and selects topicz_j, the user will request f_f with probability P( f_f| z_j). Then, the joint probability that u_k requests f_fcan be expressed asP( u_k,f_f )= P( u_k)P( f_f | u_k )= P( u_k ) ∑_ z_j ∈𝒵P( f_f | z_j) P( z_j| u_k).§.§ Learning Individual User Behavior in Requesting Contents According to maximal likelihood (ML) principle, we can learn P( u_k ), P( f_f | z_j) and P( z_j | u_k) with requests history n_k , f by maximizing the following log-likelihood function <cit.>ℒ = ∑_ilog P( u_i_ u, f_i_ f)= ∑_ u_k ∈𝒰∑_ f_f ∈ℱ n_k , flog P( u_k ,f_f )_(a) =∑_ u_k ∈𝒰∑_ f_f ∈ℱn_k , flogP( u_k ) ∑_ z_j ∈𝒵P( f_f | z_j) P( z_j| u_k)_(b),where the ith sample corresponding to the event that the i_ uth user requests the i_ fth file, and in (b) the pLSA model is applied. §.§.§ ML algorithm without pLSA modelBy maximizing the log likelihood function in (a) of (<ref>) without the pLSA model, it is not hard to obtain thatP̂( u_k ,f_f )= n_k,f/∑_k'=1^K∑_f=1^F n_k', f. Then, the active level and user preference can be learned as ŵ_k = P̂( u_k) = ∑_f=1^FP̂( u_k ,f_f ) and q̂_f|k= P̂( u_k ,f_f )/P̂( u_k) = P̂( u_k ,f_f )/∑_f=1^FP̂( u_k ,f_f ), respectively. This algorithm is actually a simple frequency-count prediction, which can serve as a baseline for learning active level and user preference. Remark 4: If we directly predict w_k and q_f|k using (<ref>), the number of parameters to estimate is KF. By using the pLSA as in (b) of (<ref>), the number of parameters is reduced from KF to K+KZ+ZF = Z(K+F)+K, whereK parameters are for learning active level, KZ parameters are for topic preference, and ZF parameters are for P( f_f| z_j). With less number of parameters to estimate, a learning algorithm can converge more quickly.§.§.§ ML algorithm with pLSA modelTo maximize the log-likelihood function in (b) of (<ref>), we first rewrite the function asℒ = ∑_ u_k ∈𝒰 n_k logP( u_k )_(a) +∑_ u_k ∈𝒰∑_ f_f ∈ℱ n_k , flog∑_ z_j ∈𝒵P( f_f | z_j) P( z_j| u_k)_(b),where n_k =∑_ f_f ∈ℱ n_k , f. It is not hard to see that the terms in (a) and (b) can be independently maximized. The active levelof u_k can be learned by maximizing term (a) in (<ref>) asŵ_k = P̂( u_k ) = n_k/∑_k'=1^K∑_f=1^F n_k' , f,which is the same as that obtained from (<ref>). The other two model parameters P( f_f | z_j) andP( z_j| u_k) can be learned by maximizing term (b) in (<ref>) using the EM algorithm as follows <cit.>.Starting from randomly generated initial values for the model parameters P( z_j| u_k) and P( f_f | z_j),1 ≤ j ≤ Z, 1 ≤ f ≤ F and 1 ≤ k ≤ K, the EM algorithm alternates two steps: expectation (E) step and maximization (M) step. In the E-step, the posterior probabilities are computed for latent variable z_j with current estimation of the parameters asP̂( z_j| u_k, f_f)= P̂( z_j| u_k)P̂( f_f | z_j)/∑_ z_j'∈𝒵P̂( z_j'| u_k) P̂( f_f | z_j'),which is the probability that f_f requested byu_k belongs to topic z_j. In the M-step, given P̂( z_j| u_k, f_f) computed by previous E-step, the parameters are updatedas P̂( f_f| z_j) =∑_ u_k ∈𝒰n_k , fP̂( z_j| u_k, f_f)/∑_ u_k ∈𝒰∑_ f_f'∈ℱ n_k , f'P̂( z_j| u_k, f_f') ,andP̂( z_j| u_k) = ∑_ f_f ∈ℱ n_k , fP̂( z_j| u_k, f_f) /n_k . By alternating (<ref>) with (<ref>), the EM algorithm converges to a local maximum of log-likelihood function. Then, the preference of the kth user for the fth file can be learned asq̂_f|k= P̂( f_f| u_k) = ∑_ z_j ∈𝒵P̂( f_f | z_j) P̂( z_j| u_k).§.§.§ Prior Knowledge Based Algorithm to Learn User PreferenceVideo files in real world website always have topic information, e.g., movies are labeled with comedy, drama and so on.Intuitively, the topic preference and active level of a user change slowly over time, and hence can be regarded as invariant. This will be validated later by real dataset. Thanks to the pLSA model, such intuition naturally yields a prior knowledge based algorithm to learn user preference by exploiting the active level and topic preference of a user learned previously during a much longer time than learning user preference, with the help of the topic information. This algorithm can be regarded as a parameter-transfer approach <cit.>. While the active level P( u_k) and topic preference P( z_j| u_k) can never be learned perfectly, we assume that they are known in order to show the potential of such transfer learning.Then, the user preference can be learned by only estimating P( f_f| z_j), which can be obtained similarly as in (<ref>),P̂( f_f| z_j) = { ∑_ u_k ∈𝒰n_k , fP̂( z_j| u_k, f_f)/∑_ u_k ∈𝒰∑_ f_f'∈ℱ n_k , f'P̂( z_j| u_k, f_f') ,f_f ∈ℱ_j0 ,f_f ∉ℱ_j , .where ℱ_j is the set of files associated with topic z_j (1≤ j ≤ Z), which is available on the video website. For instance, the movie Forrest Gump is associated with topics comedy, romance and war on the MovieLens.The detailedalgorithmis presented in Algorithm <ref>. Step 2 takesKFZ times computation of posterior probability by (<ref>), where each computation is with complexity O(Z-1) and thus totally at most O(KFZ(Z-1)). Step 3 computes (<ref>) with ZF times, each is with complexity O(K(F+1)). It is not hard to see that step 4 is with complexity O(KFZ). Hence the total time complexity for Algorithm <ref> is O(t_A3KFZ(Z+F+1)), where t_A3 is the number of iterations for step <ref>. § USER REQUEST BEHAVIOR ANALYSIS WITH MOVIELENS DATASETThe gain from caching highly depends on the user behavior in requesting contents, both collectively and individually. In this section, we first use a real dataset to analyze the connection between file catalog size and number of users in a region, as well as the active level, topic preference of each user and user preference, and validate the intuition in Section <ref>. Then, we validate the user preference model provided in Section II.§.§ Statistical Results of User Demands We use the MovieLens 1M Dataset <cit.> to reflect the request behavior, where MovieLens is a website that recommends movies for its users operated by GroupLens lab at the University of Minnesota. This dataset contains 1000209 ratings for 3952 movies provided by 6026 usersfrom the year of 2000 to 2003. Each sample of the dataset consists of user identity (ID), movie ID, ratingand timestamp. Because users typically give ratings only after watching, we translatethe rating record into the request record, i.e., when a user gives rating for a movie, we set the movie as requested by once. Except for the ratings, MovieLens also provides topic information of movies. Every movie is associated with one, two or more topics from 18 topics, which include action, adventure, animation, children's, comedy, etc. genre and are denoted as z_1, ..., z_18. For instance, Forrest Gump is associated with topics comedy (z_5), romance (z_14) and war (z_17). From the topic information provided by MovieLens, we can see that if the fth movie is not associated with the jth topic, users who select jth topic will not request the fth file, i.e., we can set P( f_f| z_j) = 0in (<ref>). To analyze temporal evolution of user behavior, we sort all the 3952 movies according to their released date in ascendant orderand then divide them into two subsets ℱ_1 and ℱ_2, where the file request matrices on ℱ_1 and ℱ_2 are N_1 ∈ℝ^6040× 1976 and N_2 ∈ℝ^6040× 1976, respectively. N_1 can reflect user behavior on previously released file subset ℱ_1, and N_2 can reflect user behavior on subsequently released file subset ℱ_2.Specifically, we analyze the following statistical results: * File catalog size: To reflect the randomness of the users in sending requests, it is the average number of files requested by a given number of randomly chosen users, which is obtained from N = [N_1N_2 ] and the average is taken over users. * Active level: P_1( u_k) and P_2( u_k) are computed using (<ref>) with N_1 and N_2, respectively. * Topic preference: Denote the topic preference of the kth user estimated on subsets ℱ_1 and ℱ_2 as p_1(𝒵| u_k) = [P_1(z_1| u_k),...,P_1(z_Z| u_k)] and p_2(𝒵| u_k) = [P_2(z_1| u_k),...,P_2(z_Z| u_k)], respectively. P_1(z_j| u_k), P_2(z_j| u_k) are computed using (<ref>) by EM algorithm with N_1 and N_2, respectively. To reflect the temporal dynamic of topic preference for the kth user, we use the metric of cosine similarity in (<ref>) to evaluate the similarity level as sim (p_1(𝒵| u_k),p_2(𝒵| u_k)). * User preference:q_f|k=∑_ z_j ∈𝒵P( f_f | z_j) P( z_j| u_k) is obtained by EM algorithm on N_1. The result obtained from N_2 or N is similar, and hence is omitted for conciseness.In Fig. <ref>(a), we show the relation between file catalog size and the number of users obtained from dataset (with legend “MovieLens”) and the fitted curves. The curve with legend “Log"fits the function f(x)=alog( bx), and the curve with legend “Power" is withf(x) =ax^ b +c. To evaluate the goodness of fit, we use the coefficient of determination (also called R-square) in linear regression, i.e., R^2 = 1 - ∑_i=1^S (y_i - f(x_i))^2/∑_i=1^S (y_i - y̅)^2, where S is the number of samples, (x_i,y_i) is the ith sample, and y̅ = ∑_i=1^S y_i/S <cit.>.R^2 ≤ 1,and the large value of R^2 indicates good fitting result.The parameters a, b and c for each function and R^2 are listed in the legends. We can see that the catalog size first increases quickly and then slowly, where “Power" function fits better than “Log" function. When the number of users is small (e.g., in a small cell), the catalog size is small, which implies that the cache hit ratio could be high with limited cache size. However, with limited number of requests due to a few associated users, the popularity is hard to predict rapidly at the small BS. When the number of users is large (e.g., in a macro cell), the catalog size increases slowly, and both fitted curves are close to the measured catalog size. In <cit.>, the authors suggest to use logarithm function to compute the catalog size without validation. Here, the result shows that“Log" is reasonable when the number of users in the considered area is large, say K ≥ 100, at least for MovieLens dataset. In Fig. <ref>(b), we show the active levels of users, where the user indices are ranked in descending order according to P_1( u_k). We can see that the distribution of active levels is skewed, which indicates that majority requests are generated by a small number of users. Besides, the distribution of active levels from the two subsets of dataare similar, where the cosine similarity is 0.87. This validates that the active level of a user changes slightly over time.We also show the corresponding fitted distributions, where “Weibull" is with function f(x) =abx^ b-1e^- ax^ b, “Exponential" is with function f(x) =ae^- bx, and “Zipf"is with function f(x)=ax^-β (shown only for the most active 1000 users). We can see that the curve with the real data is linear on a log-log scale for active users, but the tail (after the 1000th user) decreases quickly. The truncate tail may come from the rating behavior of users for watched movies on MovieLens website.Some users rarely give ratings, and some users do not continuously give ratings. As a result, the observed active levels of these users are very low. From the values of R^2, we can find that “Weibull" is the best fitted distribution. Nonetheless, the distribution of the most active 1000 users is very close to Zipf.In Fig. <ref>(a), we show the topic preferences of the 1st, 10th and 100th users obtained from ℱ_1, i.e., p_1(𝒵| u_1), p_1(𝒵| u_10) and p_1(𝒵| u_100). The results obtained from ℱ_2 are similar and are not shown. The labels of x-axis are ranked in descending order according to p_1(𝒵| u_1). The topic preferences of the 10th and 100th users with re-ranked x-axis according to p_1(𝒵| u_10) and p_1(𝒵| u_100) are also provided in the inner-figures.We can see that topic preference of each user is skewed, which indicates that each user hasstrong preferences towards specific topics. In fact, the topic preferences of all users in the dataset are skewed, which is not shown for consciousness. We can also see that topic preferences of different users differ. For example, the most favorite topic is comedy for the 1st and 100th user and drama for the 10th user. In Fig. <ref>(b), we show the topic preference of the 1st user and the fitted distributions in a log-log coordinate. We can see that the best fitted distribution is a Zipf distribution with parameter β = 1.05. We have also fitted distributions of topic preferences for other users, but the curves are not provided. We observe that the best fitted distributions differ for users, where Zipf distribution is the best of 1425 users, Weibull distribution is the best for 1899 users, and Exponential distribution is the best for the remaining 2702 users (but the difference in R^2 from Weibull distribution for these users is negligible). For the users whose best fitted distributions are Zipf distributions, the parameters of β differ, which approximately follow a uniform distribution in [0.5, 3]. Yet for the most favorite several topics, Zipf distribution is always the best. In Fig. <ref>(a), we show the empirical cumulative distribution function (CDF) and probability density function (PDF) of the cosine similarity between topic preferences over time of all users. We can see that 60% of all users have cosine similarity larger than 0.8, and almost 90% users among the top 1/3 active users have cosine similarity larger than 0.8 (i.e., their topic preferences change slowly in the three years). Considering that the statistical results for active users with more requests are with high confidence level, this result indicates that topic preferences can be approximated as invariant over time.[In recommendation problem, it has been shown that user preference varies over time due to the dynamic of file catalog and the user's exploration for new items <cit.>. However, the topic preference variation has never been analyzed in the literature.] This validates the intuition in Section <ref>.In Fig. <ref>(b), we show user preference of the 1st user and the fitted distributions. The user preference for the top 100 favorite files is close to Zipf distribution (a straight line in the log-logcoordinate), and the preference for less favorite files has a truncated tail. This is because a user almost does not request the files belonging to its unfavorable topics. We have also fitted the preferences of other users (but are not shown). We find that Weibull is the best fitted distribution for all users, but the parameters and the skewness of curvesdiffer. The user preferences for the top favorable 100 files of each user can be fitted with Zipf distribution, but the parameters of β differ in a range of [0.2, 0.8]. In summary, the preferences of different users differ in the favorite file set, skewness, and ranking. This is not consistent with the model that all user preferences are Zipf distributions with same parameter but with different ranks as assumed in <cit.>. Besides, we observe that the average cosine similarity of preferences among different users on dataset N_1 is as low as 𝔼_k,m [ sim( q_k, q_m) ] ≈ 0.4.[We also analyze a real video dataset of Youku in a university campus. The result shows that 𝔼_k,m [ sim( q_k, q_m) ] ≈ 0.28.]This is mainly because the interested file sets of users are less overlapped, recalling that the topic preferences of users differ. §.§ Validating Synthetic User Preference Model Now, we validate the user preference model by comparing the results obtained from data synthesized by the generative process in Section II and those from the MovieLens dataset.In Fig. <ref>(a), we first show the impact of parameter α in the kernel function. The inner-figure indicates that the synthetic user preference model can capture different levels of similarity among user preferences by adjusting α, while the Zip parameter β has negligible impact on the average cosine similarity. This seems counter-intuitive, since a more skewed popularity distribution seems to imply highly correlated user preferences. However, such an intuition comes from the implicit assumption that the users send their requests with equal probability (i.e., with identical active level), which is not true in reality. From the figure we can observe that even when β=1, α can be as small as 0.1. This is because few users are very active in sending file requests and have skewed user preference, who have large impact on content popularity according to (<ref>). We can see that the distributions of user active level are skewed, which agree well with the results in Fig. <ref>(b) obtained from the MovieLens dataset. In Fig. <ref>(b), we show the topic preference of the 1st, 10th and 100th users. The labels of x-axis are ranked according to p(𝒵| u_1), as in Fig. <ref>(a). We can see that the topic preferences of the users are skewed, and the topic preferences of the three users are with differentdistributions, which are consistent with the results in Fig. <ref>(a) obtained from the MovieLens dataset. § SIMULATION RESULTS In this section, we demonstrate the caching gain by exploiting user preference over content popularity, either perfect or predicted. Simulation results are obtained from data generated by the synthetic model in section <ref>, which can provide ground truth of the request behavior for evaluation, and from the MovieLens dataset, which can validate the gain from real data.We consider a square area with side length 500 m, where K=100 users areuniformly located. The collaboration distance r_c = 30 m. The file catalog size F=3000 (obtained from Fig. <ref>(a) for 100 users), and each user is willing to cache M=5 files (i.e., 1.67 ‰ of all files). α=0.36 in the kernel function of the synthetic model, which corresponds to the average cosine similarity 0.4 of the MovieLens dataset (obtained from Fig. 5(a)). The parameter of Zipf distribution is β =0.6, which is slightly smaller for a small area than that is observed at the Web proxyas reported in <cit.>.We divide time into two-hour periods, each consisting of a peak time and an off-peak time.[Using other values as the period does not affect the learning performance of user preferences. Yet the period can not be too short, since a frequent content placement brings extra traffic load. Numerical results under the considered setup show large opportunity of using multicast for precaching. In particular, (i) only 76 files need to be pre-downloaded to users via unicast where the other 424 files can be placed to users via multicast, (ii) 20% of the 500 files should be placed at more than 10 users, and 50% of files should be placed at more than 5 users. Besides, not all the 500 files need to be changed in each update.] The cached files at each user are updated in off-peak time. This setup is used in the sequel unless otherwise specified.§.§ Impact of Key Parameters In the sequel, we analyze the impact of user mobility, user preference similarity parameter α, collaboration distance r_c, cache size M, and Zipf parameter β on the offloading probability.We consider a widely used mobilitymodel, random walk model, where a user moves from its current location to a new location by randomly choosing a direction and speed to travel <cit.>. To compute the contact probability matrix, we consider the two-hour period where each user moves 100 seconds before changing direction and speed. The users are initially uniformly distributed, and the speed and direction of each user are uniformly chosen from [0, v_max] m/s and [0, 2π], respectively.By computing the duration that the kth and the mth user can establish D2D links, t^d_k,m, in the period of T_p = 2 hours, we can obtain the contact probability a_k,m = t^d_k,m/T_p. By increasing v_max, users may move with higher speed, and when v_max = 0, all users keep fixed. In this subsection, both user preference and content popularity are perfect. In Fig. <ref>, we show the impact of user mobility. A1 and A2 in the legend respectively represent the greedy algorithm (i.e., Algorithm 1) and local optimal algorithm (i.e., Algorithm 2), which achieve almost the same offloading probability. The offloading probabilities decrease slightly with the growth of v_max, as explained as follows. Owing to the mobility model, the average number of users that a user can establish D2D links with at any time does not change with v_max. Then, the total effective cache size seen by the user does not change with v_max. On the other hand, every user can contact with more users in the period with higher v_max. Then, the caching policy needs to be optimized by considering the preferences of more users, which reduces the cache hit ratio due to heterogeneous user preferences. Since the impact of mobility is not significant, we only consider v_max = 0 in the sequel. To obtain the results of A2 in Fig. <ref>, three iterations of step 1 (i.e., t_A2 = 3) is necessary for convergence. According to analysis in Section <ref>, the time complexity for A1 and A2 are respectively O(K^2FM(K^2+1)) and O(3KF(K^2+M)), and A2 will be KM(K^2+1)/3(K^2+M)≈ 167 times faster than A1 when K=100 and M=5.Since the proposed local optimal algorithm can achieve the same performance and is faster than the greedy algorithm, we only use A2 to obtain the caching policy in the following.In Fig. <ref>(a), we show the impact of α and r_ c. We can see that the offloading gain of S1 over S2 is high when α is small. This suggests that optimizing caching policy according to user preferences is critical when the userpreferences are less correlated. As expected, when α→ 1, the performance of the two policies coincide. The offloading gain is high for large collaboration distance, but the gain by usingS1reduces as indicated in Remark 3. This is because with the growth of r_ c, the number of users whose preferences a helper should consider for optimizing caching policy increases (when r_ c→∞, the number of users equals to K).In Fig. <ref>(b), we show the impact of β and M. As expected, with the growth of the value of M or β, the offloading probabilities increase for both S1 and S2. This is because with a given value of α, the preference of every user becomes more skewed when β increases. §.§ Offloading Gain with Learned User PreferenceIn what follows, we demonstrate the performance gain of the proposed caching policy by exploiting user preferences over that with content popularity, and the gain provided by learning with the pLSA model and the priori knowledge. To this end, we compare the following schemes: * “S1-perfect”: The proposed caching policy with perfect user preference and active level, which is the solution of problem P1.* “S2-perfect”: The existing caching policy optimized with perfect content popularity, which is the solution of problem P2 (slightly different from the policies in <cit.>). * “S1-EM”: The proposed caching policy with ŵ and Q̂ learned by the EM algorithm.* “S2-EM”: The existing caching policy with learned local popularity of the K users, which is computed with (<ref>) from the learned user preference by EM algorithm.* “S1-prior”: The proposed caching policy with Q̂ learned by Algorithm <ref>.* “S2-prior”: Theexisting caching policy with learned local popularity of the K users, which is computed from the learned user preference by Algorithm <ref> as p̂_f= ∑_ u_k ∈𝒰P̂( u_k ,f_f ).* “S1-baseline”: The proposed caching policy with learned user preference, which is obtained by the ML algorithm without pLSA model.* “S2-baseline”: Theexisting caching policy with learned local popularity of the K users, which is obtained by using the traditional frequency-count popularity prediction method in<cit.>. Such a popularity learning method is the same as the method used in <cit.>.In Fig. <ref>, we show the offloading probability achieved by these schemes during the learning procedure with the synthetic data. To compare with the results to be obtained with realistic dataset in the sequel, we set the x-axis as the accumulated number of requests. When the request arrival rate of the users in the area is 0.04 requests per second, which reflects the high traffic load scenario for files with 30 MBytes size (typical size of the YouTube videos) in <cit.>, the accumulated 40, 80 and 120 requests correspond to 22.8, 55.6, and 83.3 hours, respectively. It is shown that using pLSA and even the priori information do not help accelerate convergence of local popularity, because the simple frequency-count method already converges rapidly. Compared to the proposed caching policy with learned user preference (S1-EM, S1-prior and S1-baseline), S2 with learned local popularity (S2-EM, S2-prior and S2-baseline) converge to S2 with perfectcontent popularity (S2-perfect) more quickly. This is because the number of requests for each file from each user is much less than that from all the users in the area. Nonetheless, the proposed caching policy with learned user preference can quickly achieve higher offloading probability than S2 with learned (and even perfect) content popularity. The proposed caching policy with pLSA (both S1-EM and S1-prior) is superior to the baseline (S1-baseline), especially when the cache size at each user M is large.This is because some unpopular files will be cached with large M. For the unpopular files, the number of accumulated requests is less and user preference learning is more difficult. Besides, we can see that by exploiting prior knowledge of user active level and topic preference, S1-prior converges much faster thanS1-EM.In Fig. <ref>, we show the offloading gain with the MovieLens dataset. Because S2-EM and S2-prior perform closely to S2-baseline, we only simulate S2-baseline here. We randomly choose 100 users from the dataset (which includes both active and inactive users) and the most popular 3000 files. The timestamps of user requests are shuffled as in <cit.> to ensure the training set has the same user demand statistics as the test set.Compared with Fig. <ref>(a), we can see that the offloading probabilities for all methods on the realistic dataset are less than those with synthetic data. This is because a user requests each movie at most once in the realistic dataset translated from rating data, while a user may request a file more than one time in the synthetic data. In practice, a user may request a file more than once, e.g., the file is a favorite song of the user or an educational video.Nonetheless, we can see that the proposed caching policy with predicted user preference still achieves much higher offloading gain than existing scheme. Because the prior knowledge of topic preferences is learned on realistic dataset in Fig. <ref> rather than perfectly known as in Fig. <ref>(a), the performance gain of S1-prior over S1-EM is lower here, which however is still remarkable. Besides, we can see that S1-baseline always performs the worst, because user preference for unvisited files is hard to predict by using the ML algorithm without pLSA model when each user only requests a file at most once. However, we can still predict user preference for the unvisited files with both S2-EM and S2-prior, owing to the pLSA model.§ CONCLUSIONS In this paper, we demonstrated the caching gain by exploiting learned individual user behavior in sending request. We first showed the connection between user preference and content popularity, and provided a probabilistic model to synthesize user preference from content popularity. We then formulated an optimization problem with given user preference and active level to maximize the offloading probabilityfor cache-enabled D2D communications. Since the problem is NP-hard, a low-complexity algorithm achieving at least 1/2 optimality was proposed to solve the problem. Next, we modeled the user request behavior by pLSA, based on which the EM algorithm was used to learn the user preference and active level. We used a Movielens dataset to analyze several kind of user behavior in requesting contents, and validated the synthetical model. We find that: (i) when the number of users K in an area is large, the file catalog size is a logarithm function of K, (ii) the active level of the most active users can be modeled as Zipf distribution, (iii) the preferences for the most favorable files of each user can be modeled as Zipf distribution, but with different skewness parameters and different file sets, (iv) the user preferences are less similar, and (v) the active level and topic preference of each user change slowly over time, say in the time scale of year. Based on the 5th observation from the real dataset, we introduced a prior knowledge based algorithm to exploit the active level and topic preference previously learned, which shows the potential of transfer learning. Simulation results showed that using pLSA can quickly learn the individual user behavior, and the prior knowledge based algorithm converges even faster. Compared to existing caching policy using content popularity, the performance can be remarkably improved by the caching policy exploiting user preferences, both on the synthetic data with parameters fitted from real dataset and on the MovieLens dataset.equationsection§ PROOF OF PROPOSITION <REF> The objective function of problem P1' can be further derived asf_ off( c_k' ) = ∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_m=1,m≠ k'^K (1- a_k,mc_m,f) (1- a_k,k'c_k',f) )= 1- ∑_f=1^F∑_k=1^K w_kq_f|k∏_m=1,m≠ k'^K (1- a_k,mc_m,f)_(a)+ ∑_f=1^F c_k',f(∑_k=1^K w_kq_f|k a_k,k'∏_m=1,m≠ k'^K (1- a_k,mc_m,f))_(b),where both terms in (a) and (b) are not related to c_k',f. Then, solving the problem in (<ref>) is equivalent to solving the following problemP1'max_c_k',f∑_f=1^F c_k',f(∑_k=1^K w_k q_f|k a_k,k'∏_m=1,m≠ k'^K (1- a_k,mc_m,f))(a)=∑_f=1^F c_k',f b_k',fs.t.∑_f=1^F c_k',f≤ M, c_k',f∈{0,1}, 1 ≤ f ≤ F,where (a) is obtained by letting b_k',f = ∑_k=1^K w_kq_f|k a_k,k'∏_m=1,m≠ k'^K (1- a_k,mc_m,f). By finding file indices of the maximal M values of b_k',f (1≤ f ≤ F) to constitute the set ℐ_k', it is not hard to show that the optimal caching policy c^*_k',f can be obtained as (<ref>). To obtain c^*_k',f, we need to compute b_k',f with time complexity O(K^2F) and then choose themaximal M values of b_k',f with complexity O(FM). Finally, we can prove that the optimal solution of problem (<ref>) can be obtained with complexity O(K^2F+FM) = O(F(K^2+M)).§ PROOF OF PROPOSITION <REF>Denote the caching policy at the k'th user after the (t-1)th iteration as c^(t-1)_k'. In the tth iteration, the offloading probability before step <ref> of Algorithm <ref> is f_ off( c^(t-1)_k' ) as in (<ref>). After that step, c^(t)_k' is computed for the k'th user and the corresponding offloading probability is f_ off( c^(t)_k' ). By subtracting f_ off( c^(t-1)_k' ) from f_ off( c^(t)_k' ), we can obtain f_ off( c^(t)_k' ) - f_ off( c^(t-1)_k' ) = ∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_m=1,m≠ k'^K (1- a_k,mc_m,f) (1- a_k,k'c^(t)_k',f) ) - ∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_m=1,m≠ k'^K (1- a_k,mc_m,f) (1- a_k,k'c^(t-1)_k',f) )(a)=∑_f=1^F c^(t)_k',f b_k',f - ∑_f=1^F c^(t-1)_k',f b_k',f(b)=(max_c_k',f∑_f=1^F c_k',f b_k',f )- ∑_f=1^F c^(t-1)_k',f b_k',f≥ 0,where (a) is obtained by substituting (<ref>) and (<ref>), and (b) is obtained from (<ref>). Thus, the offloading gain is monotonically improved until convergence.To prove the optimality guarantee of the local optimal algorithm, we first convert the offloading probability into a function of a set instead of a matrix (i.e., C). Denoting f_f^k as an action that caching the fth file at the kth user. Recall that c_k,f=1 represents the kth user caching the fth file. Then, the caching policy for the kth user, c_k = [c_k,1,c_k,2,...,c_k,F], can be re-expressed as a set 𝒞_k = { f^k_f|c_k,f=1}, i.e., caching which files at the kth user. Let 𝒞 = {𝒞_1, 𝒞_2, ..., 𝒞_K }, then problem P1 is equivalent to the following problem,max_𝒞 f_(𝒞) =∑_k=1^K w_k ∑_f=1^F q_f|k(1-∏_ f_f^m∈𝒞 (1- a_k,m) ) s.t. |𝒞_k| ≤ M, 1 ≤ k ≤ K. By defining a set 𝒮 = { f_1^1,f_2^1, ... ,f_F^1, ...,f_1^K, f_2^K,..., f_F^K}, we can see that 𝒞⊆𝒮 and f_(𝒞): 2^𝒮→ R is a discrete set function on subsets of 𝒮. Let 𝒜,ℬ⊆𝒮, 𝒜⊆ℬ, and f_k'^f'∈𝒮∖ℬ.Denote the global optimal caching policy as 𝒞^* = {𝒞^*_1,𝒞^*_2,...,𝒞^*_K}, a local optimal caching policy obtained by Algorithm <ref> as 𝒞^L = {𝒞^L_1,𝒞^L_2,...,𝒞^L_K}, and caching policy at users except the kth user as 𝒞̅^̅L̅_̅k̅ = {𝒞^L_1,𝒞^L_2,.,𝒞^L_k-1,𝒞^L_k+1,..,𝒞^L_K}. Then, we can obtainf(𝒞^*) - f(𝒞^L) (a)≤ f(𝒞^*∪𝒞^L ) - f(𝒞^L) ≜f_𝒞^L(𝒞^*) (b)≤∑_k=1^Kf_𝒞^L(𝒞^*_k) (c)≤∑_k=1^Kf_𝒞̅^̅L̅_̅k̅(𝒞^*_k) (d)≤∑_k=1^Kf_𝒞̅^̅L̅_̅k̅(𝒞^L_k)(e)≤f(𝒞^L_1)+∑_k=2^Kf_∪_i=1^k-1𝒞^L_i(𝒞^L_k) = f(𝒞^L),where (a) is obtained because offloading probability is a monotone increasing function, i.e., f(𝒞^*) ≤ f(𝒞^*∪𝒞^L), (b) is obtained by the property that for set 𝒜,ℬ,𝒞⊆𝒮, we have f_𝒜(ℬ∪𝒞) ≤ f_𝒜(ℬ) + f_𝒜(𝒞), (c) and (e) are obtained by the property that for set 𝒞⊆𝒜⊆𝒮 and ℬ⊆𝒮, f_𝒜(ℬ ) ≤ f_𝒞(ℬ), and (d) is obtained because for any caching policy at the kth user denoted as 𝒞^a_k, we have f_𝒞̅^̅L̅_̅k̅(𝒞^L_k) - f_𝒞̅^̅L̅_̅k̅(𝒞^a_k) = f(𝒞^L_k ∪𝒞̅^̅L̅_̅k̅) - f(𝒞^a_k ∪𝒞̅^̅L̅_̅k̅) ≥ 0 considering 𝒞^L is the local optimum of Algorithm <ref>. Thus, we have f(𝒞^L) ≥1/2 f(𝒞^*), and Proposition <ref> follows. IEEEtran | http://arxiv.org/abs/1707.08409v2 | {
"authors": [
"Binqiang Chen",
"Chenyang Yang"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170726123913",
"title": "Caching Policy for Cache-enabled D2D Communications by Learning User Preference"
} |
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku,Kyoto 606-8502, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, JapaniTHEMS Program, RIKEN, Wako 351-0198, JapanA sanity check rules out certain types of obviously false results, but does not catch every possible error.After reviewing such asanity check for NN bound states with the Lüscher's finite volume formula<cit.>, we give further evidences for the operator dependence of plateaux, a symptom of the fake plateau problem, against the claim in <cit.>.We then present our critical commentson <cit.> by NPLQCD: (i) Operator dependences of plateaux in NPL2013<cit.> exist with the P-values of 4–5%.(ii) The volume independence of plateaux in NPL2013 does not prove their correctness. (iii) Effective range expansion (ERE) fits in NPL2013 violate the physical pole condition. (iv) Ref. <cit.> is partly based on new data and analysis different from the original ones<cit.>. (v) A new ERE in Refs. <cit.> does not satisfy the Lüscher's finite volume formula. Sanity check for NN bound states in lattice QCD with Lüscher's finite volume formula– ExposingSymptoms ofFake Plateaux – Sinya Aoki1,2Speaker, [email protected] This work is supported in part by the Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific Research (Nos. JP15K17667, JP16H03978), by a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using Post “K" Computer,and by Joint Institute for Computational Fundamental Science (JICFuS). We thank other members of HAL QCD collaboration for useful discussion. Takumi Doi3,4 TakumiIritani3 December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTIONIn the previous publications<cit.>, we pointed out that the binding energy of the two nucleon (NN) system at heavy pion masses extracted from the temporal correlation function at t≃ 1 fmin lattice QCD is unreliable due to contaminations of excited scattering states. Tab. <ref> summarizesour sanity check of NN data in recent literatures,all of which employ the plateau fitting at t≃ 1 fm. The sanity check is performed to rule out certain classes of obviously false results, but the correctness is not guaranteed even if the data pass the test. The fact that none in the table passes the test brings a serious doubt on existence of NN bound states in ^1S_0 or ^3S_1 channels claimed on the basis of these data. More sophisticated method than the simple plateau fitting, such as the variational method<cit.>, is mandatory for the reliable study ofNN systems. Our concerns mentioned in Refs. <cit.>, however,seem to be misunderstood by authors in Refs. <cit.>. Therefore we will first emphasize again the serious problem of the plateau fitting for the NN system and then provide critical comments on Refs. <cit.> in the remaining part of this report.§ FAKE PLATEAU PROBLEM-0.6cmPlateaux of the NN effective energy shift at early t≃ 1 fmcannot be trusted in general, since the fake plateaux can easily appear due to contaminations from elastic excited states. To demonstrate this, we consider the mock-up data for the effective energy shift<cit.>, described byΔ E^ eff_NN(t) = -1/alog(R(t+a)/R(t)),R(t) = e^-Δ E_NN t( 1 + b_1 e^-δ E_ el. t + c_0e^-δ E_ inel. t).with the correct energy shift Δ E_NN, where we take δ E_ el. = 50 MeV,which is the typical lowest elastic excitation energy at L≃ 4 fm (and m_N ≃ 2 GeV), and δ E_ inel.= 500 MeV, which is the mass of the heavy pion. In this situation, we naively expectthat the ground state saturation requires t ≥ 1/δ E_ el.≃ 4 fm, which however is too large to have good signals for NN systems. The strategy used in the previous studiesis tuning the source operators to satisfy b_1≃ 0, so as to achieve the ground state saturation at much smaller t before noises dominate. In practice, the source/sink operator is tunedto havea plateau in Δ E^ eff_NN(t) at much smaller t. Fig. <ref> (Left) shows three examples of Δ E^ eff_NN(t) with b_1=0 (an optimally tuned operator) and b_1=± 0.1keeping c_0=0.01,where random fluctuations are assigned to R(t) whose magnitude increases exponentially in t. Indeed the fake plateaux appear at t≃ 1 fm. If no legend of data is given as in the figure, we can not tell which are fake from the t dependence. This example clearly demonstrates that the strategy mentioned above does NOT work at all: Even if the plateau appears in Δ E^ eff_NN(t), we cannot tellwhether it is fake or not. Hereafter we will use the words,“the fake plateau problem", to remind readers of this fundamental problem. This is the serious issuefor the current NN data and nothing more is needed to doubt the validity of data in Refs. <cit.>. Therefore conclusions in these references are not valid anymore, andmore reliable methods such as the variational method<cit.> are required to obtain the definite conclusion on the NN system at heavy pion masses. § OPERATOR DEPENDENCEAs manifestations of“the fake plateau problem",there exist several symptoms, one of which is the operator dependence of the plateau. Fig. <ref> (Right) shows Δ E_NN^ eff(t) obtained from wall and smeared sources,which indicates plateaux at different values<cit.>, so that at least one of them must be fake.Authors of Ref. <cit.> claimed thatthe plateau in Δ E_NN^ eff(t) from the wall source isfake, caused bya cancellation between the NN effective energy and the nucleon effective mass, both of which seem to reach plateaux slower than Δ E_NN^ eff(t). This argument, however, does not properly address the issue in Ref. <cit.>as explained below.First of all,whetherthe plateau from the wall source is correct or not does not logically affect whether the plateau from the smeared source is correct or not. Moreoverthere is no reason for the smeared source works better than the wall source for the NN system, since the smeared source is known to suppressinelastic contributions to nucleon mass butto couple strongly to elastic scattering states with non-zero momenta.In fact, as shown in Fig. <ref> (Left), where the effective energy of ΞΞ (^1S_0) is plotted for several different sink operators,[ΞΞ has smaller statistical errors than NN. See Appendix A in Ref. <cit.> for details.] the smeared source produces several different plateaux, (almost) all of which should be fake. This clearly shows that the plateau method does not have a predictive power at all.In this conference<cit.>,Berkowitz and his collaborators pointed outa possibility that inelastic excited state contributions in the nucleon effective mass may be largely cancelled by those in the NN effective energy, so that the plateau may appear atearlier t inΔ E_NN^ eff(t) as long as contaminations from elastic excited states in the NN effective energy are negligible. This possibility, however, should be verified by more reliable methods such as the variational method<cit.>, since the simple plateau method cannot provide any information on this possibility.In Ref. <cit.>, authors also claimed that the wall source and the smeared source agree at larger t within large errors, as for data at t≥ 16 in the case of Fig. <ref> (Left). If so, however, one has to fit data at t≥ 16, leading to a different central value with much larger errors than before (solid black lines).Moreover, even if the agreement between two sources is observed, it does not guarantee that the results are reliable, since t is sill much smaller than the inverse of the typical excitation energy.It is pointless to speculate a possible scenario for results from the plateau method to be correct without clear evidences. Instead one has to show that they are indeed correct without relying on the plateau method. The issue is “the fake plateau problem", notthe wall source against the smeared source unlike the argument in Ref. <cit.>. § COMMENT ON REF. <CIT.> Authors of Ref. <cit.> claimed that the assessment in Ref. <cit.> to NPL2013<cit.> is inadequate and NPL2013 has passed all tests in Tab. <ref>. In this section, we investigate their claims one by one and conclude that they are invalid orirrelevant to address the fake plateau problem. §.§ Operator dependenceAuthors of Ref. <cit.> compared the ground state energy of the NN system in ^1S_0 and ^3S_1 channels among NPL2013<cit.>, CalLat2017<cit.> and NPL2017<cit.>, all of which employed the same gauge ensembles, and concluded that the source operator dependenceis not established, since the χ^2 test on the assumption that all results are same gives P-values (significances) ranging from 8% to 65%. However, comparing all data at onceis not an optimal way to investigate the operator dependence. Instead, it is better to perform the one-to-one comparison for a maximal detection of the dependence. Such comparisons are made for NPL2013 and CalLat2017 in Tab. <ref>, where the operator dependence of the ground state energy is clearly seen between the center of mass and the moving systems in ^1S_0 of NPL2013, between non-displaced and displaced source operators of CalLat2017, and between NPL2013 and CalLat2017 (displaced), while that of the 1st excited state is confirmed between NPL2013 and CalLat2017for ^3S_1 at L=32.Furthermore, an absence of the operator dependence for the limited sets of operators, even if it were confirmed, does not mean the absence of “the fake plateau problem". One has to prove separately that the plateau is not fake.It is also claimed in Refs. <cit.>, without any numerical evidences, that the result from the displaced source operator in CalLat2017 does not corresponds to the ground state energy, so that no operator dependence exists in CalLat2017.[The argument in Ref. <cit.> that the displaced operator does not couple to the ground state does not have solid ground, since it assumes the absence of the fake plateau problem without evidence. ]This interpretation, however, brings other problems. If the plateau of the displaced operator correspond to the excited scattering state, the negative energy shift of this state disagrees with the positive value from other sources with non-zero momentum. Thus it creates another huge operator dependence. If this plateau corresponds to the second shallow bound state, on the other hand,one has to explain why NPL2013 missed this state, andthe effective range expansion (ERE) in NPL2013 and CalLat2017 should be drastically modified as in Fig. <ref> (Right), to satisfy the physical pole condition<cit.>.As we confirmed above, NPL2013/CalLat2017 data show the source dependence, which strongly suggests thattheir extractions of energiesfrom plateaux suffer from “the fake plateau problem".Refs. <cit.>also claimed that data at the largest volume (L=48) is important to determine the binding energy in the infinite volume extrapolation. Although this statement itself is correct, the large volume data have nothing to do with the validity of data at smaller volumes discussed in Ref. <cit.>. §.§ Volume scaling of energiesShowing that NN energies in NPL2013 are almost volume independent,authors of Ref. <cit.> claimed that this volume independence rules out the possibility that the plateau structures are caused by the cancelations among excited states. Contrary to this claim, however, in general there is no logical connection between the volume independence and the correctness of the plateau. Indeed, while energies in YKU2012<cit.> are almost volume independent, the corresponding ERE is very singular as shown in Ref. <cit.> and the source/sink operator dependences are observed as seen in the previous section. This means that the volume independence does not guarantee the correctness of data, contrary to the claim of Ref. <cit.>. §.§ Physical pole conditionAuthors of Ref. <cit.> argued that ERE fits in NPL2013 satisfy the physical pole condition . d/d k^2[kδ_0(k) - (-√(-k^2))].|_k^2= -k^2_b < 0,where k^2_b correspond to thebound state pole,by showing that the slope of -√(-k^2) at k^2=-k^2_b is larger than the slope of ERE, called the effective range. This argument, however, is insufficient to check the physical pole condition, which implies that all intersection points between the ERE and the pole condition satisfy eq. (<ref>). Fig. <ref> shows kδ_0(k)/m_π as a function of (k/m_π)^2, together with the NLO ERE line (as well as its error band),taken directly from figures in NPL2013 and replotted in Ref. <cit.>. As the red line in both channels appears below the black solid line at (k/m_π)^2 ≃ 0 and (k/m_π)^2 ≃ -0.13, the ERE line must intersect with the pole condition twice,so that the deeper one inevitably violates the physical pole condition.We therefore put `No' on sanity check (iii) for NN(^1S_0) and NN(^3S_1) in the original assessment <cit.>. [Since the light red error band goes above the pole conditionat (k/m_π)^2 ≃ -0.13 for NN(^3S_1), we will change `No' to `?' in the revised version of Ref. <cit.>.]Ref. <cit.> states that “The incorrect conclusions reached in HAL potentially arise from attempting to reanalyze highly correlated data, such as the encountered in the phase-shift analysis, without making use of those correlations".[ After the lattice conference, the version 2 of <cit.> appeared without this sentence.] However, this statement is not correct, since the ERE lines in the figures are NOT our reanalysis but are taken from the figure inNPL2013, as already mentioned. § SOME OTHER CONCERNS ON REF. <CIT.>§.§ Inadequate data handllingIn Ref. <cit.>, authors stated that they made new ERE analyses to their data and claimed that data passed the sanity check. As already emphasized, our sanity check in Ref. <cit.> was applied to ERE fits in NPL2013 but not to new analysis, so that the statement in Ref. <cit.> that “ the results of Refs.[6.7][Refs.[6,7] correspond to NPL2013.] pass this check, in stark contrast to the claims of HAL"is inadequate.Moreover, we found thatvalues of the energy shift were modified from NPL2013, as summarized in Tab. <ref>, where data in NPL2013 are compared with data read off from figures in Ref. <cit.>. At this conference, we pointed outthat criticisms based on data modified from the original ones cannot be taken as face value.[ Although this data modification is mentioned in its version 2, there still remains a similar statementthat “the results of Refs.[12.13] pass this check, contrary to the claims in HAL", where Refs.[12.13] mean NPL2013.] §.§ Incorrect fitting-0.3cmThe ERE for ^1S_0 in Refs. <cit.> does not intersect with the line of the Lüscher formula at L=24, as shown in Fig. <ref> (Left).Since the χ^2 cannot be defined in this case, it is not clear how authors of Ref. <cit.> obtained the line and its errors in their ERE analysis. Since the vertical error is derived from the horizontal one, one needs to fit lattice data by the ERE with the constraint due to the Lüscher's finite volume formula, as demonstrated in Fig. <ref> (Right): Once the ERE line is fixed by given fit parameters,an intersection between the ERE and the Lüscher's formula on each L is determined (black square), and then (k/m_π)^2 of the intersection is compared with that of lattice data (blue circle) to calculate the corresponding χ^2.As the correct χ^2 cannot be defined without intersection, it seems that Ref. <cit.> did not take into account this constraint correctly. §.§ Lattice spacing and pion massNPLQCD collaboration employed the SU(3) flavor symmetric QCD configurations for various publications<cit.>. While the lattice spacing a in Refs. <cit.> (a≃ 0.15 fm) is different from a in Refs. <cit.> (a≃ 0.12 fm),the pion mass m_π in these references is always the same (m_π∼ 806 MeV). Although the lattice spacing can depend on the physical input (r_0, light hadron masses, heavy meson spectra, etc.), the pion mass (and dimensionfull quantities such as the energy shift) must bechanged accordingly.It is not clear as well how these values are obtained and how the chiral and/or continuum extrapolation are made.§ SUMMARYAs we discussed in this report, our sanity check in Tab. <ref> remains valid, contrary to the claims in Refs. <cit.>, which contain either invalid statements or inadequate data handling or both. | http://arxiv.org/abs/1707.08800v1 | {
"authors": [
"Sinya Aoki",
"Takumi Doi",
"Takumi Iritani"
],
"categories": [
"hep-lat",
"nucl-th"
],
"primary_category": "hep-lat",
"published": "20170727095044",
"title": "Sanity check for $NN$ bound states in lattice QCD with Lüscher's finite volume formula -- Exposing Symptoms of Fake Plateaux --"
} |
We establish a strict asymptotic inequality between a classof graph partition problems on the sparse and random regular graph ensembles with the same average degree.Along the way, we establish a variational representation for the ground state energy for generalized mixed p-spin glasses and derive strict comparison inequalities for such models as the alphabet changes. Sectoring in Multi-cell Massive MIMO Systems Shahram Shahsavari, Parisa Hassanzadeh, Alexei Ashikhmin, and Elza Erkip S. Shahsavari, P. HassanzadehandE. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {shahram.shahsavari,ph990, elza}@nyu.eduA. Ashikhmin is with Bell Labs, Nokia, Murray Hill, NJ, USA. Email:[email protected] December 30, 2023 ==========================================================================================================================================================================================================================================================================================================================================§ INTRODUCTIONConsider the following graph partition problem, called the unbalanced cut problem. Let G=(V,E) be a graph and let V_1,V_2⊂ V be a partition of the vertex set, V, such that V_1=αV and V_2=(1-α)V,where 0<α<1/2 is a fixed number. Let (V_1,V_2) denote the number of edges joining V_1 to V_2.We are interested in the maximum of this quantity over all such partitions, which we denote by _α(G)=max_V_1=αV V_2=(1-α)V(V_1,V_2).Observe, for example, that when α=1/2 this is the maximum bisection problem (Note that the quantity is only well-defined when α is rational).We aim to compare this quantity asymptotically between two well-knownrandom graph models. The first ensemble we consider is the sparse random graph, G(N,d/N), where each edge is added independently with probability d/N, where d is a fixed constant. The second ensemble we consider is the random d-regular graph, d, wherea d-regular graph on N vertices is selected uniformly at random.These random graphs are typically sparse: such a graph on N vertices has O(N) edges with high probability. Our main result is a strict comparison between the unbalanced cut problem on these two ensembles in the mean field approximation, i.e., for asymptotically large degrees.For any 0< α < 1/2,there is a constant C(α)>0 such thatlim inf_d→∞lim inf_N→∞_α(G(N,d/N))-_α(d)/√(d)N≥ C(α)almost surely. Thenovelty in this inequality is the fact that C(α) is strictly positive. Indeed, for the maximum bisection problem (α=1/2) it is known that that this difference is in fact zero <cit.>. From a combinatorial perspective, this inequality is surprising: intuitively, it suggests that the rigidity of the edge structure of random regular graphs, in comparison to that of graphs, has macroscopic ramifications for cut problems.The curious reader might also wonder if C(α) isin fact the sharp constant. We do not believe that this is the case. Instead, our approach yields a natural conjecture regarding the sharp constant.in terms of a minimizer of _T. We discuss this conjecture in Section <ref>. The core of our approach is a connection between the unbalanced cut problem — a priori a question of pure combinatorics— to the ground state energy of the Generalized Sherrington-Kirkpatrick model—a priori a question of statistical physics.Before presenting this connection, let us first place it in context. If instead of _α(G) one considers the minimum α-cut problem, i.e., taking a minimum in (<ref>) instead of a maximum, the inequality (<ref>) is reversed in the obvious way with no change to the proof.§.§ Background Graph partition problems are classical combinatorial optimization problems, having applications in Computer Science,Statistics and Machine Learning <cit.>. These problems are described as follows.Given a graph G= (V,E), we seek to divide the set of vertices into two or more partssuch that the number of edges between the distinct parts is optimized. For example, the well-known MaxCut problem seeks to partition the vertex set into two parts,V_1 and V_2, such that the number of edges connecting the two parts, (V_1,V_2), is maximized.Another exampleis the maximum bisection problem,_1/2(G). The study of these problems in the sparse regime has received much attention from the combinatorics community <cit.>, though they remain very challenging. Graph partition problems have also been studied extensively in the physics literatureas they are predicted to lie in a canonical class of models called spin glasses. Connections between spin glasses and combinatorial optimization problems are by now classical observations.This perspective has received a tremendous amount of attention in the physics, mathematics, and combinatorics literatures. It is impossible to provide here anywhere near a complete survey of this literature. Instead we point the reader to the texts <cit.>. Although many of the predictions of the physics literature have been verified, we are very far from understanding the full picture. For a sample of recent, rigorous results in this direction see <cit.>.In the setting of graph partition problems, this connection goes back at least to the work of Fu–Anderson <cit.>. Recently, there has been significant progress in formalizing this connection.First, Bayati–Gamarnik–Tetali <cit.> explored this connection by using a sub-additivity argument to establish the existence of a deterministiclimit for the MaxCut on sparse and random regular graphs. In this light,it is natural to study this deterministic limiting value as a function of the degree, d.In the large degree limit, or mean field approximation, it is not hard to see that the leading order contribution is of order d and is essentiallythe expected value of the objective function. Evidently, the heart of the matter is then in subsequent terms of the expansion in d. Indeed, the next term, often of order √(d), is highly nontrivial, and related to the ground state energy of mean field spin glasses. This idea was partially formalized by Dembo, Montanari, and one of the authors in<cit.>,where it was shown that asymptotically first in the vertex number and then in the degree, the normalized MaxCut, maximum bisection, and the minimum bisection of the andthe random d-regular graph ensembles are equal to second order in d. Again, the first order contribution is that of a random labelingof the vertices, d/4. The second order term, of order √(d), is (essentially) the ground state energy of the Sherrington-Kirkpatrick model <cit.>. In subsequent work, one of the authors <cit.> generalized this resultto a family of combinatorial optimization problems, where the objective is of a tensorial nature.For a general class of these problems on or random regular hypergraphs,a similar asymptotic appears, where this time the order √(d) term is the groundstate energy of a suitably chosen spin glass model.From this perspective, it is natural to believe that for a wide class of these problems, the optimal value should be the same on these two ensembles, up to o(√(d)) corrections. To this end, <cit.> derives broad sufficient conditions for the normalized maxima to have the same value, up to lower order contributions in d. In this paper, we show, surprisingly, that this belief is flawed.Theorem <ref> establishes that even a small perturbation of the maximum bisection problem has widely different behavior on and random regular graphs. Theorem <ref> is also significant for a number of conceptual reasons. First, it establishes a strict inequality between these statistics on and random regular graphs— which is difficult to establish using purely combinatorial techniques. Second,aside from solving an interesting question of combinatorics, it leads us to resolve an importantquestion ofindependent interest in the theory of spin glasses, namely the ground state energy of the generalized mixed p-spin glass model. Resolving these spin-glass questions is in fact our main technical contribution in this paper. §.§ Generalized mixed p-spin models and their connection to Theorem <ref>At the heart of thm:cut_difference is a connection between the unbalanced cut problem and what are called Generalized mixed p-spin models which were introduced by Panchenko in <cit.>. These are natural generalizations of the Ising p-spin model <cit.> to the case where the spins take values in afinite alphabetΣ⊂. (The Ising p-spin model corresponds to Σ={±1}.) More precisely, let Σ⊂ be a finite set called the alphabet, and let the configuration space be defined as Σ^N. The Hamiltonian for this model is the centered Gaussian process indexed by Σ^N with covariance H_N(σ^1)H_N(σ^2)=Nξ(R(σ^1,σ^2)),where ξ(t)=∑_p≥2β_2p^2t^2p is an even power series and R(σ^1,σ^2)=1/N∑_i=1^Nσ_i^1σ_i^2,is called the overlap. Let =max_ϵ∈Σϵ^2 and =min_ϵ∈Σϵ^2.We assume that ξ(+ϵ)<∞ for some >0 so that this process is well-defined. The application to graph-partition problems (Theorem <ref>) motivates our interestin the restricted normalized ground state energy of this process, GS_N(A_N):=max_σ∈ A_NH_N(σ)/N,where A_N⊂Σ^N. Specifically, we are interested in two cases, either A_N=Σ^N or A_N=A_N(T,ϵ_N)={σ∈Σ^N:R(σ,σ)∈(T-ϵ_N,T+ϵ_N)} ,for some ϵ_N→0 sufficiently slowly.To understand why, we will show by an application of the results of <cit.> that the proof of Theorem <ref> can be reduced to a strict comparison betweenthe limiting ground state energies of two generalized Sherrington-Kirkpatrick (SK) models— models for which ξ(t)=2 t^2. For a formal statement, we refer the reader to Lemma <ref>. For the graph, we obtainthe SK model, while for the random regular graph, we obtain a generalized SK model,both constrained in a certain natural fashion.Consequently, if an inequality sufficed, one could then use a well-knownGuerra-type <cit.> or Slepian-type <cit.> interpolation to easily obtain the desired estimate.We are interested, however, in a strict inequality asymptotically in N.To accomplish this goal, our approach is to provide a quantitative understandingof the derivative of this interpolation.This is accomplished by a fine analysis of the limiting ground state energy in generalized mixed p-spin models.The derivative is naturally related to the minimizers of a certain family of variational problems called “Parisi variational problems”, and the question pertains to the scaling of the minimizers of these problem as onetunes a certain parameter, called the temperature. This naturally leads us to a question of Γ-convergence of such variational problems <cit.>.This approach will not only yield that this constant is positive but will also yielda natural conjecture as to the sharp constant.We explain this in greater detail insec:annealing-introanddiscuss the aforementioned conjecture insec:conj1.§.§ Ground State Energies of Generalized Mixed p-spin modelsIn this section, we explain our results regarding variational representations for ground state energies of generalized mixed p-spin models. The question of the ground state energy is natural from a statistical physics perspective, and has recently received considerable attention in the mathematics literature. In the case that the configuration space is the sphere, S^N-1(√(N)), a variational formula was independently provided by Chen–Sen<cit.> and Tobasco with one of the authors <cit.>. In the case that the alphabet is Σ = {± 1}, called the Ising spin setting, a variational representation was obtained by Auffinger–Chen in <cit.>. These representations have since beenused to study a wide variety of questions<cit.> . We derive a variational representation for the normalized ground state energy of generalized mixed p-spin models as a consequence of our approach.To this end, we introduce the following notation. Let T∈[,]. Let ([0,T]) be the positive cone of Radon measures on [0,T]. Let _T⊂([0,T]) be the set of measures of the form _T={ν∈([0,T]):ν=m(t)dt+cδ_T, m(t)≥0 non-decreasing and cadlag},equipped with the weak-* topology. Note, in particular, finitness enforces ∫_0^T m(t) dt < ∞.On this space we define the ground state energy functional _T:_T×→. For ν=mdt+cδ_T, we let_T(ν,λ)=u_ν,λ(0,0)-λ T-1/2∫_0^Tξ”(s)sdν(s),where u_ν,λ is the unique weak solution to∂_tu+ξ”/2(Δ u+m(s)(∂_xu)^2)=0 (t,x)∈[0,T) ×u(T,x)=f(x,λ,c) x∈,where f(x,λ,c)= sup_ϵ∈Σ{ϵ x+(λ+ξ”(T)/2· c)ϵ^2} ifT ∈ (, ),max_ϵ^2= T{ϵ x+(λ+ξ”(T)/2· c)ϵ^2} ifT ∈{, }.(For a notion of weak solution of such PDEs see <cit.> and forbasic regularity in this setting see app:pde.) We then have the following.For any ϵ_N→0 sufficiently slowly, lim_N→∞GS_N(A_N)=inf_ν∈_T λ∈_T(ν,λ)almost surely. Furthermore lim_N→∞GS_N(Σ^N)=sup_Tinf_ν∈_T λ∈_T(ν,λ)almost surely. We note here that one can eliminate the dependence of this variational problem on c by making the substitution λ↦λ- c/2ξ”(T). We leave the problem in this form for two key reasons: first, our derivation of this result will be by way of Γ-convergence for which one must allow c to be non-zero (see the discussion in the next section); second, our main application, the proof of thm:cut_difference, will use this formula and said Γ-convergence result to characterize limit points of certain sequences of measures, which may have non-zero c. For the discussion of the physical interpretationof c see <cit.> and app:appendix-at. To return to our combinatorial motivations,let us begin by first observing that as a corollaryto thm:variational_rep and en route to proving thm:cut_difference, we also provide explicit formulas for _α to second order in the degree. In the following, we let ^1_T(ν,λ) denote the functionaleq:local-zero-temp-pfunc with Σ={± 1-(2α -1)} and ξ(t)=2t^2. Let T(α)=4α(1-α).Finally, let^2:_1×→ denote the functional𝒫^2(ν,h)=u_ν,0(0,h)-1/2∫_0^1ξ”(s)sdν(s),where u is the unique solution to (<ref>) with alphabet Σ ={+1,-1} and ξ(t)=2t^2, i.e., with final time data f(x,0,c)=|x|+2c.For any 0<α<1/2 we have that,lim_d→∞lim_N→∞_α(d)-N dα(1-α)/√(d)N = 1/4inf_ν,λ_T(α)^1(ν,λ) lim_d→∞lim_N→∞_α(G(N,d/N))-N dα(1-α)/√(d)N = 1/4inf_ν,h[ ^2(ν,h)-(2α-1)h]. §.§ An analytical approach to annealingAt the heart of the recent work regarding variational representations for ground state energies is an analytical approach to the notion of annealing.Annealing, that is, adding a temperature and sending it to zero, is natural fromthe point of view of statistical physics and underlies well-known algorithms for optimization <cit.>.The idea, roughly, is as follows. The ground state energy can be computed as the limit of an important quantity called the free energywhich is defined as follows. Recall the Hamiltonian H_N from (<ref>).The free energy at inverse temperature β is defined as F_N(β,ξ) = 1/Nlog∫_Σ^N e^β H_N(σ)dσ,where dσ is the counting measure, and the restricted free energy corresponding to a set A⊂Σ^Nand inverse temperature β is defined asF_N(β,ξ;A) = 1/Nlog∫_A e^β H_N(σ)dσ.In our setting, we are interested in A_N of the form eq:A_N-def.In these models, a variational expression for the free energy at a fixed temperature is obtained using a“Parisi-type formula”.ForΣ={± 1}, this was proved by Talagrand in <cit.> and Panchenko in <cit.>.For general alphabets, the variational problem was derived by Panchenko in <cit.>(and more recently again in <cit.>)where he showed that for any T∈ [d,D] and any _N→0 sufficiently slowlyF_N(β,ξ;A_N(T,_N)) → F(β,ξ;T)F_N(β,ξ) → F(β,ξ)where F(β,ξ;T) = βinf_ν∈ X_β, T,λ∈_β,T(ν,λ)F(β,ξ) →βsup_T∈[,] F(β,ξ;T)Here _β,T is called the local Parisi functional. For its precise definition see eq:pfunc-def. It is not difficult to see that 1/βF(β,ξ;T)β→∞⟶lim_N→∞ GS_N(A_N).Thus the question of ground state energies is related to the large β limit of these variational problems. A natural approach to the asymptotic analysis of variational problems is De Georgi's notion of Γ-convergence <cit.>. Our approach, following <cit.>, is to study the Γ-limit of _β,T. As a direct consequence, we obtain a variational representation for the limiting ground state energy, similar to <cit.>. Further, this allows us to control zero temperature asymptotics of physically relevant quantities, and derive strict comparison inequalities.We remark here that upper bounds only require the Γ-liminf inequality, and have been used in the recent progress in <cit.>.Due to the natural topology of the Γ-limit, one formally expects the need to understand how the nonlinear term — the solution in space-time of a Hamilton–Jacobi–Bellman equation, where the coefficient of the non-linearity is the variable of optimization — behaves as one allows this coefficient to become the derivative of a Dirac mass. More precisely, one needs an appropriate limiting notion of solution for such situations. (This explanation is necessarily vague, for a more precise description see sec:gamma-parisi-intro.) Auffinger–Chen <cit.> observed that in the case Σ ={±1}, the linear term in the functional exactly cancels this effect, allowing oneto avoid this issue. If one perturbs the problem by allowing the spins to take values {±1+ϵ}, however, the arguments in the literature do not apply. We are then forced to tackle the question of the limit of the non-linear term. To this end, we introduce a notion of annealed solution which yields an interpretation for the solution of the PDE in this singular regime as an appropriate zero-temperature Γ-limit. The Γ-convergence of _β,T then follows.§.§ AcknowledgementsThe authors thank Amir Dembo for introducing them to the unbalanced cut problem. The authors thank Jonathan Shi for pointing out an error in an earlier version of this manuscript. A.J. thanks Ian Tobasco for fruitful discussions,as well as the University of Toronto and Harvard University mathematics departments for their hospitality where part of this research was conducted.This research was conducted while A.J. was supported by NSF OISE-1604232 and NSERC [RGPIN-2020-04597, DGECR-2020-00199]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG). §.§ Outline of proof of Theorem 1.1 Let us now briefly outline the proof of Theorem 1.1. The starting point in our argument is Lemma lemma:gaussiancomp, which establishes that one can approximate_α on Erdős-Rényi and random regular graphs up to o(√(d)) correctionsby the ground state energies of certain Generalized SK models.We then apply a Gaussian interpolation argument in lem:int_by_parts to compare the corresponding free energies.The error term in this comparison is a quantity that depends on the minimizer of the corresponding variational problems. Finally, we send β→∞, and analyze the limiting variational problem to prove a sign on the limit of this error term in Theorem <ref>. §.§ Outline of paperThe remainder of this paper is organized as follows. In the next section, we reduce the proof of thm:cut_difference to an inequality about asymptotics of “overlaps” of spin glass models (we introduce this notion presently).In order to study this question, we introduce, in sec:gamma-parisi-intro,the Parisi boundary value problem and the notion of annealed solutions to this problem.In sec:loc-free-energy, we presentPanchenko's Parisi-type formula for the Free energy in this setting and use the notion of annealed solutions to compute its Γ-limit.We then prove pre-compactness and convergence of the minimizers of this problem.We then turn briefly in sec:analysis-zero-temp to computing the first variation of the groundstate functional _T. Finally, Section <ref> establishes the main spin glass estimate necessary for thm:cut_difference. For the benefit of the reader, we briefly present some basic analytical and topological results used in this paper in the appendix.§ THE UNBALANCED CUT PROBLEMIn this section, we establish Theorem <ref>. To this end,let G_1 ∼ G(N, d/N) and G_2 ∼ d be defined on the same probability space (Ω, ℱ, ). Recall that the random graph G(N, d/N) has vertex set [N], and the edges are added independently with probability d/N each. For ease of computation, we consider d to be drawn from the configuration model <cit.>. While this is a multi-graph in general, it is easy to see that conditioned on simplicity, the graph obtained is actually uniformly distributed. Further, the probability of the obtained graph being simple is bounded away from zero (see for example <cit.> and references therein). Thus it suffices to establish our result for the configuration model. §.§ Concentration of Our first result establishes the concentration properties ofon G_1 and G_2 around their respective expectations. For any ε>0 sufficiently small, there exists a universal constant C(d, ϵ)>0 [ | (G_1)/N - [ (G_1)/N] | > ε]≤ 5 exp[- C(d, ε) N][ | (G_2)/N - [ (G_2)/N] | > ε]≤2 exp[- N ε^2/d]. The concentration argument forrandom regular graph G_2 follows immediately upon an application of <cit.>.Next, we establish the result for the random graph G_1. To this end,let E denote the number of edges in G_1 and observe that |E| ∼Bin( N2, d/N). Thus [|E|] = (N-1)d/2. We have, [ | (G_1) - [(G_1) ] | > N ε] ≤[ | (G_1) - [(G_1) | |E| ]| > Nε/2] + [ | [(G_1) | |E| ]- [(G_1) ] | > N ε/2].:= I + II. To control I, we proceed as follows. We set ℰ = {|E| ≤[|E|] + N ε_0 }, for some ε_0>0 to be chosen appropriately. I≤[ 1(ℰ) [ | (G_1) - [ (G_1) | |E| ] | > N ε/2| |E| ]+[ ℰ^c]. ≤ 2 exp[ - N^2 ε^2/8 [ [|E|] + N ε_0 ]] + exp[ - 2/3N ε_0^2/d],where we bound the first term using the Azuma-Hoeffding inequality on the traditional edge-exposure martingale, and the second term by the Chernoff bound. To control II, let G_1' = ([N], E') ∼ G(N, d/N) be an random graph independent of G_1. We claim that | [(G_1) | |E| ] - [ (G_1') | |E'| ]| ≤| |E| - |E'| |.Given the claim, we have, using Jensen's inequality, | [ (G_1) | |E|] - [(G_1) ] | ≤_|E'||[ (G_1) | |E|]-[ (G_1') | |E'|]| ≤_|E'|[ | |E| - |E'| | ] ≤| |E| - [|E|] | + | |E'| - [ |E'| ] | ≤| |E| - [|E|] | + C √(N d),for some constant C>0, where the last inequality follows using Cauchy-Schwarz. Thus we have the bound,II < [ | |E| - [|E|]| > N ε/2 - C √(Nd)] ≤ 2 exp[- N ε^2/6d],where the last inequality follows using the Chernoff bound. This completes the proof, modulo the claim (<ref>), once we optimize over ε_0. To prove this claim, we proceed as follows.Given |E|, |E'|, we will construct a coupling (H, H') such that marginally, H and H' are distributed as G_1 conditioned to have |E| and|E'| edges respectively. Assume without loss of generality, that |E'| > |E|. Start with an empty graph on [N]. Add edges sequentially, uniformly at random. At the end of |E| steps,call the graph formed H.Continue adding edges, and at the end of |E'| steps call the graph H'. Under this construction, | (H) - (H') | ≤| |E'| - |E| | almost surely. Taking the expectation of this inequality with respect to the joint law of (H, H') and applying Jensen's inequality yields (<ref>), as desired.§.§ Comparison to a Gaussian problem In light of Lemma <ref>, it suffices to compare the expectations ofon G_1 and G_2. To this end, we introduce the following notation.For any graph G= (V,E) with |V| = N, assume thatV= [N] without loss of generality. Observe that every partition V= V_1 ⊔ V_2 of a graph can be represented by a vector σ∈{± 1}^N, with the two parts being encoded as V_1 = { i : σ_i =1} and vice versa. Therefore, every partition with |V_1| = α N, |V_2| = (1-α)N corresponds to a unique vector σ∈{± 1}^N such that ∑_i σ_i = N (2α -1). We setS_N(α) = {σ∈{± 1 }^N : ∑_i σ_i = N (2 α -1) }. Next, we consider a GOE matrix J = (J_ij)_N × N and define g_i = ∑_j J_ij/√(N). For σ∈ S_N(α), we define, H_0(σ)= ∑_ijJ_ij/√(N)σ_i σ_j, H_1(σ)= ∑_ijJ_ij/√(N)σ_i σ_j - 2 (2α -1) ∑_i g_i σ_i + (2α-1)^2∑_ijJ_ij/√(N). We have the following lemma. As N→∞, we have, [ (G_1)/N]= d α (1-α) + √(d)/4N[ max_σ∈ S_N(α) H_0(σ) ] + o(√(d)).[ (G_2)/N] = d α(1-α) + √(d)/4N[ max_σ∈ S_N(α) H_1(σ)] + o(√(d)).We start with the proof of (<ref>). This follows directly from <cit.>. In this case, we havep=2, A_N = S_N(α), f : {-1, 1}^2 →ℝ, f(x,y)= 1_{x ≠ y} = (1- x y)/2 and κ_1 = 1. Further, we note that on the set A_N, the contribution from the expectation is exactly α (1- α), and this completes the proof. Next we consider (<ref>). This will be established using <cit.>. Using the same setup as above, we obtain that for random regular graphs,[ (d)/N] = d α (1-α) + √(d)/2N[ max_σ∈ S_N(α)[ ∑_ijJ_ij/√(N)1_{σ_i≠σ_j } - 2/N∑_ij g_i 1_{σ_i ≠σ_j }] ] + o_d(√(d)). We note that for σ∈ S_N(α), ∑_i σ_i = N(2 α -1) and 1_{x ≠ y} = (1- xy)/2. Plugging these into the equation above completes the proof upon noting that the last term in (<ref>) is o_N(1) with high probability.§.§ Proof of Theorem <ref>The following Theorem establishes a strict lower bound on the difference between limiting ground state energies of the two generalized SK models. The proof is deferred to section <ref>.For 0< α < 1/2, there exists a constant C_0(α)>0 such that lim inf_N →∞1/N[ [max_σ∈ S_N(α) H_0(σ) ] - [ max_σ∈ S_N(α) H_1(σ) ]] > C_0(α). Now, we note that Lemma <ref> immediately implies that [(G_1)] - [(G_2)]/N √(d) = 1/4N[ [max_σ∈ S_N(α) H_0(σ) ] - [ max_σ∈ S_N(α) H_1(σ) ]] + o_d(1).The proof of Theorem <ref> can be completed by combining thm:diff_groundstate, Lemma <ref> and (<ref>), using a simple Borel-Cantelli argument. §.§ Proof of thm:diff_groundstateFor v ∈ [0,1], consider the interpolating Hamiltonian H_v(σ) = ∑_ijJ_ij/√(N) (σ_ i - √(v) ( 2α -1) )(σ_ j - √(v) ( 2α -1)),andthe interpolating free energyF_N(v, β; α) = 1/N[ log∑_σ∈ S_N (α)exp(β H_v(σ) )]. At v=0 and v=1 these are the free energies for the Hamiltonians H_0 and H_1 respectively.It is convenient to make the change of variables σ↦τ where τ_i = σ_i - √(v)(2α-1) .Under this change of variables,H_v is a generalized mixed p-spin model with ξ(t)=2t^2 andwhere the spins take values in the set Σ(v,α), such thatΣ(v,α) = { 1- √(v) ( 2 α -1) , - 1 - √(v) (2α -1) }. Furthermore, if we defineS_N(v,α) = {τ∈Σ(v,α)^N : ∑1{τ_i = 1 - √(v) (2α -1) } = N α} we can equivalently writeF_N(v, β ; α) = 1/N[ log∑_τ∈ S_N(v,α)exp(β H_v (τ))], where we abuse some notation and index the interpolating Hamiltonian by the new spins τ.For τ, τ' ∈ S_N(v,α), we define the overlap as usualR(τ, τ') = 1/N∑_i=1^Nτ_i τ_i'. We defineT(v,α) = α (1 - √(v) (2 α -1 ))^2 + (1 - α) (1 + √(v)(2 α -1 ))^2. and note that for α≠ 1/2 and v > 0,τ∈ S_N(v,α) τ^2 = N T(v, α).In other words, for α≠ 1/2 and v>0, S_N(v,α) specifies sets with a constant “self-overlap". Note that at thisjuncture, we immediately obtain Corollary <ref>. This follows by combining (<ref>), (<ref>), and (<ref>)with thm:variational_rep in the case v=1 and Theorem <ref> in the case v=0. We will need the following results to complete the proof.First, we obtain the following explicit expression for the derivative of the interpolating free energy in v using integration by parts. To this end, we will need the following definition. For any function f: S_N(v,α) →ℝ, we define the expectation under Gibbs measure of the interpolated Hamiltonian ⟨ f ⟩_v = ∑_τ∈ S_N(v,α) f(τ) exp(β H_v (τ))/∑_τ∈ S_N(v,α)exp(β H_v (τ)). For each N ≥ 1, and α∈ (0, 1/2), we have,1/β∂/∂ v F_N(v, β; α) = - 2 β (2α -1)^2 (1- √(v))/√(v)[ T(v) - ⟨ R_12⟩_v]. The proof will be deferred to the end of the subsection. Next, we will need the following crucial theorem. For all α∈ (0, 1/2) and v ∈ (0,1), there exists an explicit constant C_1(v,α)>0 such that lim inf_β→∞lim inf_N →∞β[T^2 - ⟨ R_12^2 ⟩_v] > C_1(v,α).The proof of this estimate will require the full machinery of the method of annealing. We establish this result in Section <ref>. Before turning to these results, however,we establish thm:diff_groundstate. Central to our approach is the following well-known elementary observation(see, e.g., <cit.>).The following inequality holds for any A_N ⊂Σ^N and β >0. | 1/β F_N(β, ξ; A_N)- GS_N(A_N) | ≤log |Σ|/β.We observe that the thesis of thm:diff_groundstate can be equivalently formulated aslim inf_N→∞lim inf_β→∞1/β [F_N(0, β ; α) - F_N(1, β; α)] > C_0(α) >0 . In light of Lemma <ref>, it suffices to establish lim inf_β→∞lim inf_N →∞1/β [F_N(0, β; α) - F_N (1, β ;α)] > C_0(α) . To this end, we have,1/β[ F_N(1, β; α) - F_N(0, β; α) ] = ∫_0^11/β∂/∂ v F_N(v,β;α)v.Using (<ref>) and Lemma <ref>, it suffices to establish that for all v ∈ (0,1),lim inf_β→∞lim inf_N →∞β[ T - ⟨ R_12⟩_v] > C_0(v, α) >0. Note that T ≥ | R_12| and thus (T - ⟨ R_12⟩_v ) ≥ (T^2 - ⟨ R_12^2 ⟩_v)/ 2 T. Thus the proof of thm:diff_groundstate follows upon applying Theorem <ref>. We establish Lemma <ref> in the rest of this section. The result will follow directly by integrating by parts <cit.>. To see this, we begin by observing that 1/β∂/∂ v F_N(v, β ; α) = 1/N[ ⟨∂/∂ v H_v (σ) ⟩_v]. We note that∂/∂ v H_v(σ) = - (2α -1)/√(v)∑_ijJ_ij/√(N) (σ_i - √(v) (2α -1)). Moreover, we note that for τ∈ S_N(v, α), ∑_i τ_i = N (2α -1) ( 1- √(v)). Therefore, we have,[ H_v(σ^1) ∂/∂ v H_v(σ^2) ] = - N2 ( 2α -1)^2 (1 -√(v)) /√(v) R(τ^1, τ^2) . Thus we have, by Gaussian integration-by-parts for Gibbs averages (see, e.g., <cit.>), that 1/β∂/∂ vF_N(v , β;α) = - 2 β( 2α -1)^2 (1 -√(v)) /√(v)[ ( T - ⟨ R(τ^1, τ^2) ⟩_v )]. This completes the proof.§ ANNEALED SOLUTIONS OF PARISI INITIAL VALUE PROBLEMSIn the study of mean field spin glasses, a central role is played by the Parisi boundary value problem, which is defined as follows. For β>0, let X_β,T⊂_T denote the set of measures of the form X_β,T={ν=βμ[0,t]dt,μ∈([0,T])} .Equip the product space _T× with theproduct topology, where we recall that _T was given the weak-* topology.On this space define for every (t,x)∈[0,T)×the functional _β,T(ν,λ;t,x) given by_β,T(ν,λ;t,x)=u^β_ν,λ(t,x)ν=βμ([0,t])dt+∞ otherwise,where u_ν,λ^β(t,x) is the weak solution tothe Parisi boundary value problem∂_tu+ξ”/2(Δ u+βμ([0,s])(∂_xu)^2)=0u(T,x)=f_β(x,λ)with boundary data f_β(x,λ)= 1/βlog∫_{σ^2=T}e^β(ϵ x+λϵ^2)dϵ T∈{,} 1/βlog∫_Σe^β(ϵ x+λϵ^2)dϵ otherwise.Here dσ is the counting measure on Σ.A central role in our study will be played by _β,T(ν,λ;t,x).More precisely, we will be interested in the limit of this functional as β→∞ along sequences (ν_β,λ_β) where ν_β∈ X_β. The main issue in understanding this limit is as follows.On _T×,a typical convergent sequence, (ν_β) with ν_β∈ X_β, satisfiesν_β→ν = m(t)dt+cδ_T.Thus one must have a method of interpreting u^β_ν_β,λ_β(t,x) in this limit. A naive approach does not suffice: in this limit the coefficient in front of the non-linearity formally converges to an expression of the form f(t) = m(t)+cδ'_T where m is some cadlag non-decreasing function and δ'_T is the distributional derivative of the Dirac mass at T. Evidently,care must be taken in this limiting procedure.To this end we introduce a notion of solution that respects this mode of convergence called an annealed solution. Themain idea is that the singular contribution is, effectively, a change of initial data.More precisely, we are led to the following definition. We say that ϕ_ν,λ(t,x) is an annealed solution to eq:zero-temp-pde, if for every (t,x)∈ [0,T)×, we have that * (lim inf condition) If (ν_β,λ_β)→(ν,λ), then for u^β_ν_β,λ_β the solution of eq:ppde-finite-beta, we have thatlim inf u^β_ν_β,λ_β(t,x)≥ϕ_ν,λ(t,x). * (lim sup condition) There is a sequence (ν_β,λ_β)→(ν,λ) such that for u^β_ν_β,λ_β, the solution of (<ref>), we have thatlim u^β_ν_β,λ_β(t,x) = ϕ_ν,λ(t,x).The goal of this section is to interpret the weak solution, u, of eq:zero-temp-pde, as anannealed solution. For every ν and λ, the weak solution to (<ref>), u_ν,λ(t,x),is an annealed solution. Before turning to the proof of this result, let us pause to make two comments.We note here that the definition of annealed solution is such that the non-linear terms in (<ref>) will Γ-converge.Although we only use this result at the point (0,0), our argument extends to general (t,x) with no change. Furthermore, working with general (t,x) clarifies the proof substantially. Moreover, we imagine that this notion will be useful in future research. Indeed, it is known <cit.> that the solutionu^β and its derivatives have physical interpretations, where the values at a given point in space-time are related to Gibbs averages of natural quantities.We expect the notion of annealed solutions to provide insights into the zero-temperature behavior of these physical quantities. The proof ofthm:annealed-soln-thm is in two parts. In the following, we will omit the dependence on (t,x) and T whenever it is clear for readability. We will frequently make use of basic regularity of u^β and u. For the former see <cit.> and the latter see app:pde. We divide the proof of this theorem into two lemmas.We begin in lem:gamma-limsup below, by proving the lim sup condition. We then turn to the lim inf condition in lem:gamma-liminf.For every every (t,x)∈ [0,T)×, and every (ν, λ) ∈_T×ℝ, there exists a sequence (ν_β,λ_β)→(ν,λ) such that lim u^β_ν_β,λ_β(t,x) = u_ν,λ(t,x)where u_ν,λ is the weak solution to (<ref>).Our goal is to construct such a recovery sequence.To this end,fix ν and λ. Let λ_β=λ. To construct ν_β,we remind the reader of thefollowing construction from <cit.>. Let ν∈_T and let c_β(ν)=c if ν({1})=c>0 and c_β(ν)=β^-1 otherwise. For β sufficiently large,there exists a q_β∈ (0,T) with the following properties: * ∫_q_β^T m(t)dt+c_β=β(T-q_β) * q_β→ T * m(q_β)≤β.In particular, ifμ_β=m(t)/βt<q_β 1 t≥ q_β then dν_β=βμ_β([0,t])dt∈ X_β and ν_β→ν.Let us now show that this sequence allows us to recover the value of the function. To this end, we solve the Parisi PDE eq:ppde-finite-beta in the interval [q_β, T] using the Hopf-Cole transform.This yields, u^β_ν_β,λ_β(q_β,x)= 1/βlog∫exp[β( x + (λ+β/2(ξ'(T)-ξ'(q_β))^2))]d.Since u^β=u^β_ν_β,λ_β and u=u_ν,λ solve the same PDE in the time interval (0,q_β), their difference, w= u^β - u, solves the boundary value problem, ∂_t w + ξ”(t) /2( Δ w + m(t) (∂_x u^β + ∂_x u ) ∂_x w ) = 0,(t,x) ∈ (0, q_β) ×ℝ,w(q_β, x) = u^β(q_β, x) - u(q_β, x) x ∈.Since ∂_x u^β and ∂_x u are both in C_tC^∞_x and m(t)∈ L^∞([0,q_β]) for s<q_β<T,this is a linear heat equation andwe have the representation w(t,x)= _Z_t = x [w(q_β, Z_q_β)], Z_t= ξ”(t) m(t) (∂_x u^β + ∂_x u)t+ √(ξ”(t)) W_t .Thus we immediately obtain |u^β(t,x) - u(t,x) | =| w(t,x) | ≤ u^β(q_β, ·) - u(q_β, ·) _∞.The proof of the lim sup condition will be complete once we establish that the right-hand side vanishesas β→∞. To this end, we note that u^β(q_β, ·) - u(q_β, ·) _∞≤ u^β(q_β, ·) - u(T, ·) _∞ +u (T, ·) - u(q_β, ·) _∞.Let us first show that by (<ref>) and (<ref>)the first term vanishes.Indeed, u^β(q_β,x)-u(T,x) ≤sup_{ x + (λ+(∫_q_β^T m(t)dt +c_β)ξ'(T)-ξ'(q_β)/2(T-q_β)·^2)} - u(T,x) +o_β(1)≤/2(∫_q_β^T m(t)dt +c_β)ξ'(T)-ξ'(q_β)/T-q_β-cξ”(T) +o_β(1)where the error terms o_β(1) are uniform in x. This goes to zero as β→∞ by lem:recovery-sequence.It remains to bound the second term. To see this,note that by Itô's lemma,u(q_β, x) = _Z̃_q_β = x[ u(T, Z̃_T) ],where Z_s is the Itô processZ̃_s = ξ”(s) m(s) ∂_x u( s, Z̃_s)s + √(ξ”(s)) W_s .Thus since u is Lipschitz in space uniformly in time,ũ_λ, ν(q_β, ·) - ũ_λ, ν(T, ·) _∞≤∂_x ũ_λ, ν_∞| Z̃_T - Z̃_q_β | → 0as β→∞ using elementary properties of diffusions, as q_β→ T as β→∞.We now turn to the lim inf-condition. For this we recall the dynamic programmingformulationof these PDEs. Let _t^T be the space of all bounded processes on [t,T] that are progressively measurable with respect to the filtration of Brownian motion. In the following, for a measure μ, we let μ(s)=μ([0,s]). The weak solution u_ν,λ^β of eq:ppde-finite-beta corresponding to ν = βμ(s)ds solvesu^β_ν,λ(t,x) = sup_α∈_t^T_X_t^α =x( f_β(X_T^α,λ) - β/2∫_t^Tξ”(s)βμ(s)α^2_sds)where X^α solvesdX^α_s = ξ”(s)βμ(s)α_sds+√(ξ”(s))dW_swith initial data X_t=x. Furthermore, any optimal control, α^*, satisfiesμ(s)α_s^* =μ(s)∂_x u_ν,λ^β(s,X_s)a.s.where X_s solves:dX_s = ξ”(s)βμ(s)∂_x u_ν,λ^β(s,X_s)ds+√(ξ”(s))dW_s. Furthermore, the weak solution u of eq:zero-temp-pde corresponding to ν = m(s)ds+cδ_Tsolvesu_ν,λ(t,x) = sup_γ∈ℬ_t^T_X̃_t^γ = x [ u_ν,λ(T, X̃_T^γ) - 1/2∫_t^Tξ”(s) m(s) γ^2_ss ],where X̃^γ has initial data X̃^γ_t =x and solves the SDEX̃^γ_s = ξ”(s) m(s) γ_ss + √(ξ”(s)) W_s. This result is an immediate consequence of the verification argument <cit.>.For u^β, this is immediateafter recalling that u is smooth in space and weakly differentiable in time, with bounded derivatives <cit.>.See <cit.> for a proof that applies essentially unchanged to our setting. For u, the same proof applies after observing that the solution has the regularity from lem:ppde-reg-zero-temp. The only point to note is that one should apply Itô's lemma for times t<T and then pass to a limit as t→ T.Similar arguments appear frequently in the literature, see, e.g., <cit.>, so we omit it. Before turning to the lower bound, note the following.Consider the function ψ(x,y)=max_∈Σ[x+^2y ].For every y, this function is differentiable in x Lebesgue a.e. Furthermore, the derivative is continuous in x Lebesgue a.e. and is given by ∂_xψ(x,y)=_*,where _* is the unique solution to ψ(x,y)=_*x+_*^2y. Fix y. Then x↦ψ(x,y) is convex in x. Thus by Alexandrov's theorem <cit.>, it is differentiable in x Lebesgue a.e. with a derivative that is continuous in x Lebesgue a.e.. Combining this with Danskin's envelope theorem <cit.>, we see that the derivative is given by∂_xψ(x,y)=_*,where _* is such that ψ(x,y)=_*x+_*^2y.(Implicit in this argument is that there is a unique such _* for almost every x, by another application of Alexandrov's theorem.)With these in hand we can now prove the lower bound.For every (t,x)∈ [0,T)× and every (ν, λ) ∈×ℝ, if (ν_β ,λ_β) ∈×ℝ with (ν_β, λ_β) → (ν, λ), thenlim inf u^β_ν_β,λ_β(t,x)≥ u_ν,λ(t,x)where u_ν,λ is the weak solution to (<ref>). We focus on the case T∈(,). The case T∈{,} is the sameafter taking λ=0 everywhere, as in this case the functional is constant in λ.Let ϕ(x) = argmax_∈Σ[ x + ( λ + c/2ξ”(T) )^2].Observe that by lem:psi-diff, ϕ∈ L^∞(). Thus for any M>0, we can consider a sequence of smooth,compactly supported function ϕ_n such that ϕ_n→ f strongly in L^2([-M,M]) and almost everywhere, where f is as in (<ref>). Furthermore, one may take ϕ_n such thatϕ_n_∞≤ Cfor some C>0 that does not depend on M or n.Fix γ∈ℬ_0^T,τ∈(t,T), and for each n, consider the controlα_s^τ,n = γ_sifs ≤τ ϕ_n(X̃^γ_s)if s > τ.Note that α_s^τ,n is a bounded, progressively measurable control.Further, the continuity of ϕ_n ensures the left-continuity of α^τ,n_s as s ↑ T.By topological properties of _T, see lem:convergence,we have that for each n,∫_t^Tξ”(s) (α_s^τ,n)^2 ν_β(s) β→∞→∫_t^τξ”(s) γ_s^2 m(s)s +∫_τ^Tξ”(s) ( ϕ_n(X̃^γ_s) )^2ν(s)=: D_τ,n(ν)We then let τ↑ T through the continuity points of ν to derive D_τ(ν) →∫_t^Tξ”(s) γ_s^2 m(s)s + c (ϕ_n(X̃^γ_T) )^2 ξ”(T) almost surely. Similarly, we have thatlim_β→∞X_T^α^τ,n →∫_0^τξ”(s) γ_s m(s)s + ∫_τ^Tξ”(s) ϕ_n(X̃^γ_s) ν(s) + ∫_0^T√(ξ”(s)) W_s + x =: Z_T(τ,ν),almost surely. As before, we let τ↑ T to obtain, Z_T(τ, ν) →X̃_T^γ + c ϕ_n(X̃^γ_T) ξ”(T).By the dynamic programming principle (<ref>),lim inf_β→∞ u^β_ν_β,λ_β(t,x) ≥lim inf_β→∞_X_t^α^τ,n=x[f_β(X_T^α^τ,n,λ_β) - 1/2∫_0^Tξ”(s) (α_s^τ,n)^2 ν_β(s) ].If we combining these results with the fact that f_β(x,λ) ≥ f(x,λ,0),we may send β→∞, followed by τ↑ T and use the dominated convergence theorem to conclude that lim inf_β→∞ u^β_ν_β,λ_β(t,x)≥lim inf_β→∞_X_t^α^τ,n=x[ sup_∈Σ[ X_T^α^τ,n + λ_β^2 ] - 1/2∫_0^T ξ”(s) (α_s^τ,n)^2 ν_β(s) ]≥[ζ_n],where we define ζ_n= [ sup_∈Σ(X̃_T^γ + c ϕ_n(X̃^γ_T) ξ”(T))+λ^2) - 1/2∫_0^Tξ”(s) γ_s^2 m(s)s - c/2 (ϕ_n(X̃^γ_T) )^2 ξ”(T) ].Finally, it remains to send n →∞. We observe that [ζ_n] = [ζ_n 1(X̃^γ_T ∈ [-M, M] ) ] + [ζ_n 1 (X̃^γ_T ∉ [-M,M] )].Conditionally on the event X̃^γ_T ∈ [-M, M], ϕ_n(X̃^γ_T) →ϕ(X̃^γ_T) almost surely as n →∞. We define ζ_∞ = sup_Σ[ ( X̃_T^γ + c ϕ(X̃^γ_T) ξ”(T)) + λ^2] - 1/2∫_0^Tξ”(s) γ_s^2 m(s)s - c/2 (ϕ(X̃^γ_T) )^2 ξ”(T).By the dominated covergence theorem, we may send first n→∞ and then M→∞ to obtain,lim_M →∞lim_n→∞[ζ_n 1(X̃^γ_T ∈ [-M, M]) ] = [ζ_∞].Similarly, since ϕ_n and γ are bounded and λ,ν are fixed, we see that ζ_n is uniformly bounded, so thatby Hölders inequality,[ζ_n1 (X̃^γ_T ∉ [-M,M] ) ] ≤ζ_n_∞[X̃^γ_T ∉ [-M, M]].We let M →∞ to control this term and obtain the lower boundlim inf_β→∞ u^β_ν_β,λ_β(t,x)≥[ζ_∞].Finally, after recalling the choice of ψand maximizing in γ, adirect computation yields that sup_γ[ζ_∞] = u_ν,λ(t,x).The result then follows.§ LOCAL FREE ENERGIES: THEIR DERIVATIVES AND ZERO TEMPERATURE LIMITS Our starting point for this section is equationeq:loc-fe. Following <cit.>, we refer to F(β,ξ;T) as the local free energy.The functional _β,T:_T×→ in eq:loc-fe is called the local Parisi functionaland is given by _β,T(ν,λ)=_β,T(ν,λ;0,0)-λ T-β/2∫_0^Tξ”(s)sμ([0,s])ds,where _β,T is defined in eq:finite_temp_functional. In particular, recall that _β,T is infinite on _T∖ X_β,T, so that _β,T is infinite as well.In this section, we study basic properties of this functional. First, in sec:diff-fe, we compute the derivative of the variational problem eq:loc-fe in β and relate this to the minimizer of _β,T. In sec:gamma-conv-local-fe, we then compute the Γ-limit of the sequence (_β,T)_β as β tends to infinity and establish aprecompactness theorem regarding the corresponding sequence of minimizers. We end this section by applying this Γ-convergence result to prove thm:variational_rep. §.§ Differentiability of the Local Free energyWe begin by showing that eq:loc-fe is differentiable in β and to provide an expression for this derivative in terms of the minimizer of thevariational problem. For the purposes of this section, it is convenient to restrict _β,T to the space X_β,T, defined in (<ref>). Furthermore, it is convenient toview this as a function of μ where ν =βμ([0,s])ds. To this end, we denote P_β,T(μ,λ)= β_β,T(ν,λ/β) = v_μ,λ(0,0)-λ T-β^2/2∫_0^Tξ”(s)sμ([0,s])dswhere ν = βμ([0,s])ds and v_μ,λ(t,x)=β u_ν,λ/β(t,x/β), where u^β solves (<ref>).In this notation,F(β,ξ;T)= inf_μ,λP_β,T(μ,λ)When it is clear, we will also omit dependence on T in our notation. The main result of this section is the following.We have that∂_β F(β,ξ;T) = β∫ (ξ(T)-ξ(t) )dμwhere for T>0, μ is the unique minimizing measure of P_β,T. In the case that Σ={±1}, this result was obtained earlier in <cit.>. An alternative proof was provided in <cit.>. Both of thesearguments use in an essential way that the initial data satisfies (∂_x f)^2 +∂_x^2 f = const. which does not hold in the general setting.The results of<cit.> hold in the more general setting that Σ is an arbitrary compact set and where the integral in this definition with respect to any measure ρ with support Σ.This result also extends to this setting with no change, however for the sake of exposition and with an eye toward our eventual application, we do not consider this setting here. We begin the proof of thm:beta-deriv-overlap with the following lemmas. The first result is regarding the convexity of P_β,T and the uniqueness (or non-uniqueness) of minimizers. This was first proved in <cit.>, see also <cit.>.If T∈ (,), the map P_β,T is strictly convex in (λ,μ). If T∈{,}, and {σ^2=T}=2, the map is strictly convex in μ. If T=0, the map is identically zero.Otherwise the map is convex and the functional is uniquely minimized at μ=δ_0. We defer the proof of this to the end of the section.We note here that since u_ν,λ has the representation (<ref>), v_μ,λ has the representationv_μ,λ(t,x) = sup_α∈_T[ β f_β (X^α_T/β,λ/β) - β^2/2∫_t^T ξ”(s)μ(s)α_s^2ds]with an optimal control α^*_s = ∂_x v_μ,λ(t,x)and optimal trajectory X̂_s which solvesdX̂_s = β^2ξ”(s)μ(s)∂_x v_μ,λ(t,x)ds + √(β^2ξ”(t))dW_t.Consequently, we have the following characterizationof minimizers of P_β,T. In the following letG_μ,λ(t)=∫_t^Tβ^2ξ”(s)( (∂_x v_μ,λ)^2(s,X̂_s) - s)ds.If T∈(,), there is a unique minimizing pair (μ,λ). Furthermore this pairsatisfies μ(G(s)=min_t∈[0,T] G(t)) = 1, ∂_λ v_μ,λ(0,0)= T. Finallywe have that for every q in the support of the optimizing μ, ∂_x v_μ,λ(q,X̂_q)^2 = q.If T=, or T= with >0 then the above still holds, with the exception of eq:fixed-point-T. If T=0, all measures minimize and the above conditions are vacuous.Note that in the case T∈{,}, if we instead work with the definition of f_β(x,λ) that does not change in T, we see that eq:fixed-point-T still holds, except in a limiting sense.We defer this proof to the end of the section as well. In the subsequent discussion we will be repeatedly differentiating an optimization problem. These results generally go by the name of“envelope theorems”. The two envelope theorems we shall use are Danskin's theorem <cit.>, as well as the following lemma from <cit.>. Let K be a metric space and I = [a,b) be a half open interval. Let f be a real valued function on K× I and g(y)=sup_x∈ K f(x,y). Suppose that there is a K-valued continuous function a(y) on I such that g(y)=f(a(y),y) and ∂_y f is jointly continuous on K× I. Then g is right differentiable with derivative ∂_y f(a(y),y)In the following, it is useful to equip the space _T with the L^2 metric and ([0,T]) with the metric d(μ,ν) = ∫μ([0,t])-ν([0,t])dt which metrizes the weak-* topology on this set.Let us now prove the main result of this section.The following argument is a modification of <cit.>. Fix β,λ. For ease of notation, let v=v_μ,λ, and similarly for u when it is unambiguous, and also let g(x)=β f_β(x,λ/β). Let us take T>0 as the case T=0 is vacuous, by lem:strict-convexity-finite-beta. Since u^β has the variational representationeq:dpp-finite-beta, we may apply lem:wei-kuo to obtain∂_β v(t,x)= ∂_x v(T,X̂_T)(2β∫_t^Tμ(s)ξ”(s)∂_x v(s,X̂_s)ds+∫_t^T √(ξ”)dW_s)- β∫_t^Tμ(s)ξ”(s)(∂_x v(s,X̂_s))^2ds.(We use here, implicitly, that the map β↦ v^β is continuous.) Since w=∂_x v weakly solves the heat equation∂_t w+L w =0with initial data w(T,·)=∂_x g, where L=β^2ξ”/2(∂_x^2 +2μ∂_x v ∂_x) is the infinitesimal generator ofeq:local-field, it follows that ∂_x v(s,X̂_s) is a martingale:∂_x v(s,X̂_s)=∫_0^s ∂_x^2 v(s',X̂_s')√(β^2ξ”(s'))dW_s'+∂_x v(0,0).Thus ∂_β v(0,0)=β{∫_0^Tμ(s)ξ”(s) (∂_x v)(s,X̂_s)^2ds + ∫_0^Tξ”(s)∂_x^2 v(s,X̂_s)ds}.Integrating the second term by parts,using Itô's lemma and the fact that ξ'(0)=0, we see that∫ _0^T ξ”(s)∂_x^2v(s,X̂_s)ds = ξ'(T) ∂_x^2 v(T,X̂_T) + β^2∫_0^T μ(s) ξ'(s)ξ”(s)(∂_x^2 v(s,X̂_s) )^2ds.Integrating the first term by parts and applying Itô's lemma again, we see that ∫_0^Tμ(s) ξ”(s)(∂_x v)^2(s,X̂_s)ds= ξ'(T) (∂_x v)^2(T,X̂_T)-∫_0^T ξ'(s) (∂_x v)^2(s,X̂_s)dμ -β^2∫_0^T μ(s) ξ'(s) ξ”(s)(∂_x^2 v(s,X̂_s))^2 ds.Combining these results, we obtain∂_β v(0,0) = β{ξ'(T)( ∂_x^2 v(T,X̂_T)+(∂_x v)^2(T,X̂_T)) - ∫_0^Tξ'(s)(∂_x v)^2(s,X̂_s)dμ}.Differentiating P_β,T in β, applying eq:fixed-point-q, and integrating by parts, we see that at the optimal μ_*, ∂_β P_β,T(μ_*,λ) = β{ξ'(T)[( ∂_x^2 v_μ_*,λ(T,X̂_T)+(∂_x v_μ_*,λ)^2(T,X̂_T)) -T ] +∫_0^Tξ(T)-ξ(t)dμ_*(t)}.Let us now show that at (μ_*,λ_*),( ∂_x^2 v_μ_*,λ_*(T,x)+(∂_x v_μ_*,λ_*)^2(T,x)) =T.To this end, observe that if we define π_x,λ(d) by π_x,λ(d)=e^ x + ^2 λ/∫ e^ x+^2 λdd,then we have∂_x g = , ∂_x^2 g=^2-^2,and ∂_λ g =^2where · denotes expectation with respect to π_x,λ. Thus∂_x^2 g + (∂_x g)^2 = ∂_λ g.In the case that T= or T=, if we take λ in the above expression to either -∞ or+∞ respectively, then we obtain the desired expression. It remains to consider the case T∈(,). By a classical differentiable dependence argument(see, e.g., <cit.>), v is classically differentiable in λ, and w=∂_λ v weakly solves the Cauchy problem∂_t w + L w = 0 with boundary data w(T,·)=∂_λ g. Thus ∂_λ v_μ_*,λ_*(0,0) = ∂_λg̅(X̂_T),where g̅ is g evaluated at λ=λ_*. As a result eq:combines-to-T then follows from eq:fixed-point-T at λ =λ_*,the optimal λ.Combining these results with Danskin's envelope theorem <cit.>, we see that∂_β F = ∂_β P_β,T(μ_*,λ_*)= β∫_0^T ξ(T)-ξ(t)dμ_*(t)as desired. Let us begin with the first case T∈(,). The proof in the case T∈{,} and {σ^2=T}=2 is identical. This proof is verbatim the argument in <cit.>.Take (μ_0,λ_0),(μ_1,λ_1) distinct in ([0,T])×.Let μ_θ = θμ +(1-θ)ν and define λ_θ analogously.Let α^θ be the optimal control for the Parisi PDE corresponding to (μ_θ,λ_θ).Consider the processes Y_t and Z_t which solve dY_t = β^2ξ”(t)μ_0(t)α_t^θ dt+√(β^2ξ”(t))dW_tanddZ_t =β^2ξ”(t)μ_1(t)α_t^θ dt+√(β^2ξ”(t))dW_t,with initial data Z_0=Y_0=0. Therefore by the strict convexity of g(λ,x) = β f_β(x/β,λ/β) in the pair (λ,x), and the dynamic programming principle eq:dpp-finite-beta, v_μ_θ,λ_θ(0,0)= [g(λ_θ,X̂_T^α^θ)-β^2/2∫_0^T ξ”(s)μ_θ(s)(α_s^θ)^2ds ]≤θ[g(λ_1,Z_T)- β^2/2∫_0^T ξ”(s)μ_1(s)(α_s^θ)^2ds] +(1-θ)[ g(λ_0,Y_T) -β^2/2∫_0^T ξ”(s)μ_0(s)(α_s^θ)^2ds]≤θ v_μ_1,λ_1(0,0)+(1-θ)v_μ_0,λ_0(0,0). Furthermore the first inequality is strict if either λ_1≠λ_0 or P(Y_T≠ Z_T)>0.Thus it remains to show that this probability is positive provided μ_0≠μ_1.To this end, observe that it suffices to show that(Y_T-Z_T)>0.By Itô's lemma, (Y_1-Z_1)=∫_[0,T]^2Δ_sΔ_t K(s,t)ds dt, where Δ_s = ξ”(s)(μ_0(s)-μ_1(s)) andK(s,t)=[(α_s^θ-α_0^θ)·(α_t^θ-α_0^θ)],so it suffices to show that K is positive definite.By Itô's Isometry, K(s,t)=p(t∧ s)wherep(s) =∫_0^sβ^2ξ”(t)∂_x^2 v(t,X_t^θ)dt. By eq:f-derivs, ∂_x^2 f>0. Thus by Itô's lemma,we have the maximum principle:∂_x^2 v(t,x) = _X_t=x( ∂_x^2 f(X_T)+ ∫_t^Tξ”(s)μ(s)∂_x^2v(s,X_s)ds)>0 . This immediately implies that p is strictly increasing, so that K is positive-definite as desired. In the remaining case, σ^2=T=1, and one can explicitly solve the PDE to find that P_β,T(μ,λ)=β^2/2∫_0^T ξ”(s)μ(s)(T^2-s)ds which is evidently maximized at μ= δ_0 and uniquely so if and only if T^2>0. In the subsequent it will be useful to define the following log-momentgenerating function,ψ(θ)=log∫ e^θ^2d.Observe that, ψ is continuous and monotone, with lim_θ→∞ψ(θ)/θ=,lim_θ→-∞ψ(θ)/θ=.Note that in the case that =, ψ is constant.For every μ,λ,β,ξ, and T∈(,), we have thatv_μ,λ≥ψ( λ).By the parabolic comparison principle (lem:ppde-reg-zero-temp), we see thatv_μ,λ(t,x)≥ v_δ_T,λ(t,x). Thusv_μ,λ(0,0)≥log∫ e^( B_β^2ξ'(T)+λ^2)d,where B_t is a standard Brownian motion run until time t. Using again the convexity of the map x↦log∫ e^ x+λ^2d,(see eq:f-derivs) we have that this is lower bounded by v_μ,λ(0,0)≥log∫ e^λ^2d=ψ(λ)as desired.That a minimizer is unique follows by lem:strict-convexity-finite-beta.That a minimizer exists can be seen as follows. Firstly, μ∈([0,1]) which we may equip with the weak-* topology.Thus it suffices to show that we may restrict λ to a compact set if T∈(,), as in the other case we may take λ=0. To this end, suppose first that T∈(,).Observe that for any μ, we then have that by lem:mgf-boundP_β,T(μ,λ)≥ψ(λ)-λ T-∫ sξ”(s)ds.Thus the limit of the right hand side as λ→±∞ is infinity. Thus we may restrict to a compact set. The result then follows by (lower semi)continuity of P_β,T.Let us now proveeq:G-minimizing.Let γ_t=(μ_t,λ) be a path such that μ_tis weakly differentiable on (0,1) and right weakly differentiable at t=0 in the sense thatlim_t→ 0^+μ_t-μ/t = μ̇weak-* as measures for some signed measure μ̇. By the same argument as in <cit.>, we have that P_β,T(μ_t,λ) is right differentiable at t=0, and d/dt^+P_β,T(μ_t,λ)=∫ G(t) dμ̇.The only difference is to notice that since ∂_x f from is uniformly bounded in (x,λ) by (<ref>), ∂_x u is uniformly bounded in (t,x,λ) as well, using lem:ppde-reg-zero-temp. Thus by the first order optimality conditions forconvex functions,∫ G(t) d μ̇≥0for all such paths. Taking μ̇= μ_1-μ_0 yields eq:G-minimizing. To obtain (<ref>), we differentiate the variational formula in λ and use that v is classically differentiable in λ as explained above (<ref>).It remains to prove eq:fixed-point-q. Since G is differentiable, we see thatξ”(q)/2( (∂_x v)^2(q,X_q)-q)=0.This yields eq:fixed-point-q for q≠0, and for q=0 if ξ”(0)≠ 0.To see this for the point q=0, it suffices to show that(∂_x v(0,0))^2 ≤inf(μ).To this end, let q_0=inf(μ). Observe that by eq:G-minimizing,G(q_0)≤ G(q_0+)for >0 sufficiently small. Averaging this inequality and using the definition of G, we see that there is somet∈(q_0,q_0+) such that(∂_x v(t,X_t))^2 ≤ t.As observed in eq:max-princ, ∂_x^2 u>0. So by Itô's isometry, t↦ (∂_xv)^2(t,X_t) is strictly increasing. Thus (∂_x v)^2(0,0)=(∂_x v)^2(0,X̂_0)≤ t≤ q_0+.Sending →0 yields eq:support-lower-bound. The remaining cases can be proved in an identical fashion. §.§ Γ-convergence of the local free energyWe now turn to proving the Γ-convergence of the local free energy functional. We begin by recalling the notion of sequential Γ-convergence.Let X be a topological space. We say that a sequence of functionals F_n : X → [- ∞, ∞] sequentially Γ-converges to F : X → [-∞, ∞] if* The Γ-lim inf inequality holds: For every x and sequence x_n → x,lim inf_n →∞ F_n(x_n) ≥ F(x).* The Γ-lim sup inequality holds: For every x, there exists a sequence x_n → x such thatlim sup_n →∞ F_n(x_n) ≤ F(x). For a sequence of functionals F_β indexed by a real parameter β, we say that F_β sequentially Γ-converges to F if for any sequence β_n →∞, the sequence F_β_n sequentially Γ converges to F.For every T∈[,], we have that _β, TΓ→_T. Recall from (<ref>) that we may write _β,T in the form_β,T(ν,λ)=_β,T(ν,λ; 0,0) +ℓ_1(λ)+ℓ_2(ν),where ℓ_i are both linear functionals that do not vary in β.For (t,x) ∈ [0,T]×ℝ, set _T(ν,λ;t,x) = u_ν,λ(t,x).For (t,x) ∈ [0,T]×ℝ, recall the functional ℱ_β,T(·, ·; t,x) from (<ref>), and note that Theorem <ref> immediately implies that ℱ_β,T(·, ·; 0,0) Γ→ℱ_T(·, ·; 0,0). The desired conclusion follows immediately using (<ref>) and the stability of Γ-convergence under continuous perturbations <cit.>. Our interest in the Γ-convergence of these functions is of course to understand convergence of minima and minimizers. To this end, we need a precompactness theorem for such minimizers. If T∈(,), then _β,T has a unique sequence of minimizers (ν_β,λ_β) which is precompact. Furthermore any limit point of this sequence converges to a minimizer of _T(ν,λ). If T= or , then the family ν_β is precompact, we may take λ=0, and any limit point of this sequence is such that (ν,λ) is a minimizing pair.Let us now turn to the precompactness theorem thm:conv-minimizers-thm.Before we begin the proof,we need the following theorem regarding the compactness of λ.There is a β_0(Σ) such that if β≥β_0,T∈(,), and _β,T(λ,ν)≤ M, then λ≤M+1/min{-T,T-}.Observing that the last term in (<ref>) is negative, M≥ u_ν,λ(0,0)-λ T.In this case,lem:mgf-bound implies that M ≥1/βψ(λβ)-λ T. Observe that there is a c>0 that depends only on Σ such that1/βψ(βλ)≥max{λ,λ} -c/β.Taking β_0=c^-1 and re-arranging yields the result.We are now in the position to prove the precompactness theorem. That there is a unique minimizing pair for finite β is proved in lem:optimality-finite-beta. We begin by studying the precompactness of this sequence.Let us first show that the collection of minimizing ν_β are precompact.To this end, observe that, by eq:loc-fe-conv,F(β) is the point-wise limit of free energies.As functions, these are convex and smooth in β. Furthermore, they satisfy ∂_β F(β;T) = lim_N→∞ F'_N(β;T)=lim_N→∞1/N⟨ H_N(σ)⟩≤ C(ξ).Here ⟨·⟩ denotes integration with respect to the Gibbs measure,η({σ})∝ e^β H_Nand we use thatthe expect normalized maximum of H_N is bounded by a function of ξ alone <cit.>.By thm:beta-deriv-overlap and Fubini's theorem, it then follows that∫ξ'(t)βμ(t)dt≤ C(ξ) By the Harris-FKG inequality, this implies that the total variation norm of ν_β satisfies ν_β≤ C'(ξ). Thus the minimizing ν_βare pre-compact. We now study the pre-compactness of λ_β.Suppose first that T∈(,). In this case,there is some M such thateventually _β,T(ν_β,λ_β)≤ M. Similarly by the above estimate, we may assume that ν_β≤ C. The result then follows byeq:coercive-lambda-bound. The case T= and T= are obvious. §.§ Variational representation for the Ground State EnergyWith the above in hand, the proof ofTheorem <ref> is essentially immediate. cor:gamma-conv-main-thm establishes Γ-convergence of the functional 𝒫_β,T to 𝒫_T.Furthermore, the minimizers of 𝒫_β,T are pre-compact by Theorem <ref>. Thus by the fundamental theorem of Γ-convergence, the minima converge, i.e., inf_ν∈𝒜_T, λ∈ℝ𝒫_β,T(ν, λ)→inf_ν∈𝒜_T, λ∈ℝ𝒫_T(ν,λ)as β→∞. Lemma <ref> implies 1/β F_N(β, ξ; A_N) - log |Σ|/β≤GS_N(A_N) ≤1/β F_N(β, ξ; A_N) + log |Σ| /β.We let N→∞, followed by β→∞ and use (<ref>) to derive that lim_N →∞GS_N(A_N) = inf_ν∈𝒜_T, λ∈ℝ𝒫_T(ν,λ) =: E(ξ;T).Another application of Lemma <ref> implies that |1/β F(β, ξ; T) - E(ξ; T) | ≤log |Σ|/β.Thus the family {1/βF(β , ξ; ·) : β >0 } is uniformly convergent, and thus the supremum converges, sup_T1/β F(β, ξ ; T) →sup_T E(ξ ; T)when β→∞. Finally, using Lemma <ref>, we havelim inf_N →∞ GS_N(Σ^N)≥lim inf_N →∞ F_N(β, ξ ) - log |Σ|/β≥lim inf_N →∞ F_N(β, ξ; T) - log |Σ|/β = F(β,ξ; T)- log |Σ|/β.We let β→∞ and take supremum over T to derive the lower bound lim inf_N →∞ GS_N(Σ^N) ≥sup_T E(ξ; T).To derive the upper bound, we observe thatlim sup_N →∞ GS_N(Σ^N) ≤lim sup_N →∞ F_N(β, ξ) + log |Σ|/β = F(β, ξ) + log |Σ|/β.It is easy to see that F(β, ξ) = sup_T F(β, ξ; T) and then we let β→∞ to obtain lim sup_N →∞ GS_N(Σ^N) ≤sup_T E(ξ, T).§ ANALYSIS OF THE ZERO TEMPERATURE PROBLEMIn this section, we briefly turn to calculating the first variation of the functional _T from (<ref>)Fix ν_0,ν_1∈ with ν_1({T})=ν_0({T}) and λ. Let ν_θ=(1-θ)ν_0+θν_1. We have thatd/dθ|_θ=0_T(ν_θ)=1/2∫_0^Tξ”(t)( u_x(t,X_t)^2-t)d(ν_1-ν_0). Let m_θ=(1-θ)m_0+θ m_1. Let X_T^α,m_θ be the process in lem:dpp corresponding to m_θ with initial data x=0. Consider the auxiliary function Ξ:_T×[0,1]→Ξ(α,θ) =[ψ(X_T^α,m_θ,λ+ξ”(T)/2c)-1/2∫_0^Tξ”(s)m_θ(s)α_s^2ds].Since ψ is continuous, it is clear that Ξ is jointly continuous in (α,θ).Consider the function ψ(x,y)=max_∈Σ[x+^2y ].Recall from lem:psi-diff that ψ is a.e. differentiable with a.e. continuous derivative which satisfies∂_xψ(x,y)=_*,where _* is such that ψ(x,y)=_*x+_*^2y.Thus ∂_θΞ(α,θ) is jointly continuous in the pair θ,α. By lem:dpp, if we letα_*(θ)=u_x^θ(s,X_s^θ),then these achieve optimality in (<ref>). Furthermore, the map θ↦α_*(θ) is continuous by lem:continuity. Thus by lem:wei-kuo, we have that d/dθ|_θ=0_T(ν_θ)=1/2∫_0^Tξ”(t)( u_x^2-t)(m_1(t)-m_0)dt,as desired. Our next result is a convexity property of the zero temperature functional _T, and should be compared to Lemma <ref>.For λ∈ℝ and ν∈𝒜, ν(t) = m(s)t + c δ_T,the ground state Parisi functional _T is convex in (m,c,λ). Combining Lemmas <ref> and <ref> immediately yields the following corollary.Let λ_*, ν_* be any minimizer of _T. Set ν_*(s) = m_*(s)s + c_* δ_T. Consider any measure ν_1(s) = m_1(s)s + c_* δ_T and define the path ν_θ = θν_1 + (1- θ) ν_*. Then we have, /θ^+_T(λ_*, ν_θ) |_θ=0≥ 0. § PROOF OF THEOREM <REF> We prove Theorem <ref> in this section.Recall the interpolating free energy F_N(v,β;α) from(<ref>). Further, recall that by Panchenko's theorem <cit.>, see (<ref>),F_N(v,β ; α )→ F(v,β; α),where F(v,β; α) is the local free energy (<ref>) corresponding to ξ(t)=2 t^2, Σ=Σ(v,α) andT=T(v,α).[We note here that we may take _N =0 in (<ref>). The “fattening” by _N was necessary in <cit.>, only because they needed to work with self-overlaps which were possibly un-realizable, e.g., irrational. In our setting, however, we are implicitly in the regime where the set with self-overlap T is non-empty infinitely often in N. ]Our proof of Theorem <ref> will crucially use the ground state energy functional (<ref>). We adapt the ground state energy functionalto this setting for the convenience of the reader. Fix v ∈ [0,1] and define the functional 𝒫(·, · ;v) : ℝ×𝒜_T →ℝ such that 𝒫(λ, ν; v) = - λ T + ũ_λ, ν(0,0) - 2 ∫_0^T s ν(s),where we set T:= T(v) as in (<ref>), and where forν(s) = m(s)s + c δ_T, ũ_λ,ν solves ∂_t ũ_λ, ν + 2 ( Δũ_λ,ν + m(t) (∂_x ũ_λ,ν )^2 )=0,(t,x)∈ [0,T]× ũ_λ,ν(T, x) = f̃(x,λ,c).where f̃(x,λ,c) = | x - 2 ( λ + 2c )√(v) (2 α -1) | - √(v) (2 α -1 ) x + ( λ + 2c) ( 1 + v(2 α -1)^2 ).Recall the local Parisi functional 𝒫_β,T(ν, λ) (<ref>), and the pre-compactness of its minimizers, as established in Theorem <ref>.Next, we will establish the following crucial property about the minimizers.Fix any α∈ (0, 1/2) and v ∈ (0,1). Let (ν_*, λ_*) be any limit point of the minimizers (ν_β, λ_β). Then ν_* ≠ 0. We defer the proof of this result to the end of the section. Given these results, we are now in a position to establish Theorem <ref>. ByGaussian integration of parts for Gibbs measures <cit.>, we observe that∂_β F_N(v,β ; α) =2 β[T^2 - ⟨ R_12^2 ⟩_v]. We note that for any fixed v, α, F_N( v, β; α ) is convex in β and converges to F(v, β; α ), which is differentiable in β by Theorem <ref>. Thus by Griffith's lemma for convex functions,∂_β F_N(v, β; α) →∂_β F(v, β; α). Using Theorem <ref>, we have∂_β F(v, β; α) =2 β∫ (T^2 - x^2 ) μ_β(x), where μ_β is the minimizer of the local free energy functional 𝒫_β,T. The minimizers of 𝒫(·, ·; v) are functions of v, but for ease of notation, we will keep this dependence implicit. We set ν_β(t) = βμ_β([0,t])t, where μ_β is unique minimizer of 𝒫_β,T. Let (ν_*, λ_*) be any limit point of (ν_β, λ_β).Recall that by Theorem <ref>, such a limit point exists.Using lem:convergence andLemma <ref>, we have,that for any subsequence converging to this limit point,lim_k →∞β_k∫ (T^2 - x^2) μ_β_k(x)=∫ 2 x ν_*(x)>0.This observation implies lim inf_β→∞β∫ (T^2 - x^2) μ_β(x)>0 . SettingC_1(α) = lim inf_β→∞β∫ (T^2 - x^2) μ_β(x) >0 gives us the desired constant, and completes the proof. It remains to prove Lemma <ref>. We outline this in the rest of the section.Fix v ∈ (0,1) and assume for the sake of contradiction that ( 0,λ_*)is a limit point of (ν_β,λ_β). By Theorem <ref>, (λ_*, 0) is a minimizer of 𝒫(·, ·; v) and λ_* is finite.For any probability measure μ on [0,T], we can construct the path on measures ν_θ =(1- θ) ν_1, where we set ν_1(s) = μ([0,s])s. In this case, if we apply Fubini's theorem toLemma <ref>, we obtain/θ^+𝒫( ν_θ, λ; v) |_θ=0= ∫ G_vμ, where G_v is defined asG_v(t)=∫_t^Tξ”(s)( (∂_x ũ)^2(s,X_s) - s)ds,where ũ is the solution to (<ref>) corresponding to 0.As 0 as assumed to be a minimizer, an application of Corollary <ref> implies∫ G_v μ≥ 0. Further, μ is an arbitrary probability measure on [0,T],and thus G_v(s ) ≥ 0 on [0,T]. We will establish that G_v(T) = G'_v(T) = 0 whilelim_t ↑ T G”_v(t) = ∞. Thus G_v(t) is negative for t sufficiently close to T, and this yields a contradiction.To this end, we note that the definition of G_v immediately implies that G_v(T) =0. We will next establish that [(∂_xũ)^2(T, B_4T)] =T,which implies G'_v(T) =0. To this end, note that the weak derivatives of ũ satisfy the relation(∂_x ũ)^2 (T ,x) = ∂_λũ(T,x).Now, note that λ_β are pre-compact, and thus bounded, implying that λ_* is finite.Differentiating the functional in λ as in lem:optimality-finite-beta, and setting this to zero,we obtain T = ∂_λ_*ũ(0,0) = [∂_λ_*ũ(T, B_4T) ] = [(∂_x ũ)^2(T, B_4T)],where the second equality follows from the observation that ∂_λ_*ũ satisfies the heat equation with boundary data ∂_λ_*ũ(T, x). Finally, we prove that lim_t ↑ T G_v”(t) = ∞. We have, for t < T, G_v”(t) = 4 ( / t [( ũ_x)^2(t, B_4t)] - 1 ).Using Itô Lemma's we immediately obtain that for such t,/ t[(ũ_x)^2(t, B_4t)] =4 [ (ũ_xx)^2(t,B_4t)].Note that ũ_xx solves the heat equation ( ∂_t+4/2Δ )ũ_xx = 0ũ_xx(T, · ) =2δ_2 λ_* a(v)(·)in the sense of distributions, where a(v) = √(v)(2α -1). In particular, by a standard argument <cit.> (or an explicit computation) we have,ũ_xx(t,x) = 2/√(8π (T-t) )exp( - (x- 2 λ_* a(v) )^2/8(T-t)),for t < T. Finally, this immediately implies [(ũ_xx)^2(t, B_4t)] = 1/2π (T-t)[exp( - (B_4t - 2 λ_* a(v))^2/4(T-t)) ] = 1/2 π√(T^2 - t^2)exp(-( λ_* a(v) )^2 /T+t).We let t ↑ T to complete the proof. § A CONJECTURE REGARDING THE SHARP CONSTANT IN THM:CUT_DIFFERENCEIn this section, we record the conjecture regarding the sharp constant in Theorem <ref>. To this end , define Σ(v,α)= { 1- √(v) ( 2 α -1) , - 1 - √(v) (2α -1) },T(v,α)= α (1 - √(v) (2 α -1 ))^2 + (1 - α) (1 + √(v)(2 α -1 ))^2 ,and then wehave the following. We have thatlim_d→∞lim_N→∞_α(G(N,d/N))-_α(d)/√(d)N=K(α),whereK(α)=2(2α-1)^2∫_0^1 (1 -√(v))/√(v)ν_v([0,T(v,α)])dv.and where ν_v is a minimizer of _T(v,α). The motivation for this conjecture is as follows. By lem:int_by_parts one formally expects thatlim_d→∞lim_N→∞ _α(G(N,d/N))-_α(d)/√(d)N=lim_β→∞lim_N→∞ 2(2α-1)^2∫1-√(v)/√(v)βT(v)-R_12_v dv,In the physics literature, a basic tenet of the replica symmetry breaking method is<cit.> that in generic situations we have the correspondencelim_N→∞T-R_12=β∫ (T-x)dμ_β(x),where μ_β is such that βμ_β dt the minimizer of _β,T from (<ref>).The conjecture then comes from combining this correspondence with thm:conv-minimizers-thm.The question as to when this correspondence holds is a major open problem in the mathematical study of mean field spin glasses. For references in this direction see <cit.>.§ BASIC PROPERTIES OF THE PARISI PDEIn this section, we briefly review the properties of solutions to Parisi-type PDEs. There is a unique weak solution to eq:zero-temp-pde, which satisfies* ∂_x u∈ L_t^∞L_x^∞ with ∂_x u_L_t,x^∞≤∂_x f_L_t,x^∞≤ C(Σ)* For any T_0<T, u is continuous in space time, with smooth bounded spatial derivatives satisfying ∂_x^nu_L^∞([0,T_0]×ℝ )≤ C(T_0,T,Σ)and is weakly differentiable in time with ∂_t∂_x^nu_L^∞([0,T_0]×ℝ)≤ C(T_0,T,Σ).Furthermore, if u,v are two solutions corresponding to μ and ν respectively where μ,ν∈ are of the form μ=m_1dt+cδ_1 ν=m_2dt+cδ_1thenu-v_L_t,x^∞≤ C(ξ)m_1-m_2_L^1and we have the parabolic comparison principle:if m_1≤ m_2 pointwise for all t then u≤ v. Finally, the same results hold for the weak solutions u to (<ref>). This is a standard argument, see, e.g., <cit.>. For the reader's convenience we briefly sketch the main points. We begin first with the existence for a dense class of ν's. Assume that ν=m(t)dt+cδ_1. The existence for m which is a bounded step function with finitely many jumps, can be seen by an application of the Cole-Hopf transformation. That the derivative is bounded in space for such solutions can be seen either by explicit differentiation or the maximum principle. This yields∂_x u_L_t,x^∞≤∂_xf_L^∞_x.By lem:psi-diff,f is differentiable in x Lebesgue a.e.and∂_xf ∈ L^∞. Furthemore,∂_xf=argmax_ϵ∈Σ{ xϵ+(λ+c/2ξ”(T))ϵ^2}a.e. which is bounded by a constant that depends at most on Σ. Observe furthermore, that the regularity claims in this setting can be seen by explicit differentiation. We now prove the Lipschitz estimate in m for m as above and the corresponding comparison principle. This follows by the same argument as in <cit.>. Indeed, if w=u-v then w solves w_t+ξ”/2(w_xx+m_1(u_x+v_x)w_x+(m_1-m_2)v_x^2) =0with initial data w(T,x)=0. Since u_x,v_x are uniformly lipschitz on any subinterval of the form [0,T_0]⊂[0,T), and m_i(t) are both uniformly bounded on such sub intervals, we see that we may solve the SDE dX_t=ξ”m_11/2(u_x+v_x)(t,X_t)+√(ξ”)dW_t,where W_t is a standard Brownian motion. Observe that w has the same regularity as u and v. In particular, by the smoothing property of the heat equation, we have that on [0,T_0]× w,w_x,w_xx∈ C_b([0,T_0],), and w is weakly differentiable in time with w_t∈ L^∞ which is Lipschitz in x uniformly in t. Thus we may apply Itô's lemma (see, e.g., <cit.>) to obtain w(t,x)=_X_t=x(∫_t^T_01/2ξ”(s)(m_1-m_2)v_x^2)≤ C(Σ,ξ)m_1-m_2_L^1.Similarly, sending t→0 and T_0→ T, yields the desired conclusions. We now show the existence for general m. By an extension argument, if m_n→ m in L^1, then u_n→ u for some function u. To see that u is a weak solution observe that it suffices to show that m_n(∂_xu_n)^2→ m(∂_xu)^2 in the sense of distributions. This follows since ∂_xu_n are uniformly bounded. To prove uniqueness, observe that by a similar Duhamel's principle argument to <cit.> using the modified heat kernel estimates from <cit.>, we have that ∂_xw_L^∞([t,T]×ℝ)≤u_x+v_x_L^∞_t,x∫_t^T∂_xw_L^∞([s,T] ×ℝ)C(Σ,ξ)/√(s-t)m(s)ds.Since m(t) is monotone and blows up at most at T, the integrand is integrable, so that by Gronwall's inequality, ∂_xw=0. Thus we have the existence and uniqueness of u and the regularity of u_x. The regularity of the higher spatial derivatives follows by the smoothing property of the heat semigroup, and the regularity in time follows from rearranging eq:zero-temp-pde and using the fact that on T_0<T, m is bounded. The map ν↦ u_x is continuous in the topology of pointwise convergence. In particular, if ν_n→ν weak-*, then u^n→ u and ∂_xu^n→∂_xu uniformly on compacta. By the same Duhammel's principle and parabolic Bootstrapping argument as in<cit.>, there are a continous function {F_i(x,y)}_i∈[3] such that for every T_0<T, ∂_x^iu_L^∞([0,T_0]×) ≤ F_i(ν(T_0),ν({1})).Since the maps ν↦ν([0,t]) and ν↦ν({1}) are upper semi-continuous in the weak-* topology on , we see that if ν_n→ν, weak-* the family {u^n} and {∂_xu^n} are uniformly lipschitz in space. Furthermore, since they weakly solve the Parisi PDE and the (spatially) differentiated form of this equation, we see that they are also uniformly lipschitz in time. Thus the families are both equicontinuous. Thus u^n→ u and ∂_xu^n→∂_xu uniformly on compacts by the Arzela-Aiscoli theorem. § THE WEAK-* TOPOLOGY ON _TIn the preceding section, we frequently work with the space _T equipped with the weak-* topology. We provide here certain basic properties of this space.Suppose that ν_β→ν with ν_β = m_β(t)dt+c_βδ_T and ν=m(t)dt+cδ_T. We then have the following. (i) The measure dm_β converges vaguely to dm. That is, for every t<T,that is a continuity point of m,m_β(t)→ m(t). (ii) Let q_β→1be such that m_β(q_β^-)→ m(1^-), where the - denote the left limit, and suppose that m(1^-)<∞. Thenc = lim_[q_β,T]β(T-t)dm_β.(iii) For any f∈ C^1([0,T]),limβ∫ f(T)-f(t)dm_β = ∫ f' dν. (iv) For any bounded Borel measurable ψ with lim_s↑ Tψ(s)=ψ(T),∫ψ(s)dν_β(s)→∫ψ(s)dν(s). The first three points were proved in <cit.>. It remains to prove the last point. Let ψ be as above.Without loss of generality, assume that ψ_∞≤1. Observe that by (1), since ν_β→νweak-*, m_β(t) → m(t) Lebesgue almost surely on [0,T). Furthermore,for any t ∈ [0,T), lim sup_β→∞ m_β(t) ≤ m(t).Thus, for any t ∈ [0, T),we have that, ∫_0^t ψ(s) m_β(s)s →∫_0^t ψ(s) m(s)s, as β→∞, by the dominated convergence theorem. Therefore, | ∫_0^Tψ(s) ν_β(s) - ∫_0^Tψ(s) ν(s) | ≤|∫_0^tψ(s) ν_β(s) - ∫_0^tψ(s) ν(s)| + |∫_t^Tψ(s) ν_β(s) - ∫_t^Tψ(s) ν(s) |.By (<ref>),lim sup_β→∞| ∫_0^Tψ(s) ν_β(s) - ∫_0^Tψ(s) ν(s) | ≤lim sup_β→∞|∫_t^Tψ(s) ν_β(s) - ∫_t^Tψ(s) ν(s) |.Finally, by triangle inequality, we have, |∫_t^Tψ(s) ν_β(s) - ∫_t^Tψ(s) ν(s) | ≤ν_β([0,T]) max_t ≤ s ≤ T | ψ(s) - ψ(T)| + | ν_β((t, T])- c | + ∫_t^T m(s)s.We let β→∞, and let t ↑ Tto conclude that |∫_t^Tψ(s) ν_β(s) - ∫_t^Tψ(s) ν(s) |→ 0as β→∞. This completes the proof.§ CONSTRAINED GROUND STATES ON THE DISCRETE HYPERCUBEIn this section, we derive a variational representation for the ground state of a mixed Ising spin glass, with an additional magnetization constraint. For σ∈{± 1}^N, we set m(σ) = 1/N∑_iσ_i.For any a ∈[-1,1]∩ℚ and ϵ_N→0, we have, for ξ convex,lim_N→∞max_σ∈{±1}^Nm(σ)∈[a-ϵ_N,a+ϵ_N] H(σ)/N=inf_ν,h[𝒫^2(ν,h)-a· h],where 𝒫^2 is given in (<ref>).The proof of this result follows in a few steps. For a∈[-1,1], defineG_N(a;η) =max_σ∈{±1}^Nm ∈ [a- η, a +η] H(σ)/N, E_N(h) =max_σ∈{±1}^N[H(σ)/N+hm(σ)].As shown in <cit.>, we have lim_N →∞ E_N(h)=E(h)=inf_ν𝒫^2(ν,h).We also have the following result which is folklore in the spin glass literature. We include a proof for the convenience of the reader. For any η>0 and a∈[-1,1], we have lim_η→0lim_N→∞G_N(a;η)=G(a)is well defined. Furthermore, G(a) is concave and continuous for all a∈[-1,1]. The result follows by a modification of the Guerra-Toninelli argument <cit.>. We first show existence. Fix η>0, and considerG_N+M(a;η)=max_m∈ [a- η, a+ η]H_N+M(σ)/N+M.For N,M sufficiently large, {σ∈{± 1}^N+M:m(σ) ∈ [a-η, a+ η] } is non-empty, so that this is well defined.We decompose σ∈{± 1}^N + M as σ = (ρ, ϵ), with ρ∈{± 1}^N, ϵ∈{± 1}^M. Consider now the interpolating Hamiltonian for σ=(ρ,ϵ), defined byH_N+M,t(ρ,ϵ)=√(t)H_N+M(σ)+√(1-t)(H_N(ρ)+H_M(ϵ)),for t∈ [0,1], where we view the Hamiltonians at different N's to be independent. For σ^i=(ρ^i,ϵ^i) with i=1,2, let R_12=1/N+M(σ^1,σ^2), R_12^ρ=1/N(ρ^1,ρ^2), R_12^ϵ=1/M(ϵ^1,ϵ^2).For any β>0 defineϕ_β(t)=1/β(N+M)log∑_m∈ [a-η, a+η]e^β H_N,t(σ).This satisfiesϕ'_β(t)=1/N +M⟨∂_tH_N,t⟩ =β/N+M ⟨ C_11-C_12⟩with C_12=[∂_tH_t(σ^1)H_t(σ^2)]=(N+M)ξ(R_12)-Nξ(R_12^ρ)-Mξ(R_12^ϵ),where the second equality follows using Gaussian integration by parts. Note that C_11=0. Furthermore, since R_12=λ R_12^ρ+(1-λ)R_12^ϵ with λ=N/N+M, we have ξ(R_12) ≤λξ(R_12^ρ)+(1-λ)ξ(R_12^ϵ)by convexity of ξ. Thus ϕ'_β(t)≥ 0, so that ϕ_β(0)≤ϕ_β(1). Sending β→∞, this yieldsmax_m∈ [a-η, a+ η]H_N+M(σ)/N+M≥max_m∈ [a- η, a+ η]H_N+H_M/N+MNow note that since m(σ)=λ m(ρ)+(1-λ)m(ϵ), we have that {σ=(ρ,ϵ):m(ρ)∈ [a-η, a+ η],m(ϵ)∈ [a-η,a+η]}⊆{σ:m(σ) ∈ [a-η, a+η]}Consequently, G_N+M(a;η)≥λ G_N(a;η)+(1-λ)G_M(a;η). Thus the sequence NG_N(a;η) is super-additive, so that G_N(a;η) has a limit.To establish continuity of G(·; η), fix a,b ∈ [-1,1], and η >0. For each σ∈{ m(σ) ∈ [a-η, a+ η]}, let π(σ) denote a configuration in {m(σ) ∈ [b - η, b+ η]} which minimizes the Hamming distance. Similarly, for each σ∈{m(σ) ∈ [b-η, b+ η]}, let π'(σ) denote the configuration with magnetization in [a-η, a+ η] which minimizes the Hamming distance. Recall that Var(H_N(σ) - H_N(τ) )= 2N (ξ(1) - ξ(R_12 )), where N · R_12 = (σ, τ). Thus 1/Nmax{max_m∈[a-η, a+ η]Var( H_N(σ) - H_N(π(σ))) , max_m∈[b-η, b+ η]Var( H_N(σ) - H_N(π'(σ))) }≤ C(ξ') |a-b|uniformly in N, where C(ξ') is a universal constant dependent on ξ'.We have, |G_N(a;η) - G_N(b; η)| ≤1/Nmax_m∈[a-η, a+ η] | H_N(σ) - H_N(π(σ)) | + 1/N𝔼max_m∈[b-η, b+ η]| H_N(σ) - H_N(π'(σ)) |.Note that for any collection of dependent, centered gaussians { Z_i: 1≤ i ≤ r}, 𝔼[max |Z_i|] ≤ 2 𝔼[max Z_i] ≤ 2 √(2 σ^2 log r), where σ^2 = max_1≤ i ≤ rVar(Z_i). Applying this bound individually to the two terms in the display above, with r ≤ 2^N, we get |G_N(a;η) - G_N(b; η)| ≤ C'(ξ') √(|a-b|) for some universal constant C'(ξ')>0, uniformly in N.Finally, we let N →∞, and obtain the continuity of G(·;η).To obtain concavity, let λ∈[0,1] be rational and work along a subsequence such that λ N is an integer. By the same interpolation argument, we obtain that G_N(λ a+(1-λ)b;η)≥λ G_λ N(a;η)+(1-λ)G_(1-λ)N(b;η).Sending N→∞ yieldsG(λ a+(1-λ)b; η)≥λ G(a; η)+(1-λ)G(b; η).Using the continuity of G(·; η), we obtain the concavity ofG(·;η) for each η. Finally, since the map η↦ G_N(a;η) is increasing, we see η↦ G(a;η) is as well. Thus the pointwise limit G(a)=lim_η→0 G(a;η) is well-defined, continuous, and concave. Since G(a) is continuous, we may apply a standard covering argument (along with the definitions of G_N and E_N) to obtainE(h)=max_a∈[-1,1]{ha+G(a)}.Extending G(a) by -∞ off of [-1,1], we have, by concavity of G(a), G(a)=inf_h[E(h)-ha].The result then follows by (<ref>).plain | http://arxiv.org/abs/1707.09042v3 | {
"authors": [
"Aukosh Jagannath",
"Subhabrata Sen"
],
"categories": [
"math.PR",
"math.CO"
],
"primary_category": "math.PR",
"published": "20170727205702",
"title": "On the unbalanced cut problem and the generalized Sherrington-Kirkpatrick model"
} |
[email protected] prove that the category of (rigidified) Breuil-Kisin-Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil-Kisin-Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a “p-adic Serre group”. Breuil-Kisin-Fargues modules with complex multiplication Johannes Anschütz December 30, 2023 ========================================================§ INTRODUCTION In <cit.> L. Fargues introduced an analog, called Breuil-Kisin-Fargues modules, of Breuil-Kisin modules (cf.<cit.>) over Fontaine's first period ringA_inf=W(𝒪_C^♭)of Witt vectors of the ring of integers 𝒪_C^♭ inside C^♭ where C denotes a non-archimedean, complete and algebraically closed extension of _p and C^♭=_x↦ x^pC its tilt. To define Breuil-Kisin-Fargues modules let ξ∈ A_inf be a generator of the kernel of Fontaine's mapθ A_inf→𝒪_C(cf. <Ref>) and letφ A_inf→ A_inf be the Frobenius of A_inf (induced by the Frobenius of 𝒪_C^♭). Concretely, a Breuil-Kisin-Fargues modules (M,φ_M) is then a finitely presented A_inf-module M together with an isomorphismφ_Mφ^∗(M)[1/φ(ξ)]≅ M[1/φ(ξ)]such that M[1/p] is finite projective over A_inf[1/p] (cf. <Ref>). The study of Breuil-Kisin-Fargues modules, which are mixed-characteristic analogs of Drinfeld's shtuka, was taken over in <cit.> and <cit.>. More precisely, in <cit.> to every proper smooth formal scheme 𝔛 over 𝒪_C and every i≥ 0 there is associated a Breuil-Kisin-Fargues moduleH^i_A_inf(𝔛)interpolating, at least rationally, various cohomology groups attached to 𝔛. Namely, Breuil-Kisin-Fargues modules admit various realizations (cf. <Ref>): Let (M,φ_M) be a Breuil-Kisin-Fargues module and denote by k the residue field of 𝒪_C. We can associate to M * its “étale realization” T:=(M⊗_A_infW(C^♭))^φ_M=1, a finitely generated _p-module,* its “crystalline realization” D:=M⊗_A_infW(k), a finitely generated W(k)-module equipped with a φ-semilinear isomorphism φ_D:φ^∗(D[1/p])≅ D[1/p] after inverting p,* and its “de Rham realization” V:=M⊗_A_inf,θ𝒪_C,a finitely generated 𝒪_C-module. If (M,φ)=H^i_A_inf(𝔛) for a proper smooth formal scheme 𝔛/𝒪_C, then the main result of <cit.> shows that these various realizations are, at least after inverting p, given by the étale cohomology H^i_ét(X,_p) of the generic fiber X:=𝔛_C of 𝔛, the crystalline cohomology H^i_(𝔛_0/W(k))[1/p] of the special fiber 𝔛_0⊆𝔛, and the deRham cohomology H^i_(X/C) of X. Thus we see that Breuil-Kisin-Fargues modules have a “motivic flavour”. But the general picture surrounding motives is looking for the existence of a tensor functor from smooth projective schemes into some Tannakian category, and the category of Breuil-Kisin-Fargues modules is not Tannakian. In fact, it is even not abelian (cf. <cit.>). To remedy this we follow an idea in <cit.> and introduce rigidifications of Breuil-Kisin-Fargues modules. Fix a section k→𝒪_C/p of the projection 𝒪_C/p→ k. If (M,φ_M) is a Breuil-Kisin-Fargues module then a rigidification for (M,φ_M) (cf. <cit.> and <Ref>) is an isomorphismα M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_of φ-modules over B^+_ inducing the identity when base changed to W(k)[1/p]. The first main theorem of this paper is the following (the result is already stated in <cit.>).The categoryBKF_rigof rigidified Breuil-Kisin-Fargues modules is abelian.We remark that the analogous statement for Breuil-Kisin modules is true, but much simpler. The problems for Breuil-Kisin-Fargues modules arise as the ring A_inf is highly non-noetherian. But luckily we can profit from <cit.>, where enough commutative algebra of A_inf-modules is developed. From <Ref> it is not difficult to deduce that the _p-linear categoryBKF_rig^∘:=_p⊗__pBKF_rigof rigidified Breuil-Kisin-Fargues modules up to isogeny is Tannakian (cf. <Ref>). The statement is known for Breuil-Kisin modules, and again it is much simpler. We investigate the Tannakian category BKF_rig^∘ a bit: for example, we prove that it is “connected” (cf. <Ref>) and of homological dimension 1 (cf. <Ref>).In <Ref> we start to classify rigidified Breuil-Kisin-Fargues modules admitting “complex multiplication”, i.e., Breuil-Kisin-Fargues modules whose Mumford-Tate group in the Tannakian category BKF_rig^∘ is a torus (cf. <Ref> and <Ref>). More concretely, a rigidified Breuil-Kisin-Fargues module (M,φ,α) up to isogeny admits CM if and only if there exists a commutative, semisimple _p-algebra E and an injection E↪End_BKF_rig^∘((M,φ,α)) of _p-algebras such thatdim__pE=rk_A_inf(M) (cf. <Ref>). Using the crucial theorem about different descriptions of (finite free) Breuil-Kisin-Fargues modules (cf. <Ref>) due to Fargues/Kedlaya-Liu/Scholze we can then prove our main theorem about the classification of Breuil-Kisin-Fargues modules admitting CM.For every finite dimensional commutative, semisimple _p-algebra E there exists a (natural) bijection between isomorphism classes of (rigidified) Breuil-Kisin-Fargues modules (up to isogeny) admitting CM by E and functions ΦHom__p(E,C)→. Moreover, we can write down for given pair (E,ΦHom__p(E,C)→) the corresponding (rigidified) Breuil-Kisin-Fargues module (up to isogeny) explicitly (cf. <Ref>).We remark that a similar result has been obtained by Lucia Mocz in the case of Breuil-Kisin modules associated with p-divisible groups (cf. <cit.>).Finally, let 𝒯⊆BKF^∘_rig be the full Tannakian subcategory spanned by rigidified Breuil-Kisin-Fargues modules up to isogeny admitting CM and let D__pbe the pro-torus over _p with group of characters the coinduced discrete Galois moduleX^∗(D__p)=Coind^Gal(_p/_p)(cf. <Ref>). We prove the following description of the category 𝒯 of Breuil-Kisin-Fargues modules admitting CM.The étale realization defines an equivalence𝒯≅Rep__p(D__p)of Tannakian categories. This theorem can be understood as the computation of a “p-adic Serre group”, analogous to the case of rational Hodge structures (cf. <cit.>).In order to prove <Ref> we develop in <Ref> some language concerning CM-objects and reflex norms in an arbitrary Tannakian category to formalize the known case of rational Hodge structures admitting CM.§.§ AcknowledgementThe author wants to thank Lucia Mocz heartily for sharing her notes <cit.> on Breuil-Kisin modules with CM, which eventually led to the question of defining and classifying Breuil-Kisin-Fargues modules with CM answered in this paper. Moreover, the author wants to thank Bhargav Bhatt, Peter Scholze and Sebastian Posur for discussions surrounding this paper. Especial thanks go to Peter Scholze for providing the hint to the remark following <cit.> (i.e., <Ref>) which lead to the construction of the Tannakian category BKF^∘_rig.§ FORMAL CM-THEORYIn this section we write down a general theory of “CM-objects” in a Tannakian category 𝒯, mainly fixing terminology. Concretely this means that we reformulate the maximal torus quotient of the band of the category 𝒯 internal in 𝒯. We will apply this theory to the case that 𝒯=BKF^∘_rig is the category of rigidified Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>). The formalism of this section is a straightforward translation of CM-theory for rational Hodge structures or categories of (CM) motives to the case of general Tannakian categories. Therefore we expect it to be known in principle and do not claim any originality. We advise the reader to have a look at <cit.>.Let k be a field (later it will assumed to have characteristic 0) and let 𝒯 be a Tannakian category over k (not necessarily assumed to be neutral). For an object X∈𝒯 we denote by ⟨ X⟩^⊗ the full Tannakian subcategory spanned by X.An object X∈𝒯 is called a CM-object, or to admit CM, if the connected component of the band of the Tannakian category ⟨ X⟩^⊗ is multiplicative, i.e., for every fiber functorω⟨ X⟩^⊗→Bun_S,where S/k is a scheme, the connected component G^∘ of the group schemeG:=Aut^⊗(ω)is a multiplicative group scheme over S.Equivalently, the condition can be required only for the case S=(k^') is the spectrum of a field extension k^'/k, or even only for one fiber functor over some field extension k^'/k. Moreover, if k^'/k is a finite field extension and 𝒯_k^' the base change of 𝒯 from k to k^' (cf. <cit.>), then X∈𝒯 is a CM-object if and only if k^'⊗_k X∈𝒯_k^' is a CM-object, because a fiber functor ω⟨ X⟩^⊗→Vec_k^'' where k^'' is a field extension of k containing k^' extends to a fiber functor ω^'⟨ k^'⊗_k X⟩^⊗ (cf. <cit.>).If k is of characteristic 0, then a connected multiplicative group scheme of finite type is automatically a torus. Definition <Ref> is a formalization of the definition of a CM-rational Hodge structure (cf. <cit.>). Namely, let V be a (polarisable) rational Hodge structure and let 𝕊:=Res_ℂ/ℝ𝔾_m be the Deligne torus. Then V is called of CM-type if the “Mumford-Tate group” of V, i.e., the minimal closed subgroup G⊆GL(V) overcontaining the image of the morphism h𝕊→GL(V) induced by the Hodge structure V_ℂ=⊕_p,q∈V^p,q on V, is a torus (cf. <cit.>). This definition agrees with ours as the Mumford-Tate group of V is precisely the automorphism group of the canonical fiber functorω⟨ V⟩^⊗→Vec_. Let T/k be an affine group scheme of finite type which is an extension of a finite discrete group G and multiplicative group T^0, i.e., there exists an exact sequence (of fppf-sheaves)1→ T^0→ T→ G→ 1.Then every V∈_k(T) is a CM-object.Let V∈_k(T) and set 𝒯:=⟨ V⟩^⊗ to be the Tannakian subcategory generated by V. Let ω𝒯→Vec_k be the restriction of the canonical fiber functor. Then the canonical morphismT→Aut^⊗(ω)is faithfully flat and of finite presentation. This implies that this morphism is open and hence the connected component T^0 of T surjects onto the connected component Aut^⊗(ω)^∘ of Aut^⊗(ω). In particular, the group Aut^⊗(ω)^∘ is multiplicative, i.e., V is a CM-object.Let 𝒯 be a Tannakian category over k and let X∈𝒯 be a CM-object. Then every Y∈⟨ X⟩^⊗ is again a CM-object. Moreover, if X,Y∈𝒯 are CM-objects, then X⊕ Y and X⊗ Y are CM-objects as well. For the first statment we may replace 𝒯 by ⟨ X⟩^⊗, and, after enlarging k, assume moreover that 𝒯 is neutral. As X is a CM-object, the category 𝒯≅_k(T) is thus equivalent to the category of representations of an affine group scheme T, which is an extension of a finite discrete group by a multiplicative group as in <Ref>. <Ref> implies that Y∈𝒯=⟨ X⟩^⊗ is again a CM-object.Now let X,Y∈𝒯 (with 𝒯 arbitrary) be CM-objects. Then X⊗ Y∈⟨ X⊕ Y⟩^⊗ and thus it suffices to proof that X⊕ Y is again a CM-object by what has already been shown. But if ω⟨ X⊕ Y⟩^⊗→Bun_Sis a fiber functor, and we let ω_1, resp. ω_2, be the restrictions of ω to ⟨ X⟩^⊗⊆⟨ X⊕ Y⟩^⊗, resp. ⟨ Y⟩^⊗⊆⟨ X⊕ Y⟩^⊗, then the canonical morphismAut^⊗(ω)↪Aut^⊗(ω_1)×Aut^⊗(ω_2)is a closed immersion. In particular, the connected component of Aut^⊗(⊗) is again multiplicative. Let 𝒯 be a Tannakian category over k. We denote by 𝒯_CM⊆𝒯 the full subcategory of CM-objects of 𝒯. By <Ref>, the category 𝒯_CM is a Tannakian subcategory. In general, it is not closed under extensions, e.g., for representations of unipotent groups.Let η𝒯→𝒯^' be an exact tensor functor and let X∈𝒯 be a CM object. Then η(X)∈𝒯^' is a CM object. In particular, η induces an exact tensor functor η𝒯_CM→𝒯^'_CM on the full subcategories of CM-objects.Let ω^'⟨η(X)⟩^⊗→Vec_k^'be a fiber functor where k^'/k is a field extension. Set ω:=ω^'∘η. The morphismη^∗ H:=Aut(ω^')→ G:=Aut(ω)is then injective. As the connected component of G is multiplicative, the connected component H^∘ of H will therefore be multiplicative as well. The following direct corollary can be useful to prove that certain Breuil-Kisin-Fargues modules admit CM.Let T/k be a torus and let ω_k(T)→𝒯 be an exact tensor functor. Then for every object V∈_k(T) the object ω(V) has CM. This is a special case of <Ref> as any object in the category Rep_k(T) of representations of T admits CM (cf. <Ref>). We now want to give a more explicit definition of CM objects. From now on assume that k has characteristic 0. Recall that every object X∈𝒯 in a Tannakian category 𝒯 has a rankrk(X)∈End(1_𝒯)≅ kdefined as the trace of the identity on X. In general, the rank is an endomorphism of the unit object 1_𝒯. But as we assumed that k is of characteristic 0, the rank of X equals the dimension (over k^') of ω(X) for one, or equivalently any, fiber functor ω𝒯→Vec_k^' for k^'/k some field extension.We call a Tannakian category 𝒯 connected if its band is connected. For a neutral Tannakian category 𝒯≅_k(G), where G/k is an affine group scheme, this is equivalent to saying that G is connected.Let 𝒯 be a neutral, connected Tannakian category over k (where k is assumed to have characteristic 0). Then an object X∈𝒯 admits CM if and only if there exists a commutative, semisimple k-algebra E of dimension rk(X) and an injection (of k-algebras)E↪End_𝒯(X). Assume that X is a CM-object. Then, replacing 𝒯 by ⟨ X⟩^⊗, we may assume that 𝒯≅_k(T) for T/k a torus (as 𝒯 is neutral and connected). In other words, X corresponds to a representation V of the torus T. Decomposing X into a sum of simple objects, we may assume that X is simple. Over an algebraic closure k of k the representation X splits into a direct sum X⊗_kk≅ X_χ_1⊕…⊕ X_χ_n of eigenspaces for distinct characters χ_i T_k→𝔾_m. As X is simple all eigenspaces over k must be one-dimensional. In particular, we see that the endomorphism algebra End_𝒯(X) of X, the formation of which commutes with extensions of k, is a commutative semisimple algebra of dimension the rank of X. Actually, End_𝒯(X) is a field in this case as X is simple.Conversely, assume that there exists an injection E↪End_𝒯(X) of a commutative, semisimple k-algebra E such that dim_kE=rk(X). We may pass to a finite extension of k (cf. the remarks after <Ref>) and assume that E≅∏_i=1^nk is isomorphic to copies of k. The idempotents in E define a decomposition X≅⊕_i=1^n X_i with X_i of rank 1 over k. Hence, replacing X by one of the X_i, we may assume that E=k and rk(X)=1 (cf. <Ref>). But as 𝒯 is neutral, i.e., 𝒯≅_k(G) for some affine group scheme G, a representation ρ G→𝔾_m of rank 1 defines a CM-object X∈𝒯 because the automorphism group scheme of the canonical fiber functor on ⟨ X⟩^⊗ is given by the image ρ(G) of ρ which is multiplicative. In general it can happen that for a simple CM-object X in a (non-neutral) Tannakian category in characteristic 0 the endomorphismsEnd_𝒯(X)do not form a field. For example, this happens if 𝒯 is the category of isocrystals over _p. We record the following terminology. If X∈𝒯 is an object in the Tannakian category 𝒯 and ι E↪End_𝒯(X) an injection of a commutative semisimple k-algebra E such that rk(X)=dim_kE, we say that X admits CM by E.Now we assume that the Tannakian category 𝒯 is neutral (of characteristic 0) and we fix a fiber functorω_0𝒯→Vec_k.Moreover, we assume that for some field extension C/k the base extensionω_0⊗_k C𝒯→Vec_Cis equipped with a filtration, i.e., we fix a filtered fiber functorω𝒯→FilVec_Csuch that ω≅ω_0⊗_k C. We assume furthermore that C is algebraically closed. Of course this situation models the case of rational Hodge structures. But it will apply as well to the case of (rigidified) Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>).In this situation we can define the type of a CM-object.Let E/k be a commutative, semisimple k-algebra and let 𝒯 be a Tannakian category over k, equipped with fiber functors as above. For a pair (X,ι) with X∈𝒯 an object, necessarily admitting CM, and an injection ι E→End_𝒯(X) such that n:=rk(X)=dim_kE we define the type ΦHom_k(E,C)→of (X,ι) to be the unique function such that for every i∈gr^i(ω(X))≅∏_τ∈Φ^-1(i)C_τas a representation of E⊗_kC≅∏_τ∈Hom_k(E,C) C_τ.If E/k is a commutative semisimple algebra and ΦHom_k(E,C)→ a function, then we call (E,Φ) a CM-type (over k). Note thatHom_k(E,C)≅Hom_k(E,k)where k⊆ C denotes the algebraic closure of k in C. In particular, the Galois group Gal(k/k) acts naturally on the setHom_k(E,C). Let (E,Φ) be a CM-type. Then the reflex field E_Φ⊆ C of (E,Φ) is defined to be the fixed field E_Φ⊆k of the stabilizer of ΦHom_k(E,C)→. If we write E=∏ E_i as a product of fields and thus accordingly Φ=∐Φ_i for functions ΦHom_k(E_i,C)→, then the reflex field E_Φ of (E,Φ) is the composite (in k) of the reflex fields E_Φ_i of the CM-types (E_i,Φ_i).Let E be a commutative semisimple algebra over k and set T:=Res_E/k𝔾_m to be the Weil restriction of the torus 𝔾_m over E. Then the groupX_∗(T):=Hom_k(𝔾_m,k,T_k)of cocharacters of T is isomorphic (as a Galois module) to the module{ΦHom_k(E,C)→}of types Φ. Indeed, given a type ΦHom_k(E,C)→ we get the associated cocharacterμ_Φ𝔾_m,k→ T_k≅∏_τ∈Hom_k(E,C)𝔾_m,k, t↦ (t^Φ(τ))_τof T over k. By definition the reflex field E_Φ of the type (E,Φ) is the minimal subfield of k over which the cocharacter μ_Φ is defined. In particular, we obtain a cocharacterμ_Φ𝔾_m,E_Φ→ T_E_Φ.In the end, we obtain the reflex norm of (E,Φ) as the compositionr_ΦRes_E_Φ/k(𝔾_m)Res_E_Φ/k(T_E_Φ) Twhere the second morphism denotes the natural norm map.Let L/k be a finite field extension contained in k. Then we denote byL^∗:=Res_L/k(𝔾_m)the Weil restriction of the multiplicative group 𝔾_m over L to k. The character group of the torus L^∗ is naturally isomorphic to the Galois module[Hom_k(L,k)].Concretely, let τ L→k be an embedding (over k) and let R/k be a k-algebra. Then the character χ_τ L^∗_k→𝔾_m,k is given on R-points byL^∗_k(R)=(R⊗_k L)^×(R⊗_kk)^×R^×where the right arrow denotes the multiplication R⊗_kk→ R of the k-algebra R.Let k⊆ L_1⊆ L_2⊆k be a tower of field extensions. Then there are natural norm mapsN_L_2/L_1 L_2^∗→ L_1^∗which on R-valued points for a k-algebra R are given byL_2^∗(R)=(R⊗_k L_2)^× →(R⊗_k L_1)^×=L_1^∗(R)x ↦ det_R⊗_k L_1(x|R⊗_k L_2)where the determinant is taken of the multiplication by x on the finite free R⊗_k L_1-module R⊗_k L_2. We define the pro-torusD_k:=_L⊆kL^∗where the transition maps are given by the norms N_L_2/L_1 for L_1⊆ L_2. In particular, the group of characters of D_k is given by_L⊆k[Hom_k(L,k)]where for L_1⊆ L_2 the transition morphism[Hom_k(L_1,k)]→[Hom_k(L_2,k)]send τ L_1→k to the sum∑_τ^' L_2→kτ^'_|L_1=ττ^'.In particular, we see that the norm morphisms of toriN_L_2/L_1 L^∗_2→ L^∗_1are surjective (as fppf-sheaves).Let G:=Gal(k/k). The character group X^∗(D_k) of the pro-torus D_k is canonically isomorphic to the coinduced moduleCoind^G:={f G→ | fhas open stabilizer inG }This is a general statemenet about discrete G-modules for G a profinite group. Namely, for H⊆ G open,Coind^G_H:={f G→ | fis constant onH-cosets}is isomorphic to the free abelian group [G/H] viaf↦∑_g∈ G/Hf(g)gandCoind^G≅_H⊆ GopenCoind^G_Hby the continuity requirement in the definition of Coind^G. Moreover, one checks that the transition maps agree under the isomorphism Coind^G_H≅[G/H]. Recall that we have fixed a fiber functor ω_0𝒯→Vec_ksuch that its base extension ω:=ω_0⊗_k C𝒯→Vec_Cunderlies a filtered fiber functor.There is a natural tensor functorr𝒯_CM→Rep_k(D_k),which can be explicitly described as follows: Let X∈𝒯_CM be a CM-object with CM-type (E,Φ) and let E_Φ⊆k be the reflex field of (E,Φ). Then the reflex normr_Φ E_Φ^∗→ E^∗ defines an action of the torus E_Φ^∗ on ω_0(X), which defines the action of D_k on ω_0(X) as D_k↠ E_Φ^∗ surjects onto E_Φ^∗. Let G:=Aut^⊗(ω_0) be the automorphism group of the fiber functor ω_0. Then the graded fiber functorgr(ω)𝒯→GrVec_C, X↦⊕_n∈gr^n(ω(X)) defines a cocharacterμ𝔾_m,k→ G_kover k (this is spelled out more generally in <cit.>). Let T be the maximal torus quotient of G, i.e., T=Aut^⊗(ω_0_|𝒯_CM)is the Tannakian fundamental group of the Tannakian category 𝒯_CM⊆𝒯 of CM-objects in 𝒯. Composition with the canonical morphism G→ T yields the cocharacterμ𝔾_m,k→ T_kof T (which we denote by the same letter). On character groups this corresponds to a morphismμ^∗ X^∗(T)→of abelian groups. By the universal property of coinduction this defines a Galois-equivariant morphismμ^∗ X^∗(T)→Coind^Gal(k/k), χ↦ (g↦χ∘α_g^-1∘μ)where for g∈Gal(k/k) the morphism α_g T_k→ T_k denotes the action of g on T_k. In particular, we obtain (using <Ref>) the canonical tensor functor r𝒯_CMRep_k(T)→Rep_k(D_k)as the morphism induced by the morphism (over k)μ^∗ D_k→ Tassociated with μ^∗ X^∗(T)→Coind^Gal(k/k)≅ X^∗(D_k). We now identify the functor r with the construction given in the lemma. Let X∈𝒯 be a CM-object with CM-type (E,Φ). Without loss of generality let X be simple. By naturality of the construction in 𝒯 we may further assume that 𝒯=⟨ X⟩^⊗. ThenE=End_𝒯(X)(cf. the proof of <Ref>) and we getT=^⊗(ω_0)≅ E^∗with T acting on X via the action of E^∗. Indeed, T equals the centralizer C_End_k(ω_0(X))(E^∗) of E^∗ in the endomorphisms of ω_0(X), which itself is E^∗. The containment T⊆ C_End_k(ω_0(X))(E^∗) is clear and equality follows after base change to k where X splits into 1-dimensional representations with distinct characters. Let E_Φ be the reflex field of (E,Φ) (which is given by the stabilizer of μ), and set H:=Gal(k/E_Φ)⊆ G:=Gal(k/k). In particular, μ𝔾_m,k→ T_k=E^∗_k is already defined over E_Φ. We have to show that the diagramD_k@->>[r][rrd]^μ^∗E^∗_Φ[r]^-Res_E_Φ/k(μ) Res_E_Φ/k(T_E_Φ) [d]^NormT=E^∗commutes. On character groups it gives rise to the diagramCoind^G Coind^G_H()[l]Coind^G_H(X^∗(T))[l]_μ^∗X^∗(T)[llu]^μ^∗[u]where the arrows without a label are the canonical ones. Here we note that μ^∗ X^∗(T)→ defines an H-equivariant map by the definition of the reflex field (note that the G-module of maps ΨHom_k(E,C)→ is canonically isomorphic to the cocharacter group X_∗(T) of T because T=E^∗). Using the universal property of coinduction it suffices to check that the diagram commutes after composition with the canonical mapCoind^G()→, f↦ f(1). But then the commutativity follows directly from the definitions. In the case of rational Hodge structures the above functor yields the classical (connected) Serre group (cf. <cit.>) as a quotient of the pro-torus D_. In the case of Breuil-Kisin-Fargues modules we will show that the functor r in <Ref> is an equivalence using the following criterion. We define the full Tannakian subcategory𝒯_0⊆𝒯consisting of all objects X∈𝒯 such thatgr^i(ω(X))=0for i≠ 0, whereω𝒯→FilVec_Cis our fixed filtered fiber functor. Moreover, for a finite field extension E/k and an embedding τ E→ C we denote by Φ_τHom_k(E,C)→ the map sending τ to 1 and all τ^'≠τ to 0. Clearly, the reflex field of (E,Φ_τ) is τ(E)⊆k⊆ C. Assume that the canonical functor Vec_k→𝒯_0 is an equivalence. Then the functorr𝒯_CM→Rep_kD_kfrom <Ref> is fully faithful. Assume moreover that for every finite field extension E/k and every embedding τ E→ C there exists an object X∈𝒯 with CM by (E,Φ_τ). Then the functor r is essentially surjective. If this is the case, then furthermore 𝒯_CM is generated by objects X∈𝒯 with CM by (E,Φ_τ) for E/k a finite field extension and τ E→ C an embedding. We can assume 𝒯_CM=𝒯. DefineT:=Aut^⊗(ω_0)as the torus representing the tensor automorphisms of the fixed fiber functorω_0𝒯→Vec_k.Then 𝒯≅Rep_k(T) and the functor r in <Ref> is given by the morphismμ^∗ D_k→ T(in the notation of <Ref>) corresponding to the morphismμ^∗ X^∗(T)→ X^∗(D_k)≅Coind^G()on character groups. Letμ𝔾_m,k→ T_kbe the cocharacter induced by the (graded fiber functor associated with the) filtered fiber functor ω and letμ^∗ X^∗(T)→be the induced morphism on characters. The assumption that the subcategory 𝒯_0 is equivalent to the category Vec_k of k-vector spaces implies that the kernel Ker(μ^∗) contains no non-trivial orbit under the Galois group G:=Gal(k/k). Indeed, if the orbit under some non-trivial χ∈ X^∗(T) under G lies in ker(μ^∗), then the sum∑_g∈ G/Stab_G(χ)gχdefines a representation X∈Rep_k(T)≅𝒯 of T over k such that gr^i(ω(X))=0 for i≠ 0. In other words, X∈𝒯_0. But this is a contradiction as T acts by assumption trivially on any object in 𝒯_0. Thus the kernel Ker(μ^∗) does not contain a non-trivial G-orbit. The kernel Ker(μ^∗) contains the kernel Ker(μ^∗). Moreover, Ker(μ^∗) is stable under G and hence trivial as it does not contain a non-trivial G-orbit, as well. Hence, the morphismX^∗(T)→Coind^G()is injective and therefore the morphism D_k→ T surjective (as a morphism of fppf-sheaves). This implies that the functor r𝒯→Rep_k(D_k) is fully faithful. The character group Coind^G()≅_L/k[Hom_k(L,C)] of D_k is generated (as a Galois module) by embeddings τ E→ C for finite field extensions E/k (more precisely, by the canonical embedding L⊆ C for L:=τ(E)). Let X∈𝒯_CM be an object with CM by (E,Φ_τ). Then the reflex field E_Φ_τ of the CM-type (E,Φ_τ) is given byE_Φ_τ:=τ(E)⊆k⊆ C.Moreover, the reflex normr_Φ_τ E_Φ_τ^∗E^∗is induced by the inverse of τ E≅ E_Φ_τ. Indeed, let H=Gal(k/E_Φ_τ)⊆ G be the stabilizer of Φ_τ. Then the reflex norm E_Φ_τ^∗→ E^∗ corresponds to the G-invariant morphism[Hom_k(E,C)]→Coind^G_H([Hom_k(E,C)])Coind^G_H≅[Hom_k(E_Φ_τ,C)]on character lattices and τ∈Hom_k(E,C) is sent to the canonical inclusion E_Φ_τ⊆ C.In particular, we can conclude that X must be simple. Namely, its image r(X)∈Rep_k(D_k)is given by the k-vector space E with D_k through its quotient E_Φ_τ^∗ acting via τ^-1 E_Φ_τ^∗→ E^∗. In particular, r(X) and hence X are simple. As in the proof of <Ref> we see that T↠ E^∗ surjects onto the torus E^∗ and thus the functorRep_k(E^∗)→𝒯is fully faithful.As the above morphism[Hom_k(E,C)]≅[Hom_k(E_Φ_τ,C)]is an isomorphism we see that [Hom_k(E_Φ_τ,C)]⊆Coind^Glies in the image of X^∗(T). As E/k was arbitrary, we can conclude thatX^∗(T)≅Coind^Gwhich implies𝒯≅Rep_k(D_k).For the last statement it suffices to note, as was mentioned above, that the character groupCoind^G()≅_k/L/k[Hom_k(L,C)]is generated by embeddings τ L→ C for L/k finite. § BREUIL-KISIN-FARGUES MODULES UP TO ISOGENYWe try to follow closely the notations in <cit.> and <cit.>. Fix a prime p. Let C/_p be a complete, non-archimedean, algebraically closed extension of _p, e.g., C=ℂ_p=_p the completion of an algebraic closure of _p. Let 𝒪_C⊆ C be the ring of integers in C and letC^♭:=_x↦ x^pCbe the tilt of C with its ring of integers𝒪_C^♭=𝒪_C^♭=_x↦ x^p𝒪_C≅_x↦ x^p𝒪_C/p.We denote by(-)^# C^♭→ C, (x_0,x_1,…)↦ x_0resp. (-)^#𝒪_C^♭→𝒪_C, (x_0,x_1,…)↦ x_0the canonical projections. Let ϵ:=(1,ζ_p,ζ_p^2,…)∈𝒪_C^♭ be a fixed system of p-power roots of unity in C^♭ (the results of this paper will be independent of the choice of ϵ). DefineA_inf:=W(𝒪_C^♭)as the ring of Witt vectors for the (perfect) ring 𝒪_C^♭. We denote by[-]𝒪_C^♭→ W(𝒪_C^♭)the canonical Teichmüller lift. When dealing with A_inf as a topological ring we always equip it with the (p,[ϖ])-adic topology for a pseudo-uniformizer ϖ∈ C^♭ (this topology is independent of the choice of ϖ).Letφ A_inf→ A_inf, ∑_i=0^∞ [x_i]p^i↦∑_i=0^∞ [x_i^p]p^ibe the Frobenius on A_inf. We define the elementsμ:=[ϵ]-1 ξ:=[ϵ]-1/[ϵ^1/p]-1=∑_i=0^p-1[ϵ^i/p-1] ξ̃:=φ(ξ)=[ϵ^p]-1/[ϵ]-1of A_inf. Letθ A_inf→𝒪_C, ∑_i=0^∞ [x_i]p^i↦∑_i=0^∞ (x_i)^#p^ibe Fontaine's map θ and setθ̃:=θ∘φ^-1.Then ξ∈ A_inf is a generator of (θ) and ξ̃∈ A_inf a generator of (θ̃). We define as usualA_as the universal PD-thickening of θ and B^+_:=A_[1/p]. Finally, Fontaine's period ring B^+_ is defined to be the ξ-adic completion of A_inf[1/p]. In particular, the map θ induces a (surjective) morphismθ B^+_→ C.In fact, the ring B^+_ is a complete discrete valuation ring with uniformiser ξ and residue field C. Its fraction field B_:=B^+_[1/ξ] is also called the field of p-adic periods.Having fixed this notation we can now define the notion of a Breuil-Kisin-Fargues module which is a “perfectoid” analog of a Breuil-Kisin module. A Breuil-Kisin-Fargues module is a finitely presented A_inf-module M with an isomorphismφ_M (φ^∗ M)[1/ξ̃]=M⊗_A_inf,φA_inf[1/ξ̃] M[1/ξ̃] such that M[1/p] is a finite projective A_inf[1/ξ̃]-module. By <cit.> a finite projective A_inf[1/p]-module is automatically free, hence in <Ref> the condition “finite projective” can be replaced by “finite free". We denote byBKFthe category of Breuil-Kisin-Fargues modules (cf. <cit.>). It is (naturally) an exact tensor category. However, it is not abelian, only pseudo-abelian.Let (M,φ_M) be a Breuil-Kisin-Fargues module and let e M→ M be an idempotent endomorphism of (M,φ_M), i.e., e commutes with φ_M. Then ker(e) and coker(e) with their induced Frobenii are Breuil-Kisin-Fargues modules. In particular, the category BKF of Breuil-Kisin-Fargues modules is pseudo-abelian.Clearly, the kernel and cokernel of e admit Frobenii. As direct summands of finitely presented (resp. finite projective) modules are again finitely presented(resp. finite projective) ker(e) and coker(e) satisfy all conditions in <Ref>, i.e., they are Breuil-Kisin-Fargues modules. We give an easy example of a Breuil-Kisin-Fargues modules. For d∈ we set A_inf{d}:=μ^-dA_inf⊗__p_p(1) with Frobenius φ_A_inf{d}φ^∗(A_inf{d})=φ(μ)^-dA_inf⊗__p_p(1)→ A_inf{d}, a↦ξ̃^d a Let (M,φ_M) be a Breuil-Kisin-Fargues module such that M is finite projective (equivalently, free) over A_inf. Then the dual (M^∨,φ_M^∨) of (M,φ_M) is defined byM^∨:=Hom_A_inf(M,A_inf)with Frobeniusφ_M^∨ (φ^∗ M^∨)[1/ξ̃]=(φ^∗ M)[1/ξ̃]^∨(M[1/ξ̃])^∨=M^∨[1/ξ̃].where we used the notation (-)^∨=Hom_A_inf[1/ξ̃](-,A_inf[1/ξ̃]) again for duals of finite projective A_inf[1/ξ̃]-modules. We recall the following lemma about general A_inf-modules.Let M be a finitely presented A_inf-module such that M[1/p] is finite projective (equivalently, free) over A_inf. Then there is a functorial (in M) exact sequence0→ M_tor→ M→ M_free→M→ 0satisfying: i) M_tor is finitely presented and perfect as an A_inf-module, and killed by p^n for some n≫ 0.ii) M_free is a finite free A_inf-module.iii) M is finitely presented and perfect as an A_inf-module, and is supported at the closed point of (A_inf). See <cit.>.In particular, a finitely presented A_inf-module M such that M[1/p] is finite free over A_inf[1/p] is perfect as an A_inf-module, i.e., admits a finite projective resolution (cf. <cit.>). Moreover, if the A_inf-module M in <Ref> is a Breuil-Kisin-Fargues module, i.e., equipped with a Frobenius φ_Mφ^∗(M)[1/ξ̃]≅ M[1/ξ̃], then the modules M_rm, M_free and M carry a natural Frobenius as well and the exact sequence in <Ref> is an exact sequence of Breuil-Kisin-Fargues modules (note that M_tor[1/p]=M[1/p]=0 is free). In fact, the existence of this Frobenius is clear for M_tor. But M_free=H^0((A_inf),M/M_tor) (cf. the proof of <cit.>) which yields the Frobenius on M_free (and thus M as well). We want to recall some equivalent descriptions of Breuil-Kisin-Fargues modules whose underlying A_inf-module is finite free (cf. <cit.>). For this we need to recall that associated to the fixed complete and algebraically closed non-archimedean field C/_p there is associated a scheme X_FF over _p, “the Fargues-Fontaine curve”, together with a distinguished point ∞∈ X_FF whose completed stalk 𝒪_X_FF,∞≅ B^+_ is isomorphic to Fontaine's period ring B^+_ associated with C (cf. <cit.>). We also recall that the adic Fargues-Fontaine curve X_FF^=Y/φ^ is uniformized by an adic space Y admitting an action of φ (cf. <cit.>). The space Y is defined to be Y:=Spa(A_inf)∖ V(p[ϖ])where ϖ∈𝒪_C^♭ is a pseudo-uniformizer (cf. <cit.> for a proof that this is actually an honest adic space, i.e., the structure presheaf is a sheaf). Additionally to Y the adic space Spa(A_inf)∖ V(p,[ϖ]) contains two points, namely * x_ given as the image of the p-adic valuation on W(k) along the canonical projection A_inf→ W(k) (this point is denoted x_L in <cit.>).* x_étale given by the image of the valuation on 𝒪_C^♭ along the morphism A_inf→ O_C^♭=A_inf/p (this point is denoted x_C^♭ in <cit.>).We setB:=H^0(Y,𝒪_Y)and B^+:=H^0((A_inf)∖V(p), 𝒪).There exists a natural φ-equivariant injection B^+→ B^+_ making B^+_ a B^+-algebra (cf. <cit.>).The category Bun_X_FF of vector bundles on the Fargues-Fontaine curve is naturally equivalent to the category of φ-modules over B^+_, i.e., to finite projective B^+_-modules M together with an isomorphism φ_Mφ^∗ M≅ M. This is proven in <cit.>. The natural functor inducing the equivalence is given by pulling back a vector bundle ℱ∈Bun_X_FF≅_X^_FF to Y, extending it along the point x_ and tensoring the global sections of this extension (which form a φ-module over B^+) over B^+ with B^+_.The following categories are equivalent: i) Finite free Breuil-Kisin-Fargues modules (M,φ_M).ii) Pairs (T,Ξ), where T is a finite free _p-module, and Ξ⊆ T⊗__pB_ is a B^+_-lattice.iii) Quadruples (ℱ,ℱ^',β,T), where ℱ and ℱ^' are vector bundles on the Fargues-Fontaine curve X_FF, and βℱ_|X_FF∖∞ℱ^'_|X_FF∖∞ is an isomorphism, ℱ is trivial, and T⊆ H^0(X_FF,ℱ) is a _p-lattice.The proof is given in <cit.>. We shortly give descriptions of the various functors involved. Let (M,φ_M) be a finite free Breuil-Kisin-Fargues module. Then the φ_M-invariantsT:=(M⊗_A_infW(C^♭))^φ_M=1form a finite free _p-module (by Artin-Schreier theory, cf. <cit.>). Note that ξ̃ is invertible in W(C^♭), hence the base extension M⊗_A_infW(C^♭) actually carries a Frobenius. Moreover, multiplication defines an isomorphismT⊗__pW(C^♭)≅ M⊗_A_infW(C^♭)under which T⊗__pA_inf[1/μ]⊆ T⊗__pW(C^♭) is mapped to M⊗_A_infA_inf[1/μ] (cf. <cit.>). Setting Ξ:=M⊗_A_infB^+_⊆ (M⊗_A_infA_inf[1/μ])⊗_A_inf[1/μ]B_≅ T⊗__pB_defines the pair (T,Ξ) in ii) associated with (M,φ_M). If Ξ⊆ T⊗__pB_ is a B^+_-lattice, where T is a finite free _p-module, then (in order to pass to iii)) one can use this lattice to modify the trivial bundle ℱ:=T⊗__p𝒪_X_FF at the distinguished point ∞∈ x_FF on the Fargues-Fontaine curve to obtain a bundle ℱ^' with an isomorphism ℱℱ^' away from ∞ (recall that 𝒪_X_FF,∞≅ B^+_). Clearly, T⊆ H^0(X_FF,𝒪_X_FF) defines a _p-lattice as H^0(X_FF,𝒪_X_FF)≅ T⊗__p_p. Moreover, this construction can be reversed giving the equivalence between the categories in ii) and iii). Thus in particular, the hard part of the theorem is the construction of a functor from the category in ii) (or iii)) to Breuil-Kisin-Fargues modules. We shortly describe how this is done. Let (ℱ,ℱ^', α,T) be given as in iii). Pulling back ℱ and ℱ^' to Y defines two φ-equivariant vector bundles ℰ,ℰ^' on Y. By triviality of ℱ the vector bundle ℰ extends (non-canonically) to the point x_étale. This extension is not unique, but depends on the choice of a _p-lattice in H^0(X,ℱ)=H^0(Y,ℰ)^φ=1. In particular, the data of T yiels a canonical extension of ℰ to x_étale. Moreover, every φ-equivariant vector bundle on Y, thus in particular ℰ^', extends uniquely to the point x_. Using the modification ℱℱ^' away from ∞ the bundles ℰ and ℰ^' are (φ-equivariantly) isomorphic when restricted to a small annulus omitting the points φ^n(∞),n∈. In particular, one obtains a vector bundle ℰ^'' on (A_inf)∖ V(p,[ϖ]) which restricts to the extension of ℰ^' near x_ and to the extension of ℰ near x_étale. Taking global sections of ℰ^'' defines the finite free A_inf-module M which is then moreover equipped with a Frobenius φ_M away from V(ξ̃), i.e., a finite free Breuil-Kisin-Fargues module. In particular, we see that the φ-module M⊗_A_infB^+_ over B^+_ is canonically isomorphic to the φ-module over B^+_ associated with ℱ^' as in <Ref>.For example, the Breuil-Kisin-Fargues modules A_inf{d} is sent to the pair(_p(d),Ξ:=ξ^-d(_p(d)⊗__pB^+_)⊆_p(d)⊗__pB_)resp., for d≥ 0, to the modification0→_p(d)⊗__p𝒪_X_FF𝒪_X_FF(d)→ t^-d𝒪_X_FF,∞/𝒪_X_FF,∞→ 0induced by the d-th power t^d of Fontaine's t=log([ϵ])∈ H^0(X_FF,𝒪_X_FF(1))=(B^+_)^φ=p.We want to stress that the categories in <Ref> are not equivalent as exact categories, i.e., the equivalences (or their inverses) are not exact. In fact, in the construction of the finite free Breuil-Kisin-Fargues module one has to extend (by taking global sections) a vector bundle on the punctured (A_inf) with the closed point removed to the whole of (A_inf) and this is not an exact operation.We define the isogeny category of Breuil-Kisin-Fargues modules BKF^∘ as BKF^∘:=_p⊗__pBKF,i.e., the category BKF^∘ has the same objects as the category BKF, but for M,N∈BKF^∘ the space of homomorphisms is given byHom_BKF^∘(M,N):=_p⊗__pHom_BKF(M,N).The category BKF^∘ is still not abelian (for the same reason as BKF, cf. <cit.>), but rigid as we now show.The isogeny category BKF^∘ of Breuil-Kisin-Fargues modules is a rigid, exact tensor category. It suffices to prove that every object admits a dual because the tensor product and the exact structure are inherited from the category BKF. Let (M,φ_M)∈BKF be a Breuil-Kisin-Kisin-Fargues module. We claim that, in the isogeny category, (M,φ_M) is isomorphic to a Breuil-Kisin-Fargues module (N,φ_N) such that N is finite free over A_inf. Let M_tor⊆ M be the torsion submodule. Then φ^∗(M_tor) resp. φ^∗(M_tor)[1/ξ̃] is the torsion submodule of φ^∗(M) resp. φ^∗(M)[1/ξ̃]. Therefore it is stable by φ_M. By lemma <Ref> the torsion submodule M_tor is actually killed by p^n for n≫ 0, and thus zero in the isogeny category. Hence, we may assume that M is actually torsion-free. Set N:=M_free as in <Ref>, which is finite free over A_inf. Then φ^∗(N)[1/ξ̃]=φ^∗(M)[1/ξ̃] and N[1/ξ̃]=M[1/ξ̃] because the cokernel of M→ N is killed by ξresp.ξ̃. In particular, the Frobenius φ_M of M defines a Breuil-Kisin-Fargues module (N,φ_M) and we are finally allowed to replace M by N, as (M,φ_M) and (N,φ_N) are isomorphic in the isogeny category. This finishes the proof as finite free Breuil-Kisin-Fargues modules clearly admit duals. We remark that the proof of <Ref> actually shows that for a Breuil-Kisin-Fargues module (M,φ_M) the exact sequence0→ M_tor→ M→ M_free→M→ 0of <Ref> can be enhanced to an exact sequence of Breuil-Kisin-Fargues modules, i.e., there are compatible Frobenii on the modules involved.In terms of pairs of a _p-lattice and a B^+_-lattice the category BKF^∘ of Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of pairs (V,Ξ) where V is a finite dimensional _p-vector space and Ξ⊆ V⊗__pB_ a B^+_-lattice. Again, we can see that this category is not abelian: The canonical morphism(_p,ξ B^+_)→ (_p,B^+_)does not admit a cokernel. However, we see that it admits kernels. But we caution the reader that it is not clear how to describe the kernel of a morphism f (M,φ_M)→ (N,φ_N) of Breuil-Kisin-Fargues modules up to isogeny, because it is not clear whether the kernel K:=Ker(MN) is finitely presented over A_inf or satisfies the property that K[1/p] is finite projective over A_inf[1/p] (cf. the proof of <Ref>).We are seeking for a Tannakian, and thus in particular abelian, category of Breuil-Kisin-Fargues-modules up to isogeny. This is possible after adding a rigidification after base change to B^+_=A_[1/p] (cf. <cit.>). Let us recall <cit.>.Let (M,φ_M) be a Breuil-Kisin-Fargues module. Then M=M⊗_A_infW(k) is a finitely generated W(k)-module equipped with a Frobenius automorphism after inverting p. Fix a section k→𝒪_C/p, which induces a section W(k)→ A_inf. Then there is a (noncanonical) φ-equivariant isomorphismM⊗_A_infB^+_≅M⊗_W(k)B^+_ reducing to the identity over W(k)[1/p]. See <cit.> which refers to <cit.>. We remark that ξ̃ is invertible in B^+_ (but ξ is not), hence the Frobenius φ_Mφ^∗M[1/ξ̃] M[1/ξ̃] defines an isomorphismφ_M⊗φ_B^+_φ^∗(M⊗_A_inf B^+_) M⊗_A_infB^+_. We fix a section k→𝒪_C/p of the projection 𝒪_C/p→ k. This yields a section W(k)→ A_inf of the projection A_inf→ W(k).A rigidified Breuil-Kisin-Fargues module is a Breuil-Kisin-Fargues module (M,φ_M) together with a φ-equivariant isomorphismα M⊗_A_infB^+_ (M⊗_A_infW(k))⊗_W(k)B^+_such that α reduces to the identity over W(k)[1/p], i.e., α⊗_B^+_W(k)[1/p]=_M⊗_A_infW(k)[1/p]. We call an isomorphism α as in <Ref> a rigidification of (M,φ_M). We denote byBKF_rigthe category of rigidified Breuil-Kisin-Fargues modules, i.e., its objects are rigidified Breuil-Kisin-Fargues modules and the morphisms are morphisms of Breuil-Kisin-Fargues modules respecting the given rigidifications. We illustrate the effect of imposing a rigidification in the case of Breuil-Kisin-Fargues modules of rank 1.Each finite free Breuil-Kisin-Fargues modules of rank 1 is isomorphic tothe Breuil-Kisin-Fargues module A_inf{d} for some d∈. For d,d^'∈ we haveHom_BKF(A_inf{d},A_inf{d^'})≅_p ifd≤ d^'0 otherwisewhileHom_BKF_rig(A_inf{d},A_inf{d^'})≅_p ifd= d^'0 otherwiseThe first statement and the computation of Hom_BKF(A_inf{d},A_inf{d^'}) follows from <Ref> (and the example following it). To proveHom_BKF_rig(A_inf{d},A_inf{d^'})=0 if d≠ d^' it suffices to show that the isocrystal (A_inf{d}⊗_A_infW(k)[1/p],φ_A_inf{d}⊗Id) has slope d because there are no morphisms between isocrystals of different slopes. But the image ofξ̃=[ϵ^p]-1/[ϵ]-1=1+[ϵ]+… + [ϵ^p-1]in W(k) is p because k does not contain non-trivial p-power roots of unity. Hence, the isocrystal (A_inf{d}⊗_A_infW(k)[1/p],φ_A_inf{d}⊗Id) is isomorphic to (W(k)[1/p],p^dφ) which is of slope d.The category BKF_rig of rigidified Breuil-Kisin-Fargues modules is again an exact tensor category with the exact structure and tensor product inherited from BKF (taking the tensor product of the rigidifications), but it is moreover abelian (cf. <Ref>) contrary to the case of the category BKF of non-rigidified Breuil-Kisin-Fargues modules.To prove this we recall the following criterion for an A_inf-module M to satisfy that M[1/p] is finite projective over A_inf[1/p]. Recall the element μ=[ϵ]-1∈ A_inf (cf. the notation in the beginning of <Ref>).Let M be a finitely presented A_inf-module. Assume i) M[1/pμ] is finite projective over A_inf[1/pμ].ii) M⊗_A_infB^+_ is finite projective over B^+_. Then M[1/p] is finite free over A_inf[1/p]. See <cit.>. We record the following corollary in the case of Breuil-Kisin-Fargues modules. Let f (M,φ_M)→ (N,φ_N) be a morphism of Breuil-Kisin-Fargues modules and let Q:=Coker(M N) be the cokernel (of A_inf-modules). Then Q[1/pμ] is a finite free A_inf[1/pμ]-module. By <cit.> there is a canonical isomorphismM⊗_A_infA_inf[1/μ]≅ T_M⊗__pA_inf[1/μ]with T_M:=(M⊗_A_infW(C^♭))^φ_M=1 and similarly for N,N⊗_A_infA_inf[1/μ]≅ T_N⊗__pA_inf[1/μ],with T_N:=(N⊗_A_infW(C^♭))^φ_N=1. In particular, we getQ⊗_A_infA_inf[1/pμ]≅ Q^'⊗__pA_inf[1/pμ]where Q^' is the cokernel of the morphism T_M⊗__p_p→ T_N⊗__p_p induced by f. Thus we see thatQ⊗_A_infA_inf[1/pμ]is finite free. We remark that in general Q (with the induced Frobenius from (N,φ_N)) will not be a Breuil-Kisin-Fargues modules, because Q[1/p] need not be a finite projective (equivalently, free) A_inf[1/p]-module. The purpose of introducing the rigidifications is exactly to ensure this freeness (cf. the remark after <cit.>).The category BKF_rig of rigidified Breuil-Kisin-Fargues modules is an abelian tensor category. It suffices to prove that BKF_rig is abelian. Let (M,φ_M,α_M) and (N,φ_N,α_N) be two rigidified Breuil-Kisin-Fargues modules with Frobenii φ_M resp. φ_N (after inverting ξ̃) and rigidifications α_M and α_N. Let f (M,φ_M,α_M)→ (N,φ_N,α_N) be a morphism of rigidified Breuil-Kisin-Fargues modules. Consider the exact sequence0→ K→ MN→ Q→ 0of A_inf-modules. We will show that the kernel K and the cokernel Q are finitely presented over A_inf and admit Frobenii φ_K, φ_Q (after inverting ξ̃) and rigidifications α_K, α_Q induced from φ_M, φ_N resp. α_M,α_N, such that (K,φ_K,α_K) and (Q,φ_Q,α_Q) are rigidified Breuil-Kisin-Fargues modules. This will then imply that the category BKF_rig is abelian. In particular, we have to prove that K[1/p] and Q[1/p] are finite free. As f commutes with φ_M resp. φ_N it is clear that K and Q are canonically equipped with Frobenii φ_K and φ_Q. Clearly, the module Q is finitely presented. By <Ref> the module Q[1/pμ] is finite projective over A_inf[1/pμ]. Moreover, as f respects the rigidificationsα_M M⊗_A_infB^+_≅M⊗_W(k)B^+_resp. α_N N⊗_A_infB^+_≅N⊗_W(k)B^+_,where M:=M⊗_A_infW(k) resp. N:=N⊗_A_infW(k),we see that Q⊗_A_infB^+_ is isomorphic to the cokernel ofM⊗_W(k)B^+_→N⊗_W(k)B^+_,which is free because the cokernel of M[1/p]→N[1/p] is free as W(k)[1/p] is a field. By <Ref> we can conclude that Q[1/p] is finite free over A_inf[1/p]. As moreover, M[1/p] and N[1/p] are finite free over A_inf[1/p], we can conclude that K[1/p] is finite projective (and hence finite free). Let C be the two-term complex (concentrated in degree 0 and 1) …→ M N→…which is a perfect complex of A_inf-modules satisfying that its cohomology if finite free over A_inf[1/p]. By <cit.> we can conclude that H^0(C)=K is finitely presented. In other words, we have proven that (K,φ_K) and (Q,φ_Q) are Breuil-Kisin-Fargues modules and are left with showing that they admit canonical rigidifications. By right exactness of tensor products this is clear for Q. But as K,M,N,Q are free after inverting p we get a commutative diagram0[r]K⊗_A_infB^+_[r][d]^α_KM⊗_A_infB^+_[r][d]^α_MN⊗_A_infB^+_[d]^α_N0[r]K⊗_W(k)B^+_[r] M⊗_W(k)B^+_[r] N⊗_W(k)B^+_whose rows are exact. In particular, we also get a canonical rigidifications α_K on K. This finishes the proof of the theorem. We are now ready to define a main player of this paper.We define the isogeny category of rigidified Breuil-Kisin-Fargues modules to be the categoryBKF_rig^∘:=_p⊗__pBKF_rig.In other words, the category BKF_rig^∘ is the Serre quotient of the abelian category BKF_rig by the full subcategory of rigidified Breuil-Kisin-Fargues modules (M,φ_M,α_M) such the underlying A_inf-module M is annihilated by p^n, n≫ 0.The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of triples (N,φ_N,α_N) where * N is a finite free A_inf[1/p]-module* φ_Nφ_A_inf[1/p]^∗(N)[1/ξ̃] N[1/ξ̃] is an isomorphism and* α_N N⊗_A_inf[1/p]B^+_≅N⊗_W(k)[1/p]B^+_ is a rigidification reducing to the identity over W(k)[1/p] with N:=N⊗_A_inf[1/p]W(k)[1/p]such that there exists a finitely presented A_inf-submodule N^'⊆ N satisfying N^'[1/p]=N which is stable under φ_N. The natural functor sending an object (M,φ_M,α_M)∈BKF_rig^∘ to its base extension (M⊗_A_infA_inf[1/p],φ_M⊗φ_A_inf[1/p],α_M) to A_inf[1/p] is essential surjective by the last condition on the existence of N^'. It is moreover fully faithful as inverting p commutes with Hom for finitely presented modules. The condition on the existence of a φ-stable A_inf-lattice N^'⊆ N ensures that for a triple (N,φ_N,α_N) as in <Ref> the “etale realization” (cf. <Ref>)(N⊗_A_infW(C^♭))^φ_N=1is a _p-vector space of dimension the rank of N. Otherwise, the isocrystal N⊗_A_infW(C^♭) (over C^♭) need not be isoclinic of slope 0.We can now record the main theorem of this section.The category BKF_rig^∘ of rigidified Breuil-Kisin-Fargues modules up to isogeny is Tannakian. Theorem <Ref> implies that also the isogeny category BKF_rig^∘ of rigidified Breuil-Kisin-Fargues modules is abelian, namely it identifies with the Serre quotient of the category BKF_rig by the full subcategory of rigidified Breuil-Kisin-Fargues modules which are p-power torsion. By (the proof of) <Ref> each object (M,φ_M)∈BKF^∘ is isomorphic (in the isogeny category) to some (N,φ_N) with N a finite free A_inf-module. Ifα_M M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_is a rigidification for (M,φ_M), then α_M also defines a rigidification of (N,φ_N) because M⊗_A_infB^+_≅ N⊗_A_infB^+_ resp. (M⊗_A_infW(k))⊗_W(k)B^+_≅ (N⊗_A_infW(k))⊗_W(k)B^+_. In particular, we see that every object in BKF_rig^∘ admits a dual and therefore the category of BKF_rig^∘ is rigid. To prove that BKF_rig^∘ is Tannakian we can either apply Deligne's criterion that Λ^n M vanishes for every M∈BKF^∘_rig and n≫ 0 (cf. <cit.> or use that there are various concrete fiber functors <Ref>. We continue by giving an alternative description of rigidified Breuil-Kisin-Fargues modules in terms of modifications of vector bundles on the Fargues-Fontaine curve.We recall that the Fargues-Fontaine curve admits an Harder-Narasimhan formalism, namely, for every vector bundle ℱ on X_FF there exists a canonical (decreasing) filtration indexed by λ∈HN^λ(ℱ)⊆ℱsuch that the associated graded piecesgr^λ(HN(ℱ))are semistable of slope λ.The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules is equivalent to the category of quadruples (ℱ,ℱ^',β,α) where ℱ, ℱ^' are vector bundles on the Fargues-Fontaine curve X_FF, ℱ is trivial, βℱ_|X_FF∖{∞}≅ℱ^'_|X_FF∖{∞} is an isomorphism, and αℱ^'≅⊕_λ∈gr^λ(HN(ℱ^')) is an isomorphism inducing the identity on all graded pieces of the Harder-Narasimhan filtration.By <Ref> (and <Ref>) the category BKF^∘ of Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of triples (ℱ,ℱ^',β) satisfying the same condition as in the statement of this theorem. Thus we are left with showing that the rigidifications can be identified under the equivalence in <Ref>. Let (M,φ) be a finite free Breuil-Kisin-Fargues module with corresponding quadruple (ℱ,ℱ^',β,T) as in <Ref>. Let N be the φ-module over B^+_ associated with ℱ^' in <Ref>. As was shown in the proof of <Ref> there exists a canonical isomorphismM⊗_A_infB^+_≅ N.Recall that we have fixed a splitting k→𝒪_C/p of the projection 𝒪_C/p→ k. Letℰ(-)φ-Mod_W(k)[1/p]→Bun_X_FFbe the exact tensor functor associated with this splitting (cf. <cit.>). Then for the vector bundle ℱ^' there is a canonical isomorphism⊕_λ∈gr^λ(HN(ℱ^')≅ℰ(N⊗_B^+_W(k)[1/p]and we see (writing M:=M⊗_A_infW(k)[1/p]≅ N⊗_B^+_W(k)[1/p]) that the set of isomorphismsα_M M⊗_A_infB^+_≅M⊗_W(k)[1/p]B^+_is in bijection with the set of isomorphismsαℱ^'≅⊕_λ∈gr^λ(HN(ℱ^').Clearly, the condition on α_M resp. α to reduce to the identity over W(k)[1/p] resp. the graded pieces of the Harder-Narasimhan filtration is preserved. This finishes the proof. We note that we actually proved that for a finite free Breuil-Kisin-Fargues module (M,φ_M) with corresponding modification (ℱ,ℱ^',αℱ_|X_FF∖{∞}≅ℱ^'_|X_FF∖{∞},T⊆ H^0(X_FF,ℱ)) as in <Ref> there is a bijection{rigidifications M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_} ≅{rigidifications ℱ^'≅⊕_λ∈gr^λ(HN(ℱ^'))}where we call an isomorphism ℱ^'≅⊕_λ∈gr^λ(HN(ℱ^')) reducing to the identity on the graded pieces of the Harder-Narasimhan filtration a rigidification of the data (ℱ,ℱ^', α,T) or just of ℱ^'. In particular, if ℱ^' happens to be semistable there exists a unique rigidification on (M,φ_M), namely the one given by the identity on ℱ^'.Using <Ref> it is possible to give an alternative proof of <Ref>. Namely, the category of semistable vector bundles of a fixed slope is abelian and therefore the cokernel of a morphism between vector bundles on the Fargues-Fontaine curve which respects fixed splittings of the respective Harder-Narasimhan filtrations is again a vector bundle. This proves (using <Ref>) that the category BKF^∘_rig is abelian. Moreover, as it is a rigid tensor category, the existence of a fiber functor (cf. <Ref>) or Deligne's criterion proves that it is moreover Tannakian.<Ref> also proves that the category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny does not depend (up to canonical isomorphism) on the choice of the section k→𝒪_C/p. We do not know how to describe the category of rigidified Breuil-Kisin-Fargues modules up to isogeny only in terms of pairs (V,Ξ) where V is a finite-dimensional _p-vector space and Ξ⊆ V⊗__pB_ a B_^+-lattice. However, we can record the following.Let f (M,φ_M,α_M)→ (M^',φ_M^',α_M^')∈BKF_rig be a morphism of finite free rigidified Breuil-Kisin-Fargues modules. Let g:(T,Ξ)→ (T^',Ξ^') be the associated morphism between pairs of a _p- and B^+_-lattice as in <Ref>. Then the cokernel of the morphismΞΞ^'is free over B^+_. By construction of (T,Ξ) resp. (T^',Ξ^') the morphism gΞ→Ξ^' is given byf⊗Id M⊗_A_infB^+_→ N⊗_A_infB^+_.Set M:=M⊗_A_infW(k) resp. N:=N⊗_A_infW(k). By assumption the diagramM⊗_A_infB^+_[r]^f⊗Id[d]^α_M⊗ B^+_N⊗_A_infB^+_[d]^α_N⊗ B^+_ M⊗_A_infB^+_[r]^f⊗Id N⊗_A_infB^+_.with both vertical morphisms isomorphisms commutes. In particular, the cokernel of g is free. Let us list some fiber functors, i.e., “realizations”, of the Tannakian category BKF_rig^∘.We recall the following lemma which is a direct corollary of the classification of vector bundles on the Fargues-Fontaine curve (cf. <cit.>).The category of φ-modules over W(k)[1/p], i.e., the category of isocrystals over k, is equivalent to the category of -graded vector bundles ℰ=⊕_λ∈ℰ^λ such that ℰ^λ is semistable of slope λ. Cf. <cit.>. Let (M,φ_M) be a Breuil-Kisin-Fargues module. We define * its “étale realization” T:=(M⊗_A_infW(C^♭))^φ_M=1, a _p-module.* its “crystalline realization” D:=M⊗_A_infW(k), a W(k)-module equipped with the φ-semilinear isomorphism φ_D:=φ_M⊗ W(k)[1/p]:φ^∗(D[1/p])≅ D[1/p] after inverting p.* and its “de Rham realization” V:=M⊗_A_inf,θ𝒪_C, a 𝒪_C-module. The realizations from <Ref> define fiber functors * ω_étBKF^∘_rig→Vec__p* ω_BKF^∘_rig→φ-Mod_W(k)[1/p]* ω_BKF^∘_rig→Vec_C on the Tannakian category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny. The functors are defined by sending a Breuil-Kisin-Fargues module to the respective realizations (with p inverted). As each Breuil-Kisin-Fargues modules is isogenous to a finite free Breuil-Kisin-Fargues module all three functors are exact. The functors given by crystalline and de Rham realization are clearly tensor functor. To prove that also the étale realization ω_ét defines a tensor functor we remark that it factors canonically into the tensor functorBKF^∘_rig→φ-Mod_W(C^♭)[1/p], (M,φ_M)↦ (M⊗_A_infW(C^♭)[1/p],φ_M⊗Id), to isocrystals over C^♭ and the tensor functorφ-Mod_W(C^♭)[1/p]→Vec__psending an isocrystal to its isoclinic part of slope 0. Actually, the étale realization ω_ét⊗__pC, base changed to C, underlies a filtered fiber functor (cf. <cit.> and <cit.>) on the category BKF^∘_rig. This nicely parallels the case of rational Hodge structures.Let (M,φ_M) be a finite free Breuil-Kisin-Fargues modules with associated étale realizationT:=(M⊗_A_infW(C^♭))^φ_M=1 and B^+_-lattice Ξ:=M⊗_A_infB^+_⊆ T⊗__pB_≅ M⊗_A_infB_.Then we define the decreasing filtration Fil^j(T⊗__pC), j∈, on T⊗__pC=T⊗__pB^+_/(T⊗__pξ B^+_) byFil^j(T⊗__pC):=Im(ξ^jΞ∩ T⊗__pB^+_→ T⊗__pC)with j∈. Written in a suitable basis e_1,…,e_n of T⊗__pB^+_ (over B^+_) there exist λ_1,…,λ_n∈ satisfying λ_1≥…≥λ_n such that the lattice Ξ⊆ T⊗__pB_ is generated by the ξ^λ_1e_1,…,ξ^λ_ne_n. Then the filtration Fil^j(T⊗__pC) is given by the image of⟨ξ^λ_1+je_1,…, ξ^λ_n+je_n⟩∩ T⊗__pB^+_in T⊗__pC, i.e., the C-subspace generated by the residue classes e_i of the e_i such that λ_i+j≤ 0. The fiber functorω_ét⊗__pCBKF^∘_rig→Vec_Cadmits a canonical filtration induced by the filtration in <Ref>, i.e., the functorωBKF^∘_rig→FilVec_C, M↦ (ω_ét⊗__pC, Fil^∙(ω_ét(M)⊗__pC) is exact. Writing down adapted bases as was done following <Ref> one sees that ω is a tensor functor. As symmetric powers and exterior powers are kernels of projectors (in characteristic 0) and the category FilVec_C of filtered vector spaces is pseudo-abelian we see that ω commutes with symmetric and exterior powers. Moreover, ω commutes with duals. Let 0→ M→ N→ Q→ 0 be an exact sequence in BKF^∘_rig. We have to proof that the sequence0→gr(ω(M))→gr(ω(N))→gr(ω(Q))→ 0is exact. Considering dimensions and taking duals it suffices to prove that the left arrow gr(ω(M))→gr(ω(N)) is injective. This is equivalent to the statement thatΛ^r(gr(ω(M)))→Λ^r(gr(ω(N)))is non-zero where r:=rk(M). As ω commutes with exterior powers we may replace the morphism M→ N by Λ^r M→Λ^r N and assume that r=1.By <Ref> Let (V,Ξ) resp. (V^',Ξ^') be the associated pairs of a _p- and a B^+_-lattice. We identify V,Ξ with their images in V^' resp. Ξ^'. Tensoring with the dual of (V,Ξ) we can assume that Ξ=V⊗__pB^+_. Pick a generator v∈ V. Then the element v⊗ 1 is an element in Ξ^' and moreover, it is part of a basis of Ξ^' by <Ref>. Therefore we can, cf. the description following <Ref>, find an adapted basis v:=e_1,…, e_n of V^'⊗ B^+_ containing v such that Ξ^'=⟨ e_1,ξ^λ_2e_2,…,ξ^λ_ne_n⟩. From the concrete description given after <Ref> it follows that the image of v in gr^0(V^'⊗__pC) is non-zero. This finishes the proof. We remark that sending a finite free Breuil-Kisin-Fargues modules M∈BKF^∘ (without fixing a rigidification) to the filtered vector space ω_ét(M)⊗__pC is not an exact operation. An counterexample is provided by the morphism(_p,ξ B^+_)→ (_p,B^+_). Let ω_?∈{ω_ét, ω_, ω_} be one of the fiber functors in <Ref>. Then we setG_?:=Aut^⊗(ω_?)as the group of tensor automorphisms of ω_?. We do not know how to describe the affine group schemes G_ét, G_ or G_ apart from their maximal torus quotients corresponding to the full subcategory of rigidified Breuil-Kisin-Fargues modules admitting complex multiplication. We record the following general result about these group schemes.The band of the Tannakian category BKF^∘_rig is connected. In particular, the group schemes G_ét, G_ and G_ are connected. It suffices to show that every object M∈BKR^∘_rig such that the smallest abelian subcategory ⟨ M⟩, i.e., the full subcategory of subquotients of a finite direct sum ⊕ M, is closed under the tensor product is a direct sum of the unit object. Assume that there exists an object like this. Set V:=ω_ét(M)=(M⊗_A_infW(C^♭)^φ_M=1[1/p]. Then the filtration of <Ref> on the étale realization ω_ét(M)⊗__pC=V⊗__pC must be trivial. Set Ξ:=M⊗_A_infB^+_⊆ V⊗__pB_. In a suitable basis e_1,…,e_n of V⊗__pB^+_ we can write Ξ=⟨ξ^λ_1e_1,…, ξ^λ_ne_n⟩ with λ_1,…,λ_n∈. Then for j∈ ξ^jΞ⊆ V⊗__pB^+_is generated by the ξ^max{λ_i+j,0}e_i. As the filtration on V⊗__pC is trivial we can conclude thatλ_i≤ 0andλ_i+1>0for all i. In other words, λ_i=0 for all i, i.e., Ξ=V⊗__pB^+_, which proves that M is a direct sum of the unit object of BKF^∘_rig. We now calculate the Ext-groupsExt^1_BKF^∘_rig(A_inf{d},A_inf{d^'})for d,d^'∈. In particular, as they will turn out to be non-zero we will be able to conclude that the category BKF^∘_rig of Breuil-Kisin-Fargues modules is not semisimple (and thus the group schemes G_ét,G_,G_ are not reductive).We recall one more period ring, namely B_e. By definition,B_e:=H^0(X_FF,𝒪_X_FF)=B_^φ=1.Moreover, there exists the “fundamental sequence of p-adic Hodge theory” involving B_e:0→_p→ B_e→ B_/(B_^+)→ 0(cf. this follows from <cit.>).Let d∈. Then Ext^1_BKF^∘_rig(A_inf,A_inf{d})≅ B_/t^d B^+_.We use <Ref> and will classify triples (ℱ,ℱ^',β) fitting into a commutative diagramm with exact rows (where a dotted arrow means an isomorphism outside ∞∈ X_FF)0 [r]𝒪_X_FF[r]@–>[d]^t^d ℱ[r]@–>[d]^β 𝒪_X_FF[r][d]^=0 0 [r]𝒪_X_FF(d) [r]ℱ^'[r]𝒪_X_FF[r] 0and isomorphismsαℱ^'≅gr^∙(HN(ℱ^'))inducing the canonical rigidifications on 𝒪_X_FF(d) resp. 𝒪_X_FF. First we note that this last requirement actually implies that ℱ^'≅𝒪_X_FF(d)⊕𝒪_X_FF.Moreover, ℱ≅𝒪_X_FF⊕𝒪_X_FF.In other words, an extension of A_inf by A_inf{d} is thus determined by an automorphism, preserving the factor 𝒪_X_FF(d),𝒪_X_FF(d)⊕𝒪_X_FFrestricting to the identity on both factors, i.e., by an element inHom_X_FF(𝒪_X_FF,𝒪_X_FF(d))≅ (B^+_)^φ=p^d,and an isomorphismβ (𝒪_X_FF⊕𝒪_X_FF)_|X_FF∖{∞}≅ (𝒪_X_FF(d)⊕𝒪_X_FF)_|X_FF∖{∞}on X_FF∖{∞} (which again must preserve the filtration and associated gradeds).We thus obtain a canonical surjection (of _p-vector spaces)Γ B_et^d⊕ (B_^+)^φ=p^d→Ext^1_BKF^∘_rig(A_inf,A_inf{d})by sending (a,b)∈ B_et^d⊕ (B_^+)^φ=p^d to the quadruple(𝒪_X_FF⊕𝒪_X_FF, 𝒪_X_FF(d)⊕𝒪_X_FF,β= [ t^d a; 0 1 ],α= [ 1 b; 0 1 ])Assume that a pair (a,b) defines a trivial extension. Then there exists c∈_p≅ H^0(X_FF,𝒪_X_FF) and c^'∈ (B^+_)^φ=p^d≅ H^0(X_FF,𝒪_X_FF(d)) such that[ t^d a; 0 1 ][ 1 c; 0 1 ] = [ 1 c^'; 0 1 ][ t^d 0; 0 1 ] [ 1 b; 0 1 ][ 1 c^'; 0 1 ] = [ 1 0; 0 1 ].In other words, b=-c^' andct^d+a=c^'=-b.For b∈ (B^+_)^φ=p^d the element b/t^d lies in B_e. Therefore the pair (b/t^dt^d,b)∈Ker(Γ)lies in the kernel of Γ. In particular, we see that the morphismΓ_|B_et^d B_et^d→Ext^1_BKF^∘_rig(A_inf,A_inf{d})is still surjective. Moreover, its kernel is given by _pt^d because Γ(a,0) is trivial if and only if a∈_pt^d. The fundamental exact sequence implies thereforeExt^1_BKF^∘_rig(A_inf,A_inf{d})≅ B_et^d/_pt^d≅ B_/t^dB^+_. Let K:=Ker(Ext^1_BKF^∘(A_inf,A_inf{d})→Ext^1_X_FF(𝒪_X_FF,𝒪_X_FF(d)))be the kernel of the natural map (in terms of modifications it sends (ℱ,ℱ^',β) to ℱ^').Analyzing the proof of <Ref> we see thatK≅ B_et^d/(B^+_)^φ=p^d. Thus imposing the condition of a rigidification enlarges this group toB_et^d/_pt^d≅ B_/t^dB^+_. For more general Ext-groups we can prove the following result (which implies <Ref>). For simplicity, we denote by X^∘_FF:=X_FF∖{∞}=Spec(B_e)the punctured Fargues-Fontaine curve.Let (M_1,φ_M_1),(M_2,φ_M_2)∈BKF^0_rig(M_1,M_2) be rigidified Breuil-Kisin-Fargues modules up to isogeny with associated modifications (ℱ_i,ℱ_i^',β_i,α_i), i=1,2, of vector bundles on the Fargues-Fontaine curve as in <Ref>. Then there exist a natural surjectionH^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')↠Ext^1_BKF^∘_rig(M_1,M_2).If ℱ_i^', i=1,2, are semistable, then the kernel of this surjection is naturally isomorphic toH^0(X_FF,ℱ_1^∨⊗_𝒪_X_FFℱ_2).We first construct a natural morphismΓ H^0(X_FF^∘,ℱ_1|X^∘_FF^'∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')→Ext^1_BKF^∘_rig(M_1,M_2).Let b∈ H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')≅Hom_X_FF^∘(ℱ_1|X^∘_FF,ℱ_2|X^∘_FF^') be an element. Then we set Γ(b) to be the extension0 [r]ℱ_2[r]@–>[d]^β_2 ℱ_2⊕ℱ_1 [r]@–>[d]^[ β_2 b; 0 β_1 ] ℱ_1[r]@–>[d]^β_10 0 [r]ℱ_2^'[r]ℱ^'_2⊕ℱ^'_1 [r]ℱ_1^'[r] 0with rigidification[ α_2 0; 0 α_1 ]ℱ^'_2⊕ℱ^'_1≅gr^∙(HN(ℱ^'_2⊕ℱ^'_1))=gr^∙(HN(ℱ^'_2))⊕gr^∙(HN(ℱ^'_1)).We claim that Γ is surjective with kernel, if ℱ_1^',ℱ_2^' are semistable, given by H^0(X_FF,ℱ_1^∨⊗_𝒪_X_FFℱ_2)≅Hom_X_FF(ℱ_1,ℱ_2),embedded into H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^') via c↦β_2∘ c_|X^∘_FF. Let (ℱ,ℱ^',β,α)be an extension of (ℱ_1,ℱ_1^',β_1,α_1) by (ℱ_2,ℱ_2^',β_2,α_2). Then ℱ≅ℱ_1⊕ℱ_2because H^1(X_FF,𝒪_X_FF)=0. Moreover, as the extension must be compatible with the rigidification the extension0→ℱ_2^'→ℱ^'→ℱ^'_1→ 0splits, i.e., ℱ^'≅ℱ^'_1⊕ℱ^'_2. We moreover see that the rigidification α must be given by a matrixα=[ α_2 a; 0 α_1 ]with a∈Hom_X_FF(ℱ^'_1,gr(HN(ℱ^'_2))). Moreover, a must map to zero on the associated graded of the Harder-Narasimhan filtrationgr(a)gr(HN(ℱ^'_1))→gr(HN(ℱ^'_2)) as α must reduce to the identity on these graded pieces. Therefore there exists an a^'∈Hom_X_FF(ℱ^'_1,ℱ^'_2) such thatα_2∘ a^' = gr(a^')∘α_1 + a. Then the extension(ℱ,ℱ^',β,α)is isomorphic to(ℱ,ℱ^', [ 1 a^'; 0 1 ]∘β, [ α_2 0; 0 α_1 ] )and thus this extension lies in the image of Γ. Now assume that for b∈ H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^') the extension Γ(b)is trivial. This means that there exists two isomorphisms [ 1 c_1; 0 1 ]ℱ_1⊕ℱ_2≅ℱ_1⊕ℱ_2,with c_1∈Hom_X_FF(ℱ_1,ℱ_2), and[ 1 c_2; 0 1 ]ℱ^'_1⊕ℱ^'_2≅ℱ^'_1⊕ℱ^'_2,with c_2∈Hom_X_FF(ℱ_1^',ℱ_2^') inducing zero on the graded pieces of the Harder-Narasimhan filtration, such that[ β_2 b; 0 β_1 ][ 1 c_1; 0 1 ]= [ 1 c_2; 0 1 ][ β_2 0; 0 β_1 ].If ℱ^'_i,i=1,2, are semistable, then c_2=0 must be zero and thus b+β_2∘ c_1=0,i.e., b=-β_2∘ c_1. This implies therefore the last statement in the lemma. We want to deduce that the category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is of homological dimension 1.The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is of homological dimension 1, i.e., for every (M,φ_M), (N,φ_N)∈BKF^∘_rig the Ext-group (in the sense of Yoneda or equivalently as spaces of homomorphisms in the derived category) vanishes for i≥ 2, i.e., Ext^i_BKF^∘_rig(M,N)=0.<Ref> implies that the functorExt^1_BKF^∘_rig(M,-) preserves surjections of rigidified Breuil-Kisin-Fargues modules. Namely, for a surjection(N_1,φ_N_1)↠ (N_2,φ_N_2)of rigidified Breuil-Kisin-Fargues modules the associated surjectionℱ_1^'↠ℱ_2^'of vector bundles on the Fargues-Fontaine curve must be split due to the preservation of the given rigidifications. For an abelian category with enough injectives (or projectives) this would finish the proof by embedding an object into an injective object and using the associated long exact sequence. However, the category BKF^∘_rig does not contain enough injectives (or projectives), hence we have to work a bit more. For this let 𝒜 be an essentially small[This means that the isomorphism classes in 𝒜 form a set.] abelian category such that Ext^1_𝒜(A,-)preserves surjections for every A∈𝒜. Consider the (fully faithful and exact) embedding𝒜→Ind(𝒜)of 𝒜 into the category Ind(𝒜) of its Ind-objects. The category Ind(𝒜) need not have to have enough injectives in general (<cit.>), but it has if 𝒜 is essentially small which we assumed (in fact it is then Grothendieck abelian, cf. <cit.>). Hence, we can conclude that (cf. <cit.>)_A^'→ B Ext^k_𝒜(A,A^')≅Ext^k_Ind(𝒜)(A,B)for A∈𝒜, B∈Ind(𝒜) where the colimit is running over all A^'∈𝒜 with a morphism to B. Now let f B^'→ B^'' be a surjection of Ind-objects. Then f can be written as a filtered colimit of surjections A^'→ A^'' with A^',A^''∈𝒜 (cf. <cit.>). Using the above formular for the Ext-groups and our assumption on 𝒜 we can conclude that for A∈𝒜 the functorExt^1_Ind(𝒜)(A,-)preserves surjections. Using embeddings into injectives (of which there are enough in Ind(𝒜) by our assumption on essential smallness) we can conclude that Ext^i_Ind(𝒜(A,-)=0for A∈𝒜 and i≥ 2. Using again the above formula for the Ext-groups we can concludeExt^i_𝒜(A,A^')=0for A,A^'∈𝒜 and i≥ 2. Applying these considerations to the category 𝒜=BKF^∘_rig proves the lemma if we can show that the category BKF^∘_rig is essentially small. But by definition Breuil-Kisin-Fargues modules are finitely presented and the isomorphism classes of finitely presented A_inf-modules form a set. As the possibilities for adding a Frobenius or a rigidification form a set, we can conclude that the isomorphism classes of Breuil-Kisin-Fargues modules form a set as required. § CM BREUIL-KISIN-FARGUES MODULESIn this section we want to apply the formal CM theory of <Ref> to the case 𝒯:=BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>).Using Fargues' theorem <Ref> the classification of CM Breuil-Kisin-Fargues modules, i.e., CM objects in the Tannakian category 𝒯 (cf. <Ref>), is actually very simple - they are uniquely determined by their CM type (E,Φ) (cf. <Ref> and <Ref>).Actually, we can prove a stronger integral statement. For this let E/_p be a commutative semisimple algebra and let 𝒪 be an order in E.A finite free Breuil-Kisin-Fargues module with CM by 𝒪 will mean a finite free Breuil-Kisin-Fargues module (M,φ_M) together with an injection 𝒪→End_BKF((M,φ_M)), such that rk(M)=rk__p(𝒪). Let E be a commutative, semisimple, finite-dimensional algebra over _p and let 𝒪⊆ E be an order in E. Then there is a natural equivalence of categories between finite free Breuil-Kisin-Fargues module with CM by 𝒪 and pairs (T,Φ) where T is a faithful 𝒪-module, finite free over _p of rank __p(𝒪), and functions ΦHom__p(E,C)→. In particular, if 𝒪=𝒪_E is the maximal order, then there is a bijection of finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E (up to isomorphism) and “types” ΦHom__p(E,C)→. By <Ref> finite free Breuil-Kisin-Fargues modules with CM by 𝒪 are equivalent to pairs (T,Ξ) with T as in the statement of this lemma and Ξ⊆ T⊗__pB_ a B^+_-sublattice, stable under 𝒪⊗__pB^+_. But B^+_ contains _p, hence𝒪⊗__pB_≅ (E⊗__p_p)⊗__pB_≅∏_Hom__p(E,_p)B_and a 𝒪⊗__pB^+_-stable sublattice will be uniquely determined by the valuation in each factor, i.e., by a functionΦHom__p(E,_p)=Hom__p(E,C)→.If 𝒪=𝒪_E is the maximal order, then moreover every faithful 𝒪_E-module T which is finite free of rank d over _p must be isomorphic to 𝒪_E. This finishes the proof. We remark that in general there are non-trivial examples of 𝒪-modules T satisfying the hypothesis in <Ref>. We thank Bhargav Bhatt and Sebastian Posur for discussions about this point. In general, one can take 𝒪≠𝒪_E and T:=𝒪_E. But there exist also less pathological examples. For example, let E=_p(p^1/4) and set 𝒪:=_p[p^2/4,p^3/4]. Then 𝒪/p≅_p[t^2,t^3]/t^4≅_p[x,y]/(x^2,xy,y^2) is not Gorentstein. In particular, 𝒪 is not Gorenstein as well (the dualizing complex commutes with (derived) base change). However, it is still Cohen-Macaulay. Hence, the dualizing module T:=ω_𝒪≅Hom__p(𝒪,_p) (cf. <cit.>) for 𝒪 yields an example. It is finite free over _p of rank rk__p𝒪 and End_𝒪(ω_𝒪)≅𝒪. As 𝒪 is not Gorenstein, ω_𝒪≠𝒪. We now analyse rigidifications of Breuil-Kisin-Fargues modules with CM.Let E be a commutative semisimple algebra over _p of dimension d. Let (M,φ_M) be a finite free Breuil-Kisin-Fargues module of rank rk(M)=d with an injection E↪End_BKF^∘((M,φ_M)). Then there exists a unique rigidificationα M⊗_A_infB^+_≅ (M⊗_Ainf W(k))⊗_W(k)B^+_which is preserved by E, i.e., E-linear. We use <Ref> (respectively the remark following it) to argue with the modification (ℱ,ℱ^',β) of vector bundles on the Fargues-Fontaine curve associated with (M,φ_M). The algebra E acts by assumption on the vector bundle ℱ^' of rank d and we must produce a unique E-linear isomorphismℱ^'≅gr^∙(HN(ℱ^')). Decomposing E (and then ℱ^' accordingly) into factors reduces to the case that E is a field. Then we claim that ℱ^' must be semistable. Indeed, each subbundle ℰ^λ⊆ℱ^' in the Harder-Narasimhan filtration of ℱ^' must be stable under E. Let K be the function field of X_FF. Then E⊗__pK is again a field (because X_FF⊗__pE is again integral, cf. <cit.>) and passing to the generic point η∈ X_FF yields a E⊗__pK-stable flag in the 1-dimensional E⊗__pK-vector space ℱ^'_η. Hence, this flag is trivial and thus ℱ^' semistable. This implies finally that there exists a unique E-linear rigidification for ℱ^', namely the identity of ℱ^'. In other words, we can write down all rigidified Breuil-Kisin-Fargues modules up to isogeny with CM by a finite-dimensional commutative, semisimple _p-algebra E in terms of pairs (V,Ξ). Namely, V must be isomorphic (as an E-module) to E and Ξ⊆ E⊗__pB_ can be constructed explicitly from the typeΦHom__p(E,C)→. We now want to write down the Breuil-Kisin-Fargues modules (M,φ_M) corresponding to a pair (E,Φ). This will require more work.We note that for a finite free Breuil-Kisin-Fargues module (M,φ_M_Φ) as in <Ref> the function ΦHom__p(E,C)→ in <Ref> is precisely the type of the rigidified Breuil-Kisin-Fargues modules up to isogeny (_p⊗__pM,φ_M_Φ)∈BKF^∘_rig with respect to the filtered fiber functorω_ét⊗__pCBKF^∘_rig→Vec_C(cf. <Ref> and <Ref>). Namely, this follows from the concrete description of the filtration on ω_ét⊗__pC (cf. the discussion after <Ref> and the proof of <Ref>).In general a Breuil-Kisin-Fargues module with CM by a commutative semisimple _p-algebra E will decompose according to the factors of E. In particular, we may focus on the case where E is a field. Hence, we fix a finite extension E of _p and denote by E_0⊆ E its maximal unramified subextension. We fix a uniformizer π∈ E. Let _p⊆ k be the algebraic closure of _p. By formal étaleness of _p over _p there exists a unique lifting_p↪𝒪_C/p of the embedding _p→ k. Concretely, _p=⋃_n≥ 0(𝒪_C/p)^φ^n=1.Taking the inverse limit over Frobenius yields a canonical embedding_p↪𝒪_C^♭.The algebraic closure of _p in _p⊗__pA_inf is contained in_p≅_p⊗__pW(_p)⊆_p⊗__pA_infwhere _p denotes the completion of the maximal unramified extension _p^un of _p. In particular, the algebraic closure of _p in _p⊗__pA_inf is given by _p^un. For an embedding of ι E_0↪_p^un we defineA_inf,𝒪_E,ι:=𝒪_E⊗_𝒪_E_0,ιA_inf,the ring of “ramified Witt vectors” (cf. <cit.>). Its elements are formal power series∑_i=0^∞ [x_i]π^iwith x_i∈𝒪_C^♭. We setA_inf,E,ι:=_p⊗__pA_inf,𝒪_E,ι=E⊗_𝒪_EA_inf,𝒪_E,ι.Let τ E→ C be an embedding and let τ_0 E_0→ C be its restriction to E_0. Then the kernel of the homomorphismθ_τ𝒪_E⊗__pA_inf→ C, e⊗ x↦τ(e)θ(x) is principal, generated by a non-zero divisor ξ_τ. In the decomposition𝒪_E⊗__pA_inf≅𝒪_E⊗_𝒪_E_0(𝒪_E_0⊗__pA_inf)≅∏_ι E_0↪ C A_inf,𝒪_E,ιthe element ξ_τ can be chosen to be(1,…,π-[τ(π)^♭],…,1)with π-[τ(π)^♭] placed in the component τ_0. Here τ(π)^♭=(τ(π),τ(π)^1/p,…) denotes a p-power compatible systems of p-power roots of τ(π)∈ C. The morphism θ_τ(C)→(E⊗__pA_inf) must factor through one component and this component must be the one corresponding to the factor A_inf,E,τ_0 because θ_τ factors over A_inf,E,τ_0. Then the statement is well-known (cf. <cit.>). Being a non-zero divisor the element ξ_τ in <Ref> is unique up to a unit. Moreover, ∏_τ∈Hom__p(E,C)ξ_τ=uξwith u∈𝒪_E⊗__pA_inf a unit. Indeed, tensoring the exact sequence0→ A_infA_inf𝒪_C→ 0with 𝒪_E yields the sequence0→𝒪_E⊗__pA_inf𝒪_E⊗__pA_inf∏_τ𝒪_E→𝒪_C𝒪_C≅𝒪_E⊗__p𝒪_C→ 0which implies that the vanishing locus ξ and ∏_τξ_τ generate the same ideal. As both elements are non-zero divisors, they differ by a unit.Let Φ__p(E,C)→ be a type. Then we define the finite free Breuil-Kisin-Fargues module with CM by 𝒪_E as M_Φ:=𝒪_E⊗__pA_inf with Frobeniusφ_M_Φ:=ξ̃_Φφwhereξ̃_Φ:=∏_τ∈Hom__p(E,C)φ(ξ_τ)^Φ(τ)with ξ_τ as in <Ref> and φ=Id⊗φ_A_inf𝒪_E⊗__pA_inf→𝒪_E⊗__pA_inf. First observe thatξ̃_τ:=φ(ξ_τ)is a generator of the morphismθ̃_τ𝒪_E⊗__pA_inf→ C, e⊗ x↦τ(e)θ̃(x)which extends the morphism θ̃ A_inf→ C with kernel ξ̃. Henceevery ξ̃_τ is a unit in 𝒪_E⊗__pA_inf[1/ξ̃] and thus φ_M_Φφ^∗(M_Φ)[1/ξ̃]→ M_Φ[1/ξ̃]is indeed an isomorphism of A_inf[1/ξ̃]-modules. Moreover, the multiplication by 𝒪_E induces a multiplication on M_Φ and thus M_Φ is a finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E. To determine the isomorphism class of M_Φ it thus suffices (cf. <Ref>) to compute the type of M_Φ (which of course will turn out to be Φ). We check independently of <Ref> that the (isomorphism class) of the module (M_Φ,φ_M_Φ) is independent of the choice of the elements ξ_τ.We recall the following lemma. If d is degree of E_0 over _p, then we set φ_E_0:=φ^d. Fix ι∈Hom__p(E_0,C). Then for every x=∑_i≥ 0^∞[x_i]π^i∈ A_inf,𝒪_E,ι with x_0≠ 0 the 𝒪_E-module P_x:={y∈ A_inf,𝒪_E,ι | φ_E_0(y)=xy}is free of rank 1. Furthermore, if x∈ A_inf,𝒪_E,ι is a unit, then a generator of P_x is a unit in A_inf,𝒪_E,ι. After tensoring with _p the first assertion is proven in the proof of <cit.>. But P_x is ϖ-torsion free and hence free over 𝒪_E of rank 1.The second part follows from the proof of <cit.>. For example, if E=_p and x=ξ̃, then y=μ spans the space P_ξ̃ asφ(μ)=[ε^p]-1=[ε^p]-1/[ε]-1([ε^p]-1)=ξ̃μ. In this case, the space P_ξ̃ does not contain a unit in A_inf.We strengthen <Ref> a bit to handle the non-connected ring 𝒪_E⊗__pA_inf as well.Let x∈𝒪_E⊗__pA_inf such that x maps to a unit in 𝒪_E/p⊗__pC^♭. Then the spaceP_x:={y∈𝒪_E⊗__pA_inf | φ(y)=xy}is free of rank 1 over 𝒪_E. Here φ denotes the Frobenius _𝒪_E⊗φ_A_inf on 𝒪_E⊗__pA_inf. If x is already a unit in 𝒪_E⊗__pA_inf, then P_x contains a unit in 𝒪_E,⊗__pA_inf. Write 𝒪_E⊗__pA_inf≅∏_ι∈Hom__p(E_0,C) A_inf,𝒪_E,ιand thus the element x=(x_ι)_ι∈Hom__p(E_0,C) accordingly. The assumption on x implies that each x_ι satisfies the assumption in <Ref>. If x∈𝒪_E⊗__pA_inf is a unit, then as well each x_ι is a unit. Moreover, the Frobenius φ permutes the factors cyclically. The powerφ^d=φ_𝒪_E_0fixes every factor and induces, for a fixed ι, the morphism φ_𝒪_E_0 on A_inf,𝒪_E,ι. Fix some ι_0 and let y_0∈ A_inf,𝒪_E,ι be a generating solution ofφ_𝒪_E(y_0)=x_ι_0 y_0 (y_0 is a unit if x is a unit). As φ permutes the factors, it is clear that we get a generating solution y∈ A_inf,𝒪_E,ι (which is a unit if x is a unit) for the equationφ(y)=xy.Moreover, this y must be unique up to multipliciation by 𝒪_E^×. The Breuil-Kisin-Fargues module (M_Φ,φ_M_Φ) in <Ref> is up to isomorphism independent of the choice of the elements ξ_τ in <Ref>. A different choice of the elements ξ_τ yields the Breuil-Kisin-Fargues module(M^'_Φ,φ_M^'_Φ):=(𝒪_E⊗__pA_inf,uξ̃_Φφ)with u∈ (𝒪_E⊗__pA_inf)^× a unit. By <Ref> we can find a unity∈ (𝒪_E⊗__pA_inf)^×such thatφ(y)=uy.Then multiplication by y will define an (𝒪_E-linear) isomorphism(M^'_Φ,φ_M^'_Φ)→ (M_Φ,φ_M_Φ). We can now describe all (rigidified) Breuil-Kisin-Fargues modules with CM.Let ΦHom__p(E,C)→ be a type. Then the finite free Breuil-Kisin-Fargues modules(M_Φ,φ_M_Φ) from <Ref> has type Φ. Moreover, every finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E with type Φ is isomorphic to (M_Φ,φ_M_Φ). If we can show that (M_Φ,φ_M_Φ) has type Φ, then the last assertion follows from <Ref>. Fix as in <Ref> elementsξ_τ∈𝒪_E⊗__pA_infgenerating the kernel ofθ_r𝒪_E⊗__pA_inf→𝒪_Cwith τ running over all embeddings τ E→ C. For simplicity we may (after multiplying one ξ_τ by a unit in 𝒪_E⊗__pA_inf) assume thatξ=∏_τξ_τ(cf. the discussion after <Ref>). Setξ̃_τ:=φ(ξ_τ)∈𝒪_E⊗__pA_infand let μ_τ∈𝒪_E⊗__pA_inf be as in <Ref> the unique (up to multiplication by 𝒪_E^×) generator forelements satisfying the equationφ(μ_τ)=ξ̃_τμ_τ(if E=_p and τ_p→ C is the unique inclusion, then ξ_τ can be taken to be ξ, which implies μ_τ=μ (up to a unit)). Note that μ_τ is a unit in E⊗__pA_inf[1/μ]. Indeed, the productν:=∏_τμ_τsatisfiesφ(ν)=ξ̃νas we assumed ξ=∏_τξ_τ. Hence, by <Ref> ν and μ differ by a scalar in E. Applying φ^-1 to the defining equation of μ_τ yields μ_τ=ξ_τφ^-1(μ_τ) and we see that φ^-1(μ_τ) is a unit in (E⊗__pA_inf)⊗_A_infB^+_ because the same is true for φ^-1(μ) (one checks θ(φ^-1(μ))=ζ_p-1≠ 0). We note that the type of M_Φ depends only on the Breuil-Kisin-Fargues modules up to isogeny E⊗_𝒪_EM_Φ. Set V:=(M_Φ⊗_A_infW(C^♭)[1/p])^φ_M_Φ=1Then we know that V is a one-dimensional E-vector space. Recall thatφ_M_Φ=ξ̃_Φφwith ξ̃_Φ:=∏_τξ̃_τ^Φ(τ).We can explicitly find a generator of V, namely the elementμ_Φ:=∏_τμ_τ^-Φ(τ)∈ E⊗__pA_inf[1/μ].Indeed,φ(μ_Φ)=ξ̃_Φ^-1μ_Φ,which impliesφ_M_Φ(μ_Φ)=μ_Φ.By definition, the Breuil-Kisin-Fargues modules (up to isogeny) (E⊗_𝒪_EM_Φ,φ_M_Φ) corresponds to the pair(V,Ξ:=M_Φ⊗_A_infB^+_⊆ V⊗__pB_)in (the isogeny version) of <Ref>.Now, the E⊗__pB^+_-lattice V⊗__pB^+_⊆ V⊗__pB_=E⊗__pB_ is generated by μ_Φ. But μ_Φ equals, up to a unit in E⊗__pB^+_, the inverse ξ_Φ^-1 of the elementξ_Φ:=∏_τξ_τ^Φ(τ)=φ^-1(ξ̃_Φ) in E⊗__pB^+_ while the E⊗__pB^+_-lattice Ξ is generated by 1∈ E⊗__pB^+_. In other words, we find thatΞ=ξ_Φ(V⊗__pB^+_)which implies that (M_Φ,φ_M_Φ) has type Φ by looking at the explicit decompositionE⊗__pB_≅∏_τ∈Hom__p(E,C)B_under which ξ_Φ maps to the element (ξ^Φ(τ))_τ. We want to finish this chapter with a concrete description of the automorphismsAut^⊗(ω_ét,CM)of the fiber functor, “the étale realization”,ω_ét,CMBKF^∘_rig,CM→Vec__pon Breuil-Kisin-Fargues modules with CM.Recall the pro-torus D__p:=__p/L/_pL^∗ which was introduced before <Ref>. The reflex norm (with respect to the “étale realization”)rBKF^∘_rig,CM→Rep__p(D__p)is an isomorphism. In particular,Aut^⊗(ω_ét,CM)≅ D__p.We want to use <Ref>. In <Ref> with have proven that every rigidified Breuil-Kisin-Fargues modules (up to isogeny) with trivial filtration on its étale realization is actually trivial, i.e., a direct sum of the unit object. This implies that the functor r is fully faithful by <Ref> and then <Ref> implies, using again <Ref>, that r is essential surjective. This finishes the proof. We remark that <Ref> implies that the category BKF^∘_rig,CM of rigidified Breuil-Kisin-Fargues modules up to isogeny admitting CM is generated by Breuil-Kisin-Fargues modules (M,φ_M) whose associated B_^+-lattice Ξ⊆ T⊗__pB_ is minuscule, i.e.,ξ (T⊗__pB^+_)⊆Ξ⊆ T⊗__pB^+_. By <cit.> these Breuil-Kisin-Fargues modules are associated to p-divisible groups over 𝒪_C. In particular, we see that the category BKF^∘_rig,CM of CM Breuil-Kisin-Fargues modules is generated (as a tensor category) by Breuil-Kisin-Fargues modules associated with p-divisible groups admitting CM. This is analogous to the case of rational Hodge structures admitting CM (cf. <cit.>).plain | http://arxiv.org/abs/1707.08857v1 | {
"authors": [
"Johannes Anschütz"
],
"categories": [
"math.NT",
"11S99, 11G15, 14F30"
],
"primary_category": "math.NT",
"published": "20170727132615",
"title": "Breuil-Kisin-Fargues modules with complex multiplication"
} |
firstpage–lastpage The [Y/Mg] clock works for evolved solar metallicity stars Based on spectroscopic observations made with two telescopes: the Nordic Optical Telescope operated by NOTSA at the Observatorio del Roque de los Muchachos (La Palma, Spain) of the Instituto de Astrofísica de Canarias and the Keck I Telescope at the W.M. Keck Observatory (Mauna Kea, Hawaii, USA) operated by the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.D. Slumstrup 1 F. Grundahl 1 K. Brogaard 1,2 A. O. Thygesen 3 P. E. Nissen 1 J. Jessen-Hansen 1 V. Van Eylen 4 M. G. Pedersen 5 Received 3 July 2017 / Accepted 20 July2017============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Wide-field high precision photometric surveys such as Kepler have produced reams of data suitable for investigating stellar magnetic activity of cooler stars. Starspot activity produces quasi-sinusoidal light curves whose phase and amplitude vary as active regions grow and decay over time. Here we investigate, firstly, whether there is a correlation between the size of starspots - assumed to be related to the amplitude of the sinusoid - and their decay timescale and, secondly, whether any such correlation depends on the stellar effective temperature. To determine this, we computed the autocorrelation functions of the light curves of samples of stars from Kepler and fitted them with apodised periodic functions. The light curve amplitudes, representing spot size were measured from the root-mean-squared scatter of the normalised light curves. We used a Monte Carlo Markov Chain to measure the periods and decay timescales of the light curves. The results show a correlation between the decay time of starspots and their inferred size. The decay time also depends strongly on the temperature of the star. Cooler stars have spots that last much longer, in particular for stars with longer rotational periods. This is consistent with current theories of diffusive mechanisms causing starspot decay. We also find that the Sun is not unusually quiet for its spectral type - stars with solar-type rotation periods and temperatures tend to have (comparatively) smaller starspots than stars with mid-G or later spectral types. techniques: photometric – stars: activity – stars: starspots – stars: rotation§ INTRODUCTION The Kepler mission was designed to search for extrasolar planet transits in stars (within a single field of view) in particular small, Earth-like planets around Sun-like stars <cit.>. It has provided insight into planet formation as well as new exoplanet discovery, which also allowed to determine occurrence rates <cit.> and further probe the statistics of exoplanet population and system architectures.Kepler has also revolutionised stellar physics. Tens of thousands of stars have 4 years worth of almost continuous, high precision photometry, allowing for a thorough study of stellar brightness modulations across different stellar ages and types. From Kepler, fields such as asteroseismology <cit.> and differential rotation studies <cit.> of main sequence stars have evolved through the study of such a large sample of stars. <cit.> (hereafter known as ) made the first large-scale surveys of stellar rotation by analysing the autocorrelation functions of stellar light curves.This unprecedented wealth of high-precision, continuous photometric data for thousands of main-sequence stars has enabled us to take a new look at our own Sun, resulting in comparisons between it and stars which are Sun-like. <cit.> <cit.> found that the Sun appears to be unusually inactive when compared to other solar-type stars, but it has since been suggested that this may in fact not be the case <cit.>. This is discussed in <ref>. In this paper our goal is to discover how Kepler observations can be used to infer the lifetimes of active regions on other stars, and to determine how the lifetime of an active region depends on its size and on the stellar photospheric temperature. We define stellar activity, and active regions, in this context as meaning phenomena that introduce surface brightness inhomogeneities, giving rise to apparent flux modulation as the star rotates. Measurements of solar irradiance as a function of wavelength show that bright faculae and dark starspots are the main contributors to solar flux modulation on timescales of order days to weeks <cit.>. These modulations have a greater amplitude when the Sun is near the maximum of its 11-year activity cycle. The solar irradiance variations are complex; solar active regions often comprise a bipolar spot group surrounded by an extended facular region of enhanced surface brightness. As an active region crosses the solar disc, the limb brightening of the faculae and foreshortening of the dark spots tends to cause a net initial flux increase. This is followed by a decrease as the spot visibility increases and the facular limb brightening declines <cit.>. A similar pattern is seen in Kepler light curves. At times of high activity, the amplitude of variability is often seen to increase with no obvious change in the mean flux level in the Kepler bandpass. Solar irradiance measurements, however, show clearly that the facular flux increase outweighs the dark spot deficit at times of high activity <cit.>.For the Sun, a range of activity levels have been observed since telescopic records began (from the Maunder Minimum to large-amplitude cycles in the mid-20th century) and there are many differing opinions on what constitutes `typical' solar activity levels <cit.>. The consensus appears to be that the average level of solar activity lies in between the extremes observed in the past 400 years. For our purpose, we will use the activity levels seen in the last 3 to 4 sunspot cycles as typical levels.Furthering our understanding of stellar activity is not only important to the stellar community; it is crucial to many other areas of investigations, particularly in the exoplanet society. The presence of starspots and other magnetic active regions can induce quasi-periodic variations over timescales of weeks to months. These activity signatures are seen as major sources of noise in the search for small exoplanets (Earths and super-Earths); spots can lead to wrong planet radius measurements <cit.>. The presence of starspots and other magnetically active regions are a real nuisance in RV exoplanet observations. As well as starspots, faculae and granulation produce signals modulated by the star's rotation. They evolve over time, giving rise to quasi-periodic signals with varying amplitudes and phases. This induces RV variations of 1-2 ms^-1 even in the quietest stars <cit.>. Stellar noise can conceal and even mimic planetary orbits in RV surveys, and has resulted in many false detections (eg. CoRoT-7d, ; Alpha Centauri Bb, ; HD166435, ; HD99492, ; HD200466, ; TW Hydra, ; HD70573, ; HIP13044, ; Kapteyn's Star, ; Gliese 667d, ; and GJ 581d ). It also significantly affects our mass estimates, which are routinely determined from RVs. A number of methods have been developed to account for activity-induced RV signals and have been quantitatively tested to review their perfomance <cit.>. Therefore, knowing the active region lifetimes can provide significant constraints for models used to determine exoplanet properties, such as mass <cit.>. Additionally, planet radii and masses are central to theoretical models of planet composition and structure <cit.> and are essential to interpreting observations of atmospheres <cit.>. When it comes to studying atmospheric transmission spectroscopy of planet atmospheres, un-occulted spots serve to increase the ratio of the area of the planet's silhouette to that of the bright photosphere, making the transit look deeper than it really is. On the other hand, un-occulted faculae have the opposite effect. Since the contrast of both faculae and spots against the quiet photosphere depends on wavelength, particular care has to be taken in the interpretation of atmospheric transmission spectroscopy <cit.>. As the effects of starspots and suppression of the granular blueshift in faculae are expected to diminish towards longer wavelengths <cit.>, forthcoming infrared RV spectrometers such as CARMENES <cit.> and SPIRou <cit.> may help to separate planetary reflex motions from stellar activity signals. However, until recently only optical spectrometers were reaching the precision needed to determine the masses of super-Earth planets but CARMENES has been achieving 2ms^-1 which is sufficient for measuring super-Earths <cit.> which would therefore suggest that others will be able to perform similarly, according to their specifications. Sunspot (and by association, starspot) decay lifetimes have been a point of interest for decades, with many theories for the cause of their decay and what function it follows. Numerical investigations such as those by <cit.> indicate that sunspot decay is consistent with a parabolic decay law, where the area of the spots decreases as a quadratic function of time. Observations of the Sun <cit.> have similarly reflected the same behaviour. This relationship would imply that the main factor in spot decay is granulation, which was first hypothesised by <cit.>. Extrapolating the physics observed to occur on the Sun, only a few attempts have been made to measure starspot decay lifetimes. These studies would allow us to test our theories for sunspot decay on other Sun-like stars. As we cannot resolve the surfaces of others stars directly and at high-resolution like we can for the Sun, their sizes over time have to be inferred from indirect indicators. <cit.> have recovered the decay lifetime of starspots from both real and simulated Kepler data. However, there has not been a large-scale survey of starspot decay lifetimes until now.In this paper, we determine the starspot lifetimes in a large sample of stars selected to have rotation periods close to 10 days and 20 days. Our technique, based on MCMC parameter estimation, allows us to determine estimates and uncertainties for the stellar rotation period and starspot lifetime of each star. We then investigate how the decay lifetimes relate to extrapolated spot sizes and whether the stellar spectral type has a role in this relationship. In <ref>, we justify the choice of stellar targets. In <ref> we describe our improvements to the method used inand how the representative measurements for spot sizes are determined. In <ref> and <ref> we outline and discuss our results and the implications they have for stellar physics and exoplanetary discovery and characterisation. § SAMPLE SELECTIONOur samples are drawn from the sample of stars analysed by . They analysed over 34,000 main sequence stars taken from the Kepler mission stellar archive at the NASA Exoplanet Archive <cit.>. All of the stars inwere less than 6500K in temperature and excluded known eclipsing binaries (EBs) and Kepler Objects of Interest (KOIs).utilised T_ eff - log g and colour-colour cuts used by <cit.> to select only main sequence stars. The boundary of 6500K was selected byto ensure that only stars with convective envelopes, that spin down during their lifetime, were included.To keep computational time to manageable levels, two samples were drawn from the +34,000stars based on the measured rotation periods. Sample 1 has a range of periods between 9.5 and 10.5 days, and sample 2 with a range of 19.5 to 20.5 days. This resulted in 1089 and 1155 stars in each respectively. Unlike inwhere they used quarters 3-14 from the Exoplanet Archive, quarters 1 to 17 were used here. This was done to extend the temporal span of the light curves as much as possible.§ METHODS §.§ Autocorrelation Function (ACF)We created ACFs in the same fashion as <cit.> who cross-correlated each Kepler light curve with itself at a series of discrete timeshifts (time lags). The correlation increases and decreases dependening on the presence of a large dominant starspot. As a light curve can be approximated as sinusoidal in shape <cit.>, a time lag at an integer multiple of the stellar rotation period correlates strongly meaning the first side lobe of an ACF corresponds to the stellar rotation period with further side lobes as harmonics of the period. The decrease in side lobe amplitude at higher time lags occurs as the light curve gradually varies in amplitude and phase due to starspot formation and decay. Therefore the decay rate of the side lobes describes the decay rate of the starspots. By visual inspection, this appears to be comparable to an exponential decay. With this knowledge, ACFs were fitted with a simple analytical function. This is an improvement on what was reported by <cit.> as it establishes further parameters of the stellar activity but also determines errors for them.Many autocorrelation algorithms require the data to be uniformly sampled in time – Kepler data is close to uniformity but has variation in exact observation times and has significant data gaps. Therefore to generate ACFs, the light curves were binned and weighted as described by <cit.>, which has the added advantage of providing error estimates. Once the ACFs were generated, they were orthogonalised by subtracting the inverse variance-weighted mean, to ensure there were no unwanted correlations between the ACF power and the time lag.The behaviour of an ACF at zero time lag ≥0 days resembles the displacement of an underdamped simple harmonic oscillator (uSHO), described byy(t) = e^-t/τ_AR( A cos(2π t/P) ) + y_0. Many ACFs have an additional `interpulse' close to half of the stellar rotation period (Fig. <ref>). This corresponds to there being another large but less dominant starspot on the opposite side of the star. Therefore the uSHO equation was adapted to include an inter-pulse term,y(t) = e^-t/τ_AR( A cos(2π t/P) + B cos(4π t/P) + y_0 ).τ_AR is the decay timescale [days] of the ACF which represents the decay timescale of the dominant starspot. P is the stellar rotation period [days^-1]. (Parameters A, B and y_0 do not represent physical properties of the star, but are needed to fit the uSHO equation.) A and B are the amplitudes of the cosine terms and y_0 is the offset of the uSHO from y=0. The stellar rotation period is taken to be the time lag at which the largest side lobe occurs at and is found by searching for all peaks in the ACF and establishing which is the highest (besides the peak at time lag = 0 days).<cit.> used a damped, stochastically-driven harmonicoscillator model to emulate the quasi-periodic behaviour of solar p-modes. They also computed the autocorrelation function of the resulting time series, obtaining an expression equivalent to eq. <ref> above. They used this as the kernel for a gaussian-process regression analysis of the waveform. Because of the N^3 computational overhead involved in Gaussian-process regression, the large number of data points in each light curve and the large number of light curves analysed here, we elected instead to perform the parametric fit to the autocorrelation functions, as described by eq. <ref>. §.§ Monte Carlo Markov Chain The uSHO equation was fitted to ACFs using a Monte Carlo Markov Chain - MCMC. An MCMC is a means to `random walk' towards the and to sample the joint posterior probability distribution of the fitted parameters. By estimating initial values for the parameters, X_θ, an initial fit of the uSHO equation is done and the likelihood, ℒ, measured through lnℒ = -χ^2/2 - ∑_i=1^N( lnσ_y_i) - N/2ln(2π)whereχ^2 =∑_i=1^N( y_i - μ/σ_y_i)^2.where N is the number of ACF points, y_i the value of the ACF points with the error σ_y_i, μ is the model ACF point value that corresponds to y_i. As the ACFs are often more distorted from the uSHO trend at higher time lags, due to interference from new starspots coming into effect, the MCMC only fits up to a time lag equivalent to 2.5× P.The parameter values are then perturbed by a small amount to a new position in parameter space and the fit and likelihood calculations are repeated. If the likelihood is higher than the previous likelihood then the step is accepted and the next step takes place from the current location in parameter space. If the likelihood is worse than previous, it may be accepted under the Metropolis-Hastings algorithm <cit.>, otherwise it will be rejected and the step is not completed and it goes back to the previous step and randomly steps again.The Metropolis-Hastings algorithm enables occasional steps in the wrong direction to ensure that an MCMC does not become trapped at a local likelihood maximum, and to enable exploration of the entire likelihood landscape. An optimum acceptance rate for an N-dimensional MCMC is approximately 0.25 <cit.>. Rates much lower or higher than this may struggle to converge. To achieve this, an optimal step size is calculated from the curvature of the χ^2-parameter space for each parameter α,σ_X_i = √(2/∂^2χ^2/∂^2α^2) ,where the exact step size per MCMC step is a Gaussian distribution using σ_X_i and centred on the previous parameter value.The initial inputs of the parameters for the MCMC are estimated from the ACF or given standard values: period in days, determined as the time lag of the largest side lobe of the ACF, representative of the rotation period; the decay time τ_AR is based on the ratio of the first and second peaks of the ACF,τ_AR = -P/log(y_i(P)/y_i(0));A is the ACF value at time lag = 0; and B and y_0 are taken to be zero.As a means to encourage the MCMC not to search for solutions in the unlikely areas of parameter space, Gaussian priors were applied to three of the parameters: amplitude A, P and logτ_AR. For τ_AR, having a Gaussian prior in log space reduces the risk of the MCMC wandering to unlikely high values. Also a hard lower limit of 1 day was included for logτ_AR to prevent a highly improbable τ_AR value.To determine whether convergence has been achieved, we adopt a likelihood rule as used by <cit.> and <cit.>. Each calculated likelihood ℒ was stored and the current likelihood compared to the median of all those previous. When ℒ falls below the median, the MCMC is considered to have achieved convergence. The MCMC then conducts another 5000 steps from which the mean and the standard deviation of each parameter are measured. This then launches a second MCMC routine using the mean and standard deviations as new initial parameters, X_θ, and step sizes (±σ_X_θ). This second MCMC explores the likelihood maximum to find the optimal parameter values. Two final tests for convergence are applied to the final 5000 steps of the second MCMC chain: we calculate the correlation length of this chain (and check that it is less than ∼5% of the total chain) and compute the Gelman-Rubin test <cit.>. Only stars that passed both of these tests are considered completed. These stars were then quickly visually inspected to remove any where the fits were obviously wrong. Additionally, a check for correlations of all the fits of the ACFs for the targets was conducted by comparing all the parameter values to one another. In Figs. <ref> and <ref>, it can be seen that there are no strong unexpected correlations. The small correlation between the two amplitude sizes is not concerning as when there is an interpulse present in an ACF this reduces the initial amplitude at zero-time lag. Therefore, the larger the interpulse amplitude, the smaller the initial amplitude. §.§ Kepler Light Curve MorphologiesThere are three distinct types of light curve morphologies (Fig. <ref>) that can be seen in the bulk of Kepler data - `Sun-like', `Beater' and `Coherent'. These are purely qualitative descriptions. On the other hand, inspecting the autocorrelation functions, a distinction can be seen. `Sun-like' stars appear to have starspot decay lifetimes that last approximately a rotational period, `Beaters' have lifetimes that last a few rotations and the `Coherent' stars have spots that persist for many rotations. Thereby taking the ratio of the activity starspot lifetime versus rotational period, τ_ AR/P_ rot (AR=Active Region, rot=rotation), we can define the ratio for each light curve morphology as ∼1 for Sun-like stars, >1 for `Beaters' and ≫1 for the `Coherent' stars. It is known from Doppler imaging studies that many very active, fast-rotating stars have large, dark polar spots <cit.>. Unless they are perfectly axisymmetric, such large polar features are likely to give rise to quasi-sinusoidal modulation. Since polar spots are generally large, we might expect them to have long lifetimes, producing modulations that would remain coherent for many rotation cycles. At the modest activity levels of most Kepler stars, however, such large polar spots are not expected to be widespread. §.§ Determining the Starspot Sizes Whilst it is possible to determine approximate spot sizes for FGK-stars from Doppler imaging <cit.>, there is currently no direct method to measure them from light curves. However, light curves do have continuous variations – these occur due to asymmetry between two sides of the star. It is worth making the point that the amplitude of solar photometric variability increases with overall activity levels through the magnetic cycle <cit.>. This implies that the power-law distribution of active-region sizes is such that the largest individual active regions dominate the modulation. If all active regions were of similar size, an increase in the number of active regions at different longitudes would cause the light curve modulation amplitude to decrease rather than increase.<cit.> Therefore, as a proxy, the root-mean-square (RMS) scatter of the light curve can be extrapolated to be representative of starspot size.RMS = √(1/N∑_i=1^Ny_i^2)N is the total number of points in the light curve and y_i the value of the flux at each data point. For a target, the 2-σ range of the RMS (which encompasses ∼95% of points) is calculated, as this encompasses the majority of the sinuous structure of the light curve but ought not include the erroneous outliers which may not have all been removed during post-observation processing.§ RESULTSGenerally the quality of the fits produced by the MCMC routine were good, though some were poorer and a couple were entirely spurious fits. Therefore all of the results were also inspected by eye and those with significantly different fits, therefore not representative, were rejected from the sample.With 1089 stars for the 9.5-10.5 day (i.e. 10 day) period sample and 1154 stars for the 19.5-20.5 day (i.e. 20 day) period sample, the ACF fitting program returned 913 (83.8% success rate) and 861 (74.6% success rate) acceptable ACF fits for the 10 day and 20 day sample respectively.In Fig. <ref> the targets have been partitioned by spectral type (from M- to F-stars) as determined from <cit.>, and are represented by different colours and symbols which are detailed in the attached key. The first row shows the how the RMS amplitude of the rotational modulation (proxy for the starspot size for a star) varies with the stellar effective temperature for each of the two samples. The second row displays how the decay lifetime depends on the effective stellar temperature. §.§ Comparison of Rotation PeriodsInthe periods were determined using an autocorrelation function routine, and these were used during sample selection. Comparing the periods fromand those generated by the MCMC (Fig. <ref>), there is some variation with the 10 day sample varying less than the 20 day sample. This range will reflect upon the difference in autocorrelation function generation as the routines used inand this paper are different, meaning variation in stellar rotations periods is to be expected. Further, as a point of interest, the residuals for the 10 day sample are asymmetric, with our algorithm generally finding longer periods than . Due to not fitting the decay envelope,will have underestimated the period, biasing the first sidelobe to a lower time lag. Therefore, the shorter the decay lifetime, the larger a discrepancy seen in Fig. <ref>. Interestingly, this becomes symmetric for the 20 day sample, but with the same trend that shorted decay lifetimes have larger range. §.§ 10 Day Period Sample For this sample, in Fig. <ref> (left-hand side), there is a distinct distribution of starspot sizes and decay lifetimes. Hotter stars with T_ eff greater than 6200K, have a smaller range of spot sizes than cooler stars. These stars also have spots which do not survive for very long. At effective temperatures above the ∼6200K boundary, the limit on decay lifetime is less than 100 days. This is up to a third of starspot lifetimes on much cooler stars.For ease of viewing, the comparison between spot size and decay lifetime has been split into the four observed spectral types in Fig. <ref>. The coolest stars (M-stars) have a large range of spot size vs. decay timescale but given the very small stellar population this is not representative. However, there are a great many more K-stars and G-stars which show a strong trend of longer decay lifetimes for larger spots. The gradient of the trend is greater for the K-stars, indicating that the hotter the star, the shorter the lifetime. Additionally, the range of the spot sizes associated with the G-stars is less than the K-stars. This limits spots to have no larger effect on the light curve than an RMS of 0.025 mag. The F-stars, like the M-stars are not very numerous in this sample. However, they do all cluster together at low decay lifetimes and small spot sizes suggesting that for this the hottest of all the targets, spots rarely reach a large size or survive very long. This would also suggest spots survive longer the bigger they are. §.§ 20 Day Period Sample The 20 day sample is similar to the previous sample with a few small differences (Fig. <ref>): the temperature above which the range of spot sizes dramatically decreases is at a lower temperature ∼5700K and spots can survive longer on cooler stars than in the 10 day sample.As for the 10-day sample, when we partition the stars by spectral type for the relationship between decay lifetime and spot size (Fig. <ref>), the coolest stars again are not well represented. For the K- and G-stars there is again a positive relationship with increasing decay lifetime and larger spots, with the trend gradient appearing to just be slightly steeper for the K-stars. However, the range of decay lifetimes and spot sizes is much more limited for these G-stars than in the other sample. The F-stars similarly cluster in the lower decay lifetime, smaller spot size area, but have a little more range than the 10 day period sample of F-stars.§.§.§ Solar ComparisonFrom investigations on stars observed by Kepler and previous surveys, there was discussion about the activity of the Sun and whether it was unusually quiet <cit.>. Comparing it to the 20 day sample (solar rotation period ∼27 days), stars with Sun-like temperatures (∼5800K) all have small light curve amplitudes indicating small spots. The amplitudes of solar variability measured by <cit.> through the solar cycle are very similar to those measured in this work for stars with solar-like rotation periods and effective temperatures. This would <cit.> indicate that the Sun is not suspiciously inactive. §.§ Spot Size and Distribution §.§.§ RMS as a Proxy for Spot SizeWe find that stars with large RMS-variations indicate spots with longer lifetimes. This could lead to two interpretations: large variations could mean that there are a few big spots dominating with smaller RMS variations meaning there are only small spots. But it could theoretically be possible that there are many spots of a similar size. There is good physical reasoning behind the hypothesis that diffusive decay takes longer to destroy big active regions than small ones. If indeed the lifetime is short for stars that have many spots of similar size, short lifetimes would also be associated with small light curve amplitudes. Implementing Occam's Razor, the simpler explanation is, however, that the solar spot-size and spot-lifetime power laws can be extrapolated to other stars, and that the same physical processes operate.§.§.§ Active-Region Lifetime as a Function of Spot Size and Effective Temperature Using the two datasets together, it is possible to generate a function using the RMS (as a spot size proxy) and effective temperature to generate an expected active region lifetime which can be used for an individual star. Orthogonalising the data by removing the mean value of each distribution and fitting a quadratic through regression to the data in log-log space, the following relation is determined:log_10τ_AR = 10.9252 + 3.0123·log_10 RMS+ +0.5062·(log_10 RMS)^2- 1.3606·log_10T_ effwhere RMS is the RMS scatter of individual Kepler light curves which were normalised to a mean flux of unity, T_ eff is the stellar effective temperature in K, and τ_ AR is the resultant decay lifetime in days. If this is used as an estimate for the mean of a Gaussian prior probability distribution for logτ_ AR then the standard deviation σ of the residuals from the fit should be used as the standard deviation σ of the prior: σ( log_10τ_ AR) = 0.178623.§.§.§ Active LongitudesWhen considering active longitudes, evidence from the Kepler light curves suggests that even if spots persistently recur at active longitudes, they would tend to preserve the coherence of the light curve on timescales longer than the lifetimes of an individual active region. We cannot explicitly say whether such an effect is present, however we note that the decay timescales we obtain from the light curves of the solar-like stars are comparable with the lifetimes of the large solar spot groups.§ CONCLUSIONThe subject of this paper was to determine whether there is a relationship between the sinusoidal amplitude seen in Kepler light curves, as a proxy for starspot size, and the decay timescale of starspots. Furthermore, we sought to determine whether the lifetimes of spots of a given size depend on the stellar effective temperature.As seen within the two samples (9.5-10.5 days and 19.5-20.5 days period stars) drawn from , there are three main conclusions: * Big starspots live longer on any given star,* Starspots decay more slowly on cooler stars and* The Sun is not unusually quiet for its spectral type. Our observation that big spots generally survive longer longer on any given star is consistent with models of spot decay in which turbulent diffusion is eating the edges of the spots <cit.>. This is also consistent with our finding that spots generally survive longer on cooler stars. As the vigour of convection is temperature dependent, the turbulent diffusivity, and hence the rate of spot decay, will increase with the convective heat flux and hence with effective temperature. An analogy would be food colouring being dispersed more slowly in cool water than in boiling water.The work presented in this paper has deepened our knowledge of the connection between the light curve morphologies of Kepler stars and the physics that determine active-region lifetimes in convective stellar photospheres. This in turn can be applied to many areas which rely on light from stars, in particular when searching and analysing exoplanet host candidates.§ ACKNOWLEDGEMENTS We would like to thank our referee for their constructive comments that have improved the quality of this work.This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.HACG acknowledges the financial support of the National Centre for Competence in Research `PlanetS' supported by the Swiss National Science Foundation (SNSF) and the financial support of the University of St Andrews; and the computer support from the University of St Andrews. ACC acknowledges support from STFC consolidated grant number ST/M001296/1. RDH gratefully acknowledges support from STFC studentship grant ST/J500744/1, a grant from the John Templeton Foundation, and NASA XRP grant NNX15AC90G. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. This research was submitted as a Masters project for HACG supervised by ACC at the University of St Andrews in April 2015. 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"authors": [
"Helen A. C. Giles",
"Andrew Collier Cameron",
"Raphaëlle D. Haywood"
],
"categories": [
"astro-ph.SR",
"astro-ph.EP"
],
"primary_category": "astro-ph.SR",
"published": "20170726180022",
"title": "A Kepler Study of Starspot Lifetimes with Respect to Light Curve Amplitude and Spectral Type"
} |
http://arxiv.org/abs/1707.08531v1 | {
"authors": [
"Kostas Kleidis",
"Nikolaos K. Spyrou"
],
"categories": [
"astro-ph.CO",
"83F05"
],
"primary_category": "astro-ph.CO",
"published": "20170726164326",
"title": "Cosmological perturbations in the $Λ$CDM-like limit of a polytropic dark matter model"
} |
|
Étale Covers and Local Algebraic Fundamental Groups Charlie Stibitz=================================================== Practitioners apply neural networks to increasingly complex problems in natural language processing, such as syntactic parsing and semantic role labeling that have rich output structures. Many such structured-prediction problems require deterministic constraints on the output values; for example, in sequence-to-sequence syntactic parsing, we require that the sequential outputs encode valid trees. While hidden units might capture such properties, the network is not always able to learn such constraints from the training data alone, and practitioners must then resort to post-processing.In this paper, we present an inference method for neural networks that enforces deterministic constraints on outputs without performing rule-based post-processing or expensive discrete search.Instead, in the spirit of gradient-based training, we enforce constraints with gradient-based inference (GBI): for each input at test-time, we nudge continuous model weights until the network's unconstrained inference procedure generates an output that satisfies the constraints.We study the efficacy of GBI on three tasks with hard constraints: semantic role labeling, syntactic parsing, and sequence transduction. In each case, the algorithm not only satisfies constraints, but improves accuracy, even when the underlying network is state-of-the-art.§ INTRODUCTIONSuppose we have trained a sequence-to-sequence (seq2seq) network <cit.> to perform a structured prediction task such as syntactic constituency parsing <cit.>. We would like to apply this trained network to novel, unseen examples, but still require that the network's outputs obey an appropriate set of problem specific hard-constraints; for example, that the output sequence encodes a valid parse tree. Enforcing these constraints is important because down-stream tasks, such as relation extraction or coreference resolution typically assume that the constraints hold.Moreover, the constraints impart informative hypothesis-limiting restrictions about joint assignments to multiple output units, and thus enforcing them holistically might cause a correct prediction for one subset of the outputs to beneficially influence another. Unfortunately, there is no guarantee that the neural network will learn these constraints from the training data alone, especially if the training data volume is limited.Although in some cases, the outputs of state-of-the-art systems mostly obey the constraints for the test-set of the data on which they are tuned, in other cases they do not. In practice, the quality of neural networks are much lower when run on data in the wild (e.g., because small shifts in domain or genre change the underlying data distribution).In such cases, the problem of constraint violations becomes more significant.This raises the question: how should we enforce hard constraints on the outputs of a neural network?We could perform expensive combinatorial discrete search over a large output space, or manually construct a list of post-processing rules for the particular problem domain of interest.Though, we might do even better if we continue to “train” the neural network at test-time to learn how to satisfy the constraints on each input.Such a learning procedure is applicable at test-time because learning constraints requires no labeled data: rather, we only require a function that measures the extent to which a predicted output violates a constraint.In this paper, we present gradient-based inference (GBI), an inference method for neural networks that strongly favors respecting output constraints by adjusting the network's weights at test-time, for each input. Given an appropriate function that measures the extent of a constraint violation, we can express the hard constraints as an optimization problem over the continuous weights and apply back-propagation to tune them. That is, by iteratively adjusting the weights so that the neural network becomes increasingly likely to produce an output configuration that obeys the desired constraints. Much like scoped-learning, the algorithm customizes the weights for each example at test-time <cit.>, but does so in a way to satisfy the constraints.We study GBI on three tasks: semantic role labeling (SRL), syntactic constituency parsing and a synthetic sequence transduction problem and find that the algorithm performs favorably on all three tasks. In summary, our contributions are that we: * Propose a novel Gradient-Based Inference framework. * Verify that GBI performs well on various applications, thus providing strong evidence for the generality of the method. * Examine GBI across wide range of reference model performances and report its consistency. * Show that GBI also perform well on out-of-domain data.For all the tasks, we find that GBI satisfies a large percentage of the constraints (up to 98%) and that in almost every case (out-of-domain data, state-of-the art networks, and even for the lower-quality networks), enforcing the constraints improves the accuracy.On SRL, for example, the method successfully injects truth-conveying side-information via constraints, improving SOTA network [Since our submission, the previous SOTA <cit.> in SRL on which we apply our technique has been advanced by 1.7 F1 points <cit.>. However, this is a training time improvement which is orthogonal to our work.]by 1.03 F1 <cit.>. This improvement happens to surpass a algorithm for incorporating constraints while also being robust, in a way that is not, to cases for which the side constraints are inconsistent with the labeled ground-truth.§ CONSTRAINT-AWARE INFERENCE IN NEURAL NETWORKS Our goal is to design an approximate optimization algorithm that is similar in spirit to Lagrangian relaxation in that we replace a complex constrained decoding objective with a simpler unconstrained objective that we can optimize with gradient descent <cit.>, but is better suited for non-linear non-convex optimization with global constraints that do not factorize over the outputs. Although the exposition in this section revolves around Lagrangian relaxation, we emphasize that the purpose is merely to provide intuition and motivate design choices. §.§ Problem definition and motivationTypically, a neural network parameterized by weightsis a function from an inputto an output . The network has an associated compatibility function (;,)→_+ that measures how likely an outputis given an inputunder weights . The goal of inference is to find an output that maximizes the compatibility function and this is usually accomplished efficiently with feed-forward greedy-decoding. In this work, we want to additionally enforce that the output values belong to a feasible set or grammar ^ that in general depends on the input. We are thus interested in the following optimization problem:[max (,,) ∈^ ]Simple greedy inference are no longer sufficient since the outputs might violate the global constraints (i.e., ∉^).Instead, suppose we had a function (,^)→_+ that measures a loss between outputand a grammar ^ such that (,^)=0 if and only if there are no grammatical errors in . That is, (,^)=0 for the feasible region and is strictly positive everywhere else. For example, if the feasible region is a CFL,could be the least errors count function <cit.>.We could then express the constraints as an equality constraint and minimize the Lagrangian:[λmin max (,,) + λ(,^) ;]However, this leads to optimization difficulties because there is just a single dual variable for our global constraint, resultingintractable problem and thus leading to brute-force trial and error search.Instead, we might circumvent these issues if we optimize over a model parameters rather than a single dual variable. Intuitively, the purpose of the dual variables is to simply penalize the score of infeasible outputs that otherwise have a high score in the network, but happen to violate constraints.Similarly, network's weights can control the compatibility of the output configurations with the input. By properly adjusting the weights, we can affect the outcome of inference by removing mass from invalid outputs—in much the same way a dual variable affects the outcome of inference.Unlike a single dual variable however, the network expresses a differentpenalty weight for each output. And, because the weights are typically tied across space (e.g., CNNs) or time (e.g., RNNs) the weights are likely to generalize across related outputs.As a result, lowering the compatibility function for a single invalid output has the potential effect of lowering the compatibility for an entire family of related, invalid outputs; enabling faster search.In the next subsection, we propose a novel approach that utilizes the amount of constraint violation as part of the objective function so that we can adjust the model parameters to search for a constraint-satisfying output efficiently.§.§ Algorithm Instead of solving the aforementioned impractical optimization problem, we propose to optimize a “dual” set of model parameters _λ over the constraint function while regularizing_λto stay close to the originally learned weights W. The objective function is as follows:[ _λmin (,,_λ)(,^) + α - _λ; where =(,,_λ) ]Although this objective deviates from the original optimization problem, it is reasonable because by definition of the constraint loss (·), the global minima must correspond to outputs that satisfy all constraints.Further, we expect to find high-probability optima if we initialize _λ=. Moreover, the objective is intuitive: if there is a constraint violation inthen (·)>0 and the gradient will lower the compatibility ofto make it less likely. Otherwise, (·)=0 and the gradient of the energy is zero and we leave the compatibility ofunchanged. Crucially, the optimization problem yields computationally efficient subroutines that we exploit in the optimization algorithm.To optimize the objective, the algorithm alternates maximization to findand minimization w.r.t. _λ (Algorithm <ref>). In particular, we first approximate the maximization step by employing the neural network's inference procedure (e.g., greedy decoding, beam-search, or Viterbi decoding) to find thethat approximately maximizes , which ignores the constraint loss .Then, given a fixed , we minimize the objective with respect to the _λ by performing stochastic gradient descent (SGD).Sinceis fixed, the constraint loss term becomes a constant in the gradient; thus, making it easier to employ external black-box constraint losses (such as those based on compilers) that may not be differentiable.As a remark, note the similarity to REINFORCE <cit.>: the decoder outputs asactions and the constraint-loss asa negative reward.However, GBI does not try to reduce expected reward andterminates upon discovery of an output that satisfies all constraints. Furthermore, GBI also works on sequence-tagging problem, SRL (Section <ref>), wherenext output does not depend on the current output, which is far from REINFORCE setting.§ APPLICATIONSThere are multiple applications that involve hard-constraints and we provide two illustrative examples that we later employ as case-studies in our experiments: SRL and syntactic parsing. The former exemplifies a case in which external knowledge encoded as hard constraints conveys beneficial side information to the original task of interest while the latter studies a case in which hard constraints are inherent to the task of interest. Finally, we briefly mention sequence transduction as framework in which constraints may arise.Of course, constraints may in general arise for a variety of different reasons, depending on the situation. We provide example-based case studies for each application in Appendix <ref>, <ref>. §.§ Semantic Role LabelingAs a first illustrative example, consider SRL. SRL focuses on identifying shallow semantic information about phrases. For example, in the sentence “it is really like this, just look at the bus sign” the goal is to tag the arguments given “is” as the verb predicate: “it” as its first argument and the prepositional phrase “like this” as its second argument.Traditionally SRL is addressed as a sequence labeling problem, in which the input is the sequence of tokens and the output are BIO-encoded class labels representing both the regimentation of tokens into contiguous segments and their semantic roles.Note that the parse tree for the sentence might provide constraints that could assist with the SRL task. In particular, each node of the parse tree represents a contiguous segment of tokens that could be a candidate for a semantic role. Therefore, we can include as side-information constraints that force the BIO-encoded class labeling to produce segments of text that each agree with some segment of text expressed by a node in the parse tree.[The ground-truth parse spans do not always agree with the SRL spans, leading to imperfect side information.] To continue with our example, the original SRL sequence-labeling might incorrectly label “really like this” as the second argument rather than “like this.” Since according to the parse tree “really” is part of the verb phrase, thus while the tree contains the spans “is really like this” and “like this” it does not contain the span “really like this.”The hope is that enforcing the BIO labeling to agree with the actual parse spans would benefit SRL.Based on the experiments, this is indeed the case, and our hypothetical example is actually a real data-case from our experiments, which we describe later. The (,^x) for SRL factorizes into per-span constraints _i.For ith span s_i, if s_i is consistent with any node in the parse tree,_i(s_i,^x)=0, otherwise _i(s_i,^x)=1/n_s_i where n_s_i is defined as the number of tokens in s_i. Overall, (,,_λ)(,^x)= ∑_i=1^k (s_i,^x)(,s_i,W_λ) where k is number of spans on output . §.§ Syntactic parsingAs a second illustrative example, consider a structured prediction problemof syntactic parsing in which the goal is to input a sentence comprising a sequence of tokens and output a tree describing the grammatical parse of the sentence.Syntactic parsing is a separate but complementary task to SRL.While SRL focuses on semantic information, syntactic parsing focuses on identifying relatively deep syntax tree structures. One way to model the problem with neural networks is to linearize the representation of the parse tree and then employ the familiar seq2seq model <cit.>.Let us suppose we linearize the tree using a sequence of shift () and reduce () commands that control an implicit shift reduce parser. Intuitively, these commands describe the exact instructions for converting the input sentence into a complete parse tree: the interpretation of the symbolis that we shift an input token onto the stack and the interpretation of the symbolis that we start (or continue) reducing (popping) the top elements of the stack, the interpretation of a third symbol is that we stop reducing and push the reduced result back onto the stack. Thus, given an input sentence and an output sequence of shift-reduce commands, we can deterministically recover the tree by simulating a shift reduce parser. For example, the sequenceencodes a type-free version of the parse treefor the input sentence “the ball is red”. It is easy to recover the tree structure from the input sentence and the output commands by simulating the shift reduce parser.Of course in practice, reduce commands include the standard parts of speech as types (NP, VP, etc).Note that for output sequences to form a valid tree over the input, the sequence must satisfy a number of constraints. First, the number of shifts must equal the number of input tokens m_, otherwise either the tree would not cover the entire input sentence or the tree must contain spurious symbols. Second, the parser cannot issue a reduce command if the stack is empty. Third,at the end of the parser commands,the stack must have just a single item, the root node. The constraint loss (,^x) for this task simply counts the errors of each of the three types. (Appendix <ref>)As a minor remark, note that other encodings of trees, such as bracketing (of which the Penn Tree Bank's S-expressions are an example), are more commonly used as output representations for seq2seq parsing (ibid).However, the shift-reduce representation described in the above paragraphs is isomorphic to the bracketing representations and as we get similar model performance to single seq2seq modeon the same data (ibid.), we chose the former representation to facilitate constraint analysis.Although output representations sometimes matter, for example, BIO vs BILOU encoding of sequence labelings, the difference is usually minor <cit.>, and breakthroughs in sequence labeling have been perennially advanced under both representations. Thus, for now, we embrace the shift reduce representation as a legitimate alternative to bracketing, pari passu.§.§ Synthetic sequence transductionFinally, although not a specific application per se, we also consider sequence transduction as it provides a framework conducive to studying simple artificial languages with appropriately designed properties.A sequence transducer T:_S→_T is a function from a source sequence to a target sequence.As done in previous work, we consider a known T to generate input/output training examples and train a seq2seq network to learn T on that data <cit.>. The constraint is simply that the output must belong to _T and also respect problem-specific conditions that may arise from the application of T on the input sentence.We study a simple case in Section <ref>. § EXPERIMENTSIn this section we study our algorithm on three different tasks: SRL, syntactic constituency parsing and a synthetic sequence transduction task. All tasks require hard constraints, but they play a different role in each. In the transduction task they force the output to belong to a particular input-dependent regular expression,in SRL, constraints provide side-information about possible true-spans and in parsing, constraints ensure that the outputs encode valid trees. While the SRL task involves moretraditional recurrent neural networks that haveexactly one output per input token, the parsing and transduction tasksprovide an opportunity to study the algorithmon various seq2seq networks .We are interested in answering the following questions (Q1) how well does the neural network learn the constraints from data (Q2) for cases in which the network is unable to learn the constraints perfectly, can GBI actually enforce the constraints(Q3) does GBI enforce constraints without compromising the quality of the network's output.To more thoroughly investigate Q2 and Q3, we also consider: (Q4) is the behavior of the method sensitive to the reference network performance, and (Q5) does GBI also work on out-of-domain data. Q3 is particularly important because we adjust the weights of the network at test-time and this may lead to unexpected behavior. Q5 deals with our original motivation of using structured predictionto enhance performance on the out-of-domain data.To address these various questions, we first define some terminology to measurehow well the model is doing in terms of constraints. To address (Q1) we measure the failure-rate (i.e., the ratio of test sentences for which the network infers an output that fails to fully satisfy the constraints).To address (Q2) we evaluate our method on the failure-set (i.e., the set of output sentences for which the original network produces constraint-violating outputs) and measure our method's conversion rate; that is, the percentage of failures for which our method is able to completely satisfy the constraints (or “convert”).Finally, to address (Q3), we evaluate the quality (e.g., accuracy or F1) of the output predictions on the network's failure-set both before and after applying our method. §.§ Semantic Role LabelingWe employ the AllenNLP <cit.> SRL network with ELMo embeddings, which is a multi-layer highway bi-LSTM that produces BIO output predictions for each input token <cit.>. For data we use OntoNotes v5.0, which has ground-truth for both SRL and syntactic parsing <cit.>. We evaluate GBI on the test-set (25.6k examples), out of which consistent parse information is available for 81.25% examples (we only include side-information in terms of constraints for this subset).We repeat the same experimental procedure over multiple networks, SRL-X, while varying the portion (X%) of the training dataset.In Table <ref>, we see that GBI is able to convert 42.25 % of failure set, and this boosts the overall F1 measure by 1.23 point over the SOTA network (SRL-100) which does not incorporate the constraints (they report 84.6 F1, we obtain a similar 84.4 F1 with their network, and achieve 85.63 after enforcing constraints with our inference). Further, to address (Q1) we measure the sentence-level failure rate as well as span-level disagreement rate (i.e., the ratio of predicted spans in a sentence that disagree with the spans implied by the true syntactic parse of the sentence). To address (Q2) we evaluate our method on the failure set (i.e., the set of sentences for which disagreement rate is nonzero) and measure our method's avgerage disagreement rate. Finally, to address (Q3), we evaluate the quality (F1 and exact match) of the output predictions on the network's failure-set both before and after applying our method.From Table <ref>, we can see that by applying GBI on SRL-100, the avgerage disagreement rate on the failure set goes down from 44.85% to 24.92% which results in an improvement of 11.7 F1 and 19.90% in terms of exact match on the sameset.These improvements answer Q1-3 favorably.To enforce constraints during inference,proposed to employ constrained-decoding. For the sake of a fair comparison with GBI, we consider decoding as used in <cit.> and report results for the SRL-X networks. We see from Table <ref>, that the GBI procedure consistently outperforms decoding on all evaluation metrics, thus demonstrating the superiority of the approach. §.§ Syntactic parsingWe now turn to a different task and network: syntactic constituency parsing. We investigate the behavior of the constraint inference algorithm on the shift-reduce parsing task described in Section <ref>. We transform the Wall Street Journal (WSJ) portion of the Penn Tree Bank (PTB) into shift-reduce commands in which each reduce command has a phrase-type (e.g., noun-phrase or verb-phrase) <cit.>. We employ the traditional split of the data with section 22 for dev, section 23 for test, and remaining sections 01-21 for training. We evaluate on the test set with evalb[<http://nlp.cs.nyu.edu/evalb/>] F1. In each experiment, we learn a seq2seq network on a training set and then evaluate the network directly on the test set using a traditional inference algorithm to perform the decoding (either greedy decoding or beam-search).In order to study our algorithm on a wide range ofaccuracy regimes (section <ref>),we train many networks with different hyper-parameters producing models of various quality, from high to low, using the standard split of the WSJ portion of the PTB.In total, we train five networksfor this study, that we describe below. We train our two best baseline models (et1,2) using a highly competitive seq2seq architecture for machine translation, GNMT <cit.> with F1 scores, 86.78 and 87.33, respectively.And, to study a wider range of accuracies, we train a simpler architecture with different hyper parameters and obtain nets (Net3-5). For all models, we employ Glorot initialization, and basic attention <cit.>. See Table <ref> for a summary of the networks, hyper-parameters, and their performance.We report the behavior of the constraint-satisfaction methodin Table <ref> for Net1-2,and in Table <ref> for Net3-5. Across all the experimental conditions (Table <ref>, <ref>), the conversion rates are high, often above 80 and sometimes above 90 supporting Q2.Note that beam search alone can also increase constraint satisfaction with conversion rates reaching as high as 51.74% (164/317) in the case of Net3 with beam size 9. However, as the quality of the model increases, the conversion rate becomes minuscule; in the case of Net1,2 the conversion rate is less than 14% with beam 9; in Net1 converting 26 out of 187 and in Net2 converting just 1 out of 287 instances from failure set.In order to address question Q3—the ability of our approach to satisfy constraints without negatively affecting output quality—we measure the F1 scores on the failure-sets both before and after applying the constraint satisfaction algorithm. Since F1 is only defined on valid trees, we employ heuristic post-processing to ensure all outputs are valid.Note that an improvement on the failure-set guarantees an improvement on the entire test-set since our method produces the exact same outputs as the baseline for examples that do not initially violate any constraints. Consequently, for example, the GNMT network improves (Net2) on the failure-set from 73.54 to 79.68 F1, resulting in an overall improvement from 86.54 to 87.57 F1(entire test-set). These improvements are similar to those we observe in the SRL task, andprovide additional evidence for answering Q1-3 favorably. We also measure how many iterationsof our algorithmit takes to convert the examples that have constraint-violations.Across all conditions, it takes 5–7 steps to convert 25% of the outputs, 6–20 steps to convert 50%, 15–57 steps to convert 80%, and 55–84 steps to convert 95%.§.§ Simple Transduction Experiment In our final experiment we focus on a simple sequence transduction task in which we find that despite learning the training data perfectly, the network fails to learn the constraint in a way that generalizes to the test set.For our task, we choose a simple transducer, similar to those studied in recent work <cit.>. The source language _S is ^⋆ and the target language _T is ^⋆. The transducer is defined to map occurrences ofin the source string toin the target string, and occurrences ofin the source string toin the target string. For example, T()↦.The training set comprises 1934 sequences of length 2–20 and the test set contain sentences of lengths 21-24. We employ shorter sentences for training to require generalization to longer sentences at test time. We employ a32 hidden unit single-layered, attention-less, seq2seq LSTM in which the decoder LSTM inputs the final encoder state at each decoder time-step. The network achieves perfect train accuracy while learning the rules of the target grammar _T perfectly, even on the test-set. However, the network fails to learn the input-specific constraint that the number of 's in the output should be three times the number of 's in the input.This illustrates how a network might rote-memorize constraints rather than learn the rule in a way that generalizes. Thus, enforcing constraints at test-time is important. To satisfy constraints, we employ GBI with a constraint loss , a length-normalized quadratic (3x_a-y_a)^2/(m+n) that is zero when the number of 's in the output (y_a) is exactly three times the number in the input (x_a) with m,n denoting input, output, respectively.GBI achieves a conversion rate of 65.2% after 100 iterations, while also improving the accuracy on the failure-set from 75.2% to 82.4%.This synthetic experiment provides additional evidence in support of Q2 and Q3, on a simpler small-capacity network.§.§GBI on wide range of reference models The foregoing experimental results provide evidence that GBI is a viable method for enforcing constraints.However, we hitherto study GBI on high quality reference networks such as SRL-100.To further bolster our conclusions, we now direct our investigation towards lower quality networks to understand GBI's viability under a broader quality spectrum. We ask, how sensitive is GBI to the reference network's performance (Q4)? To this end, we train poorer quality networks by restricting the amount of available training data or employing simpler architectures.For SRL, we simulate low-resource models by limiting the training data portion to 1%, 10%, 40%, and 70% resulting in F1 score range of 67.28-83.55. Similarly, for syntactic parsing, we train additional low-quality models et3-5 with a simpler uni-directional encoders/decoders, and on different training data portions of 25%, 75%, and 100% (Table <ref>). We evaluate GBI on each of them in Table <ref>, <ref> and find further evidence in support of favorable answers toQ2 (satisfying constraints) and Q3 (improving F1 accuracy) by favorably answering Q4. Moreover, while not reported fully due to page limits, we examined both tasks with over 20 experiments and different baseline networks in combination with different inference strategies, and we found GBI favorable in all but one case (but by just 0.04 comparing without GBI).We also study whether GBI is compatible with better underlying discrete search algorithms, in particular beam search for seq2seq.As we seen in column 2 of Table <ref>, that although beam-search improves the F1 score and reduces the percentage of violating constraints, GBI further improves over beam-search when using the latter in the inner-loop as the decoding procedure.In conclusion, improving the underlying inference procedure has the effect of decreasing the number of violating outputs, but GBI is still very much effective on this increasingly small set, despite it intuitively representing more difficult cases that even eludes constraint satisfaction via beam search inference. §.§Experiments on out-of-domain dataPreviously, we saw how GBI performs well even when the underlying network is of lower quality. We now investigate GBI on actual out-of-domain data for which the model quality can suffer. For SRL, we train a SOTA network with ELMo embedding on the NewsWire (NW) section of the OntoNotes v5.0 English PropBank corpus and then test on the other genres provided in the corpus:BC, BN, PT, TC, WB. The failure rate on the within genre data (test set of NW) is 18.10%. We can see from Table <ref>, the failure rate for the NW trained SRL network in general is higher for out-of-genre data with the highest being 26.86% for BC (vs. 18.10% NW). Further, by enforcing constraints, we see significant gains on the failure set in terms of F1 score across all genres (ranging from 9.39-16.5 F1), thus, providing additional evidences for answering Q5.As we did for SRL, we train a GMNT seq2seq model on the WSJ NW section in OntoNotes v5.0 Treebank [The PTB (40k instances) and OntoNotes (30k instances) coverage of WSJ are slightly different.] which shares the same genre classification with PropBank. The F1 on the within-genre data (test set of WSJ) is 85.03, but the F1 onthese genres is much lower, ranging from the mid-forties on BC (46.2–78.5 depending on the subcategory)to the low-eighties on BN (68.3–81.3. depending on the subcategory).Indeed, we find that overall the F1 is lower and in some cases, like WB, the failure rate is much higher (17.6% for WB vs. 11.9% for WSJ).Following the same experimental protocol as on the PTB data, we report the results in Table <ref> (aggregating over all subcategories in each genre).We see that across all genres, the algorithm has high conversion rates (sometimes close to 100%), and that in each case, enforcing the constraints improves the F1.Again, we find support for Q2, Q3 and Q5. §.§ Robustness and Runtime analysisWe perform additional experiments to analyze the robustness and runtime of GBI.First, to measure robustness, we consider a variant of the SRL task in which we include noisy constraints, and compare GBI to (Appendix <ref>).We find that in this case, performs significantly worse than the baseline, while GBI improves over the same baseline, thus showing the robustness of GBI.In terms of runtime, GBI is generally faster than , though, the difference is less clear on smaller evaluation sets (Appendix <ref>).In the case study with noisy constraints, the runtimes are similar; however, GBI has much better accuracy, showing similar gains as the noise-free setting.Lastly, in appendix <ref>, we discuss GBI's trade off between runtime and accuracy by varying the max epoch M.§ RELATED WORK Recent work has considered applying neural networks to structured prediction; for example, structured prediction energy networks (SPENs) <cit.>.SPENs incorporate soft-constraints via back-propagating an energy function into “relaxed” output variables. In contrast, we focus on hard-constraints and back-propagate into the weights that subsequently control the original non-relaxed output variables via inference.Separately, there has been interest in employing hard constraints to harness unlabeled data in training-time for simple classifications <cit.>. Our work instead focuses on enforcing constraints at inference-time. More specifically, for SRL, previous work for enforcingconstraints have focused on constrained decoding <cit.> or integer linear programming <cit.>. For parsing, previous work in enforcing hard constraints has focused on post-processing <cit.> or building them into the decoder transitions <cit.>or search constraints <cit.>. Finally, as previously mentioned, our method highly resembles dual decomposition and more generally Lagrangian relaxation for structured prediction <cit.>. In such techniques, it is assumed that a computationally efficient inference algorithm can maximize over a superset of the feasible region (this assumption parallels our case because unconstrained inference in the neural network is efficient, but might violate constraints). Then, the method employs gradient descent to concentrate this superset onto the feasible region.However, these techniques are not directly applicable to our non-linear problem with global constraints.§ CONCLUSIONWe presented an algorithm for satisfying constraints in neural networks that avoids combinatorial search, but employs the network's efficient unconstrained procedure as a black box to coax weightstowards well-formed outputs. We evaluated the algorithm on three tasks including SOTA SRL, seq2seq parsing and found that GBI can successfully convert failure sets while also boosting the task performance. Accuracy in each of the three tasks was improved by respecting constraints. Additionally, for SRL, we employed GBI on a model trained with similar constraint enforcing loss as GBI's <cit.>, and observe that the additional test-time optimization of GBI still significantly improves the model output whereas does not. We believe this is because GBI searches in the proximity of the provided model weights; however, theoretical analysis of this hypothesis is left as a future work. aaai§ APPENDIX§ GBI VS. CONSTRAINED DECODING In Table <ref> of Appendix <ref>, we provide an example data-case that shows how our algorithm solves the initially violated shift-reduce parse output.For simplicity we omit the phrase-types of constituency parsing and display only on the shift (), reduce () and stop reducing commands (), and color them red if there is an error.The algorithm satisfies the constraint in just 12 iterations, and this results in a perfectly correct parse.What is interesting about this example is that the original network commits a parsing mistake early in the output sequence.This type of error is problematic for a naive decoder that greedily enforces constraints at each time-step.The reason is that the early mistake does not create a constraint violation until it is too late, at which point errors have already propagated to future time-steps and the greedy decoder must shift and reduce the last token into the current tree, creating additional spurious parse structures.In contrast, our method treats the constraints holistically, and uses it to correct the error made at the beginning of the parse. See Table <ref> for a comparison of how the methods fix the constraints.Specifically, the constraint violation is that there were not enough shift and reduce commands to account for all the tokens in the sentence.Rather than fixing the constraint by inserting these extra commands at the end of the sequence as the greedy decoder must do, GBI inserts them at the beginning of the sequence where the initial mistake was made, thereby correcting the initial mistake.Moreover, this correction propagates to a mistake made later in the sequence (viz., the the sequence of three reduces after the four shifts) and fixes them too.This example provides evidence that GBI can indeed enforce constraints holistically and that doing so improves the output in a global sense. § EXAMPLE-BASED CASE STUDY § CONSTRAINT FUNCTIONS Here we define the specific constraint loss function (,^x) for each task.Note that a common theme is that we normalize the constraint loss by the length of the sequence so that it does not grow unbounded with sequence size.We recommend this normalization as we found that it generally improves performance.§.§ Semantic Role labelingThe (,^x) for SRL factorizes into per-span constraints _i. For the ith span s_i, if s_i is consistent with any node in the parse tree,_i(s_i,^x)=0, otherwise _i(s_i,^x)=1/n_s_i where n_s_i is defined as the number of tokens in s_i. Overall, (,,_λ)(,^x)=∑_i=1^k (s_i,^x)(,s_i,W_λ)where k is number of spans on output . More precisely, for a span s to be “consistent with a parse node” we mean the following.Let t_i∈ T be a node in the parse tree T and let s^t_i be the span of text implied by the descendents of the node t_i.Let S^T={s^t_i} be the set of spans implied by all nodes in the parse tree T.We say that a span of text s is consistent with the parse tree T if and only if s ∈ S^T.§.§ Syntactic Parsing Let m_, n be the number of input and output tokens, respectively, _i=1^n(b(i)) be the function that counts the number of times proposition b(i) is true for i=1,…,n. Now, define the following loss(,^x) =1/m_ + n{|m_ - _i=0^n(y_i=s)| + ∑_i^nmax(0,_j=0^i(y_j=r) - _j=0^i(y_j∈{s,!}))}.The first term provides loss when the number or shifts equals the number of input tokens, the second term provides loss when attempting to reduce an empty stack and the third term provides loss when the number of reduces is not sufficient to attach every lexical item to the tree.§.§ TransductionFor the transducer we chose for our experiment, _S is ^⋆ and the target language _T is ^⋆. The transducer is defined to map occurrences ofin the source string toin the target string, and occurrences ofin the source string toin the target string. For the provided transduction function, the number of 's in the output should be three times the number of 's in the input. To express this constraint, we define following constraint loss , a length-normalized quadratic(,^x)= (3x_a-y_a)^2/(m+n)that is zero when the number of 's in the output (y_a) is exactly three times the number in the input (x_a) with m,n denoting input length, output length, respectively.§ ANALYZING THE BEHAVIOR OF DIFFERENT INFERENCE PROCEDURES IN THE PRESENCE OF NOISY CONSTRAINTS Table <ref> reports the performance of GBI and in the presence of noisy constraints. We can see that the overall performance (F1-score) for drops drastically (-6.92) in the presence of noisy constraints while we still see gains with GBI (+0.47).We further analyze the improvement of GBIby looking at the precision and recall scores individually.We see that recall drops slightly for GBI which suggests that noisy constraints do inhibit predicting actual argument spans. On the other hand, we see that precision increases significantly. After analyzing predicted argument spans, we noticed that GBI prefers to predict no argument spans instead of incorrect spans in the presence of noisy constraints which leads to an increase in precision. Thus, GBI provides flexibility in terms of strictness with enforcing constraints which makes it robust to noisy constraints. On the other hand, constrained-decoding algorithmis too strict when it comes to enforcing noisy constraints resulting in a significant drop of both precision and recall.§ ANALYZING THE RUNTIME OF DIFFERENT INFERENCE PROCEDURES WITH VARYING DATASET SIZES AND GENRESTable <ref> reports the runtime for different inference procedures with varying dataset sizes. In general, we observe that GBI tends to be faster than , especially when the dataset is large enough. One exception is the BC domain where GBI is just slightly faster than A*. We hypothesize it might be due to the difficulty of the constraint violations as its failure rate is higher than usual. GBI will spend more time searching for the correct output (more iterations) if it is harder to find the solution.Also note that we explicitly set the max epoch M for GBI after which it will stop iterating to avoid pathological cases. In our SRL experiments, we have set the max epochs to be 10 (GBI-10). To study its scalability, we ran GBI with max epoch set to 30 (GBI-30). The runtime increase to 556 mins for GBI-30 as opposed to 288 mins of GBI-10.However, GBI-30 improves significantly in all accuracy metrics compared to GBI-10: overall F1 (+0.34), F1 on failure set (+3.4), exact match (+4.35%), and conversion rate (+11.24%). As can be seen from the demonstration, there is a clear tradeoff between runtime and accuracy as controlled by the maximum number of epochs M. The user can control M by the runtime constraint the system has: lower M when the serving time is most important and larger M when accuracy is more important than the serving time. | http://arxiv.org/abs/1707.08608v3 | {
"authors": [
"Jay Yoon Lee",
"Sanket Vaibhav Mehta",
"Michael Wick",
"Jean-Baptiste Tristan",
"Jaime Carbonell"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170726190010",
"title": "Gradient-based Inference for Networks with Output Constraints"
} |
Graduate Institute of Astronomy, National Central University, Chung-Li 32054, Taiwan [email protected] & [email protected] We probed the relation between properties of Seyfert nuclei and morphology of their host galaxies. We selected Seyfert galaxies from the Sloan Digital Sky Survey with redshifts less 0.2 identified by the Véron Catalog (13th). We used the “FracDev” parameter from SDSS galaxy fitting models to represent the bulge fractions of the Seyfert host galaxies. We found that the host galaxies of Seyfert 1 and Seyfert 2 are dominated by large bulge fractions, and Seyfert 2 galaxies are more likely to be located in disk galaxies whereas most of the Seyfert 1 galaxies are located in bulge-dominant galaxies. These results indicate that the types of AGNs are related to their host galaxies and can not be explained by the traditional unification model of Seyfert galaxies. § INTRODUCTIONActive galactic nucleus (AGN) galaxies are galaxies showing strong activity in their nuclei. An AGN is believed to consist of a central engine with an accretion disk and a supermassive black hole embedded in an opaque torus of dust <cit.>. The phenomena of AGN are widely explained by the accretion process of the central supermassive black hole, which would release huge energy <cit.>. Although AGNs have various types, they are considered to be similar objects and can be explained by the AGN unification model <cit.>. One of particular AGN types is called Seyfert galaxies <cit.>. <cit.> found that there were some spiral galaxies with bright central nuclei, and the spectra of the nuclei showed strong emission lines. <cit.> classified Seyfert galaxies into two subclasses according to the line widths of Balmer lines and [OIII] forbidden lines; Seyfert 1 galaxies have broader Balmer lines than the forbidden lines whereas Seyfert 2 galaxies have the same line widths of Balmer lines and the forbidden lines with line width ranging from 500 km s^-1 to 1000 km s^-1. <cit.> further divided the Seyfert galaxies into subclasses of Seyfert 1.2, Seyfert 1.5, Seyfert 1.8 and Seyfert 1.9 depending on the appearance of Hβ emission line; Seyfert 1.2 have strong broad Hβ component while Seyfert 1.8 have very weak broad Hβ component in the optical spectra. Seyfert 2 galaxies usually have a high ratio of [OIII]/Hβ; the empirical criterion for Seyfert 2 is [OIII]/Hβ ≥3<cit.>. <cit.> introduced a method, called BPT diagram, to divide star-forming galaxies and Seyfert 2 galaxies depending on the ratios of emission lines. Nowadays, the BPT diagrams with different dividing lines derived from theoretical and empirical methods are widely used in diagnosing Seyfert 2 galaxies <cit.>. According to unification model <cit.>, different types of AGNs are caused by different viewing angles. For example, Seyfert 1 galaxies are viewed face-on relative to the accretion disk and torus whereas Seyfert 2 galaxies are viewed edge-on <cit.>. <cit.> found that NGC 1068 showed broad emission lines in polarized spectroscopic observations while it was considered as a Seyfert 2 galaxies in traditional optical spectral observations. This result strongly supports the unification model of Seyfert 1 and Seyfert 2 galaxies. If the Seyfert galaxies are merely due to different viewing angles relative to torus, different types of Seyferts should be independent of their host galaxies. However, <cit.> found that Seyfert galaxies tend to be in the S0 and Sa galaxies with 27 Seyferts selected from the Véron catalogue (1985). <cit.> found that Seyfert 2 and Seyfert 1 galaxies have similar CO distributions but the host galaxies of Seyfert 2 seem to have more asymmetric morphology than those of Seyfert 1. <cit.> found that he median in morphological classes for Seyfert 1 galaxies is Sa and that for Seyfert 2 galaxies is Sb from the WPFC2 images of 256 nearest active galaxies; the subsample of these Seyfert galaxies selected from 12 μm emission also show a similar trend <cit.>. <cit.> found that the median type of Seyfert 1 galaxies is Sa while that of Seyfert 2 galaxies is Sab galaxies using 891 galaxies from a 12 μm galaxy sample. <cit.> found that there are significantly higher fraction of Seyfert 2 galaxies with a neighbor within 75 h^-1 kpc than Seyfert 1 galaxies; the authors also found that Seyfert 1 galaxies prefer to be in more dense galaxy regions than Seyfert 2 galaxies do in large scale environments. The above results seem to conflict with the traditional unification model, suggesting that the orientation in unification model is not the only reason for different Seyfert-type galaxies.In order to understand whether the different types of Seyfert galaxies are related to their host galaxies, we studied the host galaxy morphology distributions of a sample of selected Seyfert galaxies. This paper is organized as follows. In Section 2, we describe our sample selection. In Section 3, we present the distributions of host galaxy morphology for our Seyfert samples. We discuss and summarize our results in Section 4. In this paper, we used H_0=70 km s^-1 Mpc^-1, Ω_m=0.3, Λ_0=0.7, q_0=-0.55, k=0.00. § SAMPLE SELECTION We used two different methods to select our Seyfert samples from the Sloan Digital Sky Survey (SDSS) to avoid possible selection biases. For the first method of selection, we chose our Seyfert galaxies identified by the Véron Catalog (13th) in the SDSS data. We selected our Seyferts from the Table_AGN of the Véron 13th Catalog <cit.>. The table comprised Seyfert 1s, Seyfert 2s, and LINER with M_B fainter than -22.25. The Seyfert galaxies in this catalogue were divided into six subclasses <cit.> : Seyfert 1.0, Seyfert 1.2, Seyfert 1.5, Seyfert 1.8, Seyfert 1.9 and Seyfert 2 depending on the appearance of their Balmer lines. Seyfert 1 have broad Balmer and other emission lines while Seyfert 2 have the narrow Blamer and forbidden lines. We only selected Seyfert 1.0 galaxies as our Seyfert 1 samples and Seyfert 2.0 as our Seyfert 2 samples to avoid confusion. We also constrained the redshift range of the Seyfert galaxies from 0 to 0.2 to include some important emission lines within the spectral range. There are 5009 Seyfert 1 galaxies and 4204 Seyfert 2 galaxies after these selections. We extracted the host galaxies of the Seyferts from the Sloan Digital Sky Survey (SDSS) and used the observation data from SDSS Data Release 10 <cit.>. We selected SDSS sources that were classified as “GALAXY”in photometry within a 3” radius of our Seyfert samples. We only considered the sources that contain both photometric and spectroscopic information. Finally, we have 4078 Seyfert 1 galaxies and 3422 Seyfert 2 galaxies. § HOST GALAXY MORPHOLOGY To give a quantity assessment of galaxy morphology, we used the SDSS parameter, FracDev, to represent the bulge contributions of our Seyfert galaxies.F_composite=FracDev F_dev-(1-FracDev) F_expwhere F_composite is the composite model of galaxy flux, F_dev is the best fitting of the de Vaucouleurs flux of the galaxy, and F_exp is the best fitting of the exponential disk flux of the galaxy. The FracDev parameter is a coefficient of the de Vaucouleurs bulge term in the SDSS galaxy fitting and has a range from 0 to 1 <cit.>. If the value of FracDev is close to 1, the galaxy is bulge-dominant <cit.>. From the FracDev values, we can determine the bulge contributions in the host galaxies. There is an example of color images of our Seyfert galaxies as a function of FracDev in Fig. <ref>. We used FracDev_r expressing the coefficient fitting by r-band data for better quality. All FracDev in this paper present the coefficient fitting by r-band data.Fig. <ref> shows the host galaxy morphology distributions of our Seyfert galaxies. We found that Seyfert 1 and Seyfert 2 galaxies show different distributions. We did a Kolmogorov-Smirnov (K-S) test for the FracDev distributions of Seyfert 1 and Seyfert 2 galaxies. The probability that the FracDev distributions of Seyfert 1 and Seyfert 2 galaxies are drawn from the same population is P = 0.00 with K-S statistic (D) =0.26. In other words, the host galaxies of these two Seyferts have different morphology distributions and the different distributions in host morphology between the Seyfert 1 and Seyfert 2 galaxies are not biased by the different selection effect. We also compared the fractions of Seyfert galaxies for both FracDev = 1 and FracDev < 1. The fraction of Seyfert 1 with FracDev = 1 is 50% but for Seyfert 2 galaxies the fraction is 28%. This indicates that the Seyfert 2 galaxies are more likely to locate in disk galaxies whereas most of the Seyfert 1 galaxies are located in bulge-dominant galaxies. Besides, the result shows that Seyfert 1 and Seyfert 2 galaxies are dominated by galaxies with FracDev = 1. In other words, there are many Seyfert galaxies located in elliptical type galaxies, in contradiction to early studies <cit.>.We compared the morphology distributions of Seyfert galaxies with all galaxies that have redshifts less than 0.2 and M_r < -19 in the SDSS database. Fig. <ref> demonstrates that the morphology distribution of the total galaxies is different from those of the Seyfert galaxies. These results show that Seyfert galaxies are more bulge dominant than all galaxies.Furthermore, we plotted the relative ratios of the Seyfert galaxies to the total galaxies with different FracDev values in Fig. <ref>. Fig. <ref> suggests that the Seyfert 2 galaxies are more likely to emerge in FracDev ≈ 0.4 whereas the Seyfert 1 galaxies are more likely to emerge in FracDev = 1. In other words, the host galaxies of the Seyfert 1 are more likely to be early type galaxies and those of the Seyfert 2 are more likely to be late type galaxies. Fig. <ref> shows the distributions of absolute magnitudes for Seyfert 1, Seyfert 2 galaxies, and total galaxies. The Seyfert 1 and the Seyfert 2 galaxies have similar luminosity distributions, which peak at around absolute magnitudes -22 to -21; on the other hand, the total galaxies have a broader distribution extending to lower luminosities. Fig. <ref> shows the distributions of M_r and FracDev for our Seyfert galaxies. The FracDev are widely distributed over different M_r. In other words, the different morphologies between the Seyfert 1 and the Seyfert 2 galaxies are not related to the absolute magnitudes of their host galaxies.We also investigated the effect of redshift on the FracDev for our Seyfert galaxies and total galaxies. Fig. <ref> shows the distributions of redshifts and FracDev; the average FracDev values increase with redshifts for both the Seyfert galaxies and the total galaxies. We noted that at the same redshifts, the average FracDev values for the Seyfert 1 are always larger than the Seyfert 2 galaxies; however, at low redshifts, the Seyfert 2 galaxies do have more sources with small FracDev values. The average FracDev values of the total galaxies are much smaller than those of the Seyferts at low redshifts but approaching to the values of the Seyfert 2 at z ≈ 0.2. These results suggest that the host galaxies of the Seyfert 1 are more bulge-dominated than those of the Seyfert 2.§ DISCUSSION AND SUMMARY§.§ Selection effectIn order to test whether our results are affected by the selection biases of the Véron catalog, we used a different selection method to select a different Seyfert sample for comparison. We selected a new Seyfert samples independently from the SDSS DR10 using typical criteria for Seyfert galaxies without invoking the identification of the Véron catalog. We selected the new Seyfert 1 galaxies with the full width at half maximum (FWHM) of the Balmer line > 1000 km s^-1 and Hβ/[OIII] >5 <cit.>. We selected two new samples ofSeyfert 2 with different criteria of line ratios. Sample A of the new Seyfert 2 is selected with [OIII]/Hβ > 3 <cit.> with the FWHM of the Balmer lines < 1000 km s^-1; sample B of the new Seyfert 2 is selected by the criteria of <cit.> with the FWHM of the Balmer lines < 1000 km s^-1. The new Seyfert samples have redshifts between 0 and 0.2, which is the same as the redshift range of our original samples.We finally have 2102 Seyfert 1 galaxies, 36134 sample A Seyfert 2 galaxies, and 35954 sample B Seyfert 2 galaxies from the new criteria.Fig. <ref> shows the FracDev distributions of the new Seyfert 1 and the sample A Seyfert 2 galaxies. A K-S test shows that the two distributions are drawn from the same population has K-S statistic (D) =0.37 and P = 0.00.Fig. <ref> present the FracDev distributions of the new Seyfert 1 and the sample B Seyfert 2 galaxies, which have a K-S test result with K-S statistic (D) =0.35 and P=0.00. In other word, we still found that the FracDev distributions of the Seyfert 1 and the Seyfert2 are completely different even these sources are selected uniformly over SDSS without invoking any Véron information. These suggest that our results are very robust and are not affected by the biased of the Véron catalogs, which compiled the sources from literature. §.§ AGN contributions Fig. <ref> shows that Seyfert 1 and Seyfert 2galaxies have completely different morphology distributions. We note that Seyfert 1 and Seyfert 2 galaxies have different AGN strengths, which might cause some biases about the morphological fitting. To quantify the effects of the AGN strengths on the morphological fitting, we considered the contributions of the AGNs using the SDSS spectra. Because the SDSS spetra aperture is 3”, the flux inside the aperture should be dominant by AGNs. We could use the fluxes of the spectra to roughly estimate the contributions of AGNs. SpectroFlux_r is a parameter from the SDSS database and is value of spectrum flux in r-band filter.We used the parameter to be an indicator of the AGNs contributions.Fig. <ref> are the result of average AGNs contributions to their host galaxies in different FracDev ranges. We found that both Seyfert 1 and Seyfert 2 galaxies have similar trends in the ratios of AGNs contributions to their bulges and to their host galaxies. The average ratios of the AGNs to the bulges declined with FracDev and the average ratios of the AGNs to the host galaxies increase with FracDev. If the FracDev fitting was significantly affected by AGNs and different Seyferts have different effects, we should find different trends in the ratios for Seyfert 1 and Seyfert 2 galaxies. Our results of AGNs contributions to the host galaxies suggest that AGNs contributions are unrelated to the results of the host galaxy morphology distributions.§.§ Host Galaxy Morphology We examined the relation between the FracDev and de Vaucouleurs T value for a subset of our Seyfert galaxies. We constrained the redshifts of the sources to be less than 0.05 and searched for the de Vaucouleurs value of their host galaxies from NASA Extragalactic Database (NED). There are 81 Seyfert 1 galaxies and 127 Seyfert 2 galaxies with available de Vaucouleurs T values. We found that there is a good correlation between the T value and the FracDev, which is consistent with <cit.>, who found a very good correlation between the FracDev and the Hubble types for 7429 nearby normal galaxies. <cit.> also showed that the average FracDev is ≈ 1 for elliptical and S0 galaxies. These results suggest that the relation between the FracDev and the de Vaucouleurs T values for our Seyfert samples are similar to those of normal galaxies; this similarity shows that our FracDev values are not significantly affected by the central AGNs and the FracDev can be used as an indicator of the hostgalaxy morphology. The difference in the morphology distributions of the Seyfert 1 and the Seyfert 2 galaxies might be biased by the magnitudes of our Seyfert samples. Fig. <ref> shows that the Seyfert 1 and Seyfert 2 galaxies have similar luminosity distributions toward bright end. Bright galaxies usually have larger fractions of light in their bulges whereas faint galaxies have most of their light in disks <cit.>. However, Fig. <ref> shows that the Seyfert 2 galaxies have relative high ratios at medium FracDev values than the Seyfert 1 galaxies. This indicates that although the galaxy luminosities might affect the morphology distributions of the Seyfert galaxies, it can not be the main cause for the different distributions of the Seyfert 1 and the Seyfert 2. Besides, we noticed that the Seyfert 1 and Seyfert 2 galaxies have similar distributions of r-band absolute magnitude indicating that the Seyfert 1 and Seyfert 2 galaxies have similar stellar mass distributions in their host galaxies. This result suggests that the stellar mass in the different types of Seyfert galaxies is independent of the unification model.To examine the influence of galaxy luminosity on the morphology, we chose sub-samples from our Seyfert 1 and Seyfert 2 with -22 < M_r < -21 and compared the FracDev distributions for the sub-sample Seyfert 1 and Seyfert 2. The results of the Kolmogorov-Smirnov test on FracDev distributions for the sub-sample Seyfert 1 and Seyfert 2 have probability that the two distributions are drawn from same population P = 0.00 with K-S statistic (D)=0.26, indicating that the Seyfert 1 and the Seyfert 2 still have different FracDev distributions. These results suggest that the galaxy luminosity does not cause the morphology difference between Seyfert 1 and Seyfert 2.Traditional unification model <cit.> suggests that the different types of AGNs are due to different observing angles relative to the torus. The unification model suggests that Seyfert 1 and Seyfert 2 galaxies are intrinsically similar objects with different viewing angles. Seyfert 1 galaxies are viewed from face-on to the accretion disk without obscuring, whereas Seyfert 2 galaxies are viewed from edge-on and obscured by the torus. However, our results (Fig. <ref>) indicate that the morphology distributions of the host galaxies of Seyfert 1 and Seyfert 2 are different. This suggests that the differences between the Seyfert 1 and the Seyfert 2 are not only caused by the viewing angles but also might be related to their host galaxy morphologies. In the early study of the host galaxies of AGNs, people found few Seyfert AGNs residing in elliptical galaxies <cit.>. For example, <cit.> considered AGNs that were classified as Seyfert or LINER in Véron catalogue (1991) with morphological information in the Third Reference Catalogue of Bright Galaxies (RC3) and found that only 5% of their samples were elliptical galaxies. However, some other studies found AGNs residing in early-type galaxies; <cit.> found that the Hubble types of Seyfert galaxies tend to be S0 and Sa; <cit.> used spectroscopic to study the emission line in the central region of 486 nearby galaxies and found that spectroscopic AGN are dominant in early-type galaxies;<cit.> found that there are more than half of their X-ray selected AGNs residing in E/S0 type galaxies. These results suggested that the host galaxymorphologies of AGNs might prefer to be bulge-dominant galaxies. The morphology difference of the host galaxies between Seyfert 1 and Seyfert 2 was less clear. <cit.> found that among 624 Seyfert 1 galaxies, 76% of the host galaxies of Seyfert 1 are early types and 9.7% are late types; among 925 Seyfert 2 galaxies, the ratio of late-type galaxies for Seyfert 2 galaxies is 26.9% and that of early-type galaxies is 56.8%. <cit.> found that their Seyfert 1 galaxies prefer to locate in early type galaxies than Seyfert 2 galaxies do. On the other hand, Seyfert 2 galaxies were more frequently found in late type galaxies <cit.>. Our Seyfert samples are dominated by FracDev=1 for both Seyfert 1 and Seyfert 2. This indicates that most of the Seyfert galaxies are bulge-dominant.We checked the host galaxy morphology of our Seyferts by using the results of the Galaxy Zoo Project <cit.>. There are 2889 Seyfert 1 galaxies and 2826 Seyfert 2 galaxies with available information from the Galaxy Zoo. We used the voted data from the Galaxy Zoo to check the morphology of the galaxies that have FracDev = 1 <cit.>. We assumed that a galaxy with a voted-elliptical fraction > 50% has the morphology of an elliptical galaxy. We found that66% of the Seyfert 1 galaxies with FracDev = 1 show elliptical-like morphology, and 68% of the Seyfert 2 with FracDev = 1 show elliptical-like morphology. Besides, the ratio of elliptical-like galaxies for all Seyfert 1 with different FracDev values is 30% and that for Seyfert 2 is 19% from the voting rates of the Galaxy Zoo. Our results are similar to previous studies of <cit.> and <cit.>. Furthermore, we assumed that a spiral-like host for a galaxies with voted-spiral fraction >50% and FracDev <1. We have 20% spiral host in the Seyfert 1 galaxies and 30% in the Sefyert 2 galaxies. <cit.> shows that the spiral host of Type-1 AGN is 20% and Type-2 is 44%. Our results of spiral host galaxies of Seyferts show that the Seyfert 2 galaxies have more spiral host than Seyfert 1 galaxies and agree with that of <cit.>. We also investigated whether the morphology distributions of the Seyfert galaxies are affected by redshifts. We selected sub-samples of Seyfert galaxies with redshifts less than 0.05. The ratio of elliptical host galaxies in the low-redshift Seyfert 1 galaxies is 19% and that in the low-redshift Seyfert 2 is 14%. This shows that there are fewer elliptical Seyfert galaxies in low redshifts because there are fewer nearby elliptical galaxies. This explains why people found much fewer elliptical Seyferts in the earlier studies <cit.>. The different bulge distributions of Seyfert 1 and Seyfert 2 might be related to galaxy evolution. <cit.> indicated that the neighbors of Seyfert 2 galaxies are more ionized than the neighbors of Seyfert 1 galaxies;they proposed that there is an evolution sequence between Seyfert galaxies. Seyfert 2 galaxies begin from an interaction galaxies and finally transform into Seyfert 1 galaxies. The other factor to cause different bugle distributions of Seyfert 1 and Seyfert 2 galaxies might be related to ISM/orientation of host galaxy. The obscuration of dust in the host galaxy of Seyfert galaxies might affect the identification of the different Seyfert types <cit.>. We also test whether our sample is isotropic to the viewing angle of the host galaxy by comparing the axis ratio b/a of the Seyfert galaxies with FracDev =1. The results of axis ratio of Seyfert 1 and Seyfert 2 galaxies are shown in Fig. <ref>. We found that both Seyfert 1 and Seyfert 2 galaxies have peak at b/a= 0.6 - 0.7. This result indicates that our Seyfert sample is independent of the viewing angle. It has been arguing that observed type 1 and type 2 AGNs are actually drawn from different distributions of covering factors of the AGNs <cit.>.Therefore, we expect that intrinsic difference should exist in our selected samples of Seyfert 1 and Seyfer 2. However, we note that we are comparing the host of galaxies of the AGNs but not the intrinsic properties of the AGNs; in other words, any isotropic biases in selection should not affect our results about the host galaxies, unless the properties of AGNs were closely related to the host galaxies, which again is not consistent with the unification model. <cit.> found that the host galaxy morphologies of Seyfert 1 and Seyfert 2 are similar. We note that <cit.> only chose the “Face-on” source in order to classify dust features in the cores of their Seyfert galaxies. We found that they tended to select much more disk galaxies than the parent population we have. The similar distribution in the host morphology of Seyfert 1 and Seyfert 2 might be caused by the pre-selection of the “Face-on” galaxies, which might exclude most of the elliptical galaxies. Besides, the number of the Seyfert galaxies (30 Seyfert 1 and 53 Seyfert 2) in <cit.> is too small to distinct the distributions significantly. Although the distributions of host morphology of Seyfert 1 and Seyfert 2 galaxies in the samples of <cit.> are not distinguishable, they still found that the distributions of the core morphologies of Seyfert 1 and Seyfert 2 galaxies are different.Besides, <cit.> found the average color of neighbor galaxies around type-I AGN are redder than that of the neighbors around type-II AGN; they also found that the host galaxy morphology ofan AGN depends on the AGN type and the presence of a neighbor. Our results show that the host galaxies of the Seyfert 1 galaxies are dominated by galaxies with large bulge ratios whereas Seyfert 2 galaxies show host galaxies with relatively smaller bulge ratios, suggesting that different Seyferts might appear in the different evolution stages of galaxy evolution. We selected our Seyferts samples from the Véron catalog, which compiled all AGN sources from observations and literature. However, the definitions of the Seyfert 2 galaxies might be slightly different in different observations. We obtained the photometric and spectral data of these Seyfert galaxies from SDSS <cit.> and used the BPT diagram <cit.> to define a more rigorous sample of Seyfert 2 galaxies to check whether our results are affected by the contamination of starburst galaxies and LINERs in the Seyfert 2 sample. We used the definitions of <cit.> and <cit.> to divide our Seyfert 2 galaxies into three sub-samples: new rigorously defined Seyfert 2, LINERs and starburst galaxies and compare the FracDev distributions of these sub-samples with the Seyfert 1 galaxies. Fig. <ref> show that some of our original Seyfert 2 galaxies are locating in the LINER and starburst regions.Fig. <ref> shows the FracDev distributions of the new rigorously definedSeyfert 2, LINERs, starburst galaxies, and Seyfert 1 galaxies. We found that the FracDev distributions of the new-defined Seyfert 2 and the Seyfert 1 galaxies are still very different and a Kolmogorov-Smirnov test shows that the probability for the two distributions drawn from the same population is P = 0.00 0 with K-S statistic (D) =0.41. We replotted the relative ratios of the new-defined Seyfert 2 and Seyfert 1 to all SDSS galaxies with different FracDev values in Fig. <ref> and we found that the distribution pattens are still similar to Fig. <ref>. These resultsindicate that the different morphologies of Seyfert 1 and Seyfert 2 galaxies are not caused by contamination of LINERs or starburst galaxies in the Seyfert 2 sample. Besides, we found that the FracDev distributions of the Seyfert 1, the new-defined Seyfert 2 and the LINERs all show distinct peak values at FracDev=1, indicating that AGN type galaxies are more bulge-dominant; on the other hand, the FracDev values of the starburst galaxies distribute more evenly. From our results, we found that the Seyfert 1 and Seyfert 2 galaxies have different host galaxy morphology distributions; Seyfert 2 galaxies have relatively more late-type host galaxies than Seyfert 1 have. Different types of galaxies are expected to have different formation history and physical process in galaxy evolution. The different distributions of the host galaxy morphology suggests that the different types of Seyfert galaxies might be related to the processes of galaxy formation. This indicates that the differences between the Seyfert 1 and Seyfert 2 galaxies are not only affected by the viewing angles but also related to the formation process of their host galaxies.§.§ Summary § ACKNOWLEDGEMENTS This work is supported by the Ministry of Science and Technology of Taiwan (grant MOST 106-2119-M-008-016). We thank L. Kewley, P. C. Yu, M. C. Tsai, Z. Y. Chen, J. C. Huang, and Y. L. Chang for discussion and comments. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. 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"authors": [
"Yen-Chen Chen",
"Chorng-Yuan Hwang"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727100323",
"title": "Morphology of Seyfert Galaxies"
} |
TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium Lyman Laboratory of Physics, Harvard University Dilute ultracold quantum gases form an ideal and highly tunable system in which superfluidity can be studied. Recently quantum turbulence in Bose-Einstein condensates was reported [PRL 103, 045310 (2009)], opening up a new experimental system that can be used to study quantum turbulence. A novel feature of this system is that vortex cores now have a finite size. This means that the vortices are no longer one dimensional features in the condensate, but that the radial behaviour and excitations might also play an important role in the study of quantum turbulence in Bose-Einstein condensates. In this paper we investigate these radial modes using a simplified variational model for the vortex core. This study results in the frequencies of the radial modes, which can be compared with the frequencies of the thoroughly studied Kelvin modes. From this comparison we find that the lowest (l=0) radial mode has a frequency in the same order of magnitude as the Kelvin modes. However the radial modes still have a larger energy than the Kelvin modes, meaning that the Kelvin modes will still constitute the preferred channel for energy decay in quantum turbulence. Radial vortex core oscillations in Bose-Einstein condensates N. [email protected] T. Ichmoukhamedov1 J. Tempere1,2 January 15, 2018 ============================================================================ § INTRODUCTIONQuantum turbulence<cit.> (QT) was achieved in ultracold atomic gases<cit.> by shaking the condensate in conjunction with rotation<cit.> or by rapidly sweeping a laser beam through the condensate<cit.>. Since ultracold quantum gases are highly tuneable <cit.> in comparison to superfluid helium (interaction strength, number of particles, type of trapping, dimensionality, …), the realization of QT in these gases opens up new opportunities for this field of study. One example of the new opportunities offered by quantum gases is the fact that it is now possible to investigate QT in a two dimensional (2D) Bose-Einstein condensate (BEC) <cit.>. Before the experimental realization of ultracold gases, QT was extensively studied in superfluid helium <cit.>, where it was discovered already 50 years ago <cit.>. In these studies it became apparent that QT is characterized by the appearance of singly quantized vortices that are distributed in a tangled way <cit.>. These vortices appear as a consequence of the Kolmogorov decay <cit.> which causes larger eddies to break up into singly quantized vortices. How the energy further dissipates from the tangled series of quantized vortices has not yet been experimentally observed. It is theorized that the further dissipation of these tangled vortices happens via vortex reconnections and ultimately the excitation of the axial Kelvin waves <cit.> of the vortex line, leading to an energy dissipation via phonons and rotons<cit.>. These Kelvin waves were also experimentally observed in a lattice of vortices in a BEC <cit.>.In superfluid helium the size of the vortex core is in the order of nanometers<cit.>. In order to get an observation of the vortex flow experimentally micron-sized<cit.> and even sub-micron sized<cit.> solid hydrogen trackers are used. In ultracold gases however the vortex core has a size in the order of fractions of micrometers <cit.>, which yields a typical condensate size to vortex core size ratio of 10-50. Observing a vortex core in ultracold gases is typically done by imaging the condensate after expansion, using tomographical methods <cit.> for 3D reconstruction, or in situ methods <cit.>.In the present article, the starting idea is the fact that in ultracold atomic gases the size (healing length) of the vortex core is not negligible with respect to the condensate size. The fact that the size of a vortex core in an ultracold gas is non-negligible, yields new effects for the dissipation of energy in the mechanism of QT. The excitations for finite-sized vortex cores that are explored in this article are the radial oscillation modes. This oscillation mode yields an additional way, next to Kelvin modes, for energy to be dissipated by the movement of the vortex core.§ THEORETICAL DESCRIPTIONIn order to describe Bose-Einstein condensates (BECs) the standard Gross-Pitaevskii theory <cit.> is used. The condensate wave function Ψ(r,t) is described in the hydrodynamical picture:Ψ(r,t)=√(n(r,t))exp(i S(r,t)),using the real valued density n(r,t) and phase S(r,t) fields. To consider density fluctuations, this description is most useful. The velocity field is obtained from the phase field using v(r,t)=ħ/m∇ S(r,t). The magnitude of the velocity vector field v(r,t) is denoted by the scalar field v(r,t).The system that will be studied is a BEC in a cylindrical box potential with height H and radius R_cond. These homogeneous BECs can also be realized in experiments<cit.>. Using the hydrodynamic description, the kinetic and interaction energy of the homogeneous superfluid confined in r∈[0,R_cond], z∈[0,H] are given by respectively<cit.>:E_kin[Ψ] =m/2∫ d^3r[n(r,t)v^2(r,t)],E_int[Ψ] =1/2g∫ d^3r(n_∞-n(r,t))^2,where g=4πħ^2a_s/m, with a_s the s-wave scattering length and n_∞ the homogeneous bulk density. In order to calculate the different energies an analytical variational form is used for the superfluid density n(r,t) and the velocity field v(r,t). Important to note is that the energy equations (<ref>) only hold when the condensate size R_cond is large enough compared to the condensate healing length ξ. §.§ Describing the single-vortex structure (unperturbed)Since the vortex core structure n(r) has no analytical solution, even in the simplest (unperturbed, cylindrical symmetric) case, a variational function is commonly used in order do derive analytical results. A commonly used variational model is the hyperbolic tangent which yields an accurate fit for the vortex core structure<cit.>. In this work a simple cylindrical hole of the form<cit.>n(r)=n_∞Θ(r-R_v)Θ(R_cond-r)Θ(z)Θ(H-z)is used, where R_v is the size of the vortex core. This model yields the same energy as the hyperbolic tangent variational model up to an additive constant<cit.>. This means that the variational model (<ref>) yields accurate results if[A factor of R_cond/R_v equal to 10 is already sufficient in the homogeneous case for an error in energy smaller than 1%.] R_cond>>R_v. This has only been investigated for the (radial) ground state configuration, and not for the excitations. Nevertheless, the simplicity of the model allows for analytic solutions that can serve as a benchmark for future improvements of the model. Note that also H>>ξ for the variational model (<ref>), otherwise also the edge effects in the z direction will play a more important role. The velocity field of a single (singly quantized) vortex is given by:v_v(r)=ħ/mre_θ. The variational parameter for the above vortex model (<ref>) is the vortex radius R_v. Minimizing the energy of the non-perturbed vortex it can be shown that a minimal energy is achieved when:R_v=1/√(4π n_∞ a_s)=√(2)ξ,where ξ is the healing length of the condensate. Around this energy minimum all of the considered radial vortex core oscillations will be expanded. §.§ Adding perturbations to the single-vortex structureCombining the fact that the flow of a BEC is irrotational<cit.> (∇×v(r,t)=0) and that the variational model (<ref>) leads to an incompressible flow (∇·v(r,t)=0), the equation of motion for the BEC velocity field can be derived. The BEC velocity can be described by a velocity potential ϕ(r,t), obeying a potential equation ∇^2ϕ(r,t)=0.Solving the above potential equation in cylindrical coordinates, using the boundary condition v(r→∞,t)=0, results in the velocity field is given by:v_l(r,t)=Ȧ_l(t)(r/R_v)^-l-1[cos(lθ)e_r+sin(lθ)e_θ]with l∈ℕ,where the integer l labels the different oscillation modes and Ȧ_l(t) is the integration constant (fixing the size of the l^th mode). In principle[For a general velocity field consisting of a combination of different perturbation modes this is needed in order to have a complete Fourier series in the angular coordinate θ.] we should have a linear combination of cos(lθ) and sin(lθ) for the angular dependency. However for our calculations the relative phase between the different l modes will not play a role since we will be looking at small perturbations. The modes can thus be seen as a set of non interacting[The cross terms (∝Ȧ_l(t)Ȧ_k(t)) in the energy drop out in the calculations.] harmonic oscillators. Important to note is that the true boundary conditions for the velocity field of the system is given by v(r→ R_cond,t)=0, however in practice the value of R_cond is chosen to be large enough so that the boundary condition mentioned above can be implemented.§.§ Describing the radial oscillationsIn order to now introduce the radial oscillations of the vortex core, the vortex core radius (for a given mode l) is written as:R_l(θ,t)=R_v+δ R_l(θ,t),with R_v the equilibrium case (<ref>) and δ R_l(θ,t) the deformation of the vortex core due to the l^th oscillation mode. The deformation of the vortex core δ R_l(θ,t) can be derived from the velocity field using:Ṙ(θ,t)=δṘ(θ,t)=v(R_v,θ,t)·e_r.The velocity field in (<ref>) is evaluated in r=R_v, this can be done since for the perturbations r=R_v+δ R_l and R_v>>δ R_l. Since for the stable vortex, the velocity field lies along e_θ, this flow will not deform the vortex core. Using the velocity field (<ref>), the deformation of the vortex core is given by (<ref>):δ R_l(θ,t)=A_l(t)cos(lθ).Combining (<ref>) and (<ref>) then yields the deformation of the vortex core:R_l(θ,t)=R_v+A_l(t)cos(lθ).§ RESULTSUsing the description of section <ref> for the perturbed vortex core with radial oscillations, the total energy can be calculated using equations (<ref>). Given the total energy it is possible to derive the frequencies of the different oscillation modes. The obtained frequencies can then be compared with the known frequencies of the well studied Kelvin modes. §.§ Energy and frequencies of the radial oscillation modesThe total energy of the vortex core, containing radial perturbations in the vortex core radius can now be calculated by substituting (<ref>) in (<ref>), using (<ref>) for the vortex radius R_v. Since the mode l=0 (the breathing mode) will lead to logarithmic divergences, this energy should be calculated separately. Calculating the energies of the vortex core with a radial perturbation yields:E_l=0[Ψ] =E^(0)+ħ^2π n_∞/2mξ^2HA_0^2(t)+2ξ^2mn_∞π Hln(R_cond/R_v)Ȧ^2_0(t),E_l≠ 0[Ψ] =E^(0)+ħ^2π n_∞/4mξ^2HA_l^2(t)+1/lξ^2mn_∞π H[1-(R_v/R_cond)^2l]Ȧ^2_l(t),where E^(0) is the energy of the unperturbed vortex:E^(0)=ħ^2π n_∞/mHln(R_cond/R_v)+1/24πħ^2a_s/mn_∞^2π H R^2_v. In order to now determine the frequencies of the different oscillation modes, we can either write down the Hamilton-Jacobi equations, or compare (<ref>) to the Hamiltonian of the harmonic oscillator for A(t) (since we expanded A(t) only up to second order). The frequencies of the oscillation modes are then given by:ω^2_l=0 =c^2/2ξ^2[ln(R_cond/R_v)]^-1, ω^2_l≠ 0 =lc^2/2ξ^2[1-(R_v/R_cond)^2l]^-1,where c=ħ/√(2)mξ is the speed of sound. Note that in the limit ξ→ 0 (also using (<ref>)) all frequencies (<ref>) go to infinity, making these oscillation modes energetically inaccessible. For increasing values of ξ all of the frequencies (<ref>) decrease.§.§ The axial Kelvin modes (or kelvons)The Kelvin modes (or kelvons) have already been extensively studied in BECs. To ensure the vortex is confinedby an infinite potential well, Dirichlet boundary conditions are imposed, resulting in a quantization of the wavenumber for the Kelvin modes:k_n=nπ/Hwith n∈ℕ_0.The Kelvin modes have a characteristic long-wavelength dispersion relation given by<cit.>:ν_n=ħ k_n^2/2mln(1/|k_n|ξ)=n^2 cξπ^2/H^2√(2)ln(H/nπξ),where for the last equation the quantization condition (<ref>) was used. The long wavelength approximation for the Kelvin modes breaks down when the wavelength of the Kelvin modes becomes comparable to the healing length. As variations in the condensate wavefunction must occur on length scales larger than the healing length (in order to be collective excitations), this length scale sets an upper bound on the allowed k_n values (<ref>) (and thus n values):k_nξ<1 ⇔H/nπξ>1.Note from (<ref>) that the frequencies (and thus the energies) of the Kelvin modes become smaller as the condensate height H increases. §.§ Comparison of the radial oscillation modes and the axial Kelvin modesThe energies of the Kelvin and radial modes can be compared by comparing their frequencies, in other words by solving ω_l=ν_n. When both frequencies are equal, a resonance between the two different oscillation modes will occur. In order to further simplify the calculations, the dimensionless variablesh=H/nπξand x=R_v/R_condcan be used, where h is the dimensionless condensate height and x is the dimensionless inverse condensate radius. Note that x∈[0,1] and according to (<ref>) h∈[1,∞[. Using the dimensionless variables, the frequencies of the radial (<ref>) and Kelvin modes (<ref>) can be plotted, this is done in figure <ref>.In figure <ref> the dimensionless radius x (<ref>) is constrained to the interval [0,0.2]. The reason for this constraint is that the model for the vortex core (<ref>) is only valid for a small vortex radius (compared to the condensate radius).On figure <ref> the typical vortex to condensate size ratio's for a vortex in a BEC (x∈[1/50,1/10]) are indicated by a shaded area. As can be seen, in this range of x values, the radial modes will not show a resonance with the Kelvin modes. Note however that in this x range, the frequencies of the lower radial modes and the Kelvin mode are in the same order of magnitude.§ CONCLUSIONS From our calculations it can be seen that the radial modes for the vortex core will show no resonances with the Kelvin modes within the typical BEC setup (R_cond/R_v∈[10,50]). This can be seen in figure <ref>, where the frequencies of the radial and Kelvin modes are plotted in terms of condensate height and radius. This means that for low energy decay, even in ultracold gases where the vortex core has a finite size, the Kelvin modes will still have a preferred role. Note that although a strict resonance between the lower radial modes and the Kelvin modes is not possible, both modes have a frequency within the same order of magnitude. This means that including the radial modes may result in a (small) correction when included in calculation.From figure <ref> it might be tempting to state that in superfluid helium, where the vortex radius is practically zero, a resonance between the radial and Kelvin modes might be possible. Note however that liquid helium (with[Where n is the superfluid density and a_s is the s-wave scattering length.] n|a_s|^3≈ 0.5-2500) is not a dilute superfluid (where n|a_s|^3<<1 should hold). This means that the (mean field) s-wave scattering potential used in (<ref>) no longer yields an accurate description of the problem. The mean fieldGP description used here is no longer appropriate for superfluid helium, meaning that no conclusions can be drawn for liquid helium from figure <ref>.Finally it can be noted that the above study does not take finite temperature effects into account. For a finite temperature it is to be expected that both the Kelvin and radial mode energies are broadened over an energy interval of the size k_BT. This thermal blurring will lead to a (slightly) larger region in which both oscillation modes can couple. The authors acknowledge the weekly fruitful discussions with S.N. Klimin, G. Lombardi, J.P.A. Devreese and W. Van Alphen. This research was supported by the University Research Fund (BOF) of the University of Antwerp, project: 2014BAPDOCPROEX167 and FFB1550168, and by the Flemish Research Foundation (FWO-V1), project nrs: G.0115.12N, G.0119.12N, G.0122.12N and G.0429.15.N.Barenghi3 C. Barenghi, L. Skrbek, K.R. Sreenivasan, PNAS, 111, 4647-4652 (2014). Anderson B.P. Anderson, Nature 539, 36-37 (2016). Henn E.A.L. Henn, J.A.Seman, G.Roati, K.M.F. Magalhaes, V.S. Bagnato, Phys. Rev. Lett. 103, 045301 (2009). Kwon W.J. Kwon, G. Moon, J.-Y. Choi, S. W. Seo, Y.-I. Shin, Phys. Rev. A 90, 063627 (2014). White A. C. White, B.P. Anderson, V.S. Bagnato, Proc. Natl. Acad. Sci. U.S.A., 111, 4719-4726 (2014). Bloch I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). Wilson K. E. Wilson, E. C. Samson, Z. L. Newman, T. W. Neely, B. P. Anderson, Ann. Rev. Cold At. Mol. 1, 261 (2013). Barenghi C. F. Barenghi, R. J. Donnelly, and W. F. Vinen (Eds.), Quantized Vortex Dynamics and Superuid Turbulence (Springer Verlag, Berlin 2001). Fisher S.N. Fisher et al. Phys. Rev. Lett 86, 244 (2001). Barenghi2 C.F. Barenghi and D.C. Samuels, Phys. Rev. Lett. 89, 155302 (2002). Hall H.E. Hall and W.F. Vinen, Proc. R. Soc. A 238, 204 (1956). Vinen W.F. Vinen and J.J. Niemela, J. Low. Temp. Phys. 128, 167 (2002). Kobayashi M. Kobayashi and M. Tsubota, Phys. Rev. A 76, 045603 (2007). Kolmogorov A.N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the USSR Academy of Sciences 30, 299-303 (1941). Vinen2 W.F. Vinen, Phys. Rev. B 64, 134520 (2001). Thomson W. Thomson, Phil. Mag. Ser. 5, 155-168 (1880). Thomson2 W. Thomson, Proc. Roy. Soc. London 29, 71 (1879). Vinen3 W.F. Vinen, Phys. Rev. B 61, 1410 (2000). Kondaurova L. Kondaurova, V. L'vov, A. Pomyalov and I. Procaccia, Phys. Rev. B 90, 094501 (2014). Smith N. L. Smith, W. H. Heathcote, J. M. Krueger, and C. J. Foot, Phys. Rev. Lett. 93, 080406 (2004). Donnelly Russell J. Donnelly, Quantized vortices in Helium II, (Cambridge University Press, 1991). Bewley G.P. Bewley, D.P. Lathrop, K.R. Sreenivasan, Nature 441, 588 (2006). Fonda E. Fonda, K.R. Sreenivasan, D.P. Lathrop, Rev. Sci. Instrum. 87, 025106 (2016). Cooper N.R. Cooper, Rapidly rotating atomic gases, Advances in Physics, 57:6, 539-616 (2008). Rosenbusch P. Rosenbusch, V. Bretin, J. Dalibard, Phys Rev Lett. 89, 200403 (2002). Gemelke N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, Nature 460, 995 (2009).Gross E.P. Gross, Il Nuovo Cimento 20, (1961) 454-457. Pitaevskii L.P. Pitaevskii, Sov. Phys. JETP 13, (1961) 451-454. Gaunt A.L. Gaunt, T.F. Schmidutz, I. Gotlibovycj, R.P. Smith and Z. Hadzibabic, Phys. Rev. Lett. 110, 200406 (2013). Pethick C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge university press (Cambridge 2001). Verhelst N.Verhelst, S. Klimin, J. Tempere, Physica C (to be published, 2016). Ginzburg V.L. Ginzburg, L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 34, 1240 (1958). Fetter A.L. Fetter, Rev. Mod. Phys 81, (2009) 648. Pitaevskii2 L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press (Oxford, 2003). Sonin E.B. Sonin, Rev. Mod. Phys. 59, (1987) 87. | http://arxiv.org/abs/1707.08382v1 | {
"authors": [
"Nick Verhelst",
"Timour Ichmoukhamedov",
"Jacques Tempere"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170726112626",
"title": "Radial vortex core oscillations in Bose-Einstein condensates"
} |
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland We study the proximity effect in a one-dimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the “hard gap" regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.74.45.+c,71.10.Pm,73.21.Hb,74.78.NaFinite-size effects in a nanowire strongly coupled to a thin superconducting shell Jelena Klinovaja December 30, 2023 ==================================================================================§ INTRODUCTIONTopological superconductivity has been a subject of intense study in recent years <cit.> both theoreticallyand experimentally because the localized Majorana excitations of such systems obey non-Abelian statistics and can potentially be utilized for applications in quantum computing <cit.>. The most promising proposal to date for engineering Majorana bound states in nanowires combines Rashba spin-orbit coupling, proximity-induced s-wave superconductivity, and an external magnetic field applied parallel to the nanowire <cit.>. An alternative proposal which has also received a great deal of attention involves coupling a ferromagnetic atomic chain to an s-wave superconductor with strong intrinsic spin-orbit coupling<cit.>. Since the first generation of nanowire experiments <cit.>, there has been significant progress made in both the fabrication of cleaner devices as well as in the quality of the proximity-induced superconductivity <cit.>. The most significant advance in this respect has been the development of thin shells (with thickness d∼10 nm) of superconducting Al grown epitaxially on either InAs <cit.> or InSb <cit.> nanowires, thus ensuring a very strong proximity contact which has led to very hard induced gaps in the nanowires in the absence of a magnetic field.Despite the recent experimental development of these thin superconducting shells, the most comprehensive theories describing proximity-induced superconductivity in a nanowire treat the superconductor as infinitely large <cit.>. Such an assumption implies that there is a continuum of states in the superconductor, and therefore there are always states available to couple to the nanowire and open a gap. However, in reality, the superconductor has a finite level spacing δ E_s∼ħ v_F/d due to its finite size. For the thin Al shells studied experimentally (v_F∼10^6 m/s and d∼10 nm), the level spacing of the shell δ E_s∼10 meV exceeds the Fermi energy of the nanowire (∼0.1-1 meV for typical semiconducting nanowires). Thus, for the experimental system, the limit of a bulk superconductor is not the relevant one and finite-size effects are expected to play an important role in determining the strength of proximity-induced superconductivity. In this paper, we show that the finite size of the shell can be very detrimental to inducing superconductivity in the nanowire. In order to induce a sizable superconducting gap without finely tuning the thickness of the shell on the scale of the Fermi wavelength (of the superconductor), the energy scale describing tunneling between the nanowire and superconductor (γ) must be made larger than the level spacing of the shell (γ≳δ E_s). However, such strong tunneling induces a shift in the effective chemical potential of the nanowire which greatly exceeds the semiconducting energy scale. As a result, it is possible for the system to exhibit a hard gap even if the nanowire is effectively depleted; in this case, the gap is determined by the lowest subband in the superconductor rather than the nanowire. Additionally, in order to reach the topological phase, the Zeeman energy induced by an external magnetic field must counteract the large chemical potential shift. As a result, the field strength needed to reach the topological phase greatly exceeds the critical field of Al.The remainder of this paper is organized as follows. In Sec. <ref>, we describe a simple theoretical model which can be applied to the experimental geometry of a thin superconducting shell strongly coupled to a semiconducting nanowire. In Sec. <ref>, we analyze the spectrum of our model, showing that a large tunneling strength is needed to overcome the large level spacing of the superconductor and open a sizable gap in the nanowire. We consider the case when the nanowire is located near the edge of the superconductor in Sec. <ref>, while we consider the case when the nanowire is located in the middle of the superconductor in Sec. <ref>. We present a numerical tight-binding calculation to back up the theoretical analysis of our model in Sec. <ref>. In Sec. <ref>, we determine how the finite size of the superconductor affects the critical field strength needed to reach the topological phase in the nanowire. In Sec. <ref>, we relate the results of our simple model directly to the experimental setup and provide estimates for the level spacing of the superconducting shell, the tunneling strength needed to induce a sizable gap in the nanowire, and the critical field strength needed to reach the topological phase. Our conclusions are given in Sec. <ref>.§ MODEL The system we consider is displayed in Fig. <ref>(a). We consider a nanowire which is an infinitely long one-dimensional channel oriented along the y-direction (with zero width). The nanowire is tunnel coupled at a position x=x_w to a superconductor which is infinitely long in the y-direction and has finite extent d in the x-direction (the need for a finite x_w will be explained below).We consider a Hamiltonian of the formH=H_w+H_s+H_t.For now, we take a simple model for the Hamiltonian of the nanowire,H_w=∑_σ∫dk_y/2π ψ^†_σ(k_y)ξ_kψ_σ(k_y),where k_y is a conserved momentum in the direction parallel to the nanowire, ψ_σ^†(k_y) is the creation operator in the nanowire, and ξ_k=k_y^2/2m_w-μ_w (m_w and μ_w are the effective mass and chemical potential of the nanowire, respectively). The superconductor is described by a BCS Hamiltonian,H_s=1/2∫dk_y/2π∫_0^d dx η^†(k_y,x)ℋ_BCSη(k_y,x),where η(k_y,x)=[η_↑(k_y,x),η_↓^†(-k_y,x)]^T, η_σ^†(k_y,x) is the creation operator in the superconductor, and ℋ_BCS=(-∂_x^2/2m_s+k_y^2/2m_s-μ_s)τ_z+Δτ_x, with Δ the (constant in space) superconducting pairing potential and τ_x,y,z Pauli matrices acting in Nambu space. The two systems are coupled at a position x=x_w by a tunneling term which we assume preserves spin and momentum,H_t=-t∑_σ∫dk_y/2π[ψ_σ^†(k_y)η_σ(k_y,x_w)+H.c.],where t is the (spin-independent) tunneling amplitude. Such a model corresponds to local tunneling along the superconductor/semiconductor interface.In the absence of tunneling, the spectrum of the finite-sized superconductor is given by (n∈ℤ^+)E_n(k_y)=√((μ_s-k_y^2/2m_s-π^2 n^2/2m_sd^2)^2+Δ^2).When the quantization scale exceeds the gap, 1/m_sd^2≫Δ, there are very few subbands available to couple to the low-energy modes of the nanowire, as shown in Fig. <ref>(b). In this case, the relevant subbands follow a linearized formE_n(k_y)=√([(k_Fd-π n)δ E_s-k_y^2/2m_s]^2+Δ^2),where we define the level spacing δ E_s=v_F/d (v_F=k_F/m_s is the Fermi velocity of the superconductor and k_F=√(2m_sμ_s) is the Fermi momentum). As we will show explicitly, when δ E_s≫Δ, it is the level spacing which is the relevant scale (rather than Δ) in determining the strength of the proximity effect.The Hamiltonian in Eq. (<ref>) can be diagonalized by means of a Bogoliubov transformation <cit.>. The resulting Bogoliubov-de Gennes (BdG) equation is given by[ℋ_BCS+t^2δ(x-x_w)G_0^R(E,k_y)]ψ_s(x)=Eψ_s(x),where ψ_s(x) is the (Nambu spinor) wave function of the superconductor and G_0^R(E,k_y)=(E-ξ_kτ_z+i0^+)^-1 is the bare retarded Green's function of the nanowire (in the absence of tunneling). The nanowire itself enters only through the boundary condition at x=x_w [corresponding to the δ-function term in Eq. (<ref>)]. Inside the superconductor, we solve Eq. (<ref>) on both the left (x<x_w) and right (x>x_w) sides of the nanowire to giveψ_L(x)=c_1([ u_0; v_0 ])sin(k_+d)+c_2([ v_0; u_0 ])sin(k_-d),ψ_R(x)=c_3([ u_0; v_0 ])sin[k_+(d-x)]+c_4([ v_0; u_0 ])sin[k_-(d-x)], where k_±=(k_F^2-k_y^2±2im_sΩ)^1/2, u_0(v_0)=√((1± iΩ/E)/2), and Ω=√(Δ^2-E^2). The vanishing boundary conditions which we impose at the free ends of the superconductor (x=0 and x=d) are accounted for already in Eqs. (<ref>); the boundary conditions at x=x_w due to tunneling are given by <cit.>ψ_L(x_w)=ψ_R(x_w),1/k_F[∂_xψ_R(x_w)-∂_xψ_L(x_w)]=2γτ_zG_0^R(E,k_y)ψ_s(x_w), where γ=t^2/v_F is a tunneling energy scale.Imposing boundary conditions at x=x_w, the solvability condition of the resulting system of equations determines the excitation spectrum E(k_y). Assuming that μ_s≫|Ω|, we make a semiclassical expansionk_±=k_Fφ± iΩ/(v_Fφ)≡ζ± iχ,where φ=(1-k_y^2/k_F^2)^1/2 parametrizes the quasiparticle trajectory inside the superconductor (0<φ≤1). We note that the semiclassical approximation breaks down for grazing trajectories k_y≈ k_F within the superconductor, and we do not consider such trajectories. After some algebra (see Appendix <ref> for details), the solvability condition can be expressed asE^2/Γ^2(E,k_y)-Δ^2(1/Γ(E,k_y)-1)^2-[ξ_k-δμ(E,k_y)]^2=0,where we define the effective parameters Γ=(1+γ/Ωφ[cosh(2χ d)-cos(2ζ d)]{sinh(2χ d)-cos(2ζ x_w)sinh[2χ(d-x_w)]-cos[2ζ(d-x_w)]sinh(2χ x_w)})^-1,δμ=-γ/φ[cosh(2χ d)-cos(2ζ d)]{sin(2ζ d)-sin(2ζ x_w)cosh[2χ(d-x_w)]-sin[2ζ(d-x_w)]cosh(2χ x_w)}. The quantity Γ(E,k_y), which takes values 0<Γ<1 for E<Δ, renormalizes the energy and is responsible for inducing superconductivity in the nanowire, while the quantity δμ(E,k_y) corresponds to a tunneling-induced shift in the effective chemical potential of the nanowire.Note that both tunneling-induced terms vanish (δμ=0 and Γ=1) if the nanowire is taken to be strictly at the edge of the superconductor, x_w=0 or x_w=d. As a result, the system behaves as though there is no tunnel coupling [i.e., Eq. (<ref>) reduces to simply E^2=ξ_k^2]. This is a direct consequence of the fact that the nanowire was taken to have zero width. The tunneling term in Eq. (<ref>) relates the wave functions in the nanowire and superconductor at x=x_w, and the superconducting wave function vanishes at the boundaries. We choose to keep the approximation of a zero-width nanowire, as it is more consistent with previous related theories and is easier to treat analytically, and therefore the nanowire must be chosen to be located at some position x=x_w≠0 in order to have a non-vanishing tunnel coupling. We will show in Sec. <ref> that such an approximation is consistent with a numerical tight-binding calculation in which the wire can be placed strictly at the edge of the superconductor. Alternatively, if the nanowire is located strictly at the edge of the superconductor, it must be given a finite width so that the superconducting wave function does not vanish at the interface. Related calculations were carried out in Refs. <cit.>, where proximity-induced superconductivity was studied in a quasi-two-dimensional layer of finite width coupled to a semi-infinite three-dimensional superconductor.We also note that we can equivalently express the solvability condition in Eq. (<ref>) in the language of Green's functions. We can rewrite Eq. (<ref>) in the form(G^R_w)^-1=0, where G^R_w=[(G_0^R)^-1-Σ^R]^-1 is the retarded Green's function of the nanowire with a self-energy induced by the superconductor. From Eq. (<ref>), we can identify the retarded self-energy as (see also Appendix <ref>)Σ^R(E,k_y)=(1/Γ-1)(Δτ_x-E)-δμ τ_z.with Γ and δμ as defined in Eq. (<ref>).Before moving on, we pause to compare our result for the self-energy of a nanowire coupled to a finite-sized superconductor to the self-energy that has appeared extensively in the literature to describe proximitized nanowires beyond the weak coupling limit <cit.>. In these works, all based on the approach of integrating out the superconducting degrees of freedom, the superconductor is implicitly assumed to be infinitely large, with a nanowire coupled to the middle of the superconductor. In this geometry, one obtains the same self-energy as given in Eq. (<ref>), but with the vastly simplified effective parameters δμ=0 and Γ=(1+γ/Ω)^-1. We find that we recover this form for the self-energy by setting x_w=d/2 and taking the limit d→∞ in Eq. (<ref>) (the momentum dependence must also be neglected by setting φ=1). For maximum transparency in relating the current work to the previous ones, we show in Appendix <ref> how Eqs. (<ref>) and (<ref>) can be equivalently derived by integrating out the superconductor.§ EXCITATION SPECTRUM In this section, we analyze the excitation spectrum of our model in two simplified limiting cases. First, in Sec. <ref>, we consider the case when the nanowire is placed very close to the boundary of the superconductor, such that k_Fx_w≪1 (i.e., the distance between the nanowire and the edge of the system is much smaller than the Fermi wavelength of the superconductor λ_F). In Sec. <ref>, we consider the case when the nanowire is placed in the middle of the superconductor, x_w=d/2. Throughout this section, we assume that the width of the superconductor is much smaller than its coherence length, d≪ξ_s; equivalently, its level spacing is much larger than the gap, δ E_s≫Δ.§.§ Wire near edge of superconductor §.§.§ Analytical calculation of excitation gap We first look to analytically determine the excitation gap of the semiconductor/superconductor system when the wire is placed near the edge of the superconductor (k_Fx_w≪1). In this limit, and taking E<Δ (Δ is the upper bound on the size of the excitation gap), the effective parameters of Eq. (<ref>) can be simplified toΓ=(1+2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd))^-1,δμ=2γ(k_Fx_w)[1-(k_Fx_w)(k_Fd)].In Eq. (<ref>), we have additionally assumed that |sin(k_Fd)|≫Δ/δ E_s (recall that we are assuming Δ/δ E_s≪1). Therefore, Eq. (<ref>) breaks down when the thickness of the shell approaches k_Fd→π n (n∈ℤ^+). We have also neglected the momentum dependence of φ(k_y) by setting φ=1; this assumption is justified provided that k_y/k_F≪1/√(k_Fd).Because the effective parameters of Eq. (<ref>) are not functions of energy or momentum, it is particularly simple to solve for the spectrum,E^2=Γ^2(k_y^2/2m_w-μ_eff)^2+Δ^2(1-Γ)^2,where we define μ_eff=μ_w+δμ. [Remember, Eq. (<ref>) should be taken to describe the spectrum only for E<Δ.] If μ_eff>0, the spectrum of Eq. (<ref>) describes a superconductor with an induced gap of sizeE_g/Δ=1-(1+2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd))^-1which is opened around the effective Fermi momentum k_F,eff=√(2m_wμ_eff). We note that Eq. (<ref>) cannot be applied if the tunneling energy is made too large, such that k_F,eff/k_F≳1/√(k_Fd). In terms of energy scales, we find that Eq. (<ref>) breaks down when √(γ/δ E_s)≳√((m_s/m_w)/(k_Fx_w))≫1. If μ_eff<0, then Eq. (<ref>) describes the spectrum of an insulator with gap |μ_eff|. In this case, one must take into account the full momentum dependence φ(k_y) in order to calculate the gap and Eq. (<ref>) does not apply.In the limit when the induced gap is small E_g≪Δ, it is necessarily given byE_g/Δ=2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd)≪1.This result has several important implications. First, assuming that the thickness of the shell is not finely tuned on the scale of the Fermi wavelength of the superconductor [i.e., sin^2(k_Fd)∼1], we see that the induced gap can be small even if the tunneling energy greatly exceeds the gap of the superconductor (γ≫Δ). This result is purely a finite-size effect and is due to the suppression of the gap by a factor Δ/δ E_s≪1. Second, we see that the gap is additionally suppressed by a factor (k_Fx_w)^2≪1, which is a direct consequence of the smallness of the superconducting wave function in the vicinity of the edge. Finally, we note that it is still possible to induce a sizable gap if the thickness of the shell is fine-tuned to the limit sin^2(k_Fd)≪(γ/δ E_s)(k_Fx_w)^2≪1; in this limit, we find from Eq. (<ref>) that E_g≫Δ and our original expansion breaks down. This leads to a resonance behavior, with sharp peaks in the induced gap when the resonance condition sin(k_Fd)=0 is satisfied. The width of the resonance peaks is estimated as x_w√(γ/δ E_s)≪λ_F.Rearranging Eq. (<ref>), we can express the tunneling energy γ in terms of the experimentally observable quantities E_g and Δ (similarly to what is done in Refs. <cit.> for the case of a bulk superconductor). However, due to the presence of the quantities k_Fx_w and k_Fd, which would be impossible to determine experimentally, we can obtain only an order of magnitude estimate for γ for the case of a finite superconductor. Assuming that sin^2(k_Fd)∼1, we findγ_edge∼E_g/Δ-E_gδ E_s/(k_Fx_w)^2.From Eq. (<ref>), it is clear that the lower bound on the tunneling strength needed to induce a sizable gap in the system (such that E_g∼Δ) is given by the level spacing δ E_s.§.§.§ Numerical calculation of spectrumWhile we were able to solve for the excitation spectrum at energies E<Δ to determine the gap in certain limits [see Eq. (<ref>)], it is much more difficult to solve for the full spectrum. Because the full spectrum E(k_y) obeys a transcendental equation that cannot be solved analytically in general, we must resort to solving Eq. (<ref>) numerically.In Fig. <ref>, we plot the excitation gap E_g as a function of superconductor width d. We calculate the gap numerically by computing the spectrum and finding the minimum of the lowest subband, allowing us to treat values of k_Fd for which Eq. (<ref>) breaks down [namely, for sin(k_Fd)→0 and μ_eff<0]. Overall, we find very good agreement between the numerical solution for the gap and the analytical form given in Eq. (<ref>). For weak tunneling [Fig. <ref>(a)], the gap is in general very small with very sharp resonance peaks around k_Fd=π n. As the tunneling is increased, the resonance peak is broadened and the size of the gap is generally shifted to larger values [Fig. <ref>(b)]. When γ∼δ E_s/(k_Fx_w)^2, the gap is always of the same order as that of the superconductor, E_g∼Δ [Fig. <ref>(c)].To better understand the behavior of the gap as a function of γ, we plot the spectrum for various γ and fixed superconductor width (chosen to be off resonance) in Fig. <ref>. In the absence of tunneling [Fig. <ref>(a)], there is a large separation in energy between the band of the nanowire and the lowest subband of the superconductor (a consequence of the fact that δ E_s≫μ_w). As the tunneling strength is increased [Figs. <ref>(b) and <ref>(c)], the effective chemical potential of the nanowire μ_eff increases and the two lowest subbands move closer in energy; as a result, the nanowire can more efficiently couple to the superconductor and the proximity-induced gap increases. When γ∼δ E_s/(k_Fx_w)^2 [Fig. <ref>(d)], the tunneling is strong enough to overcome the large subband spacing of the superconductor. This creates significant overlap between the two lowest subbands of the system and a sizable excitation gap E_g∼Δ. §.§.§ Simple two-band modelIn this section, we present a simple two-band model which can be used to better understand the “weak tunneling" limit γ≪δ E_s/(k_Fx_w)^2. In this limit, we can safely assume that the nanowire couples only to the lowest subband of the superconductor. Taking into account only the lowest superconducting subband, we can write down a simple tunneling HamiltonianH=1/2∫dk_y/2π Ψ^†[ξ_k0 -t0;0 -ξ_k0t; -t0ξ_nΔ;0tΔ -ξ_n;]Ψ,where Ψ=(Ψ_w,Ψ_n)^T (Ψ_w describes states in the nanowire, while Ψ_n describes states in subband n of the superconductor) and t is a coupling between the two bands with dimensions of energy [note that this is not the same t which was introduced in Eq. (<ref>)]. Quantization of the superconducting bands is accounted for through ξ_n=k_y^2/2m_s-μ_n, with μ_n=δ E_s(k_Fd-π n) for n∈ℤ^+ [see Eq. (<ref>)].The corresponding spectrum is given by2 E^2= Δ^2 + ξ_k^2+ξ_n ^2 +2t^2 ±√((Δ^2 - ξ_k^2+ξ_n^2)^2+4t^2[Δ^2+(ξ_k+ξ_n)^2]).In general, |μ_n|≫μ_w,Δ,t and we can expand Eq. (<ref>) to giveE=±√(ξ_k^2+t^4/μ_n^2).In this case, the lower subband takes on a superconducting dispersion with a small induced gap E_g=t^2/μ_n≪Δ. The gap can only be enhanced when |μ_n|≲μ_w,Δ,t, which occurs only when a new superconducting subband becomes occupied, k_Fd≈π n. While we cannot solve analytically for the gap in this limit, we find by plotting the spectrum that the gap is approximately Δ.We plot the spectrum Eq. (<ref>) for different superconductor widths d in Fig. <ref>. Away from resonance [Fig. <ref>(a)], the lowest subband has a superconducting dispersion with very small induced gap [Eq. (<ref>)]. As the resonance is approached [Fig. <ref>(b)], the lowest superconducting subband becomes available to more strongly couple to the nanowire band and the gap is enhanced. On resonance [Fig. <ref>(c)], overlap between the two subbands is maximal and the full gap Δ is opened. As a new subband in the superconductor becomes occupied and moves away from the nanowire band [Fig. <ref>(d)], the gap is again suppressed by the large subband spacing in the superconductor. By plotting the spectrum, we indeed see that the excitation gap is sharply peaked as a function of d around k_Fd=π n, consistent with the resonance behavior discussed in Sec. <ref><ref> and shown in Fig. <ref>.§.§ Wire in middle of superconductorIf the wire is placed in the middle of the superconductor, x_w=d/2, the effective parameters of Eq. (<ref>) for energies E<Δ can be simplified toΓ=(1+γ/δ E_s1/2cos^2(k_Fd/2))^-1,δμ=γtan(k_Fd/2).Again, we neglect the momentum dependence of the effective parameters by setting φ=1 and assume |sin(k_Fd)|≫Δ/δ E_s.Since the parameters in Eq. (<ref>) are independent of energy and momentum, the spectrum is again given by Eq. (<ref>). Assuming that μ_eff>0, we find an excitation gapE_g/Δ=1-(1+γ/δ E_s1/2cos^2(k_Fd/2))^-1.While the gap can still be small for γ≫Δ, there are two important differences when comparing to the case when the nanowire is at the edge of the superconductor [Eq. (<ref>)]. First, the gap is no longer suppressed by the factor (k_Fx_w)^2≪1 which originated from the smallness of the superconducting wave function near the edge. Instead, without any fine-tuning of the superconductor width [cos^2(k_Fd/2)∼1], a sizable gap can be induced for γ≳δ E_s. Second, the periodicity of the gap as a function of k_Fd is twice as large. In the weak tunneling limit γ≪δ E_s, resonance peaks are half as frequent and occur near k_Fd=π(2n+1) (n∈ℤ^+). The width of each resonance peak is larger than in the case of the nanowire at the edge of the superconductor, but it is still much smaller than the Fermi wavelength, λ_F√(γ/δ E_s)≪λ_F.The excitation gap E_g is plotted as a function of d for several different tunneling strengths γ in Fig. <ref>. Similarly to Sec. <ref><ref>, we solve for the gap numerically by computing the full spectrum E(k_y) and finding the minimum. Again, all parameters are chosen the same as in Fig. <ref> (except for x_w=d/2). As previously discussed, the periodicity of the gap as a function of k_Fd is increased by a factor of two compared with the case when the nanowire is located at the edge of the superconductor. We see that the gap is maximized near k_Fd/π=2n+1 and is minimized near k_Fd/π=2n. This result can be inferred from the structure of the wave function corresponding to the lowest quantized subband of the superconductor in the absence of the nanowire. For k_Fd/π=2n+1, there is a superconducting subband available at low energies with which the nanowire can couple; because the wave function of this subband is extremal at x=d/2, the nanowire efficiently couples and a large gap is induced. For k_Fd/π=2n there is again a superconducting subband at low energies; however, the wave function of this subband has a node at x=d/2 and does not couple efficiently to the nanowire. In this case, a sufficiently large gap is opened only when the tunneling is very strong. We also find that as the tunneling strength is increased, extended plateaus emerge as a function of k_Fd. To better understand this behavior, in Figs. <ref>(a)–<ref>(c) we plot the spectrum for different γ choosing k_Fd/π=35.75. As the tunneling strength is increased from γ=0 [Fig. <ref>(a)], a superconducting gap is induced on the band which originates in the nanowire. However, at the same time, the nanowire band gets depleted. For some critical tunneling strength, the nanowire band becomes depleted completely and enters an insulating phase. The minimum excitation gap is then given by the insulating gap in the nanowire at k_y=0 [Fig. <ref>(b)]. As the tunneling strength is increased further and the insulating gap on the nanowire band exceeds Δ, the minimum excitation gap is determined by the lowest occupied subband of the superconductor [Fig. <ref>(c)]. This behavior can be understood as follows. The depletion of the nanowire band can be inferred from δμ given in Eq. (<ref>). When the wire is in the middle of the superconductor, δμ<0 for precisely half of a period, including for k_Fd/π=35.75. In the strong tunneling limit, |δμ|≫μ_n in general and μ_eff<0 (i.e., the nanowire becomes insulating) also for half of a period (and, as shown in Fig. <ref>, the plateau extends over half of a period in the strong tunneling limit). We also find that the lowest subband of the superconductor (corresponding to n=36) remains almost completely unaffected by tunneling. This is consistent with our previous analysis and results from the fact that this subband cannot efficiently couple to the nanowire because the corresponding wave function has a node at x=d/2. Rather, the nanowire band couples most efficiently to the second-lowest subband (corresponding to n=35). While the presence of the nanowire modifies this subband somewhat in the vicinity of k_y=0, it has no effect on the gap of this subband at finite k_y. These findings are summarized in Fig. <ref>(d), where we plot the excitation gap as a function of tunneling strength for k_Fd/π=35.75.Using our result for the gap [Eq. (<ref>)], we provide an order of magnitude estimate for the tunneling strength γ assuming that the superconductor width d is tuned outside of the plateau region and satisfies cos^2(k_Fd/2)∼1. In this case, we estimateγ_middle∼E_g/Δ-E_gδ E_s.Compared with Eq. (<ref>), we find that a sizable gap (E_g∼Δ) can be induced with a much smaller tunneling strength when the wire is placed in the middle of the superconductor.Finally, we compare three different cases by plotting the induced gap in the nanowire as a function of γ in Fig. <ref>. In Fig. <ref>(a), we show a direct comparison between the case of a nanowire located at the edge of the superconductor and the case of a nanowire located in the middle. We also contrast our result for a finite superconductor against that for a bulk superconductor by plotting the induced gap as a function of γ for the latter case in Fig. <ref>(b). For a bulk system, the induced gap obeys the equation <cit.>γ_bulk=E_g√(Δ+E_g/Δ-E_g).We see that to open a sizable gap in a finite superconductor, a tunneling strength which is several orders of magnitude larger than in the bulk case is needed. § TIGHT-BINDING MODELIn this section, we check our analytical results by comparing them with a numerical tight-binding model for proximity-induced superconductivity <cit.> in the geometry shown in Fig. <ref>. The system is again assumed infinite in the y-direction, so that the Hamiltonian takes a block-diagonal form in momentum k_y, H=∑_k_y H_k_y.The size of the superconductor in the x direction is N a_x (a_x,y are lattice constants), while the size of the nanowire is taken to be a single site. The Hamiltonian of the superconductor is given byH_s,k_y = ∑_σ∑_i=1^N{ [μ_s- 2 t_0 cos (k_y a_y)] c_k_y,i,σ^† c_k_y,i,σ-(t_0c_k_y,i+1,σ^† c_k_y,i,σ -Δ c_k_y ,i ,↑^† c_-k_y ,i ,↓^† + H.c.)},where c_k_y,i,σ destroys a state of momentum k_y and spin σ in the superconductor at site i, t_0 is the hopping amplitude, μ_s is the chemical potential (calculated from the bottom of the band), and Δ is the pairing potential. The Hamiltonian of the nanowire is given byH_w,k_y=∑_σ[μ_w-2t_wcos(k_ya_y)]b^†_k_y,σb_k_y,σ,where b_k_y,σ destroys a state of momentum k_y and spin σ in the nanowire, μ_w is the chemical potential, and t_w is the hopping amplitude. The nanowire is coupled to the superconductor at site j,H_t,k_y=-t∑_σ(c^†_k_y,j,σb_k_y,σ+H.c.),where t is the tunneling amplitude between the nanowire and superconductor. We assume that tunneling preserves both spin and momentum.We consider two separate cases. First, the nanowire is placed at the end of the superconducting chain (j=1). Whereas analytically we were unable to place the nanowire strictly at the edge of the superconductor due to its vanishing width, we can do so in the tight-binding formulation (the nanowire has a finite width of one site). Second, the nanowire is placed in the middle of the superconductor (j=N/2). The results of our tight-binding calculation are shown in Fig. <ref>. We plot the excitation gap as a function of superconductor width choosing μ_s=0.1t_0, μ_w=10^-4t_0, Δ=10^-5t_0, and t_w=50t_0 (all parameters are chosen to coincide with those used previously in the analytical calculation). For Fig. <ref>(a), which corresponds to j=1, we find very good agreement with the analytics of Sec. <ref> and Fig. <ref>. We note that the resonance peaks in the curve corresponding to weak tunneling [black curve in Fig. <ref>(a)] are narrower than a single site and therefore do not appear in the numerics. For Fig. <ref>(b), which corresponds to j=N/2, we find very good agreement with the analytics of Sec. <ref> and Fig. <ref>. Notably, a sizable gap E_g∼Δ is seen only when the tunneling strength t exceeds the chemical potential of the superconductor. § TOPOLOGICAL CRITERION To access the topological phase, we now assume that the nanowire has Rashba spin-orbit coupling and a Zeeman splitting that results from the application of an external magnetic field B parallel to the nanowire, corresponding to the typical setup for realizing topological superconductivity <cit.>. The Hamiltonian of the nanowire in this case is given by H_w=1/2∫dk_y/2π Ψ^†(k_y)ℋ_w(k_y)Ψ(k_y),where Ψ(k_y)=[ψ_↑(k_y),ψ_↓(k_y),ψ_↑^†(-k_y),ψ_↓^†(-k_y)]^T. The Hamiltonian density, which is a 4×4 matrix in Nambu ⊗ spin space, is given byℋ_w(k_y)=ξ_kτ_z-α k_yσ_z-Δ_Zτ_zσ_x,where α is the Rashba constant, Δ_Z=gμ_BB/2 is the Zeeman splitting (g is the g-factor of the nanowire and μ_B is the Bohr magneton), and σ_x,y,z are Pauli matrices acting in spin space. Following convention, we neglect the effect of the external magnetic field on the superconductor. Generalization of the solution of the BdG equation given in Eq. (<ref>) to the case where one additionally has to account for the spin degree of freedom is straightforward. The addition of spin-orbit coupling and Zeeman splitting simply modifies the retarded Green's function which enters the boundary conditions [Eqs. (<ref>)], G_0^R=(E-ℋ_w+i0^+)^-1. Solving the boundary conditions, we find that the self-energy given in Eq. (<ref>) still holds, with the simple replacement Δτ_x→-Δτ_yσ_y to account for the spin-singlet nature of the induced pairing <cit.>. Given the self-energy, we find that the excitation spectrum is determined from the implicit equationE^2/Γ^2=Δ_Z^2+Δ^2(1/Γ-1)^2+(ξ_k-δμ)^2+α^2k_y^2 ±2√(Δ_Z^2Δ^2(1/Γ-1)^2+(ξ_k-δμ)^2(Δ_Z^2+α^2k_y^2)).The critical Zeeman splitting needed to close the excitation gap at k_y=0, which we find by setting k_y=E=0 in Eq. (<ref>), is found to beΓΔ_Z^c =√(Γ^2(μ_w+δμ)^2+E_g^2),where we replace Δ(1-Γ)=E_g, noting that this replacement is strictly valid only when δ E_s≫Δ. [The explicit forms of δμ(0,0) and Γ(0,0) are given in Eq. (<ref>).] If the chemical potential shift δμ can be compensated by tuning the chemical potential of the wire (such that μ_w+δμ=0), then the critical field strength is Δ_Z^c∼ E_g/Γ (this is a similar criterion used, for example, to analyze the data of Ref. <cit.>; note that the critical Zeeman splitting needed is larger than the induced gap E_g). If, however, the chemical potential shift is made too large (|δμ|≫μ_w), the critical Zeeman splitting is determined solely by this shift, Δ_Z^c∼|δμ|, and the finite size of the superconductor pushes the topological threshold to significantly higher magnetic field strength. We will provide numerical estimates in the next section to argue that the latter case is more relevant to thin superconducting shells that have a large level spacing δ E_s≫Δ.§ RELATION TO EXPERIMENTS WITH EPITAXIAL SUPERCONDUCTING SHELLSIn this section, we argue that the theoretical model that we have considered to this point is applicable to recent experiments studying InAs or InSb nanowires strongly coupled to thin superconducting Al shells (see Fig. <ref>) <cit.>. We also provide an order of magnitude estimate for the level spacing of the shell, which determines the critical field strength needed to reach the topological phase in such a setup.First, in order to treat the nanowire as one-dimensional, we assume that the wave function is uniform across the entire cross-section of the nanowire (which spans roughly ∼100 nm). Second, because we have replaced the nanowire cross-section by a single point, we must also neglect the width of the superconducting shell, W∼100 nm (see Fig. <ref>). To justify this assumption, let us define a phenomenological tunneling strength that originates from the coupling between the nanowire and superconductor along this dimension [call it t(W)]. Such a tunneling should be given by the product of the nanowire and superconducting wave functions, integrated over the width,t(W)∼∫_0^Wdz ψ_w^*(z)ψ_s(z).Here, we denote the dimension along the width of the shell by z, imagining for simplicity that the shell cross-section is rectangular rather than kinked. The superconducting wave function is quantized as ψ_s∼sin(k_sz)/√(W), where k_s=nπ/W and 1/√(W) is a normalization factor. The wave function of the nanowire is uniform, but it also must be normalized, ψ_w∼1/√(W_w). Because the normalization factor of the nanowire wave function should scale with the width of the shell, W_w∼ W, the effective tunneling strength is given byt(W)∼1/W∫_0^Wdz sin(k_sz)∼1.Because the tunneling does not scale with W, this additional dimension is unimportant and can be neglected.We now turn our attention toward applying our results to make qualitative predictions about the experimental setup. The relevant geometrical parameter (which corresponds to d) is the thickness of the superconducting shell, d∼10 nm. For shells of this thickness, we estimate a level spacing δ E_s=ħ v_F/d∼10 meV, where we take v_F∼10^6 m/s for Al.Given the level spacing of the shell, we can also estimate the tunneling strength needed to induce a gap E_g∼Δ. Because the experimental systems universally exhibit sizable induced gaps (and therefore it is safe to assume that the shell thicknesses are not fine-tuned to a resonance point), we use Eq. (<ref>) to estimateγ_edge≳100 meV,where we take (k_Fx_w)^2∼0.1.Therefore, the chemical potential shift induced by tunneling is estimated from Eq. (<ref>) as|δμ(0,0)|∼γ(k_Fx_w)≳30 meV.Given that the characteristic energy scale of the nanowire is the spin-orbit energy (E_so=m_wα^2/2ħ^2), which takes typical values E_so∼0.1-1 meV in semiconducting nanowires <cit.>, the topological phase transition is controlled entirely by tunneling, Δ_Z^c∼|δμ(0,0)|≳30 meV. Such a large Zeeman energy corresponds to a critical field strength of B_c≳60 T for Al/InAs (g_InAs∼20) and B_c≳30 T for Al/InSb (g_InSb∼40). In either case, the magnetic field threshold needed to reach the topological phase greatly exceeds the field strength at which superconductivity in Al is destroyed (which occurs for B∼2 T).As discussed previously, in order to negate the effect of the tunneling-induced shift in the chemical potential, one must tune to μ_w=-δμ(0,0). However, this is difficult in practice because |δμ(0,0)|≫ E_so. If δμ(0,0)>0, it is not possible to properly tune the chemical potential without completely depleting the semiconducting band; if δμ(0,0)<0, the nanowire would have to be gated outside of the regime for which it is semiconducting.We note that there are several aspects of the experimental setup which are not accounted for by our simple model. For example, our model does not include an additional renormalization of the nanowire g-factor by the superconductor beyond that which is contained in Eq. (<ref>). This is because the g-factor in the superconductor was taken to be g_s=0 [and therefore no Zeeman term enters the self-energy of Eq. (<ref>)]. However, the additional renormalization should be small, as we estimate the effective g-factor of the wire as Γ g+(1-Γ)g_s≈Γ g given that g≫ g_s and Γ∼(1-Γ)∼1 <cit.>. We also do not account for any orbital effects in the nanowire, as the nanowire was taken to have zero width. In the experimental setup, however, the nanowire has a diameter W_w∼100 nm; compared with a typical cyclotron radius for electrons in the wire r_c=m_wv_Fw/eB∼10 nm (evaluated for a field strength B∼1 T and Fermi velocity v_Fw∼10^5 m/s), orbital effects can be non-negligible <cit.>. Additionally, we do not consider the superconductor to be disordered. It has been shown that disorder in a bulk superconductor can be detrimental to the proximity effect <cit.>; however, this does not seem to be an issue experimentally, as sizable hard proximity-induced gaps are universally observed <cit.> along with ballistic transport in the absence of a magnetic field <cit.>. Relatedly, given the fact that applied magnetic fields are small, we do not determine Δ(x) self-consistently as done, for example, for Shiba states, where the exchange interaction (effective Zeeman field) is comparable to the Fermi energy <cit.>. Finally, we do not account for the fact that there may be multiple subbands in the nanowire which contribute to transport. While this possibility seems to be excluded by the fact that typical nanowires exhibit a quantized conductance of 2e^2/h in the ballistic transport limit over a wide range of gate voltages <cit.>, it could be that with the introduction of the large tunneling energy scale, higher subbands can become important. The intersubband spacing in an InSb nanowire was estimated in the absence of a magnetic field and a superconducting shell to be ∼20 meV in Ref. <cit.>, so we cannot rule out the possibility that a |δμ(0,0)|∼30 meV simply places the chemical potential into a higher subband. However, this possibility requires a more in-depth theoretical treatment that cannot be captured by a single-band model. § CONCLUSIONS We have studied the proximity effect in a semiconducting nanowire strongly coupled to a thin superconducting shell with thickness d. We have shown that finite-size effects become detrimental to the induced superconductivity when the level spacing of the shell δ E_s∼ħ v_F/d exceeds its gap, δ E_s≫Δ. In this limit, a large tunneling strength γ≳δ E_s is needed to overcome the level spacing to induce a gap in the nanowire (we estimate γ≳100 meV for typical experimental setups with d∼10 nm). In turn, this large tunneling energy induces a very large shift in the effective chemical potential of the nanowire (δμ≳30 meV) which would be difficult to compensate by gating.In order to overcome the detrimental finite-size effects, the thickness of the superconducting shell should be made larger than its coherence length, d≫ξ_s. In this limit, the level spacing and chemical potential shift become negligible (δ E_s,δμ≪Δ), and a sizable proximity gap can be induced with a much smaller tunneling energy (γ∼Δ). However, this requirement may prove problematic when using Al, which has a very long coherence length ξ_s∼1 μm, to induce superconductivity. It would therefore be beneficial to choose a superconductor with shorter coherence length (for example, Nb has ξ_s∼10 nm).This work was supported by the Swiss National Science Foundation and the NCCR QSIT. § SOLVABILITY CONDITION AND SELF-ENERGY In this appendix, we show how to arrive at the solvability condition given in Eq. (<ref>) of the main text. The boundary conditions that must be imposed on the wave function of Eq. (<ref>) are given in Eq. (<ref>). These boundary conditions can be significantly simplified by assuming that μ_s≫Δ,E. In this limit, we can make a semiclassical approximation k_±=(k_F^2-k_y^2±2m_siΩ)^1/2≈ k_Fφ± iΩ/(v_Fφ), where φ=(1-k_y^2/k_F^2)^1/2 (0<φ≤1). The boundary conditions in Eq. (<ref>) can be written in matrix form as Mc=0, where c=(c_1,c_2,c_3,c_4)^T and M =([ u_0ϕcos(k_+x_w) v_0ϕcos(k_-x_w); v_0ϕcos(k_+x_w) u_0ϕcos(k_-x_w); -u_0sin(k_+x_w) -v_0sin(k_-x_w); -v_0sin(k_+x_w) -u_0sin(k_-x_w) ]. ⋯⋯.[ u_0(φcos[k_+(d-x_w)]+2γ/E-ξ_ksin[k_+(d-x_w)]) v_0(φcos[k_-(d-x_w)]+2γ/E-ξ_ksin[k_-(d-x_w)]); v_0(φcos[k_+(d-x_w)]-2γ/E+ξ_ksin[k_+(d-x_w)])u_0(φcos[k_-(d-x_w]-2γ/E+ξ_ksin[k_-(d-x_w)]);u_0sin[k_+(d-x_w)]v_0sin[k_-(d-x_w)];v_0sin[k_+(d-x_w)]u_0sin[k_-(d-x_w)] ]) In Eq. (<ref>), we have approximated k_±=k_Fφ outside of the trigonometric functions while keeping k_±=k_Fφ± iΩ/(v_Fφ) inside. Taking the determinant of the matrix in Eq. (<ref>), we find a solvability condition given by0 =Ωφ^2sin(k_+d)sin(k_-d){E^2-ξ_k^2-γ^2/φ^2β_+β_- +γ/φ[ξ_k(β_++β_-)+E^2/iΩ(β_+-β_-)]},where we define β_±={cos[k_±(d-2x_w)]-cos(k_± d)}/sin(k_± d). Dividing Eq. (<ref>) through by the common factor Ωφ^2sin(k_+d)sin(k_-d) and rearranging, we obtain0 =E^2(1+γ/Ωφ Im(β_+))^2-Δ^2γ^2/Ω^2φ^2[Im(β_+)]^2 -(ξ_k-γ/φ Re(β_+))^2Defining the quantities Γ={1+γ Im(β_+)/(Ωφ)}^-1 and δμ=γ Re(β_+)/φ, we arrive at the solvability condition presented in Eq. (<ref>). Substituting k_±=k_Fφ± iΩ/(v_Fφ) into the expressions for Γ and δμ, we arrive at the definitions presented in Eq. (<ref>).We also note that the solvability condition of Eq. (<ref>) can be expressed as [G_w^R(E,k_y)]^-1=0, where(G_w^R)^-1=([ E/Γ-ξ_k+δμ-Δ(1/Γ-1);-Δ(1/Γ-1) E/Γ+ξ_k-δμ ])Noting that the bare retarded Green's function of the nanowire is given by (G_0^R)^-1=E-ξ_kτ_z+i0^+, the full Green's function can be written as (G_w^R)^-1=(G_0^R)^-1-Σ^R with self-energyΣ^R=([ E(1-1/Γ)-δμΔ(1/Γ-1);Δ(1/Γ-1) E(1-1/Γ)+δμ ]),as given in Eq. (<ref>). § INTEGRATING OUT SUPERCONDUCTOR In this section, we show how to alternatively derive Eqs. (<ref>) and (<ref>) by integrating out the superconductor. For completeness, we first go through the steps of performing the integration. We start with the same model as considered in the main text, expressed in terms of the Euclidean action. The action of the nanowire is given byS_NW=1/2∫dω/2π∫dk_y/2π Ψ_w^†(G_w^0)^-1Ψ_w,where (G_w^0)^-1=iω-ξ_kτ_z is the inverse Matsubara Green's function of the nanowire in the absence of tunneling (ω is a Matsubara frequency) and Ψ_w=[ψ_↑(k_y,ω),ψ_↓^†(-k_y,-ω)]^T is a Heisenberg spinor field describing states in the nanowire. The action of the superconductor is given byS_BCS=1/2∫dω/2π∫dk_y/2π∫_0^ddx Ψ_s^†(iω-ℋ_BCS)Ψ_s,where ℋ_BCS is as defined below Eq. (<ref>) and Ψ_s=[η_↑(x,k_y,ω),η^†_↓(x,-k_y,-ω)] is a spinor field describing states in the superconductor. The tunneling action, which couples the nanowire to the superconductor at x=x_w, is taken to beS_t=-t/2∫dω/2π∫dk_y/2π∫_0^ddx[Ψ_w^†τ_zΨ_sδ(x-x_w)+H.c.].The path integral representation of the partition function is then given by𝒵=∫ D[Ψ_w^†,Ψ_w]∫ D[Ψ_s^†,Ψ_s]e^-S_w-S_BCS-S_t.In the exponential, we rewrite S_BCS+S_t =1/2∫dω/2π∫dk_y/2π{∫_0^ddx[Ψ_s^†-tΨ_w^†τ_zG_s^0(x_w,x)](iω-ℋ_BCS)[Ψ_s-tG_s^0(x,x_w)τ_zΨ_w]-t^2Ψ_w^†τ_z G_s^0(x_w,x_w)τ_zΨ_w}. In Eq. (<ref>), we introduce a function G_s^0(x,x') that must satisfy (iω-ℋ_BCS)G_s^0(x,x')=G_s^0(x',x)(iω-ℋ_BCS)=δ(x-x'); i.e. G_s^0(x,x') corresponds to the Green's function of the superconductor in the absence of tunneling. Evaluating the Gaussian path integral over superconducting fermions, we obtain an effective action describing the nanowire given byS_eff=1/2∫dω/2π∫dk_y/2πΨ^†_w[(G_w^0)^-1-Σ]Ψ_w,with the self-energy given byΣ=t^2τ_zG_s^0(x_w,x_w)τ_z. To explicitly evaluate the self-energy, we must choose the appropriate bare Green's function G_s^0(x,x') for the geometry under consideration. For our purposes, we evaluate the self-energy using the Green's function of a finite-sized superconductor satisfying vanishing boundary conditions at x=0 and x=d. The bare Green's function must satisfy the equation[iω+(∂_x^2/2m_s-k_y^2/2m_s+μ_s)τ_z-Δτ_x]G_s^0(x,x')=δ(x-x').The solution to Eq. (<ref>) can be written as the sum of a homogeneous solution G_h(x,x') and a particular solution G_p(x-x'),G_s^0(x,x')=G_h(x,x')+G_p(x-x'),with the particular solution corresponding to the bulk superconducting Green's function. We determine the bulk Green's function in real space by Fourier transformation. Defining ξ_ks=(k_x^2+k_y^2)/2m_s-μ_s, we haveG_p(x-x') =-∫dk_x/2πiω+ξ_ksτ_z+Δτ_x/Δ^2+ξ_ks^2+ω^2e^ik_x(x-x')=-1/v_FΩφ[(iω+Δτ_x)cos(ζ|x-x'|) -Ωτ_zsin(ζ|x-x'|)]e^-χ|x-x'|,where, as we have done throughout, we replace k_±=ζ=k_Fφ outside of the exponentials while keeping k_±=ζ± iχ=k_Fφ± iΩ/(v_Fφ) inside of the exponentials (in Matsubara frequency space, Ω=√(Δ^2+ω^2)). The homogeneous solution is given byG_h(x,x') =(iω+Δτ_x+iΩτ_z)[c_1e^ik_+x+c_2e^-ik_+x] +(iω+Δτ_x-iΩτ_z)[c_3e^ik_-x+c_4e^-ik_-x].The unknown coefficients, which are functions of the coordinate x', are determined by imposing the boundary conditions G_s^0(0,x')=G_s^0(d,x')=0. Solving the boundary conditions givesc_1(x') =sin[k_+(d-x')]/2v_FΩφsin(k_+d), c_2(x') =1/2v_FΩφ[i+(k_+d)]sin(k_+x'), c_3(x') =1/2v_FΩφ[-i+(k_-d)]sin(k_-x'), c_4(x') =sin[k_-(d-x')]/2v_FΩφsin(k_-d).Substituting Eqs. (<ref>)-(<ref>) into Eq. (<ref>) and setting x=x'=x_w, we find the bare Green's function G_s^0(x_w,x_w) =-1/v_FΩφ[cosh(2χ d)-cos(2ζ d)]{(iω+Δτ_x){sinh(2χ d)-cos(2kζ x_w)sinh[2χ(d-x_w)] -cos[2ζ(d-x_w)]sinh(2χ x_w)}-Ωτ_z{sin(2ζ d)-sin(2ζ x_w)cosh[2χ(d-x_w)]-sin[2ζ(d-x_w)]cosh(2χ x_w)}}. Substituting Eq. (<ref>) into Eq. (<ref>) and defining Γ and δμ as in Eq. (<ref>), we obtain a self-energy given by (recall γ=t^2/v_F)Σ=(Δτ_x-iω)(1/Γ-1)-δμ τ_z.After analytic continuation, we reproduce the retarded self-energy given in Eq. (<ref>).We note that choosing the bare Green's function to be translationally invariant and equal to the bulk superconducting Green's function, as is typically done to describe proximitized nanowires, corresponds to the geometry of a nanowire coupled to the middle of an infinitely large superconductor. In this case, the bare Green's function is simply G_s^0(x_w,x_w)=G_p(0), yielding a self-energyΣ_bulk=-γ(iω-Δτ_x)/φ√(Δ^2+ω^2). | http://arxiv.org/abs/1707.08417v2 | {
"authors": [
"Christopher Reeg",
"Daniel Loss",
"Jelena Klinovaja"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726130341",
"title": "Finite-size effects in a nanowire strongly coupled to a thin superconducting shell"
} |
A search for FRB 121102-like persistent radio-luminous sources – Candidates and implications for the FRB rate and searches Eran O. Ofek1December 30, 2023 ==========================================================================================================================Remote attestation (RA) is a popular means of detecting malware in embedded and IoT devices.RA is usually realized as an interactive protocol, whereby a trusted party – verifier –measures integrity of a potentially compromised remote device – prover.Early work focused on purely software-based and fully hardware-based techniques,neither of which is ideal for low-end devices. More recent results have yielded hybrid(SW/HW) security architectures comprised of a minimal set of features to supportefficient and secure RA on low-end devices.All prior RA techniques require on-demand operation, i.e, RA is performed in real time.We identify some drawbacks of this general approach in the context of unattended devices:First, it fails to detect mobile malware that enters and leaves the prover between successiveRA instances. Second, it requires the prover to engage in a potentially expensive (in terms of timeand energy) computation, which can be harmful for critical or real-time devices.To address these drawbacks, we introduce the concept of self-measurement where a prover deviceperiodically (and securely) measures and records its own software state, based on a pre-established schedule.A possibly untrusted verifier occasionally collects and verifies these measurements.We present the design of aconcrete technique called : Efficient Remote Attestation via Self-Measurementfor Unattended Settings,justify its features and evaluate its performance. In the process, we also define a new metric – Quality ofAttestation (). We argue that is well-suited for time-sensitive and/or safety-critical applications thatare not served well by on-demand RA. Finally, we show that is a promising stepping stone towardshandling attestation of multiple devices (i.e., a group or swarm) with high mobility.§ INTRODUCTIONIn recent years, embedded and cyber-physical systems (CPS), under the guise of Internet-of-Things (IoT), have entered many aspects of daily life, such as: homes, office buildings, public venues, factories and vehicles.This trend of adding computerized components to previously analog devices and then inter-connecting thembrings many obvious benefits. However, it also greatly expands the so-called “attack surface" and turnsthese newly computerized gadgets into natural and attractive attack targets.In particular, as recent incidentsdemonstrated, IoT devices can be infected with malware and used as bot-controlled zombies in Distributed Denial-of-Service (DDoS) attacks. Also, IoT-borne malware can snoop on device owners (by sensing) or maliciously control critical services (by actuation), as happened with Stuxnet <cit.>.One key component in securing IoT devices is malware detection, which is typically attained with Remote Attestation (RA). RA is a distinct security service that allows a trusted party,called verifier, to securely verify the internal state (including memory and storage) of a remote untrusted and potentially malware-infected device, called prover. RA is realized via an interactive protocol between prover and verifier. A typical example is described in <cit.>:(1) verifier sends an attestation request to prover, (2) prover verifies the request[Since attestation is a potentially expensive task, this verificationmitigates computational DoS attacks.] and (3) computes a cryptographic function of its internal state,then (4) sends the result to verifier, and finally, (5) verifier checks the result and decides whether prover is infected.This general approach is referred to as on-demand attestation and all current RA techniques adhere to it. In this paper, we identify two important limitations of this approach. First, it is a poor match for unattended devices, since malware that “comes and goes” (i.e., mobile malware <cit.>) can not be detected if it leaves prover by the time attestation is performed. Second, for a device working under time constraints (real-time operation) or otherwise providing critical services, on-demand attestation requires performing a possiblytime-consuming task while deviating from the device's main function(s). To address these issues, we design : Efficient Remote Attestation via Self-Measurementfor Unattended Settings. is based on self-measurements. Basically, a device (prover)measures and records its state at scheduled times. Measurements are stored in prover's insecure memory.Verifier occasionally collects and validates these measurements in order to establish the history of prover's state.Notably, with this general approach,verifier imposes only negligible real-time burden on prover. It also offers strictly better quality-of-service than prior attestation techniques, because verifier obtains prover's entire history of measurements, since the last verifier request. In other words, de-couples(1) frequency of prover checking, from (2) frequency of prover measurements, which are equivalent inon-demand attestation. Finally, simplifies RA design (in terms of required features) for prover: authentication of verifier requests is no longer needed, since computational DoS attacks do not arise.[Thisis unlike requirements in <cit.> that stipulate (potentially expensive) prover auhentication of verifier's requests.]We also introduce the new notion of Quality of Attestation () which captures: (1) how a device (prover) is attested,(2) how often its state is measured, and (3) how often these measurements are verified. It is the temporalanalogue of the concept of quality of swarm attestation (QoSA) introduced in <cit.>in the context of attesting groups of devices.NOTE: is not intended as a replacement for on-demand attestation, mainly because for some devices and some settings, real-time on-demand attestation is mandatory, e.g., immediately before or after a software update or for secure erasure/reset.Also, on-demand attestation may be more flexible, e.g., if the verifier is only interested in measuring a fraction of prover's memory. These two approaches are not mutually exclusive andmay be used together to increase , specifically, in terms of freshness of the latest measurement.The last incentive for our self-measurement approach is its suitability for highly mobile groups of devices.RA protocols developed for “swarm attestation”, e.g., <cit.>, are designed to efficiently attestgroups of interconnected devices on-demand, with a single verifier-prover interaction. However, they do not work in highly mobile swarms, since on-demand attestation requires topologyto remain essentially static during the entire attestation protocol instance – the time for which is dominated by computation on all swarm devices. Since involves virtually no real time computation for prover, it is much more suitable for high-mobility swarm settings.After overviewing the state-of-the-art in Section <ref>, we introduce and in Section <ref>.An implementation and experimental results are discussed in Sect <ref>. Issues arising in time-sensitiveapplications and partial mitigation measures are discussed in Section <ref>. Applicability of to swarm attestation is considered in Section <ref>.§ REMOTE ATTESTATION (RA)RA aims to detect malware presence by verifying integrity of a remote and untrustedembedded (or IoT) device. As mentioned earlier, it is typically realized as a protocol where trusted verifierinteracts with a remote prover to obtain an integrity measurement of the latter's state.RA techniques fall into the three main categories. (1) Hardware-basedattestation <cit.> uses dedicated hardware features such as aTrusted Platform Module (TPM) to execute attestation code in a secure environment.Even though such features are currently available in personal computers and smartphones,they are considered a relative “luxury” for very low-end embedded devices.(2) Software-based attestation <cit.>requires no hardware support and performs attestation solely based on precise timing measures.However, it limits prover to being one-hop away from verifier, so that round-trip time is either negligible or fixed.It also relies on strong assumptions about attacker behavior <cit.> and is typically only usedfor legacy devices where no other RA techniques are viable. (3) Finally, hybrid attestation <cit.>, based on a software/hardwareco-design, provides RA while minimizing its impact on underlying hardware features.<cit.> is the first hybrid RA design with minimal hardware modifications to existing microcontroller units (MCUs). It has the following key features: * Attestation code is immutable: located in and executed from ROM. * Attestation code is safe: its execution always terminates and leaks no information other than the attestation result (token). * Attestation is atomic: (1) it is uninterruptible, and (2) it starts from the first instruction and exits at the last instruction. This is realized in by using hard-wired MCU access controls and disabling interrupts upon entering attestation code. * A secret key () is stored in a secure memory location where it can be accessed only from within the attestation code: is stored in ROM and is guarded by specialized MCU rules. <cit.> extended to defend against denial-of-service (DoS) attacks that try to impersonate verifier. We refers to this extended design as . <cit.> additionally requires prover to have a Reliable Read-Only Clock (RROC), which is needed to perform verifier authentication and prevent replay, reorder and delay attacks.To ensure reliability, RROC must not be modifiable by software. Upon receiving a verifier request, ROM-resident attestation code checks the request's freshness usingRROC, authenticates it, and only then proceeds to perform attestation.The TrustLite <cit.> security architecture also supports RA for low-end devices. It differsfrom in two ways: (1) interrupts are allowed and handled securely by the CPU ExceptionEngine, and (2) access control rules can be programmed using an Execution-Aware MemoryProtection Unit (EA-MPU). TyTAN <cit.> adopts a similar approach while providingadditional real-time guarantees and dynamic configuration for safety- and security-critical applications.<cit.> is a hybrid RA design for medium-end devices devices with a Memory Management Unit(MMU). It builds upon a formally verified micro-kernel, <cit.>, to ensure memory isolation and enforce access control to memory regions.Using these formally and mathematically proven features,access control rules can be implemented in software and enforced by .Consequently, stores and attestation code in writable memory regions (e.g., flash or RAM)and configures the system such that no other process, besides the attestation process, can access those memory regions. Access control configuration in also involves the attestation process having exclusive access to its thread control blockas well as to memory regions used for -related computations.The latter ensures the protection property. To ensure atomic execution, runs the attestationprocess as the initial user-space process with the highest scheduling priority, while the rest ofuser-land processes are spawned by the attestation process, with lower priorities. Finally, hardware-enforced secure boot is used to provide integrity of and the attestation process atsystem initialization time.In this paper, we use and as the base security architecture for .However,should be equaly applicable to other on-demand RA techniques, such as TrustLite<cit.> or TyTan<cit.>.§ SELF-MEASUREMENTS As discussed in Section <ref>, all current RA techniques perform on-demand attestation, whereby prover computes verifier-requested measurements in real-time. This can be a time-consuming activity that takes prover away from its primary mission. However, prover performs no RA-related computation between verifier's requests.In contrast, divides RA into two phases. In the measurement phase, proverperforms self-measurements based on a pre-established schedule and stores the results. In the collection phase, verifier (whenever it chooses to do so) contacts prover to fetch these measurements. The collection phase is very fast since it requires practically no computation by prover. In particular, since measurements are based on a MAC computed with a key shared between prover and verifier, no extra protection is needed when prover sends these measurements to verifier. Furthermore, unlike in on-demand RA, there is nothreat of computational DoS on prover. Thus, there is no need to authenticate verifier's requests,in contrast with on-demand attestation. A prover's measurement M_t computed at time t is defined as: M_t = <t, H(mem_t), MAC_K(t, H(mem_t))> where H is a suitable cryptographic hash function and mem_t represents prover's memory at time t.The computation of H(mem_t) and MAC is done in the context of the security architecture, e.g., or .From here on, and are used to denote verifier and prover, respectively.Although assumes a symmetric key K shared between and , a public key signature scheme could be used instead, with no real impact on securityof the scheme except for higher cost of measurements. §.§ Quality of AttestationQuality of Attestation () is primarily determined by two parameters: (1) time T_M between two successivemeasurements on , and (2) time T_C between two successive requests by to collect measurements from . We assume that in most cases T_C>T_M. If it so happens that T_C≤ T_M, verifier will simply collect the same measurements more than once, which is redundant. Instead,can explicitly request to perform a measurement before the collection. In that case, 's request would have to be authenticated and checked for freshness (as in <cit.>) before the on-demand measurement is computed. These activities clearly incur additionalreal-time overhead and delays.We refer to thisvariant as .Exactly how T_C and T_M are determined clearly depends on specifics of 's missionand its deployment setting. Security impact of these parameters is intuitive.Smaller T_M implies smaller window of opportunity for mobile malware to escape detection.Smaller T_C implies faster malware detection. If either value is large, attestation becomes ineffective.Meanwhile, though low values increase , they also increase 's overall burden, in terms ofcomputation, power consumption and communication.Without loss of generality, we assume that measurements and collections occur at regular intervals.Of course, in practice this might not work in scenarios that involve critical or time-sensitive applications (see Section <ref>). In fact, it might be advantageous to take measurements at irregular intervals, as doing so might give prover a bit of an extra edge against mobile malware (see Section <ref>). Another parameter is the number of measurements (referred to as k) obtained by in each collection phase. It can range between one (only the most recent measurement) and all. In a typical setting, 's history size should be set such that each measurement is collected exactly once. That is, k = ⌈ T_C / T_M ⌉. Finally, the collection phase involves the notion of freshness, i.e., how recent is 's latest measurement.Depending on the application, maximal freshness might be required, e.g., right before or after a software update. Maximal freshness is attainable via on-demand attestation In , freshness of a measurement (denoted as f) ranges between T_M and 0, which correspond to minimal and maximal freshness, respectively. On average, we expect f=T_M/2.Figure <ref> shows an example with two malware infections. In the first, malware covers its tracks andleaves before any measurement takes place. In the second, malware persists on . Although measurementoccurs perhaps soon after infection, corrective action can be taken only after collection, thus illustrating the importance of a small T_C. Measurements and collections are shown as punctual events in Figure <ref>.Although they do take some time to complete (measurements, in particular), it is considered negligible even for low-end devices (see Section <ref>). §.§ Measurements Storage & Collection A naïve way for to store measurements is to keep track of them indefinitely.However, this will eventually consume a lot of 's storage. To this end, uses rolling measurements. A fixed section of 'sinsecure storage is allocated as a windowed (circular) buffer for n measurements. The i-th measurementis stored at location L_i n. However, it is expected thatcollects measurementssufficiently often, such that no measurement is over-written. That is, the time between successive collections should be at most T_C ≤ n · T_M.The interaction between and is very simple: asks for k latest measurements, which simply reads from the buffer and transmits. The collection phase does not involve any change of stateon and sent measurements are not encrypted. (Though recall that they are authenticated, since each measurement is computed using K). It also does not trigger any significant computation on , i.e., in contrast with on-demand attestation, no cryptographic operations are required in the collection phase. However, this is not the case in the variant mentioned in Section <ref>, where (1) 's request must be authenticated and checked for freshness, and (2) a current measurement must be computed. Self-measurements can be stored in 's unprotected storage. This allows malware (that is possibly present on ) to tamper with measurements, by modifying, re-ordering and/or deleting them. However, since malware (by design of ) cannot access K, it cannot forge measurements.Thus, it is easy to see that any tampering will be detected by at the next collection phase and malware presencewould be immediately be noticed. For the same reasons, code that handles request parsing as well as storage and transmission of measurementsdoes not need to be executed in a secure environment or stored in ROM. Code that performs self-measurement, however,must be protected by the underlying security architecture, as in on-demand attestation.Scheduling in can be implemented in a very simple and stateless manner. Let t be the value of RROC at the time of measurement M_t, and let T_M be the time between two successive measurements, as configured in . The windowed buffer slot L_i, used to store M_t, is determined by: i = ⌊ t / T_M ⌋ n.collection protocol is shown in Figure <ref>. No operation involves the underlying architecture duringcollection; only during measurements. Notation ^*L_j refers to contents of location L_j. A sample memory layout is shownin Figure <ref>. §.§ : with On-demand Attestation As mentioned in Section <ref>, may be combined with on-demand attestation to benefit fromadvantages of both approaches. This variant, , records 's state history to detectmobile malware, and uses on-demand attestation to obtain better freshness. Freshness is particularlyrelevant whenever real-time attestation is mandatory, e.g., immediately before or after a software update.The measurement phase is not modified, while the collection phase is combined with on-demandattestation request as follows. First, as part of each attestation request now computes and includes anauthentication token and specifies k. As in <cit.>, authentication of protects against computational DoS.Then, only after checking that a request is valid, computes a measurement. Finally, this real-time measurementis sent to , along with k previous measurements. This protocol is shown in Figure <ref>.This anti-DoS protection incurs an additional cost for which may interfere with its normal function. A major advantage of over and regular on-demand attestation is that no such protection is required. §.§ Security Considerations Security of the measurement process itself is based on the underlying security architecture,e.g.,or , which: (1) provides measurements code with exclusive access to ,(2) ensures non-malleability and non-interruptibility of the measurement code, and (3) performsmemory-cleanup after execution.The timestamps used in the measurement process must be based on the RROC which(by definition) can not be modified by non-physical means. This is important sincemalware should notinfluence when measurements are taken. If RROC value could be modified, the following attack scenario would become possible: malware enters at timet_0 and remains active long enough so that a measurement at time t_0 + δ (with δ < T_M) is taken.Before leaving, malware discards that measurement and resets the counter to t_0.Soon after δ (so that a measurement, valid this time, has been taken for t_0 + δ), malware returns and resetsthe counter to time elapsed since t_0. Though this example works for one T_M window, it can be extended toarbitrarily many. It requires an additional assumption that no collection took place during the presence of malware.Fortunately, RROC is already a requirement of the underlying security architecture, for a totally different reason.In , RROC helps prevent replay and computational DoS attacks on . Thus, does not require anychanges to the underlying security architecture.As mentioned earlier, measurements need not be stored in protected memory becausetampering with them is detectable and indicates malware presence on . Likewise, the code to support the collection phase does not require any protectionsince measurements are not secret (they are unique for every device and every timestamp value),and their absence or alteration is self-incriminating. §.§ Irregular Intervals A natural extension tois to use irregular measurements intervals instead of a fixed T_M. The motivation is that mobile malware that is aware of fixed scheduling knows when to enter/leave the device in order to stay undetected.One way to implement irregular intervals is to use a Cryptographically Secure Pseudo Random Number Generator (CSPRNG) iniatialized (seeded) with the secret key K. Output of the CSPRNG can be truncated such that T_M is upper- and/or lower-bounded.For example, after computing M_t_i, can set the measurement timer to:T_M^next = map(CSPRNG_k(t_i)),where map is a function that maps CSPRNG output to seconds, e.g. map : x ↦ x (U-L) + L, with U and L upper and lower bounds, respectively.The timer itself must be read-protected to ensure that T_M^next is unknown to malware potentially present on . CSPRNG code must be protected in the same way as the measurement collection.§ IMPLEMENTATION We implemented on two security architectures: and .The main difference between them is that the former targets low-end devices, and the latter – medium-end devices with a memory management unit (MMU). §.§ Implementation on Figure <ref> shows the implementation of atop thearchitecture. As in ,measurement code and reside in ROM. However, the code is invoked periodically and autonomously, whenevera scheduled timer interrupt occurs.We now examine ROM size, hardware costs and run-time on architecture. ROM Size greatly depends on the choice of MAC algorithms. We implement ROM-resident code in "C" using three MAC functions: HMAC-SHA1 <cit.>[Notethat HMAC-SHA1 is used for comparison purposes only. We exclude it in our actual implementations due to arecent collision attack in SHA1 <cit.>.], HMAC-SHA256 <cit.> and keyedBLAKE2S <cit.>. We then use open-source MSP430-gcc compiler <cit.> to compile the"C" code into an MSP430 executable. Table <ref> shows the ROM sizefor each -based approach. As expected, requires slightly lessROM than on-demand attestation. Hardware Cost:We implement the hardware part of by modifying the MSP430 architecture, using open-source OpenMSP430 core <cit.>. We modify the memory backbone module in the OpenMSP430 core to support atomicexecution of ROM code and exclusive access to . RROC is realized as a peripheralusing a 64-bit register incremented for every clock cycle.To ensure write-protection, a write-enable wire is removed in the RROC module. For timer components, we use the unmodified version of omsp_timerA module provided by OpenMSP430.Note that hardware timers are not considered to represent additional hardware cost. This is because they are common and crucial components of embedded systems. Indeed, it is unusual to find an embedded device not equipped with at least one timer. Finally, we use Xilinx ISE 14.7 <cit.> to synthesize our modifications of the MSP430 core froma hardware description language to a combination of registers and look-up tables that serve asbuilding blocks in FPGA. As expected, our synthesized results show that utilizes the same amount of registers and look-up tables as the on-demand attestation. Compared to the unmodified MSP430 core, requires roughly 13% (655 vs. 579)and 14% (1,969 vs. 1,731)additional registers and look-up tables respectively.Measurement Run-Time: Figure <ref> illustrates run-time of the measurement phase for various memory sizes.Not surprisingly, it is linearly dependent on memory size and roughly equivalent to thatof on-demand attestation.§.§ Implementation onFigure <ref> illustrates implementations of -based and on-demand attestation.We implement these two techniques on an I.MX6 Sabre Lite <cit.> development board. RROC is implemented based on the software clock approach, suggested by Brasser et al. <cit.>. Specifically, we use a short-term counter from Sabre Lite's General Purpose Timer (GPT) and ourclock code in to construct RROC. When the counter wraps around and causes an interrupt,our clock code handles it by updating higher-order bits of the clock in . Then, the clock value isconstructed by combining these bits with the GPT counter.To ensure read-only property, is given exclusive write-access to RROC components.Also, we utilize Sabre Lite's Enhanced Periodic Interrupt Timer (EPIT) to schedule execution of measurement codeWe base the code of on open-source libraries <cit.>: seL4utils, seL4vka, seL4vspace, and seL4bench. The first three provide abstractions of: process, memory managementand virtual space, respectively, while the last one is used to evaluate performance.Finally, we use <cit.> to implement th network stack: an Ethernet driver and timer drivers in .Executable Size: Table <ref> compares executable sizes of in on-demand attestation and . Results show that is only about 1% higher in terms of the executable size. This overhead mostly comes from the need for an additional timer driver.Measurement Run-time: Measurement run-time of -based in Figure <ref> follows the same trendas -based : (1) it is linear as a function of memory sizes, and (2) it is roughly equal to that of on-demand attestation.Collection Run-time: Table <ref> shows the run-time breakdown of the collection phase for each variant. Clearly, in ,run-time of the collection phase is negligible (by at least a factor of 3,000),compared to that of the measurement phase. Collection run-time in ERASMUS+OD, on the other hand, is dominated by run-time of performing on-demand attestation.§ AVAILABILITY IN TIME-SENSITIVE APPLICATIONS In some cases, it might be undesirable to interrupt execution of the 's application process in order to obtain a measurement. This is particularly the case for time-sensitive or safety-criticalapplications. As discussed in Section <ref>, measurements can take non-negligible time, e.g.,7 seconds on an 8-MHz device with 10KB RAM. Making unavailable for that long isnot appropriate.As is, pure on-demand attestation is poorly suited for such applications. At the same time, if follows a strict schedule,is also not a remedy since it suffers from the same issue. However, it can be made more flexible.One partial measure is for to be self-aware of when time-sensitive tasks occur. That way, it can schedulemeasurements at appropriate times. If this knowledge is also available to , on-demand attestation could beused if adapts to 's schedule. Another approach is to allow to abort the measurement in progress, if the need arises. However, this has some caveats: First, the security architecture needs to be adapted to allow interrupts during measurements. Protection of keys(and cleanup in case of an interrupt) is still required; thus, there is still a need for some hardware support. Second,it would be trivial for malware to abort computation of measurements in order to avoid detection, or simply pretend, when queried by , that all attempted measurements have been aborted. Therefore, must use some externalinformation or policy to decide whether there is a valid justification for each aborted measurement.To handle such situations, we consider another variant that involves lenient scheduling.Instead of performing a measurement every T_M, has a window of w× T_M where w≥ 1.Under normal conditions, behaves as usual, using the T_M window. If something causes a measurementto be aborted, it can be rescheduled to the end of the current window.These are certainly not ideal measures, the underlying problem seems quite difficult to address deterministically. As is typical for security/usability compromises, real deployment would likely involve policy-based decisions.§ SWARM ATTESTATION Some applications require attesting a group (or swarm) of interconnected embedded devices. In such a setting,it is beneficial to take advantage of interconnectivity and perform collective attestation using a dedicated protocol.Several swarm attestation techniques have been proposed. SEDA <cit.> is the first such scheme, which relies on hybrid attestation security architectures: <cit.> and TrustLite <cit.>. SEDAcombines them with a request-flooding and response-gathering protocol. SEDA was improved and furtherspecified in LISA <cit.>. Other related techniques deal with report aggregation <cit.> or physical attacks <cit.>.A concept of Quality of Swarm Attestation (QoSA) was introduced in <cit.> to capturethe level of informationthatobtains as a result of swarm attestation. This can range from binary (“is the whole swarm healthy?”) to full(state of each individual device and topology information). , as introduced in this paper, is an orthogonal measurethat captures the state of a given device in time. and QoSA can be used in concert with one another.could be used instead of on-demand attestation in the context of swarm RA protocols. In particular,self-measurements can be coupled with a collection protocol, such as LISA-α, where the latteronly relays reports and does not perform any computation. This would yield a clean and conceptually simple approach to swarm attestation, with all the benefits of .An additional advantage of using in the swarm setting is support for high mobility. Prior swarm RA techniques, suchas SEDA, SANA and LISA require swarm topology to remain almost static during the whole swarm attestation instance.This process may be long and prohibitive for applications where connectivity changes often.does not require external input and its collection phase is very fast, since it does not involveany computation; only reading and sending stored measurements. This makes a very natural andviable technique for highly-mobile swarms.Finally, related to the discussion in Section <ref>, we consider the scenario where availability of at leastone in (or a part of) a group of devices is required at all times. This cannot be guaranteed by on-demand swarm attestation,where a large part of the network may be concurrently busy. Meanwhile, with , it is trivial to establish aschedule which ensures that only a fraction of the swarm computes measurements at any given time.§ CONCLUSION We designed as an alternative to current methods that performon-demand RA for low-end devices. provides better in that it allows to detect mobile malware, which is not possible with on-demand techniques that only detect malware if it is currently on . makes it harder for malware to avoiddetection. 's other major advantage is that it requires no cryptographic computation by as part of its interaction with . This is particularly relevant in time-sensitive and criticalapplications, where 's availability is very important. We discuss partial mitigation measures for this problem.We present the new notion of Quality-of-Attestation () as a measure of temporal securityguarantees given by an attestation technique. We show that timing of measurements and timingof verifications (that are conjoined in on-demand attestation) are two distinct aspects of .They are treated as distinct parameters in . We also discuss that the possibility ofusing on-demand attestation as part of collection phase to obtain maximal freshness.We implemented on two hybrid RA architectures, and , and demonstrated its viability on both. does not require extra features or a larger ROM than what isneeded in , and each measurement is fast than on-demand attestation since no authentication of requests is needed. Finally, we show that is a promising option for highly-mobilegroups/swarms of devices, for which no current RA technique works well. 10abera2016invited Tigist Abera, N Asokan, Lucas Davi, Farinaz Koushanfar, Andrew Paverd, Ahmad-Reza Sadeghi, and Gene Tsudik. Invited: Things, trouble, trust: on building trust in IoT systems. In ACM/IEEE Design Automation Conference (DAC), 2016.seda N Asokan, Ferdinand Brasser, Ahmad Ibrahim, Ahmad-Reza Sadeghi, Matthias Schunter, Gene Tsudik, and Christian Wachsmann. SEDA: Scalable embedded device attestation. In ACM SIGSAC Conference on Computer and Communications Security (CCS), 2015.sabre-lite Boundary Devices. i.mx6 arm development board.brasser2015tytan Ferdinand Brasser, Brahim El Mahjoub, Ahmad-Reza Sadeghi, Christian Wachsmann, and Patrick Koeberl. TyTAN: tiny trust anchor for tiny devices. In ACM/IEEE Design Automation Conference (DAC), 2015.brasser2016remote Ferdinand Brasser, Ahmad-Reza Sadeghi, and Gene Tsudik. Remote attestation for low-end embedded devices: the prover's perspective. In ACM/IEEE Design Automation Conference (DAC), 2016.lisa Xavier Carpent, Karim ElDefrawy, Norrathep Rattanavipanon, and Gene Tsudik. Lightweigh swarm attestation: a tale of two LISA-s. In ACM Asia Conference on Computer and Communications Security (ASIACCS), 2017.sha1 D Eastlake 3rd and Paul Jones. Us secure hash algorithm 1 (sha1). Technical report, 2001.hydra Karim ElDefrawy, Norrathep Rattanavipanon, and Gene Tsudik. Hydra: Hybrid design for remote attestation (using a formally verified microkernel). arXiv preprint arXiv:1703.02688, 2017.smart Karim Eldefrawy, Gene Tsudik, Aurélien Francillon, and Daniele Perito. SMART: Secure and minimal architecture for (establishing dynamic) root of trust. In Network and Distributed System Security Symposium (NDSS), 2012.openmsp430 Olivier Girard. Openmsp430, 2009.darpa Ahmad Ibrahim, Ahmad-Reza Sadeghi, Gene Tsudik, and Shaza Zeitouni. DARPA: Device attestation resilient to physical attacks. In ACM Conference on Security and Privacy in Wireless and Mobile Networks (WiSec), 2016.klein2009sel4 Gerwin Klein, Kevin Elphinstone, Gernot Heiser, June Andronick, David Cock, Philip Derrin, Dhammika Elkaduwe, Kai Engelhardt, Rafal Kolanski, Michael Norrish, et al. sel4: Formal verification of an os kernel. In Proceedings of the ACM SIGOPS 22nd symposium on Operating systems principles, pages 207–220. ACM, 2009.trustlite Patrick Koeberl, Steffen Schulz, Ahmad-Reza Sadeghi, and Vijay Varadharajan. TrustLite: A security architecture for tiny embedded devices. In ACM European Conference on Computer Systems (EuroSys), 2014.sel4-libs National ICT Australia and other contributors. sel4 libraries, 2014.sel4-util-libs National ICT Australia and other contributors. util_libs, 2014.ostrovsky1991withstand Rafail Ostrovsky and Moti Yung. How to withstand mobile virus attacks. In Proceedings of the tenth annual ACM symposium on Principles of distributed computing, pages 51–59. ACM, 1991.blake2 MJ Saarinen and JP Aumasson. The blake2 cryptographic hash and message authentication code (mac). 2015.sana Ahmad-Reza Sadeghi, Matthias Schunter, Ahmad Ibrahim, Mauro Conti, and Gregory Neven. SANA: Secure and scalable aggregate network attestation. In ACM Conference on Computer and Communications Security (CCS), 2016.SCHELLEKENS200813 Dries Schellekens, Brecht Wyseur, and Bart Preneel. Remote attestation on legacy operating systems with trusted platform modules. Science of Computer Programming, 74(1):13 – 22, 2008.Seshadri:2006:SSC:1161289.1161306 Arvind Seshadri, Mark Luk, Adrian Perrig, Leendert van Doorn, and Pradeep Khosla. Scuba: Secure code update by attestation in sensor networks. In ACM Workshop on Wireless Security (WiSe), 2006.seshadri2004swatt Arvind Seshadri, Adrian Perrig, Leendert Van Doorn, and Pradeep Khosla. SWATT: Software-based attestation for embedded devices. In IEEE Symposium on Research in Security and Privacy (S&P), 2004.sha256 Secure Hash Standard. Fips pub 180-2. National Institute of Standards and Technology, 2002.sha1-atk Marc Stevens, Elie Bursztein, Pierre Karpman, Ange Albertini, and Yarik Markov. The first collision for full sha-1. URL: https://shattered. it/static/shattered. pdf, 2017.Stumpf2006 Frederic Stumpf, Omid Tafreschi, Patrick Röder, and Claudia Eckert. A robust integrity reporting protocol for remote attestation. In Workshop on Advances in Trusted Computing (WATC), 2006.msp-gcc Texas Instruments. Msp430-gcc-opensource gcc - open source compiler for msp microcontrollers, 2017.stuxnet Jaikumar Vijayan. Stuxnet renews power grid security concerns, june 2010.xilinx2013 ISE Xilinx. Design suite, 2013. | http://arxiv.org/abs/1707.09043v1 | {
"authors": [
"Xavier Carpent",
"Norrathep Rattanavipanon",
"Gene Tsudik"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170727205758",
"title": "ERASMUS: Efficient Remote Attestation via Self- Measurement for Unattended Settings"
} |
Particle acceleration and pitch angle scattering in 2D reconnection simulations School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, U.K. School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, U.K. The conversion of magnetic energy into other forms (such as plasma heating, bulk plasma flows, and non-thermal particles) during solar flares is one of the outstanding open problems in solar physics. It is generally accepted that magnetic reconnection plays a crucial role in these conversion processes. In order to achieve the rapid energy release required in solar flares, an anomalous resistivity, which is orders of magnitude higher than the Spitzer resistivity, is often used in magnetohydrodynamic (MHD) simulations of reconnection in the corona. The origin of Spitzer resistivity is based on Coulomb scattering, which becomes negligible at the high energies achieved by accelerated particles. As a result, simulations of particle acceleration in reconnection events are often performed in the absence of any interaction between accelerated particles and any background plasma. This need not be the case for scattering associated with anomalous resistivity caused by turbulence within solar flares, as the higher resistivity implies an elevated scattering rate. We present results of test particle calculations, with and without pitch angle scattering, subject to fields derived from MHD simulations of two-dimensional (2D) X-point reconnection. Scattering rates proportional to the ratio of the anomalous resistivity to the local Spitzer resistivity, as well as at fixed values, are considered. Pitch angle scattering, which is independent of the anomalous resistivity, causes higher maximum energies in comparison to those obtained without scattering. Scattering rates which are dependent on the local anomalous resistivity tend to produce fewer highly energised particles due to weaker scattering in the separatrices, even though scattering in the current sheet may be stronger when compared to resistivity-independent scattering. Strong scattering also causes an increase in the number of particles exiting the computational box in the reconnection outflow region, as opposed to along the separatrices as is the case in the absence of scattering.Particle acceleration with anomalous pitch angle scattering in 2D MHD reconnection simulations A. Borissov 1 E.P. Kontar 2 J. Threlfall 1 T. Neukirch 1 December 30, 2023 ====================================================================================================================§ INTRODUCTION Solar flare energy release is commonly attributed to magnetic reconnection, during which magnetic energy is converted into other forms, such as plasma heating, bulk plasma flows, and non-thermal accelerated particles <cit.>. Despite many years of research, the physics behind these processes is still not entirely understood. Fast magnetic reconnection is fundamentally based upon particle scattering <cit.>, which causes a restructuring of the magnetic field through diffusion of the magnetic field with respect to the plasma. With any scattering model there will be an associated resistivity. In the case of binary Coulomb collisions the associated resistivity is the Spitzer resistivity, which is typically too small to account for the high rate of energy release in solar flares <cit.>. The introduction of anomalous resistivity, caused by turbulent processes, could account for the rate of energy release during flares <cit.>. In addition, multiple flare models require the presence of particle scattering due to turbulence <cit.>, and there has been evidence for the presence of magnetohydrodynamic (MHD) turbulence in solar flares <cit.>. Vlasov or particle-in-cell simulations are required in order to capture the physics of turbulent processes in magnetic reconnection. Unfortunately these simulations are too computationally expensive to model the whole of a solar flare, so an MHD approach is often used. While MHD allows the simulation of larger lengthscales and timescales, it cannot capture the microscopic physics involved in collisionless reconnection and hence requires the specification of an anomalous resistivity affecting the electromagnetic field evolution through the magnetic induction equation and Ohm's law. In general, for non-zero resistivity a component of the electric field will be directed parallel to the magnetic field, which will result in acceleration of non-thermal particles. Acceleration due to parallel electric field is one of the main acceleration mechanisms thought to produce a non-thermal particle population which is the cause of the observed hard X-ray radiation in solar flares <cit.>. Test particle simulations of acceleration in MHD simulations of magnetic reconnection in two dimensions <cit.>, and in various scenarios in three dimensions <cit.>, have been performed, both with and without Coulomb scattering. In all cases, however, an anomalous resistivity was specified in the MHD simulation. In order to have a more consistent description of the interaction between the accelerated particles and the background an enhanced anomalous scattering rate (relative to Coulomb scattering) should be used. One possibility is the use of pitch angle scattering <cit.>, with a scattering rate that is dependent on the resistivity.In this paper we complement previous work by presenting the results of test particle simulations including pitch angle scattering in fields generated by two-dimensional (2D) MHD simulations. We examine the impact of pitch angle scattering, with varying dependencies of the scattering rate on the velocity of the particle and anomalous resistivity used in the MHD simulations. Individual trajectories as well as energy spectra and spatial distributions are produced. The layout of the remainder of this paper is as follows: in Section <ref> we describe the configuration and results of 2D MHD reconnection simulations. Section <ref> describes our modifications to the guiding centre approach to incorporate pitch angle scattering. We present the results of test particle simulations in Section <ref> along with conclusions in Section <ref>.§ MHD SIMULATIONSWe solve the standard resistive MHD equations <cit.> given by Equations <ref>-<ref>: ∂ρ/∂ t = - ∇·(ρ𝐯) , ∂v/∂ t + v·∇v = 1/ρ𝐣×𝐁-1/ρ∇ P , ∂𝐁/∂ t = -∇×𝐄, ∂ϵ/∂ t + v·∇ϵ = -P/ρ∇·𝐯+η_a/ρj^2, 𝐄+𝐯×𝐁 = η_a 𝐣, ∇×𝐁 = 𝐣,using the Lare2d code <cit.>, with normalising scales given by L̂ = 10m, B̂ = 0.03T and ρ̂= 1.67 × 10^-12kg · m^-3. The choice of ρ̂ is reflective of the coronal environment <cit.>, while B̂ is similar to that used in other simulations of magnetic reconnection <cit.>. The lengthscale is chosen to be comparable to the current sheet size, which is not well constrained for the solar corona; current sheet sizes similar to ours have been used <cit.>, however so have much larger ones <cit.>. This choice of lengthscale pushes the limits of the applicability of MHD within the solar corona, however it was used in order to achieve a compromise between the use of self-consistent electromagnetic fields (from the MHD simulation) while at the same time incorporating aspects of microscopic physics into the particle acceleration picture (without the use of kinetic simulations). In the absence of scattering, test particle energies scale with the square of length, meaning that orbit calculations performed with a given choice of lengthscale can be extrapolated by simply adjusting the energies appropriately. This is not generally the case with scattering included, as the mean free path associated with the scattering introduces a scale independent of the MHD lengthscale which impacts the particle orbits. Increasing the lengthscale substantially, without changing the scattering mean free path, would result in particle orbit computation becoming prohibitively computationally expensive. Although it would be possible to circumvent this issue by, for example, restricting the domain size within which particle orbit calculations are performed, doing so would restrict the effect of the geometrical configuration of the MHD fields on the particle simulation. The normalising scales for all other parameters come from combinations of L̂, B̂, and ρ̂ and are quoted in Table <ref>. We set the anomalous resistivity (η_a) to zero where the critical current is below a threshold value of j_crit = 1, while η_a = 1× 10^-4 where the current exceeds j_crit (values of j_crit and η_a are given in normalised units).Our simulation of 2D magnetic reconnection starts with an isothermal force-free Harris sheet whose magnetic field is perturbed in order to initiate reconnection. The equations specifying the initial conditions for the MHD simulation are given in Equations <ref>-<ref>:B_x/B̂ = tanh(y) - b_1/b_0k_ycos(k_x x)sin(k_y y), B_y/B̂ = b_1/b_0k_xcos(k_y y)sin(k_x x), B_z/B̂ = sech(y), ε/ε̂ = T_0/T̂/m_r(γ_p - 1),where b_1/b_0 = 0.3, T_0 = 10^6K, m_r = 1.2 is the reduced mass for coronal plasma normalised to the proton mass, and γ_p = 5/3 is the ratio of specific heats. We specify the initial density to be uniform at a value of 5ρ̂. Our domain has size 15 in the x-direction and 60 in the y-direction so that our choices of k_x = 2π/15, k_y = 2π/60 ensure the perturbation has one period within the domain in both directions. Periodic boundary conditions in the x-direction and closed boundary conditions in the y-direction are imposed. The magnetic field corresponding to the initial conditions is shown in Figure <ref>. We evolve the MHD simulation until the reconnection rate drops to near-zero and we use an individual snapshot from the simulation (shown in Figure <ref>) during the reconnecting phase into which we insert test particles to compute particle orbits. For simplicity we pick a single MHD snapshot as the electromagnetic field structure changes on a longer timescale than the particle evolution. We shall see in Section <ref> that the majority of the particle orbits' durations are less than 0.1ms and the MHD fields do not vary a great deal during the main reconnection phase which lasts approximately 1ms (this can be seen from the evolution of the magnetic energy in Figure <ref>, which steadily decreases between 1 and 2ms). § GOVERNING EQUATIONS FOR TEST PARTICLE EVOLUTION Charged particle evolution is governed by the Lorentz force law, dv/dt = q( E + v×B), which can, in principle, be solved numerically for the trajectory of the particle. Unfortunately the timestep required to resolve the evolution of the test particle is too small to be practical (for the magnetic field strengths typical of the corona). A common alternative is to use the guiding centre approximation when computing particle orbits <cit.>. In this approach the position of the test particle is averaged over a gyration period (this averaged position is referred to as the guiding centre). This method allows the use of longer timesteps, because the particle gyration need not be resolved temporally. The equations for the evolution of the guiding centre in prescribed electromagnetic fields are given by <cit.> and are reproduced in Equations <ref>-<ref>: Ṙ_⊥ = b/B×[ -E + μ/γ e∇ B + mU/edb/dt + mγ/edu_E/dt. . + U/γE_∥u_E + μ/γ eu_E∂ B/∂ t],mdU/dt = mγu_E·db/dt + eE_∥ - μ/γ∂ B/∂ s,where the Lorentz factor, γ, is given by: γ = √(1 + U^2 + u_E^2/c^2 + 2μ B/mc^2).Here R denotes the guiding centre position, and Ṙ_⊥ is the drift velocity of the guiding centre perpendicular to the magnetic field. The E cross B drift of the guiding centre is given by u_E = γV_E = γE×B/B^2, U = γ v_∥ = γv·b is the velocity of the guiding centre parallel to the magnetic field, and b is the unit vector in the direction of the magnetic field. The quantity ∂ B/∂ s is the rate of change of the magnetic field strength along the guiding centre trajectory. Finally, μ = mγ^2 v_⊥^2/(2B) is the magnetic moment, v_⊥ = v_totsinθ is the gyrational component of the total particle velocity, v_tot = | v|, m is the electron mass, and e the electron charge. The guiding centre approach is valid as long as the length and timescales on which the underlying fields vary are large compared with the particle gyroradius and gyroperiod. In our simulations the maximum value of the ratio of the electron gyroradius to the width of the current sheet is approximately 0.03, while the maximum value of the ratio of the electron gyroperiod to the MHD timescale is 0.007, justifying our use of the guiding centre model. We use the relativistic version of the guiding centre equations even though the particle energies we obtain are generally non-relativistic.In regions where η_a = 0 we solve Equations <ref>-<ref> with an adaptive timestep 4th order Runge Kutta scheme. The guiding centre equations conserve the magnetic moment along the particle trajectory, which cannot be true if pitch angle scattering occurs. By modifying the magnetic moment, along with self-consistently modifying U, we can introduce pitch angle scattering into the governing equations of particle motion. To account for pitch angle scattering in regions where η_a≠ 0, in addition to solving Equations <ref>-<ref> we also solve:dγ =γ̇dt, dβ =(β̇+ F_β) dt + √(2D_ββ)dW,where β = cosθ, and dW = ζ√(dt) and ζ is a normally distributed random variable. Expressions for γ̇ and β̇ are given by:γ̇= 1/2( 1 + U^2/c^2 + 2μ B/mc^2)^-1/2( 2U/c^2dU/dt + 2μ/mc^2dB/dt) ( 1 - V_E^2/c^2)^-1/2, β̇= ( 1/UdU/dt - 1/2BdB/dt) β( 1-β^2 ).Equations <ref> and <ref> follow from taking time derivatives of the expressions for γ and μ (see appendix for derivation). Although Equation <ref> may be replaced by simply updating the energy through the definition of the Lorentz factor in the guiding centre equations (Equation <ref>), this approach was implemented in order to allow generalisation of the scattering model in future work. Our initial choice of the friction and diffusion coefficients F_β and D_ββ are F_β = -βv_tot/λ and D_ββ = (1 - β^2)v_tot/λ, where the mean free path is parametrised byλ = λ_0( 1 + v/v_th)^ακ, with λ_0 = 2× 10^8 m, representing the mean free path of an electron in a plasma with coronal parameters. We integrate Equations <ref> and <ref> using an Euler scheme whose timestep is the minimum of dt_0 = 5× 10^-9 s and dt_s=1/(3ν) =λ/(3v_tot). The value for dt_0 was determined by comparing results of integrating particle trajectories between the variable timestep code (without scattering) and imposing F_β = D_ββ = 0 with the fixed timestep code. Multiple values of dt_0 were evaluated and one was chosen that could accurately reproduce the trajectory given by the variable timestep code. Although a higher order scheme would have been preferable, the dependence of the coefficients on the particle position in the grid would necessitate extra computation of spatial gradients of the fields, which would increase computation time. Furthermore the timestep must be less than the time between scattering events, hence requiring the choice of the minimum of dt_0 and dt_s. After updating β and γ, we update the magnetic moment and parallel velocity to calculate the position of the guiding centre in Equation <ref>. The position is then integrated by the Runge-Kutta scheme with the timestep used in the Euler scheme, dt_0. § RESULTS OF TEST PARTICLE CALCULATIONS §.§ Configuration of test particle code To study the effect of pitch angle scattering on particle behaviour, we initialise test particle orbits in the MHD snapshot shown in Figure <ref>, and integrate the governing equations for their evolution, detailed in Section <ref>, until the orbit leaves the computational domain. We compare the results of calculations in the presence of different scattering rates by varying the values of κ and α in Equation <ref>. The parameter α determines how the mean free path changes as a function of test particle velocity, with α > 0 resulting in a longer mean free path (and hence less scattering) at higher particle velocities, while α < 0 results in a decreasing mean free path for higher velocities. A simple scaling of the mean free path can be applied by varying κ, with higher values leading to a longer mean free path and less scattering. We introduce a dependence on the anomalous resistivity into the mean free path by setting κ = η_sp/η_a, where η_sp is the local Spitzer resistivity at the position of the guiding centre. To get an idea of the spatial dependence of the Spitzer resistivity on position in our MHD simulation, a contour plot of the ratio κ = η_sp/η_a is shown in Figure <ref>.We perform test particle simulations with the following choices of parameters: to investigate the effect of velocity-dependent scattering we choose α = ± 2,0, with κ = η_sp/η_a; to investigate the effect of anomalous resistivity we take α = 0 and κ = 10^-5, 10^-6, 2× 10^-8,η_sp/η_a. The mean free path is related to the scattering frequency by ν = v_tot/λ. In order for the guiding centre approximation to remain valid, the scattering frequency must not exceed the gyrofrequency of the test particle. This restriction on the scattering frequency is dependent on the test particle gyrational velocity, as well as the local magnetic field strength. It is difficult to predict if a test particle orbit will break this condition, however, we find that for values κ < 5× 10^-9 the scattering frequency starts to regularly exceed the gyrofrequency. In addition to performing test particle simulations with scattering included at different rates, we perform the same simulations without scattering using the variable timestep 4th order Runge-Kutta code. We refer to these simulations as the control cases. To compute test particle energy spectra, we integrate 5× 10^5 particle orbits for each of the parameter regimes mentioned above. The particle orbits are distributed with uniformly random initial positions inside a portion of the computation box. This portion is centred on the reconnection region and has a side length of 2 in normalised units (the whole computational box has a side length of 4, also centred on the reconnection region; see Figure <ref>). The initial pitch angle takes on 100 evenly distributed values between 10^∘ and 170^∘ and the initial energy takes on 50 evenly distributed values between 10 eV and 320eV (this energy range covers over 90% of the maxwellian at 10^6 K). These choices mean that there are 100 particle orbits for every combination of initial pitch angle and energy, each having a different (uniformly random) initial position. The final energy and position of each orbit is recorded as it exits the computational box. Each orbit is weighted in proportion to the plasma density at its initial position, so that the initial particle energy distribution is approximately a Maxwellian at a temperature of 10^6 K and the initial distribution of the cosine of the pitch angle is uniform. The resulting energy spectra are shown in Figure <ref>. §.§ Selected trajectoriesOur primary interests are the energy spectra obtained through many orbit calculations, however, it is initially enlightening to examine selected orbit trajectories, energy, and pitch angle evolution. Such examples reveal the general effect of pitch angle scattering on individual orbits. To do this we place test particles at two distinct initial positions, y_0 = 0 and 5m (in both cases with x = 0m), with initial pitch angle θ_0 = 90^∘ and kinetic energy is 320 eV, into the MHD snapshot. These initial conditions are chosen so that the effect of scattering is evident on orbits that drifts into the reconnection region due to the E×B drift, as well as for orbits starting within the reconnection region. The particle trajectories are calculated as described in the previous section with no scattering, scattering with κ = 10^-6 in Equation <ref>, and with κ = η_sp/η_a. The resulting trajectories, energy evolution, and pitch angle evolution are shown in Figures <ref> and <ref>.Due to the magnetic moment no longer being conserved in the case of the different scattering regimes, the orbit trajectories in Figures <ref> and <ref> differ from the control case (Figure <ref>). This is caused by terms in the guiding centre equations (Equations <ref>, <ref>) proportional to μ having a randomising effect on the particle drifts when scattering is included (as μ is no longer constant). For the particle orbit initialised in the current sheet, the more chaotic evolution of the pitch angle when κ = η_sp/η_a (see green curve in Figure <ref> in comparison to the red and black curves) suggests that this choice of κ produces stronger scattering within the diffusion region than if κ = 10^-6. When κ = 10^-6, we note that the particle orbit crosses the reconnection region multiple times (see black particle orbits in Figures <ref> and <ref>), as has been reported previously <cit.>, which is an effect that cannot happen in the absence of scattering. Orbits which enter the current sheet multiple times can traverse a greater potential drop than if they were evolving deterministically, and hence gain more energy. Due to the stochastic nature of the orbit, such behaviour and associated increased energy is not guaranteed even with identical orbit initial conditions. We anticipate that the presence of scattering will yield energy spectra containing higher maximum energies than the case without scattering, as a result of particle trajectories traversing the reconnection region multiple times. Orbits which start outside of the reconnection region are not subject to as much scattering and acceleration if they drift into the separatrices rather than the central current sheet. As a result, although some scattering is evident in the trajectory (red lines in Figure <ref>) and pitch angle evolution (Figure <ref>) of the particle orbits initialised at y = 5 m, energy changes at the end of the orbit are much less evident than for the particle orbits initialised inside the current sheet. §.§ Energy spectraIn Figure <ref> we compare the spectra produced by the control case (without scattering, black curve), with the scattering cases where κ = 10^-5, 10^-6 (in both of these we set α = 0). We note that there is a break in the spectrum of the control case. A small population of highly accelerated particles achieve energies of approximately 100 keV (approximately 0.3% of the total number of orbits, after weighting). This break in the spectrum is due to the small size of the reconnection region. When scattering is introduced, with κ = 10^-5 (red curve in Figure <ref>), there is an increase in the spread of energies obtained by the highly energised particles orbits (compared to the control case), while the general shape of the spectrum remains unchanged. The spectrum of the scattering case with κ = 10^-6 (green curve in Figure <ref>) is smoother, without any breaks in the spectrum. This suggests that scattering is much more effective for smaller values of κ. Both green and red curves in Figure <ref> contain significant numbers of particle orbits achieving energies much greater than the maximum energy achieved by any particle orbit in the control case (in both scattering regimes approximately 0.15% of particle orbits achieve energies higher than any unscattered orbit, corresponding to approximately half of the total highly accelerated population in the control case).Next we compare spectra produced with κ given by κ = η_sp/η_a to the constant κ = 10^-6 and κ = 2× 10^-8 cases, again with α = 0 (see Figure <ref>). Figure <ref> shows a restricted energy range (between 1 and 200 keV) of the same spectra. The value of κ = 2× 10^-8 is chosen to be comparable to the minimum value of η_sp/η_a (since η_sp∝ T^-3/2 this is the location in the MHD simulation with the highest temperature, i.e. in the middle of the current sheet). Since all three cases examined here include relatively strong scattering, we see that there are no breaks in any spectrum, and furthermore there are more particles with energies in the region of 10 keV in the case when κ = η_sp/η_a and κ = 2× 10^-8 than when κ = 10^-6, with fewer higher energy particles (in particular 1.6% of the total particle orbits have energies between 5 and 30 keV for the case κ = η_sp/η_a, compared to 2.3% for the κ = 2× 10^-8 case and 0.5% for the κ = 10^-6 case). The dependency of the mean free path on the resistivity leads to lower maximum energies than even a constant but lower value of κ. This is because the ratio η_sp/η_a increases drastically in the separatrices where the temperature is lower, resulting in fewer particles being scattered. The absence of scattering within the separatrices means that fewer orbits are able to repeatedly cross the acceleration region, resulting in lower energies. We also note that scattering with κ = 2× 10^-8 yields fewer particles at energies above 100keV when compared with the κ = 10^-6 case. In both cases, the maximum energy obtained by particles is still higher than for the case without scattering. Finally, in Figure <ref>c, we consider spectra produced by varying the velocity dependence of the scattering model. In Figures <ref> and <ref> we fixed α = 0 and varied values of κ. Now we set κ = η_sp/η_a and consider α = -2,0,2. There is a very small difference in the spectra above 1keV, with the α = -2 case having very slightly more particle orbits at lower energies (8.1% of total particle orbits with energies between 1 and 10 keV, as opposed to 7.9% and 7.6% for the α = 0 and α = 2 cases respectively) and fewer higher energy orbits (0.004% of total particle orbits with energies greater than 100 keV, as opposed to 0.01% and 0.04% for the α = 0 and α = 2 cases respectively). The spectrum for the α = 0 case falls in between the other two. This indicates stronger scattering occurring for large negative α. This is to be expected as the mean free path decreases for large ratios of the particle velocity to the thermal velocity, implying more scattering. The small difference between three values of α is due to the factor 1 + v_tot/v_th only varying between approximately 1 and 6 for a test particle starting at the centre of the dissipation region (where the temperature and electric field are at their maximum). Changing the mean free path by several orders of magnitude when varying κ has a much greater impact on the spectrum than a change in the velocity dependence.The presence of pitch angle scattering should decrease the rate at which particles are accelerated. In Figure <ref> we plot a histogram of orbit durations in cases without scattering (black curve), with scattering where κ = 2× 10^-8 (red curve) and κ = η_sp/η_a (green curve, in both of the scattering cases α = 0). We see that the number of particles per duration only varies between the three cases above 0.1ms durations. The number of particle orbits with duration greater than 0.1ms is about 14% for the no scattering case, rising to 16% for the scattering case where κ = η_sp/η_a and 22% when κ = 2× 10^-8. Figures <ref> - <ref> show orbit spectra with successively longer durations. We note that the spectra of the simulations including scattering extend to progressively higher energies when particle orbits with progressively longer durations are considered. This is again due to particle orbits needing multiple traverses of the current sheet in order to gain energies higher than those possible in the absence of scattering. The abrupt step in the spectra in Figures <ref>-<ref> at approximately 320 keV is due to the particle orbits which exit the computational box without having encountered the reconnection region. This happens relatively quickly (the exact orbit duration would depend on the initial pitch angle, position, and kinetic energy of each particle orbit, but in all cases occurs faster than 0.1ms) and, as such, these particle orbits are not present in Figure <ref>, resulting in a much smoother spectrum. It is interesting to note the presence of a distinct shoulder starting at energies of approxmately 20keV in Figure <ref> for the κ = η_sp/η_a spectrum. Given the already small number of particle orbits which last longer than 0.1ms, it is not surprising that this feature is not seen in the full spectrum in Figure <ref>.§.§ Particle orbit escape positionsWe now turn our attention to the impact of scattering on the final positions of each test particle orbit upon exiting the computational box. In Figure <ref> we produce histograms for the final z and y positions. We do this for the scattering model when κ = η_sp/η_a (green curve) and κ = 2× 10^-8 (red curve), in both cases with α = 0, in addition to the control case (black curve). In the control case the highly accelerated particle population primarily escapes the simulation domain between z = 200 and z = 300m causing a prominent increase seen on the right hand side of Figure <ref>. In contrast, scattering results in more spread in the final z-position. The two scattering models differ in the distribution of the particle orbits final z-position, with scattering in the κ = η_sp/η_a model resulting in a narrower range of exit locations, compared to the stronger scattering case, with κ = 2× 10^-8, seen in the broader red curve in Figure <ref>.In Figure <ref> we present a histogram of the y-value at the point where the particles exit the computation box. The two tallest peaks correspond to the separatrices, with values between them corresponding to the reconnection outflow region. We see that the κ = η_sp/η_a scattering model and the control case give very similar results, with 14% and 12% of the total particle orbits exiting within the outflow region respectively. In the case of much stronger scattering with κ = 2× 10^-8 , significantly more orbits exit within the outflow region (20% of total). Stronger scattering in the separatrices (in the case of the κ = 2× 10^-8 case) causes more particle orbits to be scattered from the separatrices into the outflow region. Since no scattering takes place in this region and the E×B drift is directed outward, the test particles are unable to re-enter the current sheet and exit the simulation box in the outflow region. For higher values of κ, or for κ = η_sp/η_a, scattering is much weaker in the separatrices, resulting in a distribution of final y-values much closer to that of the control case. § DISCUSSION AND CONCLUSIONSWe have presented a very simple model of pitch angle scattering and have shown that it can have a significant impact on test particle energy spectra. In previous studies which included the effects of collisional scattering <cit.>, it was found that repeated crossings of the reconnection region by test particles in the presence of scattering could lead to a higher energy gain than in the absence of scattering, but that the effect on energy spectra was not significant. In our work, the strong dependence of the mean free path on the anomalous resistivity is the main aspect of the model which affects the energy spectra and box escape positions. Due to this strong scattering, the spectra we obtain show a significant number of orbits gaining energies higher than is possible without scattering, which is something that is not seen in <cit.>, probably due to their use of a much lower scattering rate.If we interpret κ from Equation <ref> as the dependence of the mean free path on the anomalous resistivity, the difference between constant and spatially varying (κ = η_sp/η_a) values of κ are mainly due to their behaviour in regions away from the central current sheet. Since our MHD simulations involved a constant anomalous resistivity where the current exceeded a specified threshold, whereas the Spitzer resistivity calculated at the location of the guiding centre is dependent on temperature, our choice of κ = η_sp/η_a resulted in the scattering rate decreasing with temperature (this is a result of η_sp∝ T^-3/2). The weaker scattering in the separatrices due to the lower temperature (in comparison to the temperature inside the central current sheet) impacted the dynamics of the particles. Less scattering in the separatrices resulted in fewer orbits re-entering the current sheet multiple times. Therefore, the temperatures calculated in the MHD simulations have a significant effect on the test particle dynamics and energy spectra. The temperatures achieved in our MHD simulations are somewhat unrealistic, with a maximum temperature of 4.2× 10^9K, due to the lack of thermal conduction or radiation used. This resulted in the small values of η_sp/η_a ≈ 10^-8 in the current sheet. In future work this could be remedied by the inclusion of thermal conduction and radiation in the MHD simulation. On the other hand, <cit.> showed that thermal conduction can be significantly reduced in coronal conditions due to pitch angle scattering, leading to temperatures of the order of 10^8 K. The reduced thermal conductivity means it is reasonable that there is a large difference in temperature between the current sheet and the separatrices, resulting in a correspondingly large difference in the scattering rates and the associated test particle dynamics. We also showed that there are some small differences between an increasing and decreasing mean free path dependence as a function of the total test particle velocity, however they were negligible in comparison to the changes to the spectra as a result of varying κ.The guiding centre formalism we used relies on the test particle being able to complete at least one full gyration in order to define a guiding centre, so the scattering rate is limited by the gyrofrequency. Our choice of scattering model, in particular κ = η_sp/η_a, would result in violating this restriction for values of anomalous resistivity more than an order of magnitude greater than the ones chosen. This may be remedied by solving full particle orbits for the time that the test particle is within the diffusion region <cit.>. On the other hand, a further decrease in the anomalous resistivity in the MHD simulations would require a greater resolution, which would eventually take prohibitively long amounts of time to compute. In contrast to previous work on particle acceleration in 2D reconnection <cit.>, the maximum energies achieved in our simulations are relatively small (of the order of 100 keV). This is a result of our use of a relatively small lengthscale causing small electric field strengths and size of reconnection region. For a given test particle orbit, the energy gain is entirely dependent on the electric potential drop that it traverses, with possible additional energy losses due to scattering. The presence of scattering introduces an additional lengthscale, namely the mean free path, which means that it is no longer possible to scale the resulting energy spectra with the MHD lengthscale. Our model of scattering did not include any energy loss during collisions, hence changes in the energy spectra are purely due to the different trajectories that particles take and the potential drop that they encounter along it. The inclusion of energy loss terms can be easily accommodated by adding stochastic terms to the energy evolution equation (Equation <ref>). This may result in an optimal anomalous resistivity for the acceleration of charged particles.Finally, more complicated magnetic field topologies are likely to impact the results obtained in this paper with regards to particle trajectories and possibly energy spectra. It would be worth investigating how test particle acceleration is modified in 3D reconnection configurations such as that studied in <cit.> or in coronal structures such as a flux tube <cit.> with the addition of anomalous scattering. § CALCULATING Γ̇ AND Β̇Since the guiding centre approach does not involve the total particle velocity, instead of the usual definition of the Lorentz factor we use,γ^2= 1 + γ^2 v^2/c^2≃ 1 + U^2/c^2 + 2μ B/mc^2 + γ^2 V_E^2/c^2= 1 + U^2/c^2 + 2μ B/mc^2/1 - V_E^2/c^2,where we used the fact that the E×B is the dominant guiding centre drift. Therefore,γ = √(1 + U^2/c^2 + 2μ B/mc^2)/√(1 - V_E^2/c^2).Differentiating this with respect to time yields,γ̇ = 1/2( 1 + U^2/c^2 + 2μ B/mc^2)^-1/2( 2U/c^2dU/dt + 2μ/mc^2dB/dt) ( 1 - V_E^2/c^2)^-1/2+ 1/2( 1 - V_E^2/c^2)^-3/2( 1 + U^2/c^2 + 2μ B/mc^2)^1/22V_E/c^2dV_E/dt.Since V_E ≪ c, the second term is negligible in comparison to the first term, resulting in Equation <ref>.For β = cosθ the derivation of the time derivative, β̇, is much more straightforward. Since μ = mu^2 β^2/2B = mU^2/2B1-β^2/β^2 and dμ/dt = 0 we have:0 = dμ/dt = mU/B1-β^2/β^2dU/dt + mU^2 β/B( -2/β^3) dβ/dt - m U^2/2B^21-β^2/β^2dB/dt= 2μ/UdU/dt - 2μ/β( 1-β^2 )dβ/dt - μ/BdB/dt.Therefore, the time derivative, β̇, can be expressed as:β̇= ( 1/UdU/dt - 1/2BdB/dt) β( 1-β^2 ).A.B. would like to thank the University of St Andrews for financial support from the 7th Century Scholarship and the Scottish Government for support from the Saltire Scholarship. E.P.K.'s work is partially supported by a STFC consolidated grant ST/L000741/1. J.T. and T.N. gratefully acknowledge the support of the UK STFC (consolidated grant SN/N000609/1).aa | http://arxiv.org/abs/1709.00305v1 | {
"authors": [
"Alexei Borissov",
"Eduard P. Kontar",
"James Threlfall",
"Thomas Neukirch"
],
"categories": [
"physics.plasm-ph",
"astro-ph.SR"
],
"primary_category": "physics.plasm-ph",
"published": "20170726191255",
"title": "Particle acceleration with anomalous pitch angle scattering in 2D MHD reconnection simulations"
} |
1Abastumani Astrophysical Observatory at Ilia State University, Tbilisi, Georgia 2IGAM, Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz, Austria, Email: [email protected] 3Departament de Física, Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain 4Institute of Applied Computing & Community Code (IAC^3), UIB, Spain 5Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria 6High Altitude Observatory, National Center for Atmospheric Research, PO Box 3000, Boulder, Colorado 80307, USA. Rieger-type periodicity has been detected in different activity indices over many solar cycles. It was recently shown that the periodicity correlates with solar activity having a shorter period during stronger cycles. Solar activity level is generally asymmetric between northern and southern hemispheres, which could suggest the presence of a similar behavior in the Rieger-type periodicity. We analyse the sunspot area/number and the total magnetic flux data for northern and southern hemispheres during solar cycles 19-23 which had remarkable north-south asymmetry. Using wavelet analysis of sunspot area and number during the north-dominated cycles (19-20) we obtained the periodicity of 160-165 days in the stronger northern hemisphere and 180-190 days in the weaker southern hemisphere. On the other hand, south-dominated cycles (21-23) display the periodicity of 155-160 days in the stronger southern hemisphere and 175-188 days in the weaker northern hemisphere. Therefore, the Rieger-type periodicity has the north-south asymmetry in sunspot area/number data during solar cycles with strong hemispheric asymmetry. We suggest that the periodicity is caused by magnetic Rossby waves in the internal dynamo layer. Using the dispersion relation of magnetic Rossby waves and observed Rieger periodicity we estimated the magnetic field strength in the layer as 45-49 kG in more active hemispheres (north during the cycles 19-20 and south during the cycles 21-23) and 33-40 kG in weaker hemispheres. The estimated difference in the hemispheric field strength is around 10 kG, which provides a challenge for dynamo models. Total magnetic flux data during the cycle 20-23 reveals no clear north-south asymmetry which needs to be explained in the future.§ INTRODUCTION Short-term variation in gamma ray flares with period of 155-160 days was discovered by <cit.> during solar cycle 21.The periodicity later was detected in almost all activity indices <cit.>. <cit.> and <cit.> reported the 155-day periodicity in records of the sunspot area during cycles 14-20 and 12-21, respectively. They found that the periodicity was clearly seen during cycles 16-21, but was absent during cycles 12-15. <cit.> analyzed the records of photospheric magnetic flux and found that the periodicity appeared during cycle 21, but it was absent in cycle 22.The Rieger type periodicity is found also in historical data sets during the earlier cycles. Using two historical aurorae data sets, <cit.> tried to evaluate presence of Rieger period during the cycles 3-4. They have detected the 150 day period in one auroral dataset during 1777-1781 (cycle 3), but they could not confirm the same periodicity for the cycle 4. <cit.> investigated the occurrence of auroras during 16th and 18th centuries and found 158 and 182-185 days period for the years of 1570-72,1736-39 and 1787-90, respectively.<cit.> analysed daily number of sunspot groups between 1610 and 1995 and found near 158 day period around the maximum of solar cycle 2. After cycle 2, no strong evidence for the periodicity was found until the 20th century.Therefore, the Rieger periodicity of 154 days is not a permanent feature of solar activity, but it varies from cycle to cycle. It was also shown that the periodicity usually appears during 1-3 years near the cycle maxima and it may vary from 130 to 185 days <cit.>. Recently, <cit.> analyzed long-term sunspot data for solar cycles 14-24 and showed that the Rieger periodicity is anti-correlated with solar cycle strength: stronger cycles show shorter periods. Observed correlation suggests that the periodicity is related to the dynamo layer in the solar interior.Most promising explanation of the Rieger-type periodicity is connected to magnetic Rossby waves in the solar tachocline<cit.>. The differential rotation and toroidal magnetic field trigger the instability of spherical harmonics of magnetic Rossby waves with period of 155-160 days, which leads to the quasi-periodic emergence of magnetic flux towards the surface. The dispersion relation of magnetic Rossby waves depends on the magnetic field strength <cit.>, therefore the observed periodicity should depend on solar activity level, which fairly corresponds to observations <cit.>. Recent discovery of Rossby waves by STEREO and SDO coronal bright point observations <cit.> fully confirmed the Rossby wave scenario as a mechanism for Rieger-type periodicity.Solar activity generally shows north-south asymmetry in many indicators<cit.>, which means that the strength of the cycle is different in northern and southern hemispheres. If the Rieger-type periodicity depends on the activity strength, then it should also display the north-south asymmetry. The different periodicity in northern and southern hemispheres then may allow to estimate the difference in magnetic field strength in the dynamo layer over hemispheres, which might be a clue for the understanding of hemispheric asymmetry.Here we analyze several available hemispheric activity indices in order to find the values of the Rieger periodicity in northern and southern hemispheres separately during activity cycles which have remarkable north-south asymmetry.§ NORTH-SOUTH ASYMMETRY IN SOLAR ACTIVITY We use three different data sets to study the north-south asymmetry in the Rieger-type periodicity:1) Greenwich Royal Observatory (GRO) daily and monthly sunspot area USAF/NOAA for northern and southern hemispheres(http://solarscience.msfc.nasa.gov/greenwch.shtml), which are available during 1874-2016, 2) Kanzelhöhe Solar Observatory (KSO) and Skalnaté Pleso Observatory (SPO) hemispheric sunspot number data (http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/A+A/447/735), which are available in the interval 1945-2004 <cit.>, 3) The Mount Wilson total magnetic flux (MWTF) data which are available between 1966-2002.North-South asymmetry was also presented during the Maunder minimum (MM, 1645-1715), when the solar activity was extremely low. <cit.> and <cit.> analyzed several data sets including both direct and indirect data catalog published by Spörer nearly 130 years ago, sunspot latitudes in the butterfly diagram during MM published by Ribes and Nesme-Ribes almost 20 years ago, aurorae historical reports during MM, Cosmogenic radionuclides etc. They have calculated the asymmetry index using these data sets and confirmed a strong south-dominated hemispherical asymmetry during MM. The Spörer data are given in the paper of <cit.> and http://haso.unex.es.We are interested to seek for the Rieger periodicity in the cycles with remarkable north-south asymmetry in order to avoid statistically insignificant correlation between activity and periodicity. Therefore, we first study the long-term north-south asymmetry using GRO sunspot data from 1901 to 2016, which correspond to the cycles 14-24, because earlier data is not fully reliable <cit.>. Figure 1 (upper panel) shows monthly averaged sunspot area vs time. From coloured polygons one can see that the north-south asymmetry is remarkable near the cycle maxima in most cases and different hemisphere dominates at different phase of corresponding cycle.For example, the southern hemisphere was more active during the ascending and descending phases of cycle 14, while the northern hemisphere was dominating near the cycle maximum. Similar result was previously noticed by <cit.>, who showed that the northern hemisphere was dominant in the early phases of cycles 12 - 15 with a switch to south-dominance later in each cycle. The opposite behaviour was found during cycles 17 - 18. Therefore, full dominance of one hemisphere is not well established.Cycles 19-23 seem exceptions as the asymmetry in these cycles are very strong and can be considered as statistically significant.Due to the small value of north-south asymmetry in most cycles, it is very important to study the statistical significance. <cit.> used several data sets and estimated the statistical significance of north-south asymmetry using different statistical analysis, such as Binomial distribution, Excess, Normal approximation to the Binomial distribution and Pearson's chi-square test. Similar analysis was performed later by <cit.>. In order to find the statistical significance of north-south asymmetry (SSNSA) in the cycles 19-23 we carried out cycle-to-cycle statistical analysis using the Binomial distribution (see the Table 1)P_k=n!k!(n-k)! p^k q^n-k.where n is the total number of sunspot area, k is the sunspot area for one hemisphere, p is the probability for one hemisphere to be stronger and q is the probability of the another hemisphere. In our case p=q=0.5.When P < 0.3%, we have a highly significant result, if 0.3% <P < 5%, we have a statistically significant result, if 5%<P<10%, it is marginally significant, and when P>10%, it is a statistically insignificant result. The results in Table 1 show that the level of asymmetry and its statistical significance are high in the cycles 19-23, therefore we use only the data of these cycles for further analysis. Figure 1 shows that the cycles 19-20 are north dominated and cycles 21-23 are south-dominated.§ NORTH-SOUTH ASYMMETRY IN RIEGER-TYPE PERIODICITY As it is noted in the previous section, we have three data sets: Greenwich observatory daily sunspot area, the joint catalogue of the KSO and SPO, where one can find the daily and monthly sunspot number, as well as smoothed monthly data for both hemispheres separately <cit.> and the Mount Wilson total magnetic flux (MWTF) data (for cycles 20-23, which starts from January 1965 and runs till May 2002).We used the Morlet wavelet analysis <cit.> to find the Rieger-type timescale in the three data series. Figures 2-3 and 4-5 show the wavelets of north-dominated and south-dominated cycles, respectively. Figure 2 shows thewavelet analysis performed using GRO data for cycles 19-20. It is clearly seen that the northern hemisphere was dominant in almost whole cycle (panel a). The Rieger-type timescale in cycle 19 was of order of 158-172 days in the northern hemisphere and 172-182 days in the southern hemisphere. The cycle 20 displays the periodicity of 160-165 days in the northern hemisphere and 182-198 days in the southern hemisphere. The cycle-by-cycle global wavelets are computed and plotted alongside each wavelet in sunspot data, where blue (red) color denotes the global wavelet for cycle 19 (20). The global wavelet analysis gives peaks at 160 (180) days in the northern (southern) hemisphere in the cycle 19 and at 165 (190) days in the northern (southern) hemisphere in the cycle 20. Wavelet analysis reveals that the period of the Rieger-type duration is shorter in the northern hemisphere (by 20-25 days) than the southern one during both cycles.Figure 3 shows thewavelet analysis of KSO/SPO data during cycles 19-20. The Rieger-type timescale was of order of 158-170 days (with a peak at 165 days) in the northern hemisphere and 174-190 (with a peak at 175 days) days in the southern hemisphere in cycle 19. In the cycle 20, the Rieger periodicity was 151-156 days (with a peak at 155 days) in the northern hemisphere and 185-190 days (with a peak at 188 days) in the southern hemisphere. KSO/SPO data also show that the stronger northern hemisphere displays shorter periodicity than the weaker southern hemisphere. Hence the north-south behavior of the Rieger periodicity agree qualitatively in GRO and KSO/SPO in the cycles 19 and 20.The N-S asymmetry in the cycles 21-23 shifted to the southern hemisphere <cit.>. We performed the wavelet analysis of the south dominated cycles separately for sunspot data. Figure 4 represents the wavelet analysis of GRO data for the south-dominated cycles 21-23. The global wavelets are plotted on right-hand-side, where blue, black and red colors correspond to the cycle 21, 22 and 23, respectively. As it is expected, the weaker northern hemisphere now shows longer periodicity: 160-187 days with peak at 183 days for the cycle 21, 168-190 days with peak at 180 days for the cycle 22 and 170-185 days (peak at 175 days) in the cycle 23. The stronger southern hemisphere displays the shorter periodicity of 155-165 days with peak at 158, 160 and 160 days, for the cycles 21-23, respectively (see the table 2).The difference between hemispheric periodicity is around 15-23 days very similar to the north-dominated cycles.Figure 5 shows the wavelet analysis of KSO/SPO data for south dominated cycles 21-23. The periodicity in northern hemisphere is of the order of 180-190 days (peak at 188 days) in cycle 21, 175-190 days (peak at 177 days) in cycle 22 and 165-185 days (peak at 174 days) during cycle 23. Stronger southern hemisphere shows the period of 150-165 days with peaks at 155, 158 and 161 days for the cycles 21, 22 and 23, respectively. However, the cycle 22 displays another stronger peaks at 190 days in the southern hemisphere in both GRO and KSO/SPO data, which is out of general picture in N-S asymmetry. This interesting disagreement will be discussed later.Figure 6 presents MWTF data during cycles 20-23 with corresponding wavelet analysis. The upper panel shows that only cycle 22 displays remarkable N-S asymmetry with more active southern hemisphere. The cycles 20, 21 and 23 have almost no hemispheric asymmetry. Wavelet analysis gives the periodicity of 160-172 days (with a peak at 168 days) in the cycle 20, 160-180 days (with peak at 170 days) in the cycle 21, 165-180 day period (peak at 175) in the cycle 22 and 160-175 days (with a peak at 170 days) in cycle 23 in the northern hemisphere. The southern hemisphere shows the periodicity of 158-168 days (with a peak at 165 days) in the cycle 20, 180-190 days (with a peak at 187 days) in the cycle 21, 150-160 days (with a peak at 155 days) in the cycle 22 and 160-180 days (with a peak at 170 days) in cycle 23. In contrast with GRO and KSO/SPO data, the total magnetic flux shows no clear north-south asymmetry in the Rieger periodicity during cycles 20 and 23. The cycles 21-22 show some N-S asymmetry in magnetic flux but not as significant as in the sunspot data. The wavelet analysis of sunspot data (GRO, KSO/SPO) clearly show that the Rieger timescale is characterized by the hemispheric asymmetry: the stronger hemisphere displays shorter periodicity of the order of 160-165 days, while weaker hemisphere displays longer periodicity of the order of 175-190 days. This result fairly agrees with the finding of <cit.> that the stronger cycles generally show shorter periodicity. Here the hemisphere (e.g. northern hemisphere in cycles 19-20 and southern hemisphere in cycles 21-23) with higher activity level has shorter periodicity.In addition, activity maxima during cycles 19-20 are shifted with 1-2 years in northern and southern hemispheres (see Figures 2a and 3a). The southern hemisphere reaches its maximum before the northern hemisphere during cycle 19, while it is opposite during the cycle 20 where northern hemisphere reaches the maximum first. The north-south phase shift of solar cycles in sunspot data was studied in details by <cit.>. They showed that the shift of cycle maxima is more pronounced than the shift of minima (see Figure 5 of the paper). Our result fairly agrees with their finding. The Rieger periodicity displays the similar phase shift as it is seen on Figures 2 and 3. This is in agreement with the previous result that the Rieger periodicity in full disc data appears near the cycle maxima.On the other hand, the Rieger periodicity shows different behavior in the total magnetic flux. Here no clear north-south asymmetry is seen. <cit.> examined magnetic flux data from Mount Wilson magnetograph during 1967-1973 and reported that the total flux in the north was greater than in the south by only a 7%, therefore asymmetry is missing in the MWTF data. <cit.> studied the behavior of the total sunspot area and magnetic flux during the year 1989 and showed that there is not always positive correlation between active regions and total magnetic flux: sometimes the flux increases or decreases, while the sunspot areas remain the same. The difference between the Rieger periodicity in sunspot area/number (GRO, KSO/SPO) and total magnetic flux (MWTF) can be related with the lack of the permanent positive correlation. The lack of correlation might reflect the fact that the total magnetic flux is a sum of strong sunspot and weak plage fluxes which may have different behavior. During cycle 21, <cit.> found quasi periodic pulsations only in the strong flux, which were uncorrelated between the hemispheres until 1983, than they appear to be synchronized. <cit.> studied MWTF data for cycles 20-23 and found a correlation between impulses in strong flux and flares, but not with weak flux. On the other hand, <cit.> reported that the Rieger periodicity was not significant in the plage index. This point surely needs more detailed study.§ DISCUSSION Rieger type periodicity has been detected during last two centuries in different activity indices, which showed that it is not a permanent feature of the solar activity but varies from cycle to cycle. It was recently shown that the Rieger periodicity correlates with solar cycle strength being shorter during stronger cycles and therefore it could be related to the internal dynamo layer, where strong toroidal magnetic field is generated <cit.>. Quasi-periodic variation of the dynamo magnetic field with Rieger-type periodicity triggers corresponding variations in activity indices owing to the modulation of erupted magnetic flux. If the Rieger periodicity is the feature of the dynamo layer then it may carry information about its physical parameters.The mechanism of solar activity still remains as one of the major unsolved problems in solar physics, but the cycles are supposed to be caused by large-scale dynamo action in the solar interior <cit.>. The tachocline, thin layer between radiative and convective envelopes, is suggested to be the location of dynamo action. However, there are also dynamo models without tachocline. The magnetic field strength according to the dynamo models without tachocline is less than 10 kG, but the models with tachocline predict much stronger field (> 10 kG) <cit.>. Therefore, the estimation of the magnetic field strength is very important as it may put some limitation on dynamo models in the solar/stellar interiors.Solar activity displays different levels of activity between northern and southern hemispheres. This north-south asymmetry is generally small with weak statistical significance, but it becomes remarkable during some (more stronger) cycles. The asymmetry probably reflects the difference between dynamo magnetic field strengths in northern and southern hemispheres, but the mechanism of the difference is unknown. Even rough estimation of the difference between hemispheric magnetic fields in the dynamo layer may give us a hint to understand the triggering mechanism for the asymmetry. The strength of dynamo magnetic field in different hemispheres can be estimated from the observed Rieger periodicity in hemispheric data.We used the hemispheric data of GRO daily and monthly sunspot area, joint KSO/SPO daily and monthly sunspot numbers and the Mount Wilson total magnetic flux to find the Rieger periodicity in northern and southern hemispheres during cycles 19-23, when the north-south asymmetry of solar activity was remarkable (see Figure 1, upper panel). Figure 1 shows that the northern hemisphere was much more active during the cycles 19-20, but the southern hemisphere became stronger during the cycles 21-23. Wavelet analysis of sunspot data (GRO, KSO/SPO) revealed that the Rieger periodicity was significantly different in both hemispheres being 160-165 days in the northern hemisphere and 175-190 days in the southern hemisphere during north-dominated cycles, while it became 155-160 days in the northern hemisphere and 175-188 days in the southern hemisphere during the south-dominated cycles (see Table 2 for details). Therefore, the periodicity clearly reflects the north-south asymmetry in solar activity.<cit.> suggested that the Rieger periodicity might be caused by r-modes of rotating Sun, which are hydrodynamic (HD) Rossby waves. Then, <cit.> suggested an explanation for the periodicity in terms of equatorially trapped HD Rossby waves. However, the periodicity is usually observed in activity indices, hence the magnetic field should be clearly involved in the scenario. Zaqarashvili et al. (2010a) showed that the Rieger periodicity is related to the instability of magnetic Rossby waves due to the differential rotation and toroidal magnetic field in the dynamo layer. Therefore, the observed periodicity alongside with the dispersion relation of magnetic Rossby waves could lead to the estimation of dynamo magnetic field in individual cycles. Based on the magnetic Rossby wave theory, <cit.> estimated the magnetic field strength in the dynamo layer being ≈ 40 kG during stronger solar cycles (16-23) and ≈ 20 kG during weaker cycles (14-15 and 24).The dispersion relation of fast magnetic Rossby waves (the slow magnetic Rossby waves may lead to the long-term variation of solar cycles as suggested by <cit.>) in the dynamo layer can be written as <cit.>ω_f=-m Ω_01+s_2 + √((1+s_2)^2+4 B^2_max4 πρΩ^2_0 R^2_0 n(n+1))n(n+1),where ω_f is the frequency of fast magnetic Rossby waves, Ω_0 is the equatorial angular velocity, s_2 is the parameter of the differential rotation, ρ is the density, R_0 is the distance from the solar center to the dynamo layer, B_max is the dynamo magnetic field strength at 45 degree, m and n are toroidal and poloidal wave numbers, respectively. Only the magnetic field strength is unknown parameter in the dispersion relation, therefore it can be deduced from the observed periodicity. <cit.> showed that the spherical harmonic with m = 1 and n = 4 may confidently explain the observed periodicity for 30-50 kG magnetic field.We use the dispersion relation (Eq. 2) for estimation of magnetic field strength in the northern and southern hemispheres during cycles 19-23. The differential rotation parameters were not estimated for the northern and southern hemispheres separately for these cycles, therefore initially we set s_2=0 in the equation (2). Based on the GRO data, we calculate the maximum magnetic field strength as 48 kG (38 kG) in the northern (southern) hemisphere during north-dominated cycle 19, 45 kG (33 kG) in the northern (southern) hemisphere during north-dominated cycle 20, 49 kG (36 kG) in the southern (northern) hemisphere during south-dominated cycle 21, 48 kG (38 kG) in the southern (northern) hemisphere during south-dominated cycle 22 and 48 kG (40 kG) in the southern (northern) hemisphere during south-dominated cycle 23. These calculations show that the difference between dynamo magnetic field strengths in northern and southern hemispheres during cycles 19-23 is of the order of 10 kG, which is a quite significant value (see Figure 7). Non-zero differential rotation parameter s_2 in Eq. (2) changes the estimated value of magnetic field strength (see the Table 3), however the hemispheric difference still remains of the order of 10 kG. It must be mentioned, however, that the estimation of magnetic field strength is rather rough and future detailed analysis (including numerical simulations) is needed to increase the accuracy. Figure 7 shows that the estimated magnetic field strength does not significantly vary during cycles 21-23, while the cycle amplitude has been continuously declining. This may support the evidence that the sunspot cycle is an "interference" pattern of overlapping 22-year bands <cit.>. Moreover, it is seen from Figure 7 that the difference between southern and northern hemispheric magnetic field strengths is also decreasing, which could be a result of interaction of the bands. This point needs detailed study in the future.The estimated large difference between dynamo field strengths in the two hemispheres needs to be explained in the future. It may become as a key point to resolve the problem of solar dynamo and activity cycles. It is possible that the observed north-south asymmetry is owing to the overlapping of 11-year oscillating dynamo magnetic field with some steady field component. In this case, the steady field of 5 kG may cause required 10 kG difference in hemispheric magnetic field. <cit.> showed that the steady (non-reversing) toroidal field can be generated in the lower tachocline due to a steady dynamo in the case of low magnetic diffusivity with the strength of > 1 kG, which is in the range of required value. Then the temporal variation of the non-reversing magnetic field with longer time scales caused by slow magnetic Rossby waves below the solar tachocline <cit.> may lead to the observed variations in north-south asymmetry. This is, however, only speculation and no real physical mechanism resolving the north-south asymmetry problem exists up to now. Recent flux transport dynamo simulations have addressed this problem in terms of N/S asymmetries in surface poloidal source <cit.> and in meridional circulation <cit.>, but reason of such asymmetries in the dynamo ingredients is yet to be physically explored.It must be noted here that the sunspot number data in the cycle 22 displays the significant peak at longer period (∼ 190 days) in the southern hemisphere, which is somehow out of regularity. This long-period peak may correspond to the higher harmonic of magnetic Rossby waves. For example, if the shorter period of 158 days is caused by m=1, n=4 harmonic (as it is suggested above) then the harmonic with m=1, n=5 would give the period of ∼ 210 days, which is not far from the observed peak. The long period peaks can be seen also in other cycles and might correspond to the regular pattern. It requires further detailed study.In contrast of sunspot number/area data, total magnetic flux does not show any remarkable north-south asymmetry in the Rieger periodicity. Therefore, it seems that the total magnetic flux does not clearly manifest the north-south asymmetry. This is probably caused by the fact that used MWTF contains both, strong sunspot flux and weak plage flux, from which only the strong flux has N-S asymmetry. This is an interesting question to be answered in the future.§ CONCLUSIONS We carried out the wavelet analysis of the hemispheric sunspot area (GRO), sunspot number (KSO/SPO) and Mount Wilson Total Magnetic flux data during solar cycles (19-23) with remarkable north-south asymmetry: the northern hemisphere was dominated during cycles 19-20 and the southern one was dominated during the cycles 21-23. The analysis of sunspot area/number data showed that the Rieger type periodicity is also asymmetric with hemispheres. We obtained the periods of 160-165 days in the northern hemisphere and 180-190 days in the southern hemisphere during cycles 19-20, while 155-160 days in the northern hemisphere and 175-188 days in the southern hemisphere during the cycles 21-23. Therefore, the Rieger-type periodicity in sunspot area/number data correlates with hemispheric activity levels in the same sense as it correlates with cycle strength based on full disc data <cit.>: the hemisphere with stronger activity displays the periodicity with shorter period. Hence, the Rieger periodicity is connected to the internal dynamo layer, where the magnetic field and the solar cycles are generated. The magnetic field might be modulated by magnetic Rossby waves, which leads to the quasi-periodic emergence of magnetic flux. This scenario is fully supported by recent direct observations of Rossby waves using STEREO and SDO coronal bright point data <cit.>. In addition, activity manifests a phase shift of 1-2 years between northern and southern hemispheres, which is clearly seen during the cycles 19-20(see more detailed analysis in <cit.>). The Rieger periodicity also takes place at different times (with similar 1-2 year shift) in the two hemispheres which means that the quasi-periodic flux emergence correlates to the maximum phase of solar cycles. The obtained periodicity and the dispersion relation of magnetic Rossby waves were used to estimate the magnetic field strength in the tachocline as 45-48 kG in more active hemisphere (northern hemisphere during the cycles 19-20 and the southern one during cycles 21-23) and 32-38 kG in the weaker hemisphere. The north-south difference in the dynamo magnetic field strength is almost 10 kG, which reaches to almost 25 % of estimated magnetic field. The significant hemispheric difference of the field strength induces future challenges for dynamo models. Acknowledgements We thank J. 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"authors": [
"Eka Gurgenashvili",
"Teimuraz V. Zaqarashvili",
"Vasil Kukhianidze",
"Ramon Oliver",
"Jose Luis Ballester",
"Mausumi Dikpati",
"Scott W. McIntosh"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170726191639",
"title": "North-south asymmetry in Rieger-type periodicity during solar cycles 19-23"
} |
Chaos synchronization of identical Sprott systems by active control Marcin DaszkiewiczInstituteofTheoreticalPhysics UniversityofWroclawpl.MaxaBorna9,50-206Wroclaw,Poland e-mail:[email protected] In this article we synchronize by active control method all 19 identical Sprott systems provided in paper <cit.>. Particularly, we find the corresponding active controllers as well as we perform (as an example) the numerical synchronization of two Sprott-A models.§ INTRODUCTION In the last four decades there appeared a lot of papers dealing with so-called chaotic models, i.e., with such models whose dynamics is described by strongly sensitive with respect initial conditions, nonlinear differential equations. The most popular of them are: Lorenz system <cit.>, Roessler system <cit.>, Rayleigh-Benard system <cit.>, Henon-Heiles system <cit.>, jerk equation <cit.>, Duffing equation <cit.>, Lotka-Volter system <cit.>, Liu system <cit.>, Chen system <cit.> and Sprott system <cit.>. A lot of them have been applied in various fields of industrial and scientific divisions, such as, for example: Physics, Chemistry, Biology, Microbiology, Economics, Electronics, Engineering, Computer Science, Secure Communications, Image Processing and Robotics.One of the most important problem of the chaos theory concerns so-called chaos synchronization phenomena. Since Pecora and Caroll <cit.> introduced a method to synchronize two identical chaotic systems, the chaos synchronization has received increasing attention due to great potential applications in many scientific discipline. Generally, there are known several methods of chaos synchronization, such as: OGY method <cit.>, active con- trol method <cit.>-<cit.>, adaptive control method <cit.>-<cit.>, backstepping method <cit.>, <cit.>, sampled-data feedback synchronization method <cit.>, time-delay feedback method <cit.> and sliding mode control method <cit.>-<cit.>.In this article we synchronize by active control scheme all 19 identical Sprott systems provided in publication <cit.>. Particularly, we establish the proper so-called active con- trollers with use of the Lyapunov stabilization theory <cit.>. It should be noted, however, that some types of Sprott model have been already synchronized with use of the active control method in papers <cit.> and <cit.>. Apart of that there has been synchronized the Sprott systems in the framework of adaptive control scheme in articles <cit.> and <cit.>.The paper is organized as follows. In second Section we recall the main result of Sprott article <cit.>, i.e., we provide Table 1 including dynamics of all 19 Sprott chaotic systems. In Section 3 we remaind the basic concepts of active synchronization method, while in Section 4 we consider as an example the synchronization of two identical Sprott-A models. The fifth Section is devoted to the main result of the paper, and it provides in Table 2 the active controllers which synchronize all identical Sprott systems. The conclusions and final remarks are discussed in the last Section. § SPROTT SYSTEMS In this Section we recall the main result of paper <cit.>, in which there has been performed a systematic examination of general three-dimensional ordinary differential equations with quadratic nonlinearities. Particularly, it has been uncovered 19 distinct simple examples of chaotic flows (so-called Sprott systems) listed in Table 1. type 1st equation 2nd equation 3rd equationA ẋ_1=x_2 ẋ_2=-x_1+x_2x_3 ẋ_3=1-x_2^2 B ẋ_1=x_2x_3 ẋ_2=x_1-x_2 ẋ_3=1-x_1x_2 C ẋ_1=x_2x_3 ẋ_2=x_1-x_2 ẋ_3=1-x_1^2 D ẋ_1=-x_2 ẋ_2=x_1+x_3 ẋ_3=x_1x_3+ 3x_2^2 E ẋ_1=x_2x_3 ẋ_2=x_1^2-x_2 ẋ_3=1-4x_1 F ẋ_1=x_2+x_3 ẋ_2=-x_1+0.5x_2 ẋ_3=x_1^2-x_3 G ẋ_1=0.4x_1+x_3 ẋ_2=x_1x_3-x_2 ẋ_3=-x_1+x_2 H ẋ_1=-x_2+x_3^2 ẋ_2=x_1+0.5x_2 ẋ_3=x_1-x_3 I ẋ_1=-0.2x_2 ẋ_2=x_1+x_3 ẋ_3=x_1+ x_2^2-x_3 J ẋ_1=2x_3 ẋ_2=-2x_2+x_3 ẋ_3=-x_1+x_2+x_2^2 K ẋ_1=x_1x_2 -x_3 ẋ_2=x_1-x_2 ẋ_3=x_1+0.3x_3 L ẋ_1=x_2+3.9x_3 ẋ_2=0.9x_1^2-x_2 ẋ_3=1-x_1 M ẋ_1=-x_3 ẋ_2=-x_1^2-x_2 ẋ_3=1.7+1.7x_1+x_2 N ẋ_1=-2x_2 ẋ_2=x_1+x_3^2 ẋ_3=1+x_2-2x_1 O ẋ_1=x_2 ẋ_2=x_1-x_3 ẋ_3=x_1+x_1x_3+2.7x_2 P ẋ_1=2.7x_2+x_3 ẋ_2=-x_1+x_2^2 ẋ_3=x_1+x_2 Q ẋ_1=-x_3 ẋ_2=x_1-x_2 ẋ_3=3.1x_1+x_2^2+0.5x_3 R ẋ_1=0.9-x_2 ẋ_2=0.4+x_3 ẋ_3=x_1x_2-x_3 S ẋ_1=-x_1-4x_2 ẋ_2=x_1+x_3^2 ẋ_3=1+x_1Table 1. The Sprott systems.§ CHAOS SYNCHRONIZATION BY ACTIVE CONTROL - GENERAL PRESCRIPTION In this Section we remaind the general scheme of chaos synchronization of two systems by so-called active control procedure <cit.>-<cit.>. Let us start with the following master model[do/dt = ȯ.]ẋ = Ax + F(x),where x = [ x_1, x_2,… ,x_n ] is the state of the system, A denotes the n × n matrix of the system parameters and F(x) plays the role of the nonlinear part of the differential equation (<ref>). The slave model dynamics is described byẏ = By + G(y) + u,with y = [ y_1, y_2,… ,y_n ] being the state of the system, B denoting the n-dimensional quadratic matrix of the system, G(y) playing the role of nonlinearity of the equation (<ref>) and u = [ u_1, u_2,… ,u_n ] being the active controller of the slave model. Besides, it should be mentioned that for matrices A = B and functions F = G the states x and y describe two identical chaotic systems. In the case A ≠ B or F ≠ G they correspond to the two different chaotic models.Let us now provide the following synchronization error vectore = y - x,which in accordance with(<ref>) and(<ref>) obeysė = By - Ax + G(y) - F(x) + u . In active control method we try to find such a controller u, which synchronizes the state of the master system (<ref>) with the state of the slave system (<ref>) for any initial condition x_0 = x(0) and y_0 = y(0). In other words we design a controller u in such a way that for system (<ref>) we havelim_t →∞||e(t)|| =0 ,for all initial conditions e_0 = e(0). In order to establish the synchronization (<ref>) we use the Lyapunov stabilization theory <cit.>. It means, that if we take as a candidate Lyapunov function of the formV(e) = e^TPV(e)e,with P being a positive n × n matrix, then we wish to find the active controller u so thatV̇(e) = -e^TQV(e)e,where Q is a positive definite n × n matrix as well. Then the systems (<ref>) and (<ref>) remain synchronized. § THE EXAMPLE: CHAOS SYNCHRONIZATION OF (IDENTICAL) SPROTT-A SYSTEMS In accordance with two pervious Sections the master Sprott-A system is described by the following dynamics (see Table 1){[ ẋ_1 = x_2; ẋ_2 = -x_1 + x_2x_3; ẋ_3 = 1-x_2^2 , ].where functions x_1, x_2 and x_3 denote the states of the system; its slave Sprott-A partner is given by{[ ẏ_1 = y_2 + u_1; ẏ_2 = -y_1 + y_2y_3 + u_2; ẏ_3 = 1-y_2^2 + u_3 , ].with active controllers u_1, u_2 and u_3 respectively. Using (<ref>) and (<ref>) one can check that the dynamics of synchronization errors e_i = y_i - x_i is obtained as[See also formula (<ref>).]{[ė_1=e_2 + u_1;ė_2= -e_1 + y_2y_3 - x_2x_3 + u_2;ė_3=-e_2(y_2 + x_2) + u_3 . ].Besides, if we define the positive Lyapunov function by[The matrix P = 1 in the formula (<ref>).]V(e) = 1/2(e_1^2+e_2^2+e_3^2),then for the following choice of control functions{[u_1= -(e_1+e_2);u_2= e_1 -e_2 - y_2y_3 + x_2x_3;u_3= e_2(y_2 + x_2) - e_3 , ].we have[The matrix Q = 1 in the formula (<ref>).]V̇(e) = -(e_1^2+e_2^2+e_3^2).Such a result means (see general prescription) that the identical Sprott-A systems (<ref>) and (<ref>) are synchronized for all initial conditions with active controllers (<ref>).Let us now illustrate the above considerations by the proper numerical calculations.First of all, we solve the Sprott-A system with two different sets of initial conditions(x_01,x_02,x_03) = (1,0.05,0.05) ,and(y_01,y_02,y_03) = (1.05,0.15,0) ,respectively. The results are presented on Figure 1 - one can see that there exist (in fact) the divergences between both trajectories. Next, we find the solutions for the master system (<ref>) (the x-trajectory) and for its slave partner (<ref>) with active controllers (<ref>) (the y-trajectory) for initial data (<ref>) and (<ref>) respectively. Now, we see that the corresponding trajectories become synchronized - the vanishing in time error functions e_i = y_i - x_i are presented on Figure 2. Additionally, we repeat the above numerical procedure for two another sets of initial data: x_0 = (0,0.15,0.05) and y_0=(0.05,0.05,0); the obtained results are presented on Figures 3 and 4 respectively.§ CHAOS SYNCHRONIZATION OF IDENTICAL SPROTT SYSTEMS The used in pervious Section algorithm can be applied to the case of all remaining Sprott systems as well. The obtained results are summarized in Table 2, i.e., there are listed controllers u_1, u_2 and u_3 for which the proper identical Sprott systems become synchronized for arbitrary initial conditions x_01, x_02 and x_03 as well as y_01, y_02 and y_03. However, as it was already mentioned in Introduction, the control functions for Sprott-L and Sprott-M models have been provided in paper <cit.>.type 1st controller 2nd controller 3rd controllerA u_1=-(e_1+e_2) u_2 = e_1-e_2-y_2y_3+ u_3=e_2(y_2+x_2)-e_3 +x_2x_3B u_1=x_2x_3-y_2y_3-e_1 u_2=-e_1 u_3=y_1y_2-x_1x_2-e_3 C u_1=x_2x_3-y_2y_3-e_1 u_2=-e_1 u_3=(x_1+y_1)e_1-e_3 D u_1=e_2-e_1 u_2=-(e_1+e_2+e_3) u_3=x_1x_3-3(y_2+x_2)·· e_2-e_3-y_1y_2 E u_1=x_1x_3-y_1y_3-e_1 u_2=-(y_1+x_1)e_1 u_3=4e_1-e_2 F u_1=-(e_1+e_2+e_3) u_2=e_1-1.5e_2 u_3=-(y_1+x_1)e_1 G u_1=-(1.4e_1+e_3) u_2=x_1x_3-y_1y_3 u_3=e_1-(e_2+e_3) H u_1=e_2-(y_3+x_3)· u_2=-(e_1+1.5e_2) u_3=-e_1· e_3-e_1 I u_1=0.2e_2-e_1 u_2=-(e_1+e_2+e_3) u_3=-(e_1+ (y_2+x_2)·· e_2) J u_1=-(e_1+2e_3) u_2=e_2-e_3 u_3=e_1-(1+y_2+x_3)·· e_2-e_3) K u_1=e_3-e_1-y_1y_2+ u_2=-e_1 u_3=-(e_1+1.3e_3)+x_1x_2 L u_1=-(e_1+e_2+3.9e_3) u_2=-0.9(y_1+x_1)e_1 u_3=e_1-e_3M u_1=e_3-e_1 u_2=-e_1(y_1+x_1) u_3=-(1.7e_1+e_2+e_3) N u_1=2e_2-e_1 u_2=-(e_1+(y_3+x_3)· u_3=2e_1-e_2-e_3 · e_3+e_2)O u_1=-(e_1+e_2) u_2=-(e_1+e_2)+e_3 u_3=-(e_1+2.7e_2++e_3)+x_1x_2-y_1y_2 P u_1=-(e_1+2.7e_2+ u_2=e_1-(y_2+x_2)· u_3=-(e_1+e_2+e_3)+e_3) · e_2-e_2)Q u_1=e_3-e_1 u_2=e_1 u_3=-(3.1e_1+(y_2+x_2)·· e_2)-1.5e_3 R u_1=-e_1+e_2 u_2=-(e_2+e_3) u_3=x_1x_2-y_1y_2 S u_1=4e_2 u_2=-(e_1+(y_3+x_3)· u_3=-(e_1+e_3) · e_3)-e_2 Table 2. The active controllers for Sprott systems. § FINAL REMARKSIn this article we synchronize all identical Sprott systems defined in paper <cit.> with use of the active control method. Particularly, we find the corresponding so-called active controllers listed in Table 2. As an example we also study numerically synchronization of Sprott-A model defined by the formulas (<ref>) and (<ref>).It should be noted that the presented investigations can be extended in various ways. For example, one may consider synchronization of Sprott models with use of others mentioned in Introduction methods. The works in this direction already started and are in progress. 99 [1] E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963) [2] O.E. Roessler, Phys. Lett. A 57, 397 (1976) [3] A.V. Getling, "Rayleigh-Benard Convection: Structures and Dynamics", World Scientific, 1998 [4] M. Henon, C. Heiles, AJ. 69, 73 (1964) [5] J.C. Sprott, Am J Phys. 65, 537 (1997) [6] G. 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"authors": [
"Marcin Daszkiewicz"
],
"categories": [
"nlin.CD",
"hep-th"
],
"primary_category": "nlin.CD",
"published": "20170726181219",
"title": "Chaos synchronization of identical Sprott systems by active control"
} |
1Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397 2Department of High Energy Astrophysics, Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), 3-1-1 Yoshinodai, Sagamihara, 229-8510, Japan;3Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; [email protected] 4Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA We report measurements of proper motion, radial velocity, and elemental composition for 14 compact X-ray bright knots in Kepler's supernova remnant (SNR) using archivaldata.The highest speed knots show both large proper motions (μ∼ 0.11–0.14^'' yr^-1) and high radial velocities (v ∼ 8,700–10,020 km s^-1).For these knots the estimated space velocities (9,100 km s^-1 ≲ v_ 3D≲ 10,400 km s^-1) are similar to the typical Si velocity seen in SN Ia near maximum light. High speed ejecta knots appear only in specific locations and are morphologically and kinematically distinct from the rest of the ejecta.The proper motions of five knots extrapolate back over the age of Kepler's SNR to a consistent central position.This new kinematic center agrees well with previous determinations, but is less subject to systematic errors and denotes a location about which several prominent structures in the remnant display a high degree of symmetry.These five knots are expanding at close to the free expansion rate (expansion indices of 0.75 ≲ m ≲ 1.0), which we argue indicates either that they were formed in the explosion with a high density contrast (more than 100 times the ambient density) or that they have propagated through relatively low density (n_ H < 0.1 cm^-3) regions in the ambient medium. X-ray spectral analysis shows that the undecelerated knots have high Si and S abundances, a lower Fe abundance and very low O abundance, pointing to an origin in the partial Si-burning zone, which occurs in the outer layer of the exploding white dwarf for SN Ia models.Other knots show slower speeds and expansion indices consistent with decelerated ejecta knots or features in the ambient medium overrun by the forward shock. Our new accurate location for the explosion site has well-defined positional uncertainties allowing for a great reduction in the area to be searched for faint surviving donor stars under non-traditional single-degenerate SN Ia scenarios; because of the lack of bright stars in the search area the traditional scenario remains ruled out. Freely Expanding Knots of X-ray Emitting Ejecta in Kepler's Supernova Remnant Toshiki Sato1,2 and John P. Hughes3,4 Received: date / Accepted: date ===============================================================================§ INTRODUCTION Kepler's supernova (SN 1604) is one of the most well-studied young supernova remnants (SNRs) in the Galaxy.General consensus holds, even without a light echo spectrum, that Kepler's SNR is a Type Ia SN based largely on X-ray observations showing shocked ejecta with strong silicon, sulfur, and iron emission and a near absence of oxygen emission <cit.>. Multiple lines of evidence going back decades <cit.> have shown that Kepler's SNR is interacting with a dense (few particles per cm^-3), strongly asymmetric, nitrogen-rich ambient medium.More recent work shows that localized regions in the SNR show prominent oxygen, neon, and magnesium X-ray emission with nearly solar O/Fe abundance ratios that indicate an association <cit.> with a dense circumstellar medium (CSM).In these regions, infrared observations reveal strong silicate features suggestive of the wind from an O-rich asymptotic giant branch (AGB) star <cit.>.<cit.> first connected the environmental characteristics of Kepler's SNR, unusual for its location a few hundred pc above the Galactic plane, with the possibility of a high-speed (∼300 km s^-1), mass-losing progenitor star. In his model, wind material is compressed by the low density ambient medium forming a dense bow shock in the direction of motion (toward the northwest).This produces a strong brightness gradient in X-ray and radio images and high density knots in the resulting SNR (see, e.g.,for an early 2D hydro bow-shock model for the remnant). Bandiera's model is likewise consistent with X-ray expansion measurements <cit.> that show slower expansion rates in the north compared with the rest of the remnant. More recent multi-dimensional hydrodynamical models <cit.> have considered a single degenerate (SD) scenario for the progenitor to Kepler's SNR with the mass loss coming from the donor companion star to the white dwarf that exploded.However, it is important to note that no surviving red giant, AGB or post AGB donor star has been found in the central region of Kepler's SNR <cit.>.In this article, we report three-dimensional space velocities of several X-ray knots in Kepler's SNR determined from both proper motions and radial velocities using archival Chandra observations.Our results yield insights on the nature of the explosion and the ambient medium around Kepler's SNR. In 2 we present the observational results from image and spectral analysis of the several knots, including measurements of elemental composition in addition to velocity. The discussion section (3) studies the kinematics of the knots; reports a new, accurate, kinematic center for Kepler's SNR; assesses the implications of these results on the evolutionary state and nature of the ambient medium; and reexamines the question of a possible left-over companion star under the SD scenario for the explosion. The final section summarizes the article. Uncertainties are quoted at the 1σ (68.3%) confidence level unless otherwise indicated; positions are given in equinox J2000 throughout. § OBSERVATIONAL RESULTS The Chandra Advanced CCD Imaging Spectrometer Spectroscopic-array <cit.>observed Kepler's SNR four times: in 2000 (PI: S. Holt), 2004 (PI: L. Rudnick), 2006 (PI: S. Reynolds) and 2014 (PI: K. Borkowski). The total net exposure time for each of these observations is 48.8 ks, 46.2 ks, 741.0 and 139.1 ks, respectively. The time differences with respect to the long observation in 2006 are 5.96 yr (2000-2006), 1.64 yr (2004-2006) and 7.90 yr (2006-2014). We reprocessed all the level-1 event data, applying all standard data reduction steps with CALDB version 4.7.2, using a custom pipeline based on “” in CIAO version 4.8.Serendipitous point sources were used to align the images.Sources were identified in each ObsID using the CIAO taskand position offsets were computed with .All ObsIDs were matched to ObsID 6175, which was chosen as the reference because it has the longest exposure time (159.1 ks). At least 4 and as many as 17 sources, depending on ObsID, were used for the alignment.The source positions showed mean shifts in R.A. and decl. of less than 0.35^''. Using the shift values, we updated the aspect solution of the event file using . After corrections, the average residuals in the point source positions relative to the reference ObsID (#6175) are <0.25^''. §.§ Proper Motions of the Knots Figure <ref> shows a three-color image of Kepler's SNR from the third epoch observation (in 2006) after image alignment, highlighting emission from primarily O-Lyα (green), Fe L-shell (red), and Si-Heα (blue). Regions of CSM emission in Kepler's SNR (e.g., CSM1–3 in the figure) tend to appear greenish in this figure due to relatively more emission from O. Ejecta knots appear as purple (both strong Si and Fe emission, e.g., N and NE knots) or orange (weaker Si to Fe, e.g., SW knots). Some of the purplish colored regions around the edge of the SNR contain strong nonthermal emission.Figure <ref> shows a Doppler velocity map of Kepler's SNR made using three narrow energy bands in the Si-Heα line. Our recent work on Tycho's SNR <cit.> demonstrated the ability of theACIS instrument to measure radial velocities for relatively large and diffuse ejecta knots.The radial velocities of knots in Tycho's SNR show an obvious pattern indicative of a spherically expanding shell—with the highest speeds through the center and decreasing speeds towards the limb. This pattern is not seen in Kepler's SNR where instead the highest speeds (both red- and blue-shifted) appear as distinct knots lying in chains that stretch east-west in specific locations largely across the northern half of the remnant.A number of these knots (the N, NE, and SW sets) also showed large proper motions when comparing the images taken in 2000 and 2014 (see Fig. <ref>).We also identified three CSM knots and the northwest ejecta knots (Ej1, 2, 3, and 4, which we combined into two separate knots Ej1-2, Ej3-4) whose proper motions were recently measured with the Hubble Space Telescope <cit.>. None of the N, NE, and SW knots showed any evidence for optical emission in the HST images, while the CSM and Ej knots all did. We selected 14 knots for proper motion and radial velocity analyses.To measure proper motions, we extracted image cut-outs about each knot for the 4 epochs.The image from the long observation in 2006 was used as the fitting model for each knot and was shifted in R.A. and decl. to obtain the 2D proper motion shifts. We employed the C-statistic, a maximum likelihood statistic for Poisson distributed data <cit.>, C = -2Σ_i,j(n_i.j ln m_i,j-m_i,j- ln n_i.j!) where n_i,j are the counts in pixel (i,j) of the image in each epoch, and m_i,j are the model counts from the 2006 image scaled by the relative number of total counts (over the 0.6–2.7 keV band) from the entire SNR.We estimate the errors on the proper motion shifts using Δ C = C - C_ min, which is similar to χ^2 <cit.>. The fitting errors are subdominant to the systematic errors in image alignment, which we determine by fitting the positions of seven serendipitous point sources using the same method. The systematic errors (σ_ x, σ_ y) are estimated to be (0.17^'', 0.18^'') for 2000, (0.26^'', 0.20^'') for 2004 and (0.21^'', 0.32^'') for 2014.Images of Δ C for each knot are shown in small frames on Figure <ref>, clearly indicating the high proper motion of the ejecta knots, and Table <ref> presents numerical results. We obtained acceptable fits for all knots (reduced C = 0.95–2.04). For the NE, N and SW sets, we found large proper motions (∼0.08–0.14^'' yr^-1), comparable to values from around the rim <cit.>. In contrast, the CSM knots have small proper motions (≲ 0.04^'' yr^-1), consistent with being either ejecta knots significantly decelerated by interaction with the CSM or features in the ambient medium overrun by the forward shock.Our X-ray proper motion of the Ej3-4 knot is consistent with the Hα result <cit.>, but the proper motion of knot Ej1-2 is smaller by about a factor of two in the X-rays than in Hα (0.069–0.083^'' yr^-1).Knot Ej3-4 is detached from the main shell of X-ray emitting ejecta (and relatively distant from any slow moving optical knots); the agreement between theand HST proper motions indicates that the X-ray knot is driving the arc-shaped Hα shock here.X-ray knot Ej1-2, however, is closer to the (slower moving) main ejecta shell and is partially superposed on bright optical radiative knots that appear to be slowly moving.We suspect that the discrepancy between the motion of the Hα shock and knot Ej1-2 is due to some contamination of the X-ray knot by slower moving ejecta from the main shell or X-ray emission associated with the optical radiative knots. §.§ X-ray Spectroscopy of the Knots We extracted spectra in each epoch (2000, 2004, 2006 and 2014) from the knot and background regions defined in Figure <ref>, accounting for position shifts due to the proper motion. The spectra were fitted in the 0.6–2.8 keV band using an absorbedmodel in XSPEC 12.9.0 (AtomDB v3.0.3).An additional Gaussian model was included to account for a feature at ∼1.2 keV from missing Fe-L lines in the atomic database <cit.>. Among the four spectra from the different epochs for each knot, fitted model parameters (temperature, ionization age, abundances, and radial Doppler velocity) were linked.Our spectral fits explicitly allow the ionization timescale, temperature, and redshift to be free parameters, so line centroid variations due to changes in the thermodynamic state are explicitly included in our fits, the derived values, and the uncertainty on the fitted redshift.To reduce the complexity of our fits, we fixed a number of our model parameters to the best-fit values determined by <cit.>. Specifically we fixed the column density to the value N_H = 6.4 × 10^21 cm^-2 using the abundance table from <cit.> and the photon index of the power-law model to 2.64, allowing the normalization parameter to be free.We assumed no H and He in the shocked SN Ia ejecta (for the N, NE, SE and Ej knots), and fitted the Ne, Mg, Si and S abundances as free parameters. We fixed the abundances of [O/C]/[O/C]_⊙, [Ar/C]/[Ar/C]_⊙, [Ca/C]/[Ca/C]_⊙ and [Fe/C]/[Fe/C]_⊙ to be 0.46, 37, 67.41 and 25.28, respectively. For the Ej knots, we had to thaw the O abundance to obtain good fits. Our fits for these knots also showed more neon and magnesium emission compared to the other ejecta knots, implying that the Ej knots show a mix of both ejecta and CSM components.For the CSM knots, we used only an absorbedmodel.Here we include hydrogen and helium in the plasma, and fixed the abundances of He, C, O, Ar, Ca and Fe to the solar values.The nitrogen abundance is fixed to [N/H]/[N/H]_⊙ = 3.5 as expected for the N-rich CSM of Kepler's SNR, while the Ne, Mg, Si and S abundances are allowed to be free parameters. Figure <ref> shows example spectra and best fitting models; the small insert figure shows the clear effect of the Doppler shifting on the Si-line of the knot spectra.The first four panels of figure <ref> plot the abundances of the various knots from our spectral fits.The abundances of all the N knots plus NE1 and NE2 are quite similar and show strong enhancements of Si and S compared to Fe as well as the lighter elements.This is also the case for the Ej knots, but with less contrast between Si/S and the other species. The SW knots plus NE3 appear to follow a different abundance pattern, while the CSM knot abundances are close to the solar ratios for Ne and Mg, but show some enhancement (by factors of a few) in Si and S (albeit with large uncertainties).The last panel of this figure shows integrated yields from a variety of published SN Ia explosion models (see the figure caption for details and citations) that indicate that the ejecta knot abundances we find are broadly consistent with theoretical expectations.However, it is highly unlikely that the knots should contain material with the spatially integrated yields, but rather they should reflect the composition of the location in the exploding white dwarf where they were formed. Indeed the high abundances of Si and S relative to Fe (especially for NE1 and NE2 and the N knots) indicate that these knots formed at the outer, partially burned layer of the exploding white dwarf.The presence of Fe and low O abundance further restrict their origin to the partial Si-burning regime (e.g., mass coordinate range of 0.7–0.9 M_⊙ in W7, ).A future study will use the fitted abundances in more detail to better identify where these knots formed in the explosion.Our fits are able to accurately represent the various knot spectra across the range of significant compositional differences we obtain. We also find that the best-fitting electron temperatures (kT_ e) and ionization ages (n_ et) differ from knot to knot (kT_ e∼ 0.2 - 1 keV, n_ et∼7×10^9-5×10^11 cm^3 s).In some cases these are correlated in interesting ways.For example the O abundance and ionization age in knot Ej1-2 are larger than those in the adjacent Ej3-4 knot: [O/C]/[O/C]_⊙ = 19 -23 and n_ et = (1.5 -1.6)×10^11 for Ej1-2, [O/C]/[O/C]_⊙ = 4-5 and n_ et = (5.9-6.9)×10^10 for Ej3-4.These spectral results argue for a significant CSM interaction for knot Ej1-2, in agreement with the argument put forward above to explain the proper motion differences.Radial velocity determination can be sensitive to the background subtraction. To test this, we replaced the individual local background regions with an annular (r =2.4^'–3.5^') blank sky region surrounding the remnant.The spectral fits were as good, but the precise values of the measured speeds differed, on average, by ∼1,500 km s^-1 with the local background results being higher than the blank sky background in nearly all regions (exceptions were the CSM and Ej regions). This trend is expected because using local background regions removes contaminating emission with a different velocity (from, e.g., the other hemisphere of the remnant) projected across the knot spectral extraction region. Such contamination tends to reduce a knot's observed speed compared to its actual speed.The numerical accuracy of our velocity measurements with the ACIS-S detector is limited by ACIS gain calibration uncertainties[See http://web.mit.edu/iachec/ for the current calibration status].For example, <cit.> showed a discrepancy in the radial velocity measurements of ∼500–2,000 km s^-1 between the ACIS-S and ACIS-I detectors for a set of knots similar to those we study here, which we argued was likely a result of uncertain gain calibration. Still, the large velocities we measure for most knots (> 5,000 km s^-1) remain significant even given the level of systematic uncertainty due to instrumental effects mentioned here and background subtraction discussed in the previous paragraph.The two rightmost columns of Table <ref> summarize the radial velocity fits.The N and SW knots have high radial speeds (5,590 km s^-1 < v < 10,020 km s^-1) that are ∼2–3 times higher than the ejecta knot speeds quoted by <cit.> from HST proper motions: ∼ 1,600–3,000 km s^-1.On the other hand, the CSM and Ej knots show relatively low speeds (< 2,300 km s^-1). § DISCUSSION§.§ Undecelerated Ejecta Knots and the Kinematic Center of Kepler's SNR Given the known age of Kepler's SNR (401.7 yr at the mean time of the third epoch observation) we can use our proper motion vectors to extrapolate the position of each knot back to its location at the time of explosion. Initially, we assume that each knot moves without any deceleration. Figure <ref> (left panel) shows the 2006 locations (small boxes), the distance traveled (green lines) and the initial locations in 1604 (green circles) of each of the knots. Five knots extrapolate back to a consistent position (for values see Table <ref>), which we identify as the kinematic center of the explosion. This result agrees well with other estimates for the explosion center, but, since it relies on knots that are nearly undecelerated, it is much less sensitive to systematic errors due to the spatial variation of the expansion rate across Kepler's SNR. We use our kinematic center to determine the expansion index (m in the relation r∝ t^m) as m = μ× t / r, which depends only on each knot's measured proper motion (μ), its distance from the expansion center (r) and the remnant's age (t).The expansion indices (see Fig. <ref>) range from low values, m < 0.25, indicating significant deceleration to high values, m > 0.75, indicating little to no deceleration.For a power-law evolution of radius with time as we use here, it is possible to estimate the effects of deceleration on the distance traveled by the knot and thereby obtain a more accurate estimate for the kinematic center. We assume that the expansion index is constant with time, which is only a good approximation for knots with high expansion indices; so we restrict our analysis to the same five knots introduced in the preceding paragraph.We integrate the time evolution of velocity v = v_f (t/t_f)^m-1 over the age of the SNR (t_f) to obtain the simple result for the distance traveled: r=v_f t_f / m.We start with the results from the previous paragraph, which yielded a kinematic center and an estimate of m for each knot. The distance each knot has moved is now redetermined assuming decelerated motion according to its m value (although m was restricted to values of 1 or less) and the equation introduced a few lines above.Averaging these values leads to a new estimate for the kinematic center.This process was iterated until the individual m values and the location of the kinematic center converged, which took about 30 iterations. The kinematic center shifts slightly south from the case of undecelerated motion above (see Table <ref>), but the difference between the two estimates is not highly significant, only ∼1 σ (∼6^'').Note how the m valueshave changed only slightly as well. Combining the proper motions and radial velocities allows us to determine the 3-dimensional (3D) space velocities of these X-ray knots.The distance to Kepler's SNR is not well known with estimates ranging from ∼4.0 kpc to > 7 kpc; here we use a value of 5 kpc which is consistent with H1 absorption measurements <cit.> and recent optical proper motion measurements of Balmer shocks <cit.>.Figure <ref> presents the scatter plot of 3D space velocity versus expansion index.There is a clear trend for low space velocity knots to have small expansion indices, while the high space velocity knots tend to have large expansion indices. It is notable that the three knots with the highest space velocities also have large expansion indices; for these specific knots we determine space velocities of 9,100 km s^-1 ≲ v_ 3D≲ 10,400 km s^-1 and expansion indices of 0.75 ≲ m ≲ 1.0.Thus not only are these knots expanding at nearly the free expansion rate, they are moving with space velocities that are comparable to the expansion speed of Si ejecta in SN Ia near maximum brightness <cit.>.We can also estimate the 3D radial locations of the knots using each knot's current projected distance from the kinematic center and the angle defined by the radial and transverse (proper motion) speeds. The angle obviously depends on the remnant's distance. As a reference for comparison, we use the maximum projected extent of Kepler's SNR from the kinematic center, which is ∼2.3^' (3.35 pc for a distance of 5 kpc); this occurs at the northwest protuberance. A knot moving with a constant speed of 10,000 km s^-1 would reach a radius of 4.1 pc in the lifetime of Kepler's SNR, and this radius would be equal to the remnant's maximum projected extent for a distance of 6.1 kpc.For the fastest moving knots (N1, N2, and N4), the estimated 3D radial locations are 1.57, 1.26, and 1.48 times the maximum projected extent assuming a distance of 5 kpc. Values are larger than the simple example given because these knots have suffered some deceleration.For a distance of 7 kpc, the radial locations are 1.16, 0.95, and 1.09, respectively, times the maximum projected extent.These considerations tend to favor distances to Kepler's SNR on the larger range of those reported.There are interesting relationships between the elemental composition of the knots and their space velocities and inferred extent of deceleration. The N series of knots show both high speed and low deceleration (m>0.5), along with a clear ejecta-dominant composition showing high Si and S abundances (Fig. <ref>).Although the knots NE1 and NE2 show similar abundances, they appear to have been decelerated more (m ≈ 0.5) and are currently moving more slowly.The SW knots are kinematically similar to the N knots, but are noticeably different in composition. On the other hand, the CSM and Ej knots have low space velocities (v_ 3D < 3,000 km s^-1) together with small expansion indices (m < 0.5) and have relatively less contrast between their light (O, Ne, Mg) and heavy (Si, S, Fe) element abundances. Based on the abundance patterns, the CSM knots appear to be dominated by the ambient medium while the Ej knots are more dominated by ejecta. §.§ Global Evolution of Kepler's SNR The mean expansion index of Kepler's SNR from published studies using high resolutionimages is m∼0.5 with evidence for higher rates of expansion in the south compared to the north <cit.>.The dashed circles plotted in Fig. <ref> are centered on our new kinematic center for either undecelerated (left panel) or decelerated (right panel) motion and were chosen to match the northern and southern extents of the SNR.The north/south radii (numerical values given in Table <ref>) differ by 9^+5_-4 % for undecelerated motion (left) and 2±4 % for decelerated motion (right), where the uncertainties were estimated from the standard deviation of the radial scatter of the plotted contour about the estimated best-fit circle.<cit.> calculated models for Kepler's SNR assuming the progenitor system was a symbiotic binary (a white dwarf and a 4-5 M_⊙ AGB star) moving toward the northwest with a velocity of 250 km s^-1.As in the model of <cit.>, this produces an asymmetric wind around the progenitor with the densest regions at the stagnation point (i.e., where the momentum of the wind and ambient medium equilibrate) ahead of the star in the direction of motion.In their model A, which provides a decent description of Kepler's SNR, the forward-shock interaction with the wind bubble begins ∼300 yr after the explosion when the forward shock was at a radius of ∼2.7 pc.This encounter has a strong, immediate, effect on the expansion index in the direction toward the wind-stagnation point where the dense shell material causes the forward shock to decelerate quickly. With time, the radial asymmetry of the remnant also grows, but the decrease of the expansion index is more immediate and dramatic.As seen in Fig. 7 in <cit.>, the expansion index m at the stagnation point changes quickly from a value of ∼0.8 to ∼0.45 during the first ∼30 yr after the encounter, while the expansion in the opposite direction remains high.Over the same time frame the remnant radius in the direction of the stagnation point grows more slowly so that the remnant starts to become asymmetric, but the difference of the radii is not so large (≲2%). Thus this model is consistent both with the large north-south variation of expansion index observed in Kepler's SNR <cit.> and the modest north-south radius difference as shown in Fig. <ref> (right). In addition, we find that the bilateral protrusions at the southeast/northwest rims match well with a simple elliptical geometry (plotted as the yellow figures in Fig. <ref>).In each panel the ellipse is centered on the kinematic center for undecelerated (left) or decelerated (right) knot motion and the axis lengths and orientation are matched to the shape of the nonthermal filament on the eastern rim.Although the physical origin of these prominent structures is not yet resolved, there have been suggestions that the symmetric protrusions are related to the explosion <cit.>.If so, the notable agreement of the shape and orientation of the protrusions to a simple elliptical geometry centered on and symmetric about the kinematic center derived from the decelerated knot motions, offers further support for this being the site of the explosion. §.§ Spatial Density Variations in the Ambient Medium In one-dimensional SNR evolutionary models, a high value of the expansion parameter indicates that the remnant is interacting with a low density ambient medium. Here we derive an estimate of the density required. <cit.> investigated the dynamical evolution of SN Ia assuming an exponential ejecta density profile.An expansion parameter m ≳ 0.75 is realized in their models at a scaled time of t^'≲ 0.1 (see Fig. 2f in their paper). The scaled time is related to the pre-shock ambient medium density (n_ H), and the remnant's age, explosion energy (E_51 in units of 10^51 ergs), and ejected mass (M_ ej) as n_ H≈ 0.24 (t^')^3E_51^-3/2 (M_ ej/M_ ch)^5/2cm^-3 for the remnant's age of 401.7 yr.Assuming typical values for the explosion energy (E_51 = 1) and ejected mass (M_ ej = M_ ch = 1.4M_⊙), allows us to convert the scaled time for nearly undecelerated motion to an upper-limit on the ambient medium density (assumed uniform) of n_ H < 2.4 × 10^-4 cm^-3.This value is comparable to the expected density at the remnant's location above the Galactic plane in the absence of stellar mass loss.However, a compact knot, overdense with respect to its surroundings, undergoes a considerably different type of evolution than does an idealized spherically symmetric distribution of ejecta. To investigate this scenario, we follow <cit.> who simulated the evolution of clumped ejecta in Tycho's SNR to understand the conditions under which an ejecta knot could survive as it propagates out to, and possibly deforms, the forward shock.In this scenario a compact ejecta knot enters the reverse shock at some time and propagates outward through the high-pressure zone of shocked ejecta. The impact of the reverse shock on the knot drives a transmitted shock into the knot crushing it; in time this shock exits the knot which sends a rarefaction wave through it.Meanwhile instabilities develop along the knot boundary due to shear flow and rapid local accelerations that result in the destruction of the cloud after a few cloud-crushing times.This key timescale is given by t_ cc = χ^1/2 r_ cloud/ v_ shock <cit.>, where χ is the density contrast of the knot with respect to the intercloud medium, r_ cloud is the initial radius of the cloud, and v_ shock is the velocity of the shock in the intercloud medium.Note that this timescale was originally developed for the case of an interstellar cloud impacted by the forward shock of a SN explosion, but is applicable to the closely analogous knot/reverse shock situation we have here.The survivability of an ejecta clump depends on its size (relative to the size of the high pressure shocked ejecta zone) and density contrast with respect to the rest of the ejecta.Smaller clumps with higher density contrast survive for longer.Moreover the drag on a high speed clump depends sensitively on the density contrast, in the sense that a higher density contrast produces relatively less drag.<cit.> therefore find that in order for there to be undecelerated clumps of ejecta near the limb of Tycho's SNR some 400 year after explosion, the density contrast needs to be high, χ>100.However, another option for increasing the survivability of an ejecta knot is to have it interact with the reverse shock when that shock is still forming during its early evolutionary phase. For example Fig. 8 ofshows a knot that survives and continues expanding out to deform the forward shock from its initial interaction with the reverse shock at a scaled time of t^' = 0.217 through to t^'∼ 0.8.The same scaling applies here as in the first paragraph of this section so a value of the scaled time of ∼0.8 corresponds to an ambient medium density of n_ H∼ 0.1 cm^-3.Such a low density would additionally allow for clumps with lower density contrast to survive to the current age.Thus, to summarize, the presence of undecelerated knots in Kepler's SNR requires either that those knots were generated with a high initial density contrast (χ>100) or that the ambient medium contains lower-density (n_ H∼ 0.1 cm^-3) windows or gaps through which potentially lower density contrast knots have propagated.The evidence that Kepler's SNR is embedded in a dense environment is strong.Yet our work now suggests that the ambient medium could be structured including both higher and lower density regions. One possibility, as briefly explored by <cit.>, might be that the donor star's wind has sculpted a dense disk-like structure with lower densities perpendicular to the disk plane.Another option could rely on an “accretion wind” from the accreting white dwarf <cit.>, which has the potential to blow a large, low density cavity that allows for ejecta to expand rapidly <cit.>.If the progenitor system to Kepler's SNR had a bipolar outflow <cit.>, the ambient density along the polar axis could be much lower than elsewhere. §.§ Implications for the Left-Over Companion Star Our new kinematic center allows us to reopen the search for a surviving donor star under the SD scenario for the progenitor to Kepler's SNR.The most extensive study to date was carried out by <cit.>.These authors identified two dozen stars from HST with V-band luminosities greater than ∼10 L_⊙ assuming they lie at the distance of Kepler's SNR.We indicate these stars on Fig. <ref> with cyan colored circles. <cit.> use ground-based optical spectroscopy to measure radial velocities for these stars, although some of the candidates (E, B, P, H, and K) were blended in the ground-based data, and in addition it was not possible to obtain a reliable radial velocity measurement for star I.The radial velocities were compared to two velocity distributions: one for field stars based on the Besançon model <cit.> of galactic dynamics and the other based on the distribution of radial velocities for SD donors ranging from main-sequence to giant stars <cit.>. A probability was assigned to each star based on a Monte Carlo simulation.None of the stars were significant outliers with respect to the Besançon model, although a number were inconsistent with the expected donor distribution. <cit.> consider candidates E1, E2, K1, L, and N to be the most notable for further follow-up, since they have L > 20 L_⊙ and a modest probability for being consistent with the expected radial velocity distribution for a donor star.We can now assess the probability of positional agreement between the explosion center and each candidate.The 3-σ limit on the allowed distance is 15^'' to which we add 3.5^'' to account for the donor's possible proper motion (<200 km s^-1).Thirteen of the HST candidate stars fall within this area, including stars L (the most luminous candidate with L_V = 86 L_⊙), I (no radial velocity measurement), and G (second highest donor probability based on radial velocity), although the later two stars are of modest luminosity (L_V = 7 L_⊙ and 9L_⊙, respectively).Although there remains no obvious donor candidate for the traditional SD-scenario, our new kinematic center has ruled out many of the interesting candidates suggested for additional follow-up and has significantly reduced the search area for donor candidates for modified SD-scenarios <cit.>.§ CONCLUSIONS Most of our key results are based on the proper motion analysis of the four available epochs (from 2000 to 2014) ofACIS-S X-ray observations of Kepler's SNR.We have discovered five X-ray–emitting knots with no detectable optical emissionthat are moving with nearly undecelerated motion (i.e., with expansion indices > 0.75).The proper motion of these knots extrapolate back, over the age of the remnant, to a consistent and accurate center when their (modest) deceleration is included. A number of prominent structures in the remnant display a notable symmetry about the new kinematic center. For example the similarity between the northern and southern radii suggests that the forward shock of the remnant has encountered the northern density enhancement fairly recently, within the last 100 years or so, as some models argue <cit.>.The symmetric shape, extent and orientation of the southeast/northwest protrusions about the kinematic center add evidence to arguments that these protrusions may be related to the explosion process itself.Our spectral analysis provides information on the composition of the knots. We report on three knots with near-solar abundances that show little proper motion or radial velocity. The Ej knots were selected because their associated Hα shocks had measured proper motions from HST data <cit.>.In the X-ray band these knots have metal-enhanced abundances but with a larger abundance of low-Z species (O, Ne, Mg) compared to Si, S, and Fe than the other X-ray ejecta knots studied here.The Ej knots show high amounts of deceleration and low 3D velocities. Beyond this, however, there is no simple relationship between knot composition and motion.The N and NE series of knots share similar abundance patterns with high Si and S abundances, some Fe, and very low O abundances but have a range of expansion indices (from 0.45 to 0.95).On the other hand the SW knots show 3D speeds and expansion indices that fall between the NE and N knots, but their spectra show considerably less enhancement of the Si and S abundances compared to Fe.This strongly indicates that the origin of the knots (traced by composition) is independent of the kinematics of the knots (traced by the expansion index). The measurement of radial velocities from spectral analysis of Chandra ACIS data is subject to systematic uncertainty from detector (gain uncertainty) and analysis (background subtraction) effects, so we summarize the key results from these measurements separately here.We find that the 3D space velocities of the highest speed knots are in the range 9,100 km s^-1 ≲ v_ 3D≲ 10,400 km s^-1, which is similar to the expansion speed of Si-rich ejecta seen in the optical spectra of SN Ia near maximum light.We also find a correlation between 3D speed and expansion index; the sense of the correlation is not surprising: higher speeds correlate with higher expansion indices and vice versus.We looked into the conditions that would allow for the existence of high speed, undecelerated knots in Kepler's SNR some 400 years after explosion, using the article by <cit.> as a useful guide.One option would require that the knots formed with a high density contrast (more than 100 times the density of the interknot medium); another would require that knots with potentially lower density contrast propagated through an ambient medium with a relatively low density (<0.1 cm^-3). We favor the latter interpretation since the generation of high-density-contrast clumps in a SN Ia explosion seems less plausible to us than the possibility that the environment of Kepler's SNR contains gaps or windows of lower density gas.Our new kinematic center has well-defined positional uncertainties which have allowed us to refine the search for possible surviving donor stars under the single-degenerate (SD) scenario for SN Ia.As shown before <cit.> there are no viable candidates for a traditional SD-scenario.Nevertheless our new position for the explosion center rules out several interesting candidates suggested by others for further follow-up and has greatly reduced the area to be searched for fainter donor stars under more exotic SD scenarios.Our work has added new important pieces of evidence to the enigmatic remnant of Kepler's SN that bear on both the nature of the explosion and the structure of the ambient medium.Further study of the detailed composition of the new high-speed ejecta knots should allow us to identify the conditions of the burning front where they formed during the explosion. Mapping the positions and velocities of other high-speed ejecta knots in Kepler's SNR should allow us to determine the extent of the low density regions within the ambient medium. This will be important information for understanding the progenitor system. T.S. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 16J03448.This research was also supported in part by NASA grant NNX15AK71G to Rutgers University.J.P.H. acknowledges the hospitality of the Flatiron Institute which is supported by the Simons Foundation.We also thank Drs. Manabu Ishida, Yoshitomo Maeda, Ryo Iizuka, Saurabh Jha, Carles Badenes, and James Stone for helpful discussion and suggestions in preparing this article.yahapj[Audard et al.(2001)]2001A A...365L.329A Audard, M., Behar, E., Güdel, M., et al. 2001, , 365, L329[Badenes et al.(2007)]2007ApJ...662..472B Badenes, C., Hughes, J. 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"authors": [
"Toshiki Sato",
"John P. Hughes"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726190028",
"title": "Freely Expanding Knots of X-ray Emitting Ejecta in Kepler's Supernova Remnant"
} |
[email protected] Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-University Munich, 80333 Munich, GermanyDepartments of Physics, Yale University, New Haven, Connecticut 06520, USARaymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel The coupling between a 2D semiconductor quantum well and an optical cavity gives rise to combined light-matter excitations, the exciton-polaritons. These were usually measured when the conduction band is empty, making the single polariton physics a simple single-body problem. The situation is dramatically different in the presence of a finite conduction band population, where the creation or annihilation of a single exciton involves a many-body shakeup of the Fermi sea. Recent experiments in this regime revealed a strong modification of the exciton-polariton spectrum. Previous theoretical studies concerned with nonzero Fermi energy mostly relied on the approximation of an immobile valence band hole with infinite mass, which is appropriate for low-mobility samples only; for high-mobility samples, one needs to consider a mobile hole with large but finite mass. To bridge this gap we present an analytical diagrammatic approach and tackle a model with short-ranged (screened) electron-hole interaction, studying it in two complementary regimes. We find that the finite hole mass has opposite effects on the exciton-polariton spectra in the two regimes: In the first, where the Fermi energy is much smaller than the exciton binding energy, excitonic features areenhanced by the finite mass. In the second regime, where the Fermi energy is much larger than the exciton binding energy, finite mass effects cut off the excitonic features in the polariton spectra, in qualitative agreement with recent experiments. Fermi-edge exciton-polaritons in doped semiconductor microcavities with finite hole mass Moshe Goldstein Received: date / Accepted: date ======================================================================================== § INTRODUCTION When a high-quality direct semiconductor 2D quantum well (QW) is placed inside an optical microcavity, the strong coupling of photons and QW excitations gives rise to a new quasiparticle: the polariton. The properties of this fascinating half-light, half-matter particle strongly depend on the nature of the involved matter excitations.If the Fermi energy is in the semiconductor band gap, the matter excitations are excitons. This case is theoretically well understood<cit.>, and the first observation of the resulting microcavity exciton-polaritons was already accomplished in 1992 by Weisbuch et al. <cit.>. Several studies on exciton-polaritons revealed remarkable results. For example, exciton-polaritons can form a Bose-Einstein condensate <cit.>, and were proposed as a platform for high-T_c superconductivity <cit.>.The problem gets more involved if the Fermi energy is above the conduction band bottom, i.e., a conduction band Fermi sea is present. Then the matter excitations have a complex many-body structure, arising from the complementary phenomena of Anderson orthogonality <cit.> and the Mahan exciton effect, entailing the Fermi-edge singularity <cit.>. An experimental study of the resulting “Fermi-edge polaritons” in a GaAs QW was first conducted in 2007 by Gabbay et al. <cit.>, and subsequently extended by Smolka et al. <cit.> (2014). A similar experiment on transition metal dichalcogenide monolayers was recently published by Sidler et al. <cit.> (2016).From the theory side, Fermi-edge polaritons have been investigated in Ref. <cit.>. However, in these works only the case of infinite valence band hole mass was considered, which is the standard assumption in the Fermi-edge singularity or X-ray edge problem.Such a model is valid for low-mobility samples only and thusfails to explain the experimental findings in <cit.>: there, a high-mobility sample was studied, for which an almost complete vanishing of the polariton splitting was reported. Some consequences of a finite hole mass for polaritons were considered in a recent treatment <cit.>, but without fully accounting for the so-called crossed diagrams that describe the Fermi sea shakeup, as we further elaborate below.The aim of the present paper is therefore to study the effects of both finite mass and Fermi-edge singularity on polariton spectra in a systematic fashion. This is done analytically for a simplified model involving a contact interaction, which nethertheless preserves the qualitative features of spectra stemming from the finite hole mass and the presence of a Fermi sea. In doing so, we distinguish two regimes, with the Fermi energy μ being either much smaller or much larger than the exciton binding energy E_B.For the regime where the Fermi energy is much larger than the exciton binding energy, μ≫ E_B, several treatments of finite-mass effects on the Fermi-edge singularity alone (i.e., without polaritons) are available, both analytical and numerical. Without claiming completeness, we list <cit.>. In our work we have mainly followed the approach ofRef. <cit.>, extending it by going from 3D to 2D and, more importantly, by addressing the cavity coupling which gives rise to polaritons. For infinite hole mass the sharp electronic spectral feature caused by the Fermi edge singularity can couple with the cavity mode to create sharp polariton-type spectral peaks <cit.>. We find that the finite hole mass cuts off the Fermi edge singularity and suppresses these polariton features.In the opposite regime of μ≪ E_B, where the Fermi energy is much smaller than the exciton binding energy, we are not aware of any previous work addressing the modification of the Fermi-edge singularity due to finite mass.Here, we propose a way to close this gap using a diagrammatic approach. Interestingly, we find that in this regime the excitonic singularities are not cut off, but are rather enhanced by finite hole mass, in analogy to the heavy valence band hole propagator treated in <cit.>. This paper has the following structure: First, before embarking into technical details, we will give an intuitive overview of the main results in Sec. <ref>. Detailed computations will be performed in subsequent sections:In Sec. <ref>, the full model describing the coupled cavity-QW system is presented. The key quantity that determines its optical properties is the cavity-photon self-energy Π, which we will approximate by the electron-hole correlator in the absence of a cavity. Sec. <ref> shortly recapitulates how Π can be obtained in the regime of vanishing Fermi energy, for infinite and finite hole masses. Then we turn to the many-body problem in the presence of a Fermi sea in the regimes of small (Sec. <ref>) and large Fermi energy (Sec.<ref>). Using the results of the previous sections, polariton properties are addressed in Sec. <ref>. Finally, we summarize our findings and list several possible venues for future study in Sec. <ref>. § SUMMARY OF RESULTS In a simplified picture, polaritons arise from the hybridization of two quantum excitations with energies close to each other, the cavity photon and a QW resonance <cit.>. The resulting energy spectrum consists of two polariton branches with an avoided crossing, whose light and matter content are determined by the energy detuning of the cavity mode from the QW mode. While the cavity photon can be approximated reasonably by a bare mode with quadratic dispersion and a Lorentzian broadening due to cavity losses, the QW resonance has a complicated structure of many-body origin. The QW optical response function is rather sensitive to nonzero density of conduction band (CB) electrons.Roughly, it tends to broaden QW spectral features, which contribute to the spectral width of polariton lines.A more detailed description of the polariton lines requires finding first the optical response function Π(, Ω) of the QW alone (without polaritons). Here,and Ω are, respectively, the momentum and the energy of an incident photon probing the optical response. The imaginary part of Π(,Ω), A(, Ω) = -Im[Π(, Ω)]/π, defines the spectral function of particle-hole excitations in the QW. In the following, we discuss the evolution of A(, Ω) as the chemical potential μ is varied, concentrating on the realistic case of a finite ratio of the electron and hole masses. We assume that the temperature is low, and consider the zero-temperature limit in the entire work. In addition, we will limit ourselves to the case where the photon is incident perpendicular to the QW, i.e. its in-plane momentum is zero, and study A(Ω) ≡ A(Q=0, Ω). The sought optical properties of the coupled cavity-QW system are determined by the retarded dressed photon Green's function <cit.>: D^R(Q,Ω) = 1/Ω - ω_Q + i0^+ - Π(Q,Ω) ,where ω_Q is the (quadratic) cavity mode dispersion, and Π(Q,Ω) is the retarded photon self-energy. This dressed photon is nothing but the polariton. The spectral function corresponding to (<ref>) is given by 𝒜(Q,ω) = -1/πIm[D^R(Q,ω)].𝒜 determines the absorption respectively reflection of the coupled cavity-QW system, which are the quantities typically measured in polariton experiments like <cit.>, <cit.>.In the absence of free carriers (μ is in the gap), a CB electron and a hole in the valence band (VB) create a hydrogen-like spectrum of bound states. In the case of a QW it is given by the 2D Elliot formula (see, e.g., <cit.>). Being interested in the spectral function close to the main exciton resonance, we replace the true Coulomb interaction by a model of short-ranged interaction potential of strength g [see Eqs. (<ref>) and (<ref>)]. As a result, there is a single bound state at an energy E_G - E_B(g), which we identify with the the lowest-energy exciton state. Here, E_G is the VB-CB gap, and energies are measured with respect to the minimum of the conduction band. A sketch of A(Ω) is shown in Fig. <ref>.For μ>0, electrons start to populate the CB. If the chemical potential lies within the interval 0<μ≪ E_B, then the excitonic Bohr radius r_B remains small compared to the Fermi wavelength λ_F of the electron gas, and the exciton is well defined. Its interaction with the particle-hole excitations in the CB modifies the spectral function A(Ω) in the vicinity of the exciton resonance. The limit of an infinite hole mass was considered by Nozières et al. <cit.>: Due to particle-hole excitations of the CB Fermi sea, which can happen at infinitesimal energy cost, the exciton resonance is replaced by a power law spectrum, see inset of Fig. <ref>. In terms of the detuning from the exciton threshold,ω = Ω - Ω_T^exc , Ω_T^exc = E_G + μ - E_B,the spectral function, A_exc(ω) = - Im[Π_exc(ω)]/π, scales as: A_exc(ω)|_M = ∞∼θ(ω) E_B/ω(ω/μ)^α^2 , ω≪μ.The effective exciton-electron interaction parameter α was found by Combescot et al. <cit.>, making use of final-state Slater determinants. In their work, α is obtained in terms of the scattering phase shift δ of Fermi level electrons off the hole potential, in the presence of a bound state, as α = |δ/π-1|. For the system discussed here this gives <cit.>:α = 1/|ln(μ/E_B)|. We re-derive the result for α diagrammatically (see Sec. <ref>), in order to extend the result of Combescot et al. to the case of a small but nonzero CB electron-VB hole mass ratio β, whereβ = m/M.While the deviation of β from zero does not affect the effective interaction constant α, it brings qualitatively new features to A(Ω), illustrated in Fig. <ref>. The origin of these changes is found in the kinematics of the interaction of the exciton with the CB electrons. Momentum conservation for finite exciton mass results in phase-space constraints for the CB particle-hole pairs which may be excited in the process of exciton creation. As a result, the effective density of states ν(ω) of the pairs with pair energy ω (also corresponding to the exciton decay rate) is reduced from ν(ω) ∼ω at β=0 <cit.> to ν(ω) ∼ω^3/2 when ω is small compared to the recoil energy E_R=βμ. A smaller density of states for pairs leads to a reduced transfer of the spectral weight to the tail; therefore, the delta function singularity at the exciton resonance survives the interaction with CB electrons, i.e. β >0 tends to restore the exciton pole, and one finds: A_exc(ω)|_M<∞ = A_exc,incoh.(ω) θ(ω) + β^α^2 E_B δ(ω),A_exc,incoh.(ω) ∼ E_B α^2/√(ωβμ)β^α^2 ω≪βμ α^2/ω(ω/μ)^α^2 βμ≪ω≪μ.The main features of this spectral function are summarized in Fig. <ref>:As expected, the exciton recoil only plays a role for small frequencies ω≪βμ, while the infinite mass edge singularity is recovered for larger frequencies. The spectral weight of the delta peak is suppressed by the interaction. For β→ 0 and α 0, we recover the infinite mass result, where no coherent part shows up. If, on the opposite, α^2 → 0 but β≠ 0, the weight of the delta peak goes to one: The exciton does not interact with the Fermi sea, and its spectral function becomes a pure delta peak, regardless of the exciton mass. A partial survival of the coherent peak at α,β≠ 0 could be anticipated from the results of Rosch and Kopp <cit.> who considered the motion of a heavy particle in a Fermi gas of light particles. This problem was also analyzed by Nozières <cit.>, and the coherent peak can be recovered by Fourier transforming his time domain result for the heavy particle Green's function. At this point, let us note the following: for μ > 0, the hole can bind two electrons with opposite spin, giving rise to trion features in the spectrum. We will not focus on those, since, for weak doping, their spectral weight is small in μ (more precisely, in μ/E_T, where E_T ≪ E_B is the trion binding energy), and they are red detuned w.r.t. the spectral features highlighted in this work. In the regime of μ≫ E_B ≫ E_T, trions should be neglible as well. Some further discussion of trion properties can be found in Appendix <ref>. Upon increase of chemical potential μ, the CB continuum part (inset of Fig. <ref>) starts building up into the well-known Fermi-edge singularity (FES) at the Burstein-Moss <cit.> shiftedthreshold, Ω_T^FES = E_G + μ. For finite mass (β≠ 0), the FES will however be broadened by recoil effects (see below). At the same time, the delta function singularity of Eq. (<ref>) at the absorption edge vanishes at some value of μ. So, at higher electron densities, it is only the FES which yields a nonmonotonic behavior of the absorption coefficient, while the absorption edge is described by a converging power law with fixed exponent, see Eq. (<ref>). This evolution may be contrasted to the one at β=0. According to <cit.>, the counterparts of the absorption edge and broadened FES are two power law nonanalytical points of the spectrum which are present at any μ and characterized by exponents continuously evolving with μ.A more detailed discussion of the evolution of absorption spectra as μ increases from small to intermediate to large values is presented in Appendix <ref>.Let us now consider the limit μ≫ E_B, where the FES is the most prominent spectral feature, in closer detail. In the case of infinite hole mass (β=0), and in terms of the detuning from the FES threshold, ω = Ω - Ω_T^FES, Ω_T^FES = E_G + μ,the FES absorption scales as <cit.>: A_FES(ω)|_M = ∞∼θ(ω) (ω/μ)^-2g,as illustrated in the inset of Fig. <ref>. In the above formula, the interaction contribution to the treshold shift, which is of order gμ, is implicitly contained in a renormalized gap E_G.What happens for finite mass? This question was answered in <cit.>: As before, the recoil comes into play, effectively cutting the logarithms contributing to(<ref>). Notably, the relevant quantity is now the VB hole recoil, since the exciton is no longer a well defined entity.The FES is then replaced by a rounded feature, sketched in Fig. <ref>, which sets in continuously:A_FES(ω)|_M<∞∼(ω/βμ)^3 β^-2g·θ(ω)ω≪βμ (√((ω - βμ)^2 + (βμ)^2)/μ)^-2g βμ≪ω≪μ.Eq. (<ref>) can be obtained by combining and extending to 2D the results presented in Refs. <cit.>.The maximum of Eq. (<ref>) is found at the so-called direct threshold, ω_D = βμ (see Fig. <ref>(a)). This shift is a simple effect of the Pauli principle:the photoexcited electron needs to be placed on top of the CB Fermi sea. The VB hole created this way, with momentum k_F, can subsequently decay into a zero momentum hole, scattering with conduction band electrons [see Fig. <ref>(b)]. These processes render the lifetime of the hole finite, with a decay rate ∼ g^2 βμ. Within the logarithmic accuracy of the Fermi edge calculations, this is equal to βμ, the cutoff of the power law in Eq. (<ref>) (See Sec. <ref> for a more detailed discussion). As a result, the true threshold of absorption is found at the indirect threshold, ω_I = 0. Due to VB hole recoil, the CB hole-electron pair density of states now scales as ν(ω) ∼ω^3, leading to a similar behavior of the spectrum, see Fig. <ref>.We note that at finite ratio β = m/M, raising the chemical potential μ from μ≪ E_B to μ≫ E_B results in a qualitative change of the threshold behavior from a singular one of Eq. (<ref>), to a converging power law, see the first line of Eq. (<ref>). Simultaneously, a broadened FES feature appears in the continuum, at ω>0.The difference in the value of the exponent in the excitonic result [Eq. (<ref>)], as compared to the FES low-energy behavior [Eq. (<ref>) for ω≪βμ], can be understood from the difference in the kinematic structure of the excitations:In the exciton case, the relevant scattering partners are an exciton and a CB electron-hole pair. In the FES case, one has the photoexcited electron as an additional scattering partner, which leads to further kinematic constraints and eventually results in a different low-energy power law. In the frequency range βμ≪ω≲μ, the physics is basically the same as in the infinite hole mass case (β=0). There, the behavior near the lowest threshold (which is exciton energy for μ≪ E_B and the CB continuum for μ≫ E_B) is always ∼ω^(1-δ/π)^2-1=ω^(δ/π)^2-2δ/π. But in the first case (μ≪ E_B), δ∼π - α is close to π (due to the presence of a bound state), so the threshold singularity is in some sense closeto the delta peak , ∼Im[1/(ω+i0^+)], that one would have for μ=0, whereas in the second case (μ≫ E_B), δ∼ g is close to zero, so the threshold singularity is similar to a discontinuity. Having discussed spectral properties of the QW alone, we can now return to polaritons. Their spectra A_p (ω) can be obtained by inserting the QW polarization as photon self-energy.While a full technical account will be given in Sec. <ref>, the main results can be summarized as follows:In the first case of study, of μ≪ E_B and finite β, the polaritons arise from a mixing of the cavity and the sharp exciton mode. The smaller the hole mass, the more singular the exciton features, leading also to sharper polariton features. Furthermore, the enhanced exciton quasiparticle weight pushes the two polariton branches further apart. Conversely, in the singular limit of infinite hole mass, the pole in the exciton spectrum turns into the pure power law familiar from previous work, resulting in broader polariton features.A comparison of the infinite and finite hole mass versions of the polariton spectra A_p(ω) when the cavity photon is tuned into resonance with the exciton is presented in Fig. <ref>. Notably, the above effects are rather weak, since the exciton is a relatively sharp resonance even for infinite hole mass.In the second case, μ≫ E_B, the matter component of the polaritons corresponds to the FES singularity, which is much less singular than the exciton. Consequently, the polaritons (especially the upper one, which sees the high-frequency tail of the FES) are strongly washed out already at β = 0. For finite hole mass, the hole recoil cuts off the FES singularity, resulting in further broadening of the polaritons. In addition, there is an overall upward frequency shift by βμ, reflecting the direct threshold effect.Fig. <ref> shows the two polariton spectra at zero detuning. The cutoff of the lower polariton for finite masses is even more drastic when the cavity is blue-detuned with respect to the threshold: Indeed, at large positive cavity detuning, the lower polariton is mostly matter-like, and thus more sensitive to the FES broadening. It therefore almost disappears, as seen in Fig. <ref>.§ MODELAfter the qualitative overview in the previous section, let us now go into more detail, starting with the precise model in question. To describe the coupled cavity-QW system, we study the following 2D Hamiltonian: H= H_M + H_L, H_M= ∑_kϵ_k a^†_ka_k - ∑_k[E_k +E_G] b^†_kb_k- V_0/𝒮∑_k, p, qa^†_k a_p b_k-qb^†_p - q, H_L=∑_Qω_Q c^†_Q c_Q -id_0/√(𝒮)∑_p,Q a^†_p+ Qb_pc_Q + h.c.Here, H_M, adapted from the standard literature on the X-ray edge problem <cit.>, represents the matter part of the system, given by a semiconductor in a two-band approximation: a_k annihilates a conduction band (CB) electron with dispersion ϵ_k = k^2/2m, while b_k annihilates a valence band (VB) electron with dispersion -(E_k + E_G) = -(k^2/2M + E_G). E_G is the gap energy, which is the largest energy scale under consideration: In GaAs, E_G ≃ 2eV, while all other electronic energies are on the order of meV. The energies are measured from the bottom of the conduction band. 𝒮 is the area of the QW, and we work in units where ħ = 1. Unless explicitly stated otherwise, we assume spinless electrons, and concentrate on the zero temperature limit. When a valence band hole is created via cavity photon absorption, it interacts with the conduction band electrons with an attractive Coulomb interaction. Taking into account screening, we model the interaction as point-like, with a constant positive matrix element V_0. The effective potential strength is then given by the dimensionless quantityg = ρ V_0,ρ = m/2 π,ρ being the 2D DOS. The appropriate value of g will be further discussed in the subsequent sections.Interactions of CB electrons with each other are completely disregarded in Eq. (<ref>), presuming a Fermi liquid picture. This is certainly a crude approximation. It can be justified if one is mostly interested in the form of singularities in the spectral function. These are dominated by various power laws, which arise from low-energy particle hole excitations of electrons close to the Fermi energy, where a Fermi-liquid description should be valid.The photons are described by H_L: We study lossless modes with QW in-plane momenta Q and energies ω_Q = ω_c + Q^2/2m_c, where m_c is the cavity mode effective mass. Different in-plane momenta Q can be achieved by tilting the light source w.r.t. the QW. In the final evaluations we will mostly set Q=0, which is a valid approximation since m_c is tiny compared to electronic masses.The interaction term of H_L describes the process of absorbing a photon while creating an VB-CB electron hole pair, and vice versa. d_0 is the interband electric dipole matrix element, whose weak momentum dependence is disregarded. This interaction term can be straightforwardly derived from a minimal coupling Hamiltonian studying interband processes only, and employing the rotating wave and electric dipole approximations (see, e.g., <cit.>).The optical properties of the full system are determined by the retarded dressed photon Green's function <cit.>:D^R(Q,Ω) = 1/Ω - ω_Q + i0^+ - Π(Q,Ω),where Π(Q,Ω) is the retarded photon self-energy. This dressed photon is nothing but the polariton. The spectral function corresponding to (<ref>) is given by 𝒜(Q,ω) = -1/πIm[D^R(Q,ω)].𝒜(Q,ω) determines the absorption respectively reflection of the coupled cavity-QW system, which are the quantities typically measured in polariton experiments like <cit.>. Our goal is to determine Π(Q,Ω). To second order in d_0 it takes the formΠ(Q,Ω) ≃ -id_0^2/𝒮∫_-∞^∞ dt θ(t) e^iΩ t×∑_k,p⟨0| b_k^†(t) a_k+Q(t) a^†_p+Q(0) b_p (0)|0|,⟩where |0⟩ is the noninteracting electronic vacuum with a filled VB, and the time dependence of the operators is generated by H_M. Within this approximation, Π(Q,ω) is given by the “dressed bubble” shown in Fig. <ref>. The imaginary part of Π(Q,ω) can also be seen as the linear response absorption of the QW alone with the cavity modes tuned away.Starting from Eq. (<ref>), in the following we will study in detail how Π(Q,ω) behaves as the chemical potential μ is increased, and distinguish finite and infinite VB masses M. We will also discuss the validity of the approximation of calculating Π to lowest order in d_0.§ ELECTRON-HOLE CORRELATOR IN THE ABSENCE OF A FERMI SEAWe start by shortly reviewing the diagrammatic approach in the case when the chemical potential lies within the gap (i.e. -E_G<μ<0). This is mainly done in order to set the stage for the more involved diagrammatic computations in the subsequent sections. In this regime of μ,Π is exactly given by the sum of the series of ladder diagrams shown in Fig. <ref>, first computed by Mahan <cit.>. Indeed, all other diagrams are absent here since they either contain VB or CB loops, which are forbidden for μ in the gap. This is seen using the following expressions for the zero-temperature time-ordered free Green's functions: G_c^(0)(k,Ω)= 1/Ω - ϵ_k + i0^+sign(ϵ_k-μ),G_v^(0)(k,Ω)= 1/Ω + E_G + E_k + i0^+sign(-E_G-E_k-μ),where the indices c and v stand for conduction and valence band, respectively, and 0^+ is an infinitesimal positive constant. For -E_G < μ< 0, CB electrons are purely retarded,while VB electrons are purely advanced. Thus, no loops are possible. Higher order terms in d_0 are not allowed as well. One can easily sum up the series of ladder diagrams assuming the simplified interaction V_0 <cit.>. Let us start from the case of infinite VB mass (β=0), and concentrate on energies |Ω - E_G| ≪ξ,where ξ is an appropriate UV cutoff of order of CB bandwidth. Since the interaction is momentum independent, all integrations in higher-order diagrams factorize. Therefore, the n-th order diagram of Fig. <ref> is readily computed: Π^(n)_ladder(Ω) = d_0^2 ρ (-g)^nln(Ω - E_G + i0^+/ - ξ)^n+1.Here and henceforth, the branch cut of the complex logarithm and power laws is chosen to be on the negative real axis.The geometric series of ladder diagrams can be easily summed:Π_ladder(Ω) = ∑_n=0^∞Π^(n)_Ladder(Ω) = d_0^2ρln(Ω-E_G + i0^+/-ξ)/1+gln(Ω-E_G + i0^+/-ξ).A sketch of the corresponding QW absorption A_ladder= -Im[Π_ladder]/π was already shown in Fig. <ref>.Π_ladder(Ω) has a pole, the so-called Mahan exciton <cit.>, at an energy of Ω - E_G = -E_B = -ξ e^-1/g.In the following, we will treat E_B as a phenomenological parameter. To match the results of the short-range interaction model with an experiment, one should equate E_B with E_0, the energy of lowest VB hole-CB electron hydrogenic bound state (exciton). Expanding Eq. (<ref>) near the pole, we obtain: Π_ladder(ω)= d_0^2E_B ρ/g^2 G^0_exc(ω) + 𝒪(ω/E_B), G^0_exc(ω)= 1/ω+i0^+,where ω = Ω-E_G + E_B, and we have introduced the bare exciton Green's function G^0_exc, similar to Ref. <cit.>.In this regime of μ, a finite hole mass only results in a weak renormalization of the energy by factors of 1+β, where β= m/M is the small CB/VB mass ratio. Furthermore, if finite photon momenta Q are considered, the exciton Green's function is easily shown to be (near the pole):G_exc^0(Q,ω) = 1/ω + Q^2/M_exc + i0^+,with M_exc = M + m = M (1+β).§ ELECTRON-HOLE CORRELATOR FOR SMALL FERMI ENERGY§.§ Infinite VB hole mass Let us now slightly increase the chemical potential μ, and study the resulting absorption. More precisely, we consider the regime0<μ≪ E_B ≪ξ.We first give an estimate of the coupling constant g = ρ V_0 Accounting for screening of the VB hole 2D Coulomb potential by the CB Fermi sea in the static RPA approximation, and averaging over the Fermi surface <cit.>one finds:g ∼ 1-8x/π x→ 0, ln(x)/xx→∞,where x= √(μ/E_0) with E_0 being the true 2D binding energy of the lowest exciton in the absence of a CB Fermi sea. In the regime under study we may assume E_B ≃ E_0 ≫μ, and therefore g ≲ 1 [Strictly speaking, this also means E_B ≲ξ, contradicting Eq. (<ref>). However, this clearly is a non-universal property, and we will not pay any attention to it in the following]. As a result, perturbation theory in g is meaningless. Instead, we will use μ/E_B as our small parameter, and re-sum all diagrams which contribute to the lowest nontrivial order in it.We will now restrict ourselves to the study of energies close to E_Bin order to understand how a small density of CB electrons modifies the shape of the bound state resonance; we will not study in detail the VB continuum in the spectrum (cf. Fig. <ref>). We first compute the contribution of the ladder diagrams;as compared to Eqs. (<ref>)–(<ref>), the result solely differs by a shift of energies: ω = Ω - Ω_T^exc , Ω_T^exc = (E_G + μ) - E_B.Also, the continuum now sets in when Ω equals Ω_T^FES = E_G + μ, which is known as the Burstein-Moss shift <cit.>.However, for finite μ one clearly needs to go beyond the ladder approximation, and take into account the “Fermi sea shakeup”. To do so, we first consider the limit of infinite M (β=0). In this regime, the QW absorption in the presence of a bound state for the model under consideration was found by Combescot and Nozières <cit.>, using a different approach [In fact, their computation is in 3D, but the case of infinite hole mass is effectively 1D anyway.].For finite μ, the physics of theFermi-edge singularity comes into play: Due to the presence of the CB Fermi sea, CB electron-hole excitations are possible at infinitesimal energy cost. As a result, the exciton Green's function, which we analogously to (<ref>) define as proportional to the dressed bubble in the exciton regime,Π_exc(ω) = d_0^2 E_B ρ/g^2 G_exc(ω) + 𝒪(ω/E_B),G_exc(ω) = 1/ω - Σ^exc(ω),gets renormalized by a self-energy Σ^exc( ω). This self-energy turns the exciton pole turns into a divergent power law <cit.>: G_exc(ω) ∼1/ω+i0^+·(ω+i0^+/-μ)^(δ/π -1)^2,where δ is the scattering phase shift of electrons at the Fermi-level off the point-like hole potential. One should note that no delta-peak will appear for δ/π≠ 1. A sketch of the resulting absorption A is shown in Fig. <ref>. Let us further discuss the result (<ref>). It was obtained in <cit.> using an elaborate analytical evaluation of final state Slater determinants, and actually holds for any value of μ. A numerical version of this approach for the infinite VB mass case was recently applied by Baeten and Wouters <cit.> in their treatment of polaritons. In addition, the method was numerically adapted to finite masses by Hawrylak <cit.>, who, however, mostly considered the mass effects for μ≫ E_B.However, due to the more complicated momentum structure, it seems difficult to carry over the method of <cit.> to finite masses analytically. Instead, we will now show how to proceed diagrammatically. Our analysis will give (<ref>) to leading order in the small parameter μ/E_B, or, equivalently, α = δ/π -1 (recall that by Levinson's theorem <cit.> δ=π for μ=0 due to the presence of a bound state — the exciton):G_exc(ω) ≃1/ω+i0^+(1+ α^2 ln(|ω|/μ)-iα^2πθ(ω)). The merit of the diagrammatical computation is twofold: First, it gives an explicit relation between α and the experimentally-measurable parameters μ, E_B. Second, the approach can be straightforwardly generalized to finite masses, as we show in the next subsection. Let us note that a similar diagrammatic method was also examined by Combescot, Betbeder-Matibet et al. in a series of recent papers <cit.>. Their model Hamiltonians are built from realistic Coulomb electron-hole and electron-electron interactions. As a result, they assess the standard methods of electron-hole diagrams as too complicated <cit.>, and subsequently resort to exciton diagrams and the so-called commutation technique, where the composite nature of the excitons is treated with care. However, the interaction of excitons with a Fermi sea is only treated at a perturbative level, assuming that the interaction is small due to, e.g., spatial separation <cit.>. This is not admissible in our model, where the interaction of the VB hole with all relevant electrons (photoexcited and Fermi sea) has to be treated on the same footing. Rather, we stick to the simplified form of contact interaction, and show how one can use the framework of standard electron-hole diagrams to calculate all quantities of interest for infinite as well as for finite VB mass. The results presented below then suggest that for μ≪ E_B the finite mass does not weaken, but rather strengthens the singularities, which is in line with results on the heavy hole found in <cit.>.Here we only present the most important physical ingredients for our approach, and defer the more technical details to Appendix <ref>. In the regime of interest, we can perform a low-density computation, employing the small parameter μ/E_B. Since all energies are close to E_B, the leading-order exciton self-energy diagrams is then the sum of all diagrams with one CB electron loop. One can distinguish two channels: direct and exchange, to be denoted by D and X, as depicted inFig. <ref>.All such diagrams with an arbitrary number of interactions connecting the VB line with the CB lines in arbitrary order have to be summed. Factoring out E_Bρ/g^2 · G^0_exc(ω)^2, the remaining factor can be identified as the exciton self-energy diagram. An evaluation of these diagrams is possible either in the time or in the frequency domain. Of course, both approaches must give same result. In practice, however, the time domain evaluation is more instructive and requires less approximations, which is why we will discuss it first. The frequency domain evaluation, however, is far more convenient for obtaining finite mass results, and will be discussed thereafter.The time domain approach is similar in spirit to the classical one-body solution of the Fermi-edge problem by Nozières and de Dominicis <cit.>. Since the infinite-mass hole propagator is trivial, G_v(t) = iθ(-t)e^i E_G t, the direct diagrams just describe the independent propagation of two electrons in the time-dependent hole potential. Thus, in the time domain the sum of all direct diagrams D(t) factorizes into two parts representing the propagation of these two electrons:D(t) = ∫_k_1<k_Fdk_1/(2 π)^2 i e^-i(E_G-ϵ_k_1)t B(t) C(t),where B(t), C(t) are infinite sums of convolutions (denoted by an asterisk) of the form B(t) = ∑_m=1^∞ (-V_0)^m ∫_k_2>k_Fdk_2/(2 π)^2 ... ∫_k_m>k_Fdk_m/(2 π)^2[G_c^0, R(k_1, ) ∗⋯∗ G_c^0, R(k_m, )∗ G_c^0, R (k_1, )](t),and similarly for C(t). G_c^0,R is the retarded bare CB Green's function in the time domain. Fourier-transforming, D(ω) is then given by a convolution of B(ω) and C(ω), each of which in turn reduces to simple summations of ladder diagrams. The full convolution D(ω) is difficult to compute; one can proceed by noting that B(ω), C(ω) have poles at ω≃ 0 and continuum contributions at ω≳ E_B. These are readily identified with the pole and continuum contributions of the exciton absorption, c.f. Fig. <ref>. Combining these, there are four combinations contributing to D(ω): pole-pole, pole-continuum (two possibilities), and continuum-continuum. The imaginary part of the latter, which is of potential importance for the line shape of the exciton spectrum, can be shown to vanish in our main regime of interest, ω≳ 0. It is instructive to study the pole-pole combination, which corresponds to a would be “trion” (bound state of the exciton and an additional electron) and is further discussed in Appendix <ref>. Adding to it the pole-continuum contributions we find, for small ω:D(ω) = ρ E_B/g^21/(ω + i0^+)^2Σ_exc^D(ω). This corresponds to a contribution to the exciton self-energy which reads: Σ^D_exc(ω) = -1/ρ∫_k_1<k_Fdk_1/(2π)^21/ln(ω + ϵ_k_1 - μ + i0^+/-E_B) .Before discussing this term further, we consider the contribution of the exchange diagrams, X(ω), of Fig. <ref>(b). Their structure is more involved compared to the direct channel, since these diagrams do not just represent the independent propagation of two electrons in the hole potential. However, relying on a generalized convolution theorem which we prove, the computation can be performed in the same vein as before (see Appendix <ref>), leading to the following results:First, the pole-pole contribution cancels that of the direct diagrams (see Appendix <ref>), which holds in the spinless case only (in the spinful case, the direct diagrams will come with an extra factor of two). This could be expected: trion physics is only recovered in the spinful case, where two electrons can occupy the single bound state created by the attractive potential of the hole. In a realistic 2D setup trion features will become important for large enough values of μ (see, e.g., <cit.>). Although we do not focus on trions here, let us stress that all standard results on trions can be recovered within our diagrammatic approach, if electrons and holes are treated as spin-1/2 particles; see Appendix <ref> for further details.The dominant contribution to X(ω) then arises from the pole-continuum contribution. It is given by: X(ω) = -ρ E_B/g^21/(ω + i 0^+)^2μ.Thus, the self-energy contribution to the exciton Green's function is simplyΣ_exc^X(ω) = -μ. Since it is purely real, it will essentially just red-shift the exciton pole by μ. A discussion of this result is presented in Appendix <ref>.Now, it should be noted that Σ_exc^X(ω) is not proportional to the small parameter μ/E_B – the latter effectively canceled when factoring out the bare excitons Green's function. Thus, it is inconsistent to treat Σ_exc^X(ω) as perturbative self-energy correction. Instead, one should repeat the calculation, but replace all ladders by ladders dressed with exchange-type diagrams. It can be expected, however, that the structure of the calculations will not change. The only change that should happen is the appearance of the renormalized binding energy Ẽ_B = E_B + μ, in accordance with <cit.>, as discussed in Appendix <ref>. In the following, we will assume this is accounted for, and therefore suppress all exchange diagrams.Let us now return to the direct self-energy contribution Σ_exc^D(ω), Eq. (<ref>), writing Σ_exc(ω) = Σ^D_exc(ω)henceforth. We may apply the following asymptotic expansion for the logarithmic integral (generalized from <cit.>), which will also prove useful later:∫ _0^ω dx x^n/ln^m(x) = 1/ln^m(ω)ω^n+1/(n+1) + 𝒪(ω^n+1/ln(ω)^m+1).This can be shown easily by integrating by parts and comparing orders.Based on this result we find, to leading logarithmic accuracy,Σ_exc(ω) ≃-μ/ln(μ/E_B) + ωln(|ω|/μ)/ln(μ/E_B)ln(|ω|/E_B)- iπωθ(ω)/ln^2(|ω|/E_B).This result has several interesting features. First, we see the appearance of a small parameter α≡ 1/|ln(μ/E_B)|, which can be interpreted as follows:the scattering phase-shift at the Fermi level, δ, which determines the Anderson orthogonality power law [c.f. Eq. (<ref>)] is approximately given by <cit.>δ≃π/ln(μ/E_B) + π , which holds for small Fermi energies, where δ is close to π.Therefore, δ and α are related by:α≃ 1-δ/π .The small pole shift of order αμ contained in Eq. (<ref>) could be expected from Fumi's theorem (see, e.g., <cit.> and the discussion in Appendix <ref>). We now perform an energy shiftω→ω + αμ.To leading order in α, we may then rewrite Σ_exc^D withlogarithmic accuracy asΣ_exc(ω) ≃α^2ωln(|ω|/μ) - iα^2πωθ(ω), Here, the imaginary part can be identified with the density of states of CB electron-hole excitations as function of ω, as discussed in Sec. <ref>.Upon inserting (<ref>) into the exciton Green's function (<ref>), we recover (<ref>) to leading (quadratic) order in α:G_exc(ω) ≃1/ω+i0^+(1+ α^2 ln(|ω|/μ)-iα^2πθ(ω)).As a result, our one-loop computation has given the first logarithm of the orthogonality power law, in complete analogy to the standard Fermi-edge problem (see Sec. <ref>).All higher loop contributions, evaluated to leading logarithmic order, should then add up to give the full power law; since we are more interested in finite mass effects here, we will not go into the details of this calculation. To carry the diagrammatics over to finite mass, as done in the next section, it is convenient to switch to the frequency domain. A summation of all one-loop diagrams ispossible by evaluating the series shown in Fig. <ref>.To perform the evaluation, we make use of the following simplification:To begin with, we often encounter complicated logarithmic integrals; however, the imaginary part of the integrand is just a delta function, so, upon integration, one finds step functions. Since the integrand is retarded, it is then possible to recover the full expression from the imaginary part using the Kramers-Kronig relation; the step functions then become logarithms.With that, the sum over diagrams appearing in Fig. <ref> assumes the formD(ω)= E_B/g^21/(ω + i 0^+)^2∫_k_1<k_Fdk_1/(2π)^2{I + I^3 + ...},whereI=ln(ϵ_k_1 + ω - μ+i0^+/-E_B). Summing up the geometric seriesexactly reproduces the time-domain result, Eq. (<ref>).Thus, we have established how thephoton self-energy can be calculated diagrammatically for the case of infinite VB mass M (to leading order in d_0). §.§ Finite hole massWe are now in a position to tackle finite VB mass M. Let us also consider a finite incoming momentum Q. Clearly, the one-loop criterion for choosing diagrams still holds, since we are still considering the low-density limit, μ≪ E_B. We also disregard any exchange contributions for the same reasons as for the infinite mass case. As a result, we only have to recompute the series of direct diagrams of Fig <ref>. We start with the first one. It gives:I =-E_B V_0/g∫_k_2 > k_Fdk_2/(2π)^21/(-ω + E_B+ E(k_2 - Q) + ϵ_k_2 - μ - i0^+)^21/ln(-E_B + ω - (Q - q)^2/2M_exc- ϵ_k_2 + ϵ_k_1 + i0^+/-E_B), where q = k_2 - k_1. The imaginary part of (<ref>) reads:Im[I] = -V_0/g∫_k_2 > k_Fdk_2/(2π)^2π δ(ω - (Q-q)^2/2M_exc - ϵ_k_2 + ϵ_k_1) +𝒪(μ/E_B).By Eq. (<ref>), I can be rewritten in a simpler form (ensuring retardation), valid for small ω: I ≃V_0/g∫_k_2>k_Fdk_2/(2π)^21/ω - (Q -q )^2/2M_exc - ϵ_k_2 + ϵ_k_1 + i0^+.This form can be integrated with logarithmic accuracy, which, however, only gives Re[I]. Specializing to Q ≪ k_F for simplicity, one obtains: Re[I] ≃ln(max(|ω + ϵ_k_1 - μ |, βμ)/E_B). As for the infinite mass case, the higher order diagrams of Fig. <ref>give higher powers of I. Similarly to Eq. (<ref>), one then obtains for the self-energy part, to leading logarithmic accuracy:Σ_exc(Q,ω) = -∫_k_1 < k_Fdk_1/(2π)^2·1/I. The imaginary part, which determines the lineshape of G_exc, is given byIm[Σ_exc(Q,ω)] ≃- π V_0/ρ g∫_k_1 < k_Fdk_1/(2π)^2∫_k_2 > k_Fdk_2/(2π)^2δ(ω - (Q - q)^2/2M_exc - ϵ_k_2 + ϵ_k_1)/ln^2(max(|ω + ϵ_1 - μ|, βμ)/E_B). We now apply the analogue of the logarithmic identity, Eq. (<ref>), for a 2D integral. Thus, in leading order we may simply pull the logarithm out of the integral of Eq. (<ref>) and rewrite it as Im[Σ_exc](Q,ω) ≃-π V_0/ρ gα^2 ∫_k_1 < k_Fdk_1/(2π)^2∫_k_2 > k_Fdk_2/(2π)^2δ(ω - (Q-q)^2/2M_exc - ϵ_k_2 + ϵ_k_1).The result (<ref>) is physically transparent: It is just a phase-space integral giving the total rate of scattering of an exciton with momentum Q by a CB Fermi sea electron. The prefactor is determined by the scattering phase shift δ.At least for sufficiently small momenta Q, the integral in Eq. (<ref>) can be straightforwardly computed. For the most important case Q = 0, one obtains for small energies (see Appendix <ref>): Im[Σ_exc](Q=0,ω) ∼ -α^2 1/√(βμ)θ(ω) ω^3/2, ω≪βμ, where we suppressed an irrelevant prefactor of order one.For ω≫βμ one recovers the infinite mass case as in (<ref>).Compared to the infinite mass case, where Im[Σ_exc]∼ωln(ω), the self-energy (<ref>) shows a suppression of the low-frequency scattering phase space, as seen from the higher frequency power law. Physically, the phase space suppression is understood as follows: We have found that, after accounting for the exchange diagrams, it is admissible to view the exciton as elementary particle with mass M_exc, which interacts with the Fermi sea with an effective interaction strength α [Eq. (<ref>)]. As can be seen from Fig. <ref>, scatterings of the exciton with CB electrons involving a large momentum transfer necessarily cost a finite amount of energy (the so-called recoil energy).By contrast, in the infinite mass case such scatterings could still happen at infinitesimal energy cost, since the exciton dispersion was flat. Thus, the finite-mass phase space is reduced as compared to the infinite mass case.This change eventually leads to the previously asserted reappearance of the exciton delta peak.This phase space reduction also affects the exciton spectral function, and hence the absorption: We first restrict ourselves to the leading behavior, i.e., we disregard any small renormalizations that arise from including Re[Σ_exc] or from higher-loop corrections. Inserting Eq. (<ref>) intoEq. (<ref>) we then obtain, for small energies ω:A(Q = 0, ω) ≃ - Δ^2 Im[Σ(ω)]/ω^2∼Δ^2 α^2 θ(ω)/√(βμ·ω),withΔ^2= d_0^2ρ E_B/g^2. The factor Δ (with units of energy) determines the polariton splitting at zero detuning, and will be discussed in Sec. <ref>. The 1/√(ω) divergence seen in (<ref>) was also found by Rosch and Kopp using a path-integral approach <cit.> for a related problem, that of a heavy hole propagating in a Fermi sea. In addition, Rosch and Kopp find a quasi particle delta peak with a finite weight. This peak can also be recovered within our approach upon inclusion of the correct form of Re[Σ_exc]. From Eqs. (<ref>) and (<ref>) we may infer it to be Re[Σ_exc(Q=0,ω)] = α^2 ωln(√(ω^2 + (βμ)^2)/μ),where we have rewritten the maximum-function with logarithmic accuracy using a square root.This cut-off of logarithmic singularities (which are responsible for edge power laws) by recoil effects is a generic feature of our model, and will reoccur in the regime of μ≫ E_B presented inSec. <ref>. In qualitative terms, this is also discussed in Ref. <cit.> (for arbitrary dimensions). Our results are in full agreement with this work.We may now deduce the full photon self-energy Π_exc as follows: In the full finite-mass version of the power law (<ref>), the real part of the logarithm in the exponent will be replaced by the cut-off logarithm from Eq. (<ref>). The imaginary part of this logarithm will be some function f(ω) which continuously interpolates between the finite-mass regime for ω≪βμ [given by Eq. (<ref>) times ω^-1], and the infinite mass regime for ω≫βμ.Therefore, we arrive atΠ_exc(Q = 0,ω) =Δ^2/ω+i0^+exp[α^2 (ln(√(ω^2 + (βμ)^2)/μ) - if(ω))] ,wheref(ω) = π√(ω/βμ)θ(ω) ω≪βμ π ω≫βμ.It is seen by direct inspection that (<ref>) has a delta peak at ω =0 with weight Δ^2 β^α^2. One can also asses the weight of the delta peak by comparing the spectral weights of the exciton spectral function in the infinite and finite mass cases:The weight of the delta peak must correspond to the difference in spectral weight as the absorption frequency power law is changed once β becomes finite.In the infinite mass case, the absorption scales as A_∞(ω)∼Δ^2 α^2/ω(ω/μ)^α^2θ(ω),as follows from Eq. (<ref>) above.Thus, the spectral weight in the relevant energy region is given by ∫_0^βμ dω A_∞(ω) = Δ^2 β^α^2.In contrast, using Eq. (<ref>), the spectral weight of the finite mass case is ∫_0^βμ dω A(Q=0,ω) = Δ^2 α^2.For scattering phase shifts δ close to π (i.e., α→ 0), and for finite mass, β>0, a pole with weight proportional to β^α^2 [Eq. (<ref>)] at ω =0 should be present in the spectrum, if β is not exponentially small in α. This weight is exactly the same as for the heavy hole when computed in a second order cumulant expansion <cit.>.The full imaginary part of Π_exc(Q=0,ω) was already given explicitly in Eqs. (<ref>) and (<ref>), and plotted in Fig. <ref>. That plot illustrates the main conclusion of this section: For finite mass, Fermi sea excitations with large momentum transfer are energetically unfavorable, and are therefore absent from the absorption power law. As a result, the pole-like features of the absorption are recovered. §.§ Validity of the electron-hole correlator as a photon self-energyLet us now assess the validity of the expressions for the CB electron-VB hole correlator [Eqs. (<ref>) and (<ref>)] as a photon self-energy. Using them, one assumes that only electron-hole interactions within one bubble are of relevance, and electron-hole interactions connecting two bubbles (an example is shown in Fig. <ref>) can be disregarded. The regime where such an approximation is valid may be inferred from the following physical argument:Electronic processes (i.e. electron-hole interactions) happen on the time scale of Fermi time 1/μ. On the other hand, the time scale for the emission and reabsorption of a photon (which is the process separating two bubbles) is given by 1/ρ d_0^2 (where d_0 is the dipole matrix element). If the second scale is much larger than the first one, electrons and holes in distinct bubbles do not interact. Thus, the our approach is valid as long asρ d_0^2 ≪μ.Under this condition, the following physical picture is applicable: an exciton interacts with the Fermi sea, giving rise to a broadened exciton, which in turn couples to the cavity photons. When Eq. (<ref>) is violated, one should think in different terms: excitons couple to photons, leading to exciton-polaritons. These then interact with the Fermi sea. The second scenario is, however, beyond the scope of this paper. The above discussion is likewise valid for the regime of large Fermi energy, which is studied below. § ELECTRON-HOLE CORRELATOR FOR LARGE FERMI ENERGYWe now switch to the opposite regime, where μ≫ E_B, and excitons are not well-defined. For simplicity, we also assume that μ is of the order of the CB bandwidth.Hence, E_B ≪μ≃ξ. Within our simplified model, the finite mass problem in 3D was solved in <cit.>. This treatment can be straightforwardly carried over to 2D <cit.>. To avoid technicalities, we will, however, just show how to obtain the 2D results in a “Mahan guess” approach <cit.>, matching known results from <cit.>. To this end, we will first recapitulate the main ingredients of the infinite mass solution. §.§ Infinite hole massThe FES builds up at the Burstein-Moss shifted threshold Ω_T^FES = E_G + μ. Its diagrammatic derivation relies on a weak-coupling ansatz: The parameter g = ρ V_0 is assumed to be small. As seen from Eq. (<ref>), this is indeed true for μ≫ E_0.In principle, below the FESthere will still be the exciton peak; however, this peak will be broadened into a weak power law, and thus merge with the FES. For finite mass (see below), the position of the would-be exciton may even be inside FES continuum, which makes the exciton disappear completely. What is more, the exciton weight, being proportional to E_B, is exponentially small in g (since μ≃ξ). We may therefore safely disregard the exciton altogether (see also discussion in Appendix <ref>).To leading order in gln(ω/μ), the dominant contribution comes from the so called “parquet” diagrams, containing all possible combinations of ladder and crossed diagrams <cit.>.The value of the pure ladder diagrams is given by Eq. (<ref>), with Ω - E_G replaced by ω = Ω -Ω_T^FES. The lowest-order crossed diagram is shown in Fig. <ref>. With logarithmic accuracy the contribution of this diagram is easily computed: Π_crossed =-1/3d_0^2ρ g^2 [ln(ω/μ)]^3.This is -1/3 times the contribution of the second order ladder diagram, c.f. Eq. (<ref>). Thus, the ladder and crossed channels partially cancel each other, a feature which persists to all orders. This also shows that the FES is qualitatively different from the broadened exciton discussed in the previous section: now the exciton effects (ladder diagrams) and the Fermi sea shakeup (crossed diagrams) have to be treated on equal footing. In his original paper Mahan computed all leading diagrams to third order and guessed the full series from an exponential ansatz <cit.>. The corresponding result for the photon self-energy Π_FES(ω) readsΠ_FES(ω) = d_0^2ρ/2g(1-exp[-2gln(ω+i0^+/-μ)]).Relying on coupled Bethe-Salpeter equations in the two channels (ladder and crossed), Nozières et al. then summed all parquet diagrams, where a bare vertex is replaced by (anti-)parallel bubbles any number of times <cit.>. The result corresponds exactly toMahan's conjecture, Eq. (<ref>).By the standard FES identificationδ/π = g + 𝒪(g^3), the power law in Eq. (<ref>) coincides with the one given in Eq. (<ref>); the phase shift is now small.One should also point out that the peaks in the spectra in the regimes of small μ (Fig. <ref>) and large μ (Fig. <ref>) are not continuously connected, since the FES arises from the continuous threshold, whereas the exciton does not. Let us finally note that since μ is a large scale,Eq. (<ref>) should be a good approximation for thephoton self-energy, since the condition (<ref>) is easily satisfied. §.§ Finite hole mass As in the regime of the exciton, in the finite mass case the result (<ref>) will be modified due to the recoil energy βμ. However, it will now be the VB hole recoil (or the hole lifetime, see below) instead of the exciton recoil —the latter is meaningless since the exciton is not a well defined entity anymore. This is most crucial:Since CB states with momenta smaller than k_F are occupied, VB holes created by the absorption of zero-momentum photons must have momenta larger than k_F. Therefore, the hole energy can actually be lowered by scatterings with the Fermi sea that change the hole momenta to some smaller value, and these scattering processes will cut off the sharp features of Π_FES (ω).The actual computation of the photon self-energy with zero photon momentum, Π_FES(Q=0,ω), proceeds in complete analogy to the 3D treatment of <cit.>. Limiting ourselves to the “Mahan guess” for simplicity, the main steps are as follows.The first major modification is the appearance of two thresholds: As easily seen by the calculation of the ladder diagrams, the finite mass entails a shift of the pole of the logarithm from ω =0 to ω = βμ, which is the minimal energy for direct transitions obeying the Pauli principle. Correspondingly, ω_D=βμ is called the direct threshold.Near this threshold, logarithmic terms can be large, and a non-perturbative resummation of diagrams is required. However, the true onset of 2DEG absorption will actually be the indirect threshold ω_I=0. There, the valence band hole will have zero momentum, which is compensated by a low-energy conduction electron-hole pair, whose net momentum is -k_F. The two thresholds were shown in Fig. <ref>. It should be noted that for E_B < βμ the exciton energy ≈ω_D - E_B,is between ω_I and ω_D. Hence, in this case the exciton overlaps with the continuum and is completely lost.Near ω_I, the problem is completely perturbative. In leading (quadratic) order in g, the absorption is determined by two diagrams only. The first one is the crossed diagram of Fig. <ref>. The second one is shown in Fig. <ref>. When summing these two diagrams, one should take into account spin, which will simply multiply the diagram of Fig. <ref> by a factor of two (if the spin is disregarded, the diagrams will cancel in leading order). Up to prefactors of order one, the phase-space restrictions then result in a 2DEG absorption (see <cit.> and Appendix <ref>): A(Q = 0,ω) = d_0^2 g^2 (ω/βμ)^3 θ(ω).The phase space power law ω^3 is specific to 2D . Its 3D counterpart has a larger exponent, ω^7/2 <cit.>, due to an additional restriction of an angular integration. Let us now turn to the vicinity of ω_D, where one has to take into account the logarithmic singularities and the finite hole life-time in a consistent fashion. Regarding the latter,one can dress all VB lines with self-energy diagrams as shown in Fig. <ref>. The self-energy insertion at the dominant momentum k = k_F readsIm[Σ_VB(k_F, ω)] = 1/√(3)θ(ω)g^2 βμω^2/(βμ)^2, ω≪βμ. As can be shown by numerical integration, this expression reproduces the correct order of magnitude for ω = βμ, such that it can be safely used in the entire interesting regime ω∈ [0,βμ]. The power law in Eq. (<ref>) is again specific to 2D. In contrast, the order of magnitude of the inverse lifetime is universal,Im[Σ_VB(k_F, βμ)] ∼ g^2βμ.Disregarding the pole shift arising from Re[Σ], the self-energy (<ref>) can be used to compute the “dressed bubble” shown in Fig. <ref>. With logarithmic accuracy, the dressed bubble can be evaluated analytically. In particular, its real part reads:Re[Π_db](ω) ≃ρ d_0^2ln(√((ω - βμ)^2 + (g^2 βμ)^2)/μ).This is just a logarithm whose low-energy divergence is cut by the VB hole life time, in full analogy to Eq. (<ref>),and in agreement with Ref. <cit.>. For the computation of polariton spectra later on, itturns out to be more practical to obtain both the real and the imaginary parts of Π_db(ω) by numerically integrating the approximate form <cit.>: Π_db(ω) ≃d_0^2/(2π)^2∫_k > k_Fdk1/ω - (ϵ_k-μ) - k^2/2M +iIm[Σ̃_VB(ω - ϵ_k + μ)], Im[Σ̃_VB(x)]= g^2√(3)θ(x)x^2/(βμ)x<βμ g^2√(3)βμx>βμ, to avoid unphysical spikes arising from the leading logarithmic approximation. A correspondingplot of -Im[Π_db] is shown in Fig. <ref>. The numerical expression -Im[Π_db] simplifies to the correct power law (<ref>) in the limit ω→ 0, andapproaches the infinite mass value d_0^2ρπ for large frequencies.Higher-order diagrams will contain higher powers of the rounded logarithm (<ref>). The parameter controlling the leading log scheme now reads l≡ gln(β g^2).One can distinguish different regimes of l. The simplest is l ≪ 1, which holds in the limit g → 0 (or, put differently, if β is not exponentially small in g). In this limit, no singularity is left. The large value of the Fermi energy (small g) and the large value of the hole decay βμ have completely overcome all interaction-induced excitonic effects. A decent approximation to the 2-DEG absorption is then already given by the imaginary part of the dressed bubble. Fig. <ref> shows the corresponding absorption. The more interesting regime corresponds to gln(βg^2) ≳ 1, where arbitrary numbers of conduction band excitations contribute to the absorption alike [The regime of gln(β g^2) ≫ 1 is out of reach for the methods used in <cit.>. To study it, a consistent treatment of thedivergences is needed, similar to <cit.>. We will not attempt this here]. A non-perturbative summation is needed, which is, however, obstructed by the following fact: As found by straightforward computation, the crossed diagrams are not only cut by g^2βμ due to the hole decay, but also acquire an inherent cutoff of order βμ due to the hole recoil. A standard parquet summation is only possible in a regime where these two cutoffs cannot be distinguished with logarithmic accuracy, i.e. where β≪ g^2. For small enough g this will, however, always be the case in the truly non-perturbative regime where β must be exponentially small in g.As a result of these considerations, the logarithms of the parquet summation have to be replaced by the cut-off logarithms (<ref>), with g^2βμ replaced by βμ. The imaginary part of the logarithm is then given by the function plotted in Fig. <ref>. The resulting full photon self-energy in the non-perturbative FES regime reads:Π_FES(Q=0,ω)≃ -d_0^2ρ/2g(exp[-2g(Π_db(ω)/ρ d_0^2)] -1).A sketch of Im[Π_FES] is shown in Fig. <ref>.§ POLARITON PROPERTIES When the cavity energy ω_c is tuned into resonance with the excitonic 2DEG transitions, the matter and light modes hybridize, resulting in two polariton branches. We will now explore their properties in the different regimes. §.§ Empty conduction bandTo gain some intuition, it is first useful to recapitulate the properties of the exciton-polariton in the absence of a Fermi sea. Its (exact) Green's function is given by Eq. (<ref>), withω_Q=0 = ω_c and Π(ω) = Δ^2/(ω+i0^+), where Δ is a constant (with units of energy) which determines the polariton splitting at zero detuning. In terms of our exciton model, one has Δ =√(d_0^2ρ E_B/g^2). ω is measured from the exciton pole.A typical density plot of the polariton spectrum A_p = -Im[D^R(ω,ω_c)]/π, corresponding to optical (absorption) measurements as e.g. found in <cit.>, is shown in Fig. <ref>.A finite cavity photon linewidth Γ_c = Δ is used.The physical picture is transparent: the bare excitonic mode (corresponding to the vertical line) and the bare photonic mode repell each other, resulting in a symmetric avoided crossing of two polariton modes. For analytical evaluations, it is more transparent to consider an infinitesimal cavity linewidth Γ_c. The lower and upper polaritons will then appear as delta peaks in the polariton spectral function, at positionsω_± = 1/2(ω_c ±√(ω_c^2 + 4Δ^2)),and with weights W_± = 1/1 + 4 Δ^2/(ω_c ±√(4 Δ^2 + ω_c^2))^2.We note that the maximum of the polariton spectra scales as 1/Γ_c for finite Γ_c. Our spectral functions are normalized such that the total weight is unity. From Eq. (<ref>) it is seen that the weight of the “excitonic” polaritons (corresponding to the narrow branches of Fig. <ref>) decays as Δ^2/ω_c^2 for large absolute values of ω_c.§.§ Large Fermi energyLet us study polariton properties in the presence of a Fermi sea. Reverting the order of presentation previously taken in the paper, we first turn to the regime of large Fermi energy, E_B ≪μ.This is because for E_B ≪μ the inequality ρ d_0^2 ≪μ (<ref>) is more easily satisfied than in the opposite limit of E_B ≫μ, facilitating experimental realization. Wecompute the polariton properties using the electron-hole correlators as cavity photon self-energy.A similar approach was applied recently by Averkiev and Glazov <cit.>, who computed cavity transmission coefficients semiclassically, phenomenologically absorbing the effect of the Fermi-edge singularity into the dipole matrix element. Two further recent treatments of polaritons for nonvanishing Fermi energies are found in <cit.> and <cit.>. In the first numerical paper <cit.>, the Fermi-edge singularity as well as the excitonic bound state are accounted for, computing the electron-hole correlator as in <cit.>, but an infinite mass is assumed. The second paper <cit.> is concerned with finite mass. However, the authors only use the ladder approximation and neglect the crossed diagrams, partially disregarding the physical ingredients responsible for the appearance of the Fermi-edge power laws. We aim here to bridge these gaps and describe the complete picture in the regime of large Fermi energy (before turning to the opposite regime of μ≪ E_B).In the infinite mass limit we will use Eq. (<ref>) as the photon self-energy. It is helpful to explicitly write down the real and imaginary parts of the self-energy in leading order in g: Re[Π_FES](ω)= Δ̃(1-(|ω|/μ)^-2g), Im[Π_FES](ω)= - Δ̃· 2π g(ω/μ)^-2gθ(ω)Δ̃ ≡d_0^2ρ/2g,where we have introduced the parameter Δ̃, which determines the splitting of the polaritons, playing a similar role to Δ in the previous case of empty CB. In the following, Δ̃ will serve as the unit of energy.For a cavity linewidth Γ_c = 1Δ̃, a typical spectral plot of the corresponding "Fermi-edge polaritons" is shown in Fig. <ref>. It is qualitatively similar to the results of <cit.>. A quantitative comparison to the empty CB case is obviously not meaningful due to the appearance of the additional parameters μ (units of energy) and g (dimensionless). Qualitatively, one may say the following: The lower polariton is still a well-defined spectral feature. For zero cavity linewidth (see below), its lifetime is infinite. The upper polariton, however, is sensitive to the high-energy tail of the 2DEG absorption power law (<ref>), and can decay into the continuum of CB particle-hole excitations. Its linewidth is therefore strongly broadened. Only when the 2DEG absorption is cut off by finite bandwidth effects (i.e., away from the Fermi-edge), a photonic-like mode reappears in the spectrum (seen in the upper right corner of Fig. <ref>). For more detailed statements, one can again consider the case of vanishing cavity linewidth Γ_c. A spectral plot with the same parameters as in Fig. <ref>, but with small cavity linewidth, Γ_c = 0.01 Δ̃, is shown in Fig. <ref>(a).We first examine the lower polariton (assuming zero linewidth), which is a pure delta peak. Its position is determined by the requirementω-ω_c - Re[Π_FES(ω)] = 0.One may study the solution of this equation in three distinct regimes, corresponding to ω_c → -∞, ω_c = 0, and ω_c → + ∞.For ω_c → - ∞, the solution of Eq. (<ref>) approaches ω = ω_c, and the lower polariton acquires the full spectral weight (unity): For strong negative cavity detunings, the bare cavity mode is probed. The corresponding spectral cut is shown in Fig. <ref>(a) (continuous line). We will refrain from making detailed statements about the way the bare cavity mode is approached, since this would require the knowledge of the photon self-energy at frequencies far away from the threshold. As the cavity detuning is decreased, the lower polariton gets more matter-like. At zero detuning [see Fig. <ref>(b)], and for g not too small (w.r.t. gΔ̃/μ), the weight of the lower polariton is approximately given by 1/(1+2g). For large positive cavity detunings [see Fig. <ref>(c)], the position of the matter-like lower polariton approaches ω=0,ω∼ -ω_c^-1/(2g)asω_c →∞. The lower polariton weight also scales in a power law fashion,∼ω_c^-1-1/(2g), distinct from the excitonic regime, where the weight falls off quadratically [Eq. (<ref>)].Due to the finite imaginary part of the self-energy Π_FES(ω), the upper polariton is much broader than the lower one: the photonic mode can decay into the continuum of matter excitations. At large negative detunings [see the inset to Fig. <ref>(a)], the upper polariton has a power law like shape (with the same exponent as the Fermi-edge singularity), and for ω_c → - ∞ its maximum approaches ω = 0 from the high-energy side. As the detuning is increased (made less negative), the maximum shifts away from ω=0, approaching the free cavity mode frequency ω = ω_c for ω_c →∞. Since the weight and height are determined by the value of Im[Π_FES] at the maximum, they increase correspondingly.Let us now consider the case of finite mass. Using the finite mass photon self-energy (<ref>) instead of (<ref>), the Fermi-edge-polariton spectrum with anonzero mass-ratio of β = 0.2 is plotted in Fig. <ref>(b). Compared to the infinite mass case of Fig. <ref>(a), Fig. <ref>(b) has the following important features:(i) The boundary line separating the lower and upper thresholds is shifted to the high-energy side from ω = 0 in the infinite mass case to ω = βμ in the finite mass case, reflecting the Burstein-Moss shift in the 2DEG absorption. (ii) As opposed to the infinite mass case, the lower polariton is strongly broadened at large positive detunings. These points are borne out more clearly in Fig. <ref>(a)–(c) (dashed lines), which presents cuts through Fig. <ref>(b) at fixed detuning. The situation at large negative detuning is shown in Fig. <ref>(a): Compared to the infinite mass case, shown as full line, the polaritons are shifted towards higher energies. In addition, the shape of the upper polariton is slightly modified — its onset reflects the convergent phase-space power law ω^3 of Eq. (<ref>) found for the 2DEG absorption. This is emphasized in the inset.At zero cavity detuning [Fig. <ref>(b)], the situation of the finite and infinite mass cases is qualitatively similar. When the cavity detuning is further increased, the position of the pole-like lower polariton approaches the direct threshold at ω = βμ (indicated by the vertical dotted line). When the pole is in the energy interval [0,βμ], the lower polariton overlaps with the2DEG continuum absorption, and is therefore broadened. This is clearly seen inFig. <ref>(c): Instead of a sharp feature, there is just a small remainder of the lower polariton at ω = βμ.As a result, one may say that in the regime of the Fermi-edge singularity, i.e., large μ, the finite mass will cut off the excitonic features from the polariton spectrum– instead of the avoided crossing of Fig. <ref>, Fig. <ref>(b) exhibits an almost photonic-like spectrum, with a small (cavity) linewidth below the threshold at ω = βμ, and a larger linewidth above the threshold, reflecting the step-like 2DEG absorption spectrum of Fig. <ref>. The finite mass thus leads to a general decrease of the mode splitting between the two polariton branches. This trend continueswhen the Fermi energy is increased further. It is instructive to compare this behavior with the experimental results reported in <cit.>. There, two differential reflectivity measurements were conducted, which can be qualitatively identified with the polariton spectra. The first measurement was carried out using a low-mobility GaAs sample (which should behave similarly to the limit of large VB hole mass), and moderate Fermi energies. A clear avoided crossing was seen, with the upper polariton having a much larger linewidth than the lower one (see Fig. 2(A) of <cit.>). In the second measurement, the Fermi energy was increased further, and a high-mobility sample was studied, corresponding to finite mass. A substantial reduction of the mode splitting between the polaritons was observed (Fig. 2(C) of <cit.>). While a detailed comparison to the experiment of <cit.> is challenging, due to the approximations we made and the incongruence of the parameter regimes (in the experiment one has μ≃ E_B), the general trend of reduced mode splitting is correctly accounted for by our theory.§.§ Small Fermi energyWe now switch to the regime of of small Fermi energy discussed in Sec. <ref>, a regime in which the polariton spectra have not been studied analytically before. We again assume that the condition (<ref>), required for the approximatingthe photon self-energy by Eq. (<ref>), is fulfilled. This may be appropriate for systems with a large exciton-binding energies,e.g., transition metal dichalcogenide monolayers as recently studied in <cit.>.For infinite mass, we may use Eq. (<ref>) as photon self-energy, multiplied by a prefactor Δ^2 = d_0^2 ρ E_B/ g^2 [cf. Eq. (<ref>)], and expand the real and imaginary parts to leading order in α^2 = (δ/π-1)^2. The energy ω is now measured from the exciton pole: ω = Ω-Ω_T^exc, Ω_T^exc = E_G + μ - E_B. The corresponding polariton spectrum for a small cavity linewidth is shown in Fig. <ref>(a). Qualitatively, it strongly resembles the bare exciton case as in Fig. <ref> (note that in Fig. <ref> the cavity linewidth was chosen to be 100 times smaller than in Fig. <ref>), but with a larger linewidth of the upper polariton. This is due to the possible polariton decay into the particle hole continuum contained in the excitonic power law, Eq. (<ref>). The detailed discussion of polariton properties in the regime of μ≪ E_B parallels the previous discussion in the regime E_B ≪μ. For small negative detuning ω_c [Fig. <ref> (a)], the lower polariton is found at approximately ω = ω_c. The upper polariton has a significantly smaller weight, its shape reflects the excitonic power law of Eq. (<ref>). However, compared to the previous spectral cuts (Fig. <ref>) the upper polariton peak is much more pronounced. This results from the exciton being now pole-like,as compared to the power law Fermi-edge singularity.Increasing the detuning, weight is shifted to the upper polariton. At zero detuning [Fig. <ref>(b)], the weight of the lower polariton is only order 𝒪(α^2) larger than the weight of the upper polariton.At large positive detuning, the position of the lower polariton is found at approximately ω∼ -ω_c^-1/(1-α^2)asω_c →∞.The lower polariton thus approaches the exciton line faster than in the pure exciton case, but slower than in the Fermi-edge regime [Eq. (<ref>)]. A similar statement holds for the weight of the lower polariton, which scales as ω_c^-2-α^2.The spectrum in the finite mass case is qualitatively similar, see Fig. <ref>(b).Quantitatively, a stronger peak repulsion can be seen, which may be attributed to the enhanced excitonic quasiparticle weight in the finite mass case.A comparison of spectral cuts in the finite mass case [Fig. <ref>(a)–(c)] further corroborates this statement [especially inFig. <ref>(c)]. Indeed, one finds that the position of the lower polariton at large cavity detuning is approximately given by ω∼ -β^α^2·ω_c^-1asω_c →∞ ,i.e., the excitonic line at ω =0 is approached more slowly than in the infinite mass case, Eq. (<ref>). The corresponding weightfalls off as ω_c^-2. Thus, the lower polariton has a slightly enhanced weight compared to the infinite mass case.In addition, in the spectral cut at large negative detuning, [inset to Fig. <ref>(a)], the upper polariton appears as a sharper peak compared to the infinite mass case, which again results from the enhanced quasi particle weight of the finite mass case. § CONCLUSION In this paper we have studied the exciton-polariton spectra of a 2DEG in an optical cavity in the presence of finite CB electron density. In particular, we have elucidated the effects of finite VB hole mass, distinguishing between two regimes. In the first regime (small Fermi energy as compared to the exciton binding energy), we have found that excitonic features in the 2DEG absorption are enhanced by the exciton recoil and the resulting suppression of the Fermi edge singularity physics. In contrast, in the second regime of Fermi energy larger than the exciton binding energy,it is the VB hole which recoils at finite mass. This cuts off the excitonic features. These modifications also translate to polariton spectra, especially to the lower polariton at large cavity detuning, which is exciton-like. Our findings reproduce a trend seen in a recent experiment <cit.>.We would like to mention several possible extensions of this work. To begin with, it would be promising to study the effect of long-range interactions on the power laws, and hence on polariton spectra, from an analytical perspective. Long-range interactions are expected to be most important in the regime of small Fermi energy, leading to additional bound states and to the Sommerfeld enhancement effects <cit.>. Moreover, one should try to explore trionic features, for which it is necessary to incorporate the spin degree of freedom (to allow an electron to bind to an exciton despite the Pauli principle). Another interesting direction would be to tackle the limit of equal electron and hole masses, which is relevant to transition metal dichalcogenides, whose polariton spectra in the presence of a Fermi sea where measured in a recent experiment <cit.>. Lastly, one should address the behavior of the polariton in the regime of small Fermi energy and strong light-matter interactions. Then, not the exciton, but rather the polariton interacts with the Fermi sea, and different classes of diagrams have to be resummed to account for this change in physics. This work was initiated by discussions with A. Imamoğlu. The authors also acknowledge many helpful comments from F. Kugler, A. Rosch,D. Schimmel, and O. Yevtushenko. This work was supported by funding from the German Israeli Foundation (GIF) through I-1259-303.10. D.P. was also supported by the German Excellence Initiative via the Nanosystems Initiative Munich (NIM).M.G. received funding from the Israel Science Foundation (Grant 227/15), the US-Israel Binational Science Foundation (Grant 2014262), and the Israel Ministry of Science and Technology (Contract 3-12419), while L.G. was supported by NSF Grant DMR-1603243. § EVOLUTION OF ABSORPTION SPECTRA WITH INCREASING CHEMICAL POTENTIALIn this Appendix, we present an extended overview of how the absorption spectra evolve inbetween the controlled extremal limits of μ≪ E_B and μ≫ E_B.For μ≪ E_B, the dominant spectral feature is the exciton. For finite mass (β≠ 0), it has a coherent delta-like part and an incoherent tail, see Eq. (<ref>), while the infinite mass exciton (β = 0) is a purely incoherent power law, see Eq. (<ref>). These pronounced excitonic features are well separated from the CB continuum part at Ω_T^FES = E_G + μ (see inset to Fig. <ref>).As μ is increased, the incoherent exciton part [Eqs. (<ref>) and (<ref>)] starts to overlap with the CB continuum part. Moreover, the overall relative weight of both the coherent and incoherent portions of the exciton part of the spectrum (which are both proportional to E_B) will diminish. Still, within our simplified model which neglects CB electron-CB electron interactions, and for β=0, this exciton feature will never disappear completely, since in this model an infinite mass VB hole is simply a local attractive potential for the CB electrons, and such a potential will always have a bound state in 2D. However, for finite VB hole mass, the exciton energy (location of the coherent delta peak) will penetrate into the CB continuum when μ becomes larger than E_B/β≫ E_B (i.e., when E_B crosses the indirect threshold, see Fig. <ref>(a)). More importantly, CB electron-CB electron interactions would screen the hole potential, and will thus reduce the exciton binding energy and presumably eliminate the exciton part of the spectrum completely as soon as μ≫ E_B. To describe this situation, it has been customary in the literature <cit.> to still employ the same simplified model neglecting CB electron-CB electron interactions, but assume that the hole potential does not create a bound state for large enough μ, a practice we follow in this work as well. Then, for μ≫ E_B, one should concentrate on the remaining, CB continuum part of the spectrum, which will evolve into the Fermi-edge singularity (FES), cut off by the VB hole recoil energy for β≠ 0.A putative evolution of absorption spectra with increasing μ is sketched in Fig. <ref>.§ EVALUATION OF THE EXCITON SELF-ENERGY IN THE TIME-DOMAINIn this Appendix, we present the time-domain evaluation of the exciton self-energy diagrams of Fig. <ref>. These diagrams contain one CB electron loop only, and therefore yield the leading contribution when μ/E_B is small. We will start with the direct diagrams [Fig. <ref>(a)], and then turn to the exchange series [Fig. <ref>(b)]. §.§ Direct diagramsFirst, we note that the bare Green's functions in the time domain readG^(0)_c(k,t)= -i(θ(t) - n_k)e^-iϵ_kt, G^(0)_v(t)= iθ(-t) e^iE_Gt, with the zero temperature Fermi function n_=̨θ(k_F-k).Using these, we will evaluate the series of direct diagrams of Fig. <ref>(a). The temporal structure of a generic direct diagram is illustrated via the example of Fig. <ref>. To compute such a diagram, we make the following observation: Since the VB propagator has no momentum dependence, all VB phase factors simply add up to give a total factor of e^-iE_Gt. Then, the step functions in the VB propagators enforce time ordering for the intermediate time integrals. In the specific case shown in Fig. <ref>, 0<T_1<t_1<T_2<t_2<T_m<t_n<t with m = n =3 (m and n count the number of interaction lines above and below the dashed VB line, respectively). However, there are also diagrams with m=n=3, but with a different relative ordering of the interaction lines.Summing over all those diagrams for m and n fixed, one needs to integrate over the time ranges 0<t_1<...<t_n<t ∩ 0<T_1<...<T_m<t. This means that the time integration for the direct diagrams splits into a product of two functions, representing the propagation of a Fermi sea electron (above the VB line in Fig. <ref>) and a photoexcited electron (below the VB line) in the time-dependent potential.We are now in the position to write down the full expression for the sum of direct diagrams D to all orders in the interaction, fixing the signs with Wick's theorem:D(t) = -∫_k_1<k_Fdk_1/(2 π)^2 e^-iE_GtB̃(t) C(t),where B̃(t) =∑_m =1^∞ (-V_0)^m∫_0<T_1<⋯<T_m<t dT_1 ⋯ dT_m∫_k_2>k_Fdk_2/(2 π)^2⋯∫_k_m>k_Fdk_m/(2 π)^2G̃_c(k_1, T_1-T_m) G̃_c(k_2, T_2-T_1) ⋯G̃_c(k_m, T_m - T_m-1), C(t) = ∑_n=0^∞ (-V_0)^n ∫_0<t_1<⋯<t_n<tdt_1 ⋯ dt_n ∫_q_1>k_Fdq_1/(2 π)^2⋯∫_q_n+1>k_Fdq_n+1/(2 π)^2G̃_c(q_1,t_1) G̃_c(q_2,t_2-t_1) ⋯G̃_c(q_n+1, t-t_n),andG̃_c(k_1, T_1-T_M)= ie^-iϵ__̨1(T_1-T_M) G̃_c(p,τ)= -ie^-iϵ_pτforp≠k_1.Defining the retarded Green's function byG^0, R_c(p,τ) = θ(τ) G̃(p,τ),we can rewrite the two factors appearing in D(t) as sequences of convolutions:B(t) ≡e^-iϵ__̨1tB̃(t) =∑_m=1^∞ (-V_0)^m ∫_k_2>k_Fdk_2/(2 π)^2⋯∫_k_m>k_Fdk_m/(2 π)^2[G_c^0,R(k_1, ) ∗ G_c^0,R(k_2, ) ⋯∗ G_c^0,R(k_m, )∗ G_c^0,R(k_1, )](t), C(t) = ∑_n=0^∞ (-V_0)^n ∫_q_1>k_Fdq_1/(2 π)^2⋯∫_q_n+1>k_Fdq_n+1/(2 π)^2[G_c^0,R(q_1, ) ∗⋯∗G_c^0,R(q_n+1, )](t).Together, Eqs. (<ref>) and (<ref>)–(<ref>) correspond to Eq. (<ref>) in the main text. Fourier transforming Eq. (<ref>) results in: D(Ω) = ∫_k_1<k_Fdk_1/(2 π)^2 i ∫dν/2π B(ν) C(Ω-E_G + ϵ__̨1 - ν)_I(Ω),where we defined I(Ω) for later purpose. The Fourier transform of B(t) reads:B(ν) = ∑_m=1^∞ (-V_0)^m ∫_k_2>k_Fdk_2/(2 π)^2⋯∫_k_m>k_Fdk_m/(2 π)^2G_c^0,R(k_1,ν ) · G_c^0,R(k_2,ν ) ⋯ G_c^0,R(k_m,ν )· G_c^0,R(k_1,ν ), with retarded real frequency Green's functions:G_c^0,R(k,ν) = 1/ν - ϵ_k + i0^+.Inserting (<ref>) into (<ref>), the integrations are trivially performed. The summation over interaction lines reduces to a geometric series, yielding: B(ν)= -V_0/g1/(ν - ϵ__̨1 + i0^+)^2·1/ln(ν - μ + i0^+/-E_B),where we used ln(E_B/ξ) = -1/g, c.f. Eq. (<ref>). For the term C(Ω - E_G + ϵ__̨1 - ν) appearing in (<ref>) we analogously arrive at: C(Ω - E_G + ϵ__̨1 - ν) =ρ/g(1- 1/g ln(κ- ν + i0^+/-E_B)), κ≡Ω - E_G+ ϵ__̨1 - μ. The functions B(ν) and C(ν) are difficult to integrate, becausethey each have both a pole and a branch cut, arising from the 1/ln term. We can split these terms as follows:1/ln(ν - μ + i0^+/-E_B) =-E_B/E_B + ν - μ + i0^++ (1/ln(ν - μ + i0^+/-E_B) + E_B/E_B + ν - μ + i0^+).The first term on the right hand side of Eq. (<ref>) has just a simple pole, while the second one's only singularity is a branch cut. Using this representation, we canevaluate I(Ω) as defined in Eq. (<ref>) employing the following argument: Physically, the terms B, C represent the propagation of the two electrons in the hole potential. Comparing to the simple exciton ladder summation (see Sec. <ref>),we associate the poles of the 1/ln-terms in these functions with the exciton contribution, while the branch cut corresponds to the continuum above the indirect threshold, Ω > E_G + μ. Following these observations, let us split I(Ω) into a pole-pole, a pole-branch, and a branch-branch contribution. I_branch-branch only contributes to the continuum part of the spectrum. More importantly (as explained in the main text), employing spectral representations of the retarded functions B_branch, C_branch, it is easily shown that Im[I_branch-branch] (which is of potential importance for the lineshape of the exciton spectrum)vanishes for frequencies close to the exciton pole (ω≳ 0).It is thus not important for our purposes.Computing contour integrals, I_pole-pole is easily evaluated to give:I_pole-pole(ω) = E_B^2/g^21/(ω + i0^+)^21/E_B + ω + ϵ_k_1- μ + i0^+,where energies are measured from the exciton pole, ω = Ω - (E_G + μ) + E_B. This contribution gives rise to trionic features in the spectrum, which are shortly discussed in Appendix <ref>.Last, computing contour integrals and disregarding terms which are subleading in ω/E_B, the pole-branch contribution is found to be:I_pole-branch(ω) ≃-E_B/g^21/(ω + i 0^+)^2×(1/ln(ω + ϵ__̨1 - μ + i0^+/-E_B) + E_B/E_B + ω + ϵ__̨1 - μ + i0^+). Insertingthe Eqs. (<ref>) and (<ref>) into Eq. (<ref>), one finally arrives at Eq. (<ref>) of the main text. §.§ Exchange diagramsThe computation of the exchange diagrams, though technically sligthly more involved, essentially proceeds along the same lines. The general time-structure of an exchange diagram is illustrated in Fig. <ref>. As for the direct diagrams, the VB propagators just enforce a time ordering. In addition, there is the condition t_n>T_1. When this condition is violated, the diagram reduces to a ladder diagram, which must be excluded to avoid double counting.Taking this into account, the full expression for the sum of exchange diagrams reads:X(t) =∑_m,n=1^∞(-V_0)^m+ne^-iE_G t∫_0<T_1<⋯<T_m<tdT_1 ⋯ dT_m ∫_T_1^t dt_n ∫_0^t_n dt_n-1⋯∫_0^t_2 dt_1∫_k_1<k_Fdk_1/(2π)^2∫_k_2>k_Fdk_2/(2 π)^2⋯∫_k_m+1>k_Fdk_m+1/(2 π)^2∫_q_1>k_Fdq_1/(2 π)^2⋯∫_q_n>k_Fdq_n/(2 π)^2G̃_c(k_1, T_1 - t_n) G̃_c(k_2, T_2 - T_1) ⋯G̃_c(k_m+1, t - T_m) G̃_c(q_1, t_1)⋯G̃_c(q_n,t_n-t_n-1)To rewrite (<ref>) as a sum of convolutions, one can employ the following easily-derived formula:ℱ(∫_-∞^∞ dt_1 f(t-t_1)g(t,t_1))(Ω) = ∫_-∞^∞dω_1/2π f(ω_1) g(Ω - ω_1, ω_1), where ℱ denotes the Fourier transform, and f and g are any two well-behaved functions.Applying this result, a computation similar to the one for D(Ω) shows that the Fourier-transform of Eq. (<ref>) can be expressed as:X(Ω) = -∫_k_1< k_Fdk_1/(2π)^2∫_-∞^∞dω_1/2π (-g) ln(ω_1 - μ + i0^+/-ξ) 1/1+g ln(ω_1 - μ + i0^+/-ξ)1/Ω - E_G - ω_1 + i0^+∫_-∞^∞dω_2/2π(-g) ln(ω_2 - μ + i0^+/-ξ)1/1+g ln(ω_2 - μ + i0^+/-ξ)1/-ω_2 + Ω - E_G + i0^+1/ω_2 + ω_1 - Ω + E_G - ϵ_k_1 - i0^+.This expression can be evaluated as before, splitting it into pole-pole, pole-branch and branch-branch contributions using Eq. (<ref>). In complete analogy to the direct diagrams, the imaginary part of the branch-branch contribution can be shown not to contribute in the regime of interest to us, and we therefore disregard it completely. Straight-forwardly evaluating the pole-pole and pole-branch contributions, one ultimately arrives at Eq. (<ref>) in the main text. § TRION CONTRIBUTION TO THE EXCITON SELF-ENERGY DIAGRAMS The pole-pole contribution to the direct self-energy D(ω) [Eq. (<ref>)] physically represents two electrons tightly bound to the hole potential. Indeed, it assumes the form:D_pole-pole(ω) = ∫_k_1<k_Fdk_1/(2 π)^2 I_pole-pole(ω),where I_pole-pole is given in Eq. (<ref>). I_pole-pole can be identified with a bare trion Green's function, since it has a pole at ω = -E_B + μ-ϵ_k_1, corresponding to the binding of a second CB electron to the exciton (recall that ω is measured from the exciton threshold), where the energy ϵ_k_1 of this second electroncan be from anywhere in the Fermi sea. Evaluation of (<ref>) close to the trion resonance ω≃ -E_B leads to D_pole-pole(ω) ≃ρ/g^2ln(E_B + ω + i0^+/E_B + ω - μ).Using Eq. (<ref>) of the main text, (<ref>) gives rise to a self-energy contribution to the excitonΣ_exc= E_B ln(E_B + ω + i0^+/E_B + ω - μ).This self-energy expression fully matches usual results found in works concerned with trions <cit.>, apart from two minor differences: First, in these works the case of finite VB hole mass (of the same order as the CB mass) is considered, but reevaluation of (<ref>) for finite mass is straightforward and only results in some trivial factors involving mass ratios. Second, in the works cited above the exciton is treated as an elementary entity, and the trion binding energy is therefore an adjustable parameter. By contrast, we have started from a microscopic model which does not contain excitons, and, accounting for exchange processes, computed excitons and trions along the way. As a result, our microscopic theory yields the same binding energy E_B for excitons and trions. However, this is clearly an artefact of disregarding electron-electron interactions (which would significantly reduce the trion binding energy), and can heuristically be accounted for by replacing E_B in Eq. (<ref>) by a trion binding energy E_T ≪ E_B.Upon inserting (<ref>) into the exciton Green's function (<ref>), one finds the following spectral features: First, there is a sharp resonance, red detuned w.r.t. the trion threshold by an order of μ, and with a weight that scales as μ/E_T. This peak is commonly called the trion, or, more appropriately, attractive polaron <cit.>, since the trion bound state is not filled. Second, there is a small step-like feature for 0< E_B + ω < μ, arising from the imaginary part of (<ref>). This feature, where the trion bound state is filled and the second electron constituting the trion can come from anywhere in the Fermi sea, has smaller (but not parametrically smaller) weight than the attractive polaron, and is usually overlooked in the literature. Investigation of further trion properties is a worthwhile goal which we leave for further work. Let us close this Appendix with a technical remark: Of course, for spinless electrons a trion cannot exist in our simple model of short range VB hole-CB electron interaction, due to the Pauli principle (two electrons cannot occupy the single bound state created by the hole). In line with that, the pole-pole contribution cancels in this case between the direct and exchange diagrams. However, in the spinful case, the direct contribution will incur a factor of two, so it does not cancel with the exchange contribution, so the trion remains. § THE SELF-ENERGY CONTRIBUTION OF THE EXCHANGE DIAGRAMS The exchange contribution to the exciton self-energy, Eq. (<ref>), can be understood by the following considerations. The ground state energy of an N-particle system in the presence of an attractive delta function potential strong enough to form a bound state is lower than the N-particle ground state energy of the system without the potential by an amountΔ E = -E_B - (1-α)μ,which is the sum of the bound state energy E_B, and a second term which arises from the rearrangement of the Fermi sea, described by Fumi's theorem <cit.> [recalling that 1-α=δ/π, cf. Eq. (<ref>)]. We find that the exchange diagrams give the contribution μ, while the term αμ stems from the direct diagrams [Eq. (<ref>)].To create such an attractive potential, one has to lift one electron from the VB to the CB, which costs E_G + μ. In our treatment, the extra cost μ appearing here is contained in the shift of the pole of the ladder diagrams, Eq. (<ref>). Thus, the minimal absorption energy predicted by our model is E_G - E_B + αμ≈ E_G - E_B.At first sight this seems to contradict the experimental results (e.g., <cit.>), according to which the minimal absorption energy is E_G - E_B + μ (or 2μ for equal electron-hole masses). This is attributed to “phase-space filling effects”, or, in other words, the Burstein-Moss shift <cit.>, which precisely correspond to the shift of the ladder pole, without the Fumi contribution.The reason for this discrepancy is that our model ignores the CB electron-CB electron interaction, which would render the exciton electrically neutral and suppress the Fumi shift. Thus, as also pointed out in the literature on the X-ray edge problem, neglecting electron-electron interactions gives the right power law scalings of the spectra only, but not the correct threshold energies. Another aspect of Eq. (<ref>) is its lack of dependence on the frequency ω. In other words, the Anderson orthogonality power law of the exciton Green's function does not depend on X(ω). This could have been anticipatedby an argument based on Hopfield's rule of thumb <cit.> and the results of <cit.>. Consider the spinful case, and study the absorption spectral function for, e.g., right-hand circularly polarized light at the exciton threshold, creating a spin down electron and a spin up hole. The spectrum should have the form1/ω·ω^(1-δ_↓/π)^2+(1-δ_↑/π)^2.For the spin down electrons, the exponent is (1-δ_↓/π)^2 rather than (δ_↓/π)^2 because of the Hopfield rule: one electron is lifted from the valence band to the conduction band. For the spin up electron, no electron is lifted. However, the exciton is the secondary threshold in the spinful case (the primary one is the trion). As seen from <cit.>, the spin up exponent should therefore also be as in Eq. (<ref>). Now, in the spinful case all direct diagrams will come with a spin factor of 2, while the exchange diagrams will not. However, we see that the exponent in (<ref>) is exactly 2 times the exponent the spinless case, Eq. (<ref>), when recalling that δ_↑ = δ_↓ = δ for our spin-independent potential.This shows that the exchange diagrams should indeed not contribute to Anderson orthogonality, at least to leading order. § COMPUTATION OF PHASE-SPACE INTEGRALS FOR THE PARTICLE-HOLE PAIR DENSITY OF STATESTo clarify the different role of the recoil in the exciton (section <ref>) and FES cases (section <ref>), let us present the computation of two important phase space integrals.§.§ Exciton recoilWe start with the evaluation of the imaginary part of the exciton self-energy Im[Σ](ω) given in Eq.(<ref>), focusing on zero exciton momentum. Im[Σ] reads: Im[Σ_exc] ≃-π V_0/ρ gα^2 ∫_k_1 < k_Fdk_1/(2π)^2∫_k_2 > k_Fdk_2/(2π)^2δ(ω - (_̨2 - _̨1 )^2/2M_exc - ϵ_k_2 + ϵ_k_1) . Im[Σ_exc] can be interpreted as rate of decay of excitons into CB electron-hole pairs, or alternatively as density of state of the CB pairs. We aim to compute the leading ω-behaviour of Im[Σ_exc]. To put it short, the delta-function in (<ref>) requires _̨1, _̨2 ≃ k_F and ∡(_̨1, _̨2) ≃ 0, and these phase space restrictions pile up to give Im[Σ_exc] ∼ω^3/2. To perform the calculation in detail, we substitute x = k_2/√(2m) ,y = k_1/√(2m). Switching the integrals for convenience, we can rewrite (<ref>), to leading order in the mass ratio β, asIm[Σ_exc] = - α^2/π∫_x>√(μ)dx∫_y< √(μ) dyδ(ω -(x^2 - μ) + (y^2 - μ)- β(x- y)^2 ). First, it is obvious that (<ref>) is proportional to θ(ω), since all terms subtracted from ω in the delta function are positive, hence there cannot be any cancellations. Second, it is clearly seen that x≃√(μ), y ≃√(μ) to yield a nonzero contribution for small ω. Thus, we may linearize the dispersion relation, starting with y:y = (√(μ) + γ_y)e_y, y^2 = μ + 2√(μ)γ_y + 𝒪(γ_y^2). In doing so, we effectively disregard subleading terms of order 𝒪(ω^2/μ) in the argument of the delta function. Introducing the notationϕ=∡(x, y),c = cos(ϕ), we arrive at:Im[Σ_exc] =-α^2θ(ω)/π∫_x > √(μ) dx∫_-1^12/√(1-c^2)∫_-√(μ)^0 dγ_y (√(μ) + γ_y) δ(ω - (x^2-μ) - β x^2 + 2 β x √(μ) c - βμ_=A + γ_y (2β x c - 2β√(μ) + 2√(μ))_=B) .Since the only contribution comes from γ_y close to the upper boundary, we can write √(μ) + γ_y ≃√(μ).Using B ≃ 2√(μ), the trivial integral over γ_y then results in Im[Σ_exc] =-α^2/π∫_x > √(μ) dx∫_-1^1dc1/√(1-c^2) θ(A) . To find the leading power law in ω of this expression, we assume thatω≪βμ. Then, we rewrite θ(A) asθ(ω - (x^2-μ) - β x^2 - βμ^=C + 2 β x √(μ) c ) =θ(c - (-C/2β x √(μ) )). We now use x ≃√(μ). Thus, we can write-C/2β x√(μ)≃ 1-(ω/2βμ - x^2 -μ/2βμ) + 𝒪(ω/μ). Going back to (<ref>) givesIm[Σ_exc] =-α^2 θ(ω)/π∫_x > √(μ)dx θ (ω-(x^2 - μ)) ∫^1_1-(ω-(x^2-μ))/2βμ dc1/√(1-c^2). Using that for 0<t<1: ∫_1-t^1 1/√(1-y^2) dy = arccos(1-t) = √(2t) + 𝒪(t^3/2), we obtain Im[Σ_exc] = -2α^2θ(ω)∫_√(μ)^√(μ + ω) x dx √(ω - (x^2 - μ)/βμ).This can be integrated exactly to give: Im[Σ_exc](ω) = -2α^2/31/√(βμ)·θ(ω) ω^3/2. The numerical prefactor should be correct, but is of no parametric relevance and is set to unity for convenience, thereby giving formula (<ref>) of the main text.§.§ FES regime: VB hole recoilIn the regime of the FES, not the exciton, but the valence band hole recoils. Near the direct threshold at ω = βμ, the quantity describing the hole decay is Im[Σ_VB(k_F,ω)] as given in (<ref>), which scales differently compared to the exciton decay because the VB hole has = k_F unlike the = 0 exciton (we do not present this computation here since the power law is of not much relevance for the 2DEG absorption we are interested in; see <cit.> for details). Near the indirect threshold, the VB hole again has momentum = 0, and the resulting 2DEG absorption A(ω) as given in (<ref>) scales as ∼ω^3. This result was already presented in <cit.>, though without derivation. Since the computation is very similar to the previous one for the exciton decay, let us just sketch it: By performing frequency integrals in Figs. <ref> and <ref>, and momentum substitutions as for the exciton, one arrives at: A(ω) ∼∫_x^2>μdx∫_z^2>μdz∫_y^2<μdyδ( ω - (x^2 - μ)+(y^2 - μ)-(z^2 - μ) - β(x +z - y)^2 ),which is similar to the previous expression (<ref>) except for an additional scattering partner, the photoexcited electron (corresponding to the z-integral). Again, there can be no cancellations in the deltafunction, and the computation proceeds analogously to sec. <ref>. Effectively, the summands (x^2-μ), (y^2 - μ) and (z^2 - μ) contribute a factor of ω to A(ω). One factor is fixed by the delta function, such that in total one has ω^2. In addition, there is the hole recoil term β(x + z - y)^2. For this to be of order ω, the angles ϕ = ∡(x + z,y) and θ = ∡(x,z) have to be fixed as depicted in Fig. <ref>. The explicit computation shows that each angle restriction give a factor of √(ω), such that in total one arrives at A(ω)∼ω^3. prsty_(no_et_al) | http://arxiv.org/abs/1707.08613v2 | {
"authors": [
"Dimitri Pimenov",
"Jan von Delft",
"Leonid Glazman",
"Moshe Goldstein"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.str-el",
"published": "20170726190922",
"title": "Fermi-edge exciton-polaritons in doped semiconductor microcavities with finite hole mass"
} |
http://arxiv.org/abs/1707.08582v3 | {
"authors": [
"Shira Chapman",
"Michal P. Heller",
"Hugo Marrochio",
"Fernando Pastawski"
],
"categories": [
"hep-th",
"quant-ph"
],
"primary_category": "hep-th",
"published": "20170726180007",
"title": "Towards Complexity for Quantum Field Theory States"
} |
|
1]Richard Connor [1]Department of Computer and Information Sciences,University of Strathclyde, Glasgow, G1 1XH, United Kingdom 2]Lucia Vadicamo 3]Franco Alberto Cardillo 2]Fausto Rabitti [2]Institute of Information Science and Technologies (ISTI), CNR, Pisa, Italy [3]Institute of Computational Linguistics (ILC), CNR, Pisa, Italy [1][email protected] [2]{lucia.vadicamo, fausto.rabitti}@isti.cnr.it [3][email protected] Search [================== Metric searchisconcerned with the efficient evaluation of queries in metric spaces. In general, alarge space of objects is arranged in such a way that, when a further object is presented as a query, those objects most similar to the query can be efficiently found. Mostmechanisms rely upon the triangle inequality property of the metric governing the space. The triangle inequality propertyis equivalent to a finiteembedding property, which states that any three points of the space can be isometrically embedded in two-dimensional Euclidean space. In this paper, we examine a class of semimetric space which is finitely four-embeddable in three-dimensional Euclidean space. In mathematics this property has beenextensively studied and is generally known as the four-point property. All spaces with the four-point property are metric spaces, but they also have some stronger geometric guarantees. We coin the term supermetric [This term has previously been used in the domains of particle physics and evolutionary biology as a pseudonym for the mathematical term ultra-metric, a concept of no interest in metric search; we believe our concept is of sufficient importance to the domain to justify its reuse with a different meaning.] space as, in terms of metric search, they are significantly more tractable. Supermetric spaces include all those governed by Euclidean, Cosine[for the correct formulation of Cosine distance, see <cit.> for details], Jensen-Shannon and Triangular distances, and are thus commonly used within many domains. In previous work we have given a generic mathematical basis for the supermetric property and shown how it can improve indexing performance for a given exact search structure. Here we present a full investigation into its use within a variety of different hyperplane partition indexing structures, and go on to show some more of its flexibility by examining a search structure whose partition and exclusion conditions are tailored, at each node, to suit the individual reference points and data set present there. Among the results given, we show a newbest performance for exact search using a well-known benchmark.Keywords:Similarity Search, Metric Space, Supermetric Space, Metric Indexing, Four-point Property, Hilbert Exclusion§ INTRODUCTIONWithin any metric space, any three objects can be used to construct a triangle in 2D Euclidean space, where the objects are represented by the vertices of the triangle and the edges preserve their distances in the original space. That is, any metric space is isometrically three-embeddable in 2D Euclidean space.Some metric spaces are also isometrically four-embeddable in 3D Euclidean space, allowing the construction of a tetrahedron.Wehavepreviously shown how these spaces havefurther geometric properties which can be used to improve the performance of exact search,in particular for any search mechanism based on hyperplane partitioning.This leads to the notion of a supermetric space <cit.>,a space which is also a metric space but with further geometric properties which give stronger guarantees for search mechanisms.Furthermore, we have given a rigorous and constructive mathematical basis for assessing whether a proper metricspace has the supermetric property, and showed how this property allows the use of the Hilbert Exclusion mechanism in place of the less powerful hyperbolic exclusion <cit.>. In <cit.>, we also showed how the supermetric property could, in principle, be used to construct arbitrary partitions within a 2D plane into which many objects are projected, due to a lower-bound property which is acorollary of the four-point property.In this paper we extend initial work which appeared in <cit.> by taking the investigation to its next stage.While we previously showed how the use of the Hilbert Exclusion property gave a significant improvement in performance when used in conjunction witha particular state-of-the-art hyperplane-based indexing mechanism, the Distal Spatial Approximation Tree (DiSAT, <cit.>), we now perform a full evaluation over its performance within a fully general context of twelve different hyperplane tree indexing structures. The outcome is that a simpler data structure is found to be the most efficient in this context, and indeed gives a new best-published performance for threshold search over the SISAP benchmark data sets <cit.>; to put this result in perspective, it requires only around 40% of the number of distance calculations per query of the previous state of the art given in <cit.>.Beyond these benchmark data sets, which in this context are relatively small and tractable, we show the performance advantages hold in some larger data sets, as dimensionality and object size increase, and also for a number of different distance metrics.Further, we begin to investigate more flexible use of the planar lower bound property we first described in <cit.>. At the time of this publication we observed that the property was more general than the Hilbert Exclusion property. Now we are able to show a remarkably flexible use within a hyperplane tree built over “real-world" data sets; the significance of such data is that it is typically distributed in a non-uniform manner within the space. The non-uniformity manifests differently with each choice of reference points, and this data structure allows a different strategy to be used in each node, to maximise the advantage which can be gained.The rest of this paper is organised as follows.Section <ref> sets the detailed technical context for the work, including related work by ourselves and others. Section <ref>then explains a novel observation which is a consequence of the four-point property: tetrahedral projection onto a plane, which gives an important lower-bound property. In fact, it turns out that Hilbert Exclusion results as a simple corollary of this more general property. In this section we discuss a number of relatively deep results which are consequent to the property.Section <ref> then fully defines a completely novel indexing structure which is only possible to use in a supermetric space, where the hyperplane partition and consequent exclusion mechanism are dynamically chosen according to the distribution of data within each individual node of the tree. Section <ref> takes as its starting point the observation that the best indexing techniques are likely to be different in the supermetric context, and gives a full investigation of various hyperplane trees in order to determine the most suitable. Section <ref>analyses the extra cost required to make use of the supermetric properties for hyperplane indexing, which in fact is very small. Section <ref>examines the use of the best data structures identified through their application to a number of large and real-world data sets, in order to test their performance in more general contexts.Finally in Section <ref> we give some conclusions and outline areas of further work.§ PRELIMINARIES AND RELATED WORKTo set the context, we are interested in searching a (large) finiteset of objects S which is a subset of an infinite set U, where (U, d) is a metric space.A metric space is an ordered pair (U, d), where U is a domain of objects and d is a total distance function d:U × U →, satisfying postulates of non-negativity, identity, symmetry, and triangle inequality <cit.>. The general requirement is to efficiently find members of S which are similar to an arbitrary member of U, where the distance function d gives the only way by which any two objects may be compared - the bigger the distance d(x,y), the less similar the data objects x,y∈ U. There are many important practical examples captured by this mathematical framework, see for example <cit.>. Such spaces are typically searched with reference to a query object q ∈ U. The simplest type of similarity query is the range search query.A range searchfor some threshold t, based on a query q ∈ U, has the solution setR={s ∈ S |d(q,s) ≤ t}.Other forms of search, for example nearest neighbor search (i.e. find the k closest objects to a query), are also useful; here we are studying mostly properties of spaces in general and restrict our attention to the scenario outlined.Symbols and abbreviations used throughout this paper are summarized in Table <ref>.§.§ Metric indexingTypically, the distance function is too expensive or S is too large to allow an exhaustive search, that is a sequential scan of the entire dataset. The retrieval process is facilitated by using a metric index, one of a large family of data structures used to preprocess the data in such a way as to minimise the time required to retrieve the query result. This data structure can be expensive to build, but this cost is amortized by saving I/O and distance evaluations over several queries to the database. In general, the triangle inequality property is exploited to determine subsets of S which do not need to be exhaustively checked.Such avoidance is normally referred to as exclusion or space pruning.For exact metric search, almost all indexing methods can be divided into those which at each exclusion possibility usea single “pivot" point to give radius-based exclusion, and those which use two reference points to give hyperplane-based exclusion. Many variants of each have been proposed, including many hybrids; <cit.>, <cit.>, and <cit.>give excellent surveys. In general the best choice seems to depend on the particular context of metric and data.Here our focus is particularly onmechanisms which use hyperplane-based exclusion.The simplest such index structure is the Generalised Hyperplane Tree (GHT) <cit.>. Others include Bisector trees <cit.> and variants on them (e.g.Monotonous Bisector Trees <cit.> and Voronoi Trees <cit.>), the Metric Index <cit.>, and the Spatial Approximation Tree <cit.>. This last has variousderivatives, notably including the Dynamic SAT <cit.> and the Distal SAT (DiSAT) <cit.>.§.§ Metric Spaces and Finite Isometric Embeddings Anisometric embedding of one metric space (V,d_v) in another (W,d_w) can be achieved when there exists a mapping function f:V→ W such that d_v(x,y) = d_w(f(x),f(y)), for all x,y ∈ V. A finite isometric embedding occurs whenever this property is true for any finite selection of n points from V, in which case the terminology used is that V is isometrically n-embeddable in W.The idea of characterising a space metrically by means of “n-point relations" seems to have originated in the paper <cit.> published in 1892 by de Tilly, a Belgian artillery officer. Some of the questions raised by de Tilly were answered by some mathematicians of the late 19th century, and only in 1928 Karl Menger <cit.> provided a first systematic development of abstract distance geometry. The interest of the distance geometry is in all those of transformations of sets for which the distance of two points is an invariant. So, as highlighted by Blumenthal <cit.>, “distance geometry may operate in any kind of space in which a notion of “distance” is attached to any point-pair of the space”. Isometric 3-embedding in The first observation to be made in this context is that any metric space (U,d) is isometrically 3-embeddable in , i.e. for any three points x_1,x_2,x_3 ∈ U there exists a mapping function f:(U,d)→ (2, ℓ_2) such that ℓ_2(f(x_i),f(x_j))=d(x_i,x_j), for i,j=1,2,3. This isapparent from the triangle inequality property of a proper metric. In fact the two properties areequivalent: for any semi-metric space which is isometrically 3-embeddable in, triangle inequality also holds.It is interesting to consider the standard exclusion mechanisms of pivot-based exclusion and hyperplane-based exclusion in the light of an isometric 3-embedding in ; Figure <ref> for example shows a basis for hyperplane exclusion using only this property rather than triangle inequality explicitly. Supermetric Spaces: Isometric 4-embedding in It turns out that many useful metric spaces have a stronger property: they areisometrically 4-embeddable in , which means that for any four points in the space there exists an embedding into (3,ℓ_2) that preserves all the 42 =6 interpoint distances. In the mathematical literature, this has been referred to as the four-point property <cit.>. Wilson <cit.> shows various properties of such spaces, and Blumenthal <cit.> points out that results given by Wilson, when combined with work by Menger <cit.>, generalise to show that some spaces have the n-point property: that is, any n points can be isometrically embedded in a Euclidean(n-1)-dimensional space. We have studied such spaces in the context of metric indexing in <cit.>, where we develop in detail the following outcomes: * Any metric space which is isometrically embeddable in a Hilbert space[ A Hilbert space H can be thought of as a generalization of Euclidean space to any finite or infinite number of dimensions. It is an inner vector space which is also a complete metric space with respect to the distance function induced by the inner product. This means that it has an inner product <·,·>:H× H →ℂ that induces a distance d(·,·)=√(<·,·>) such that every Cauchy sequence in (H,d) converges to a point in H (intuitively, there are no “points missing” from H). ] has the n-point property, and so the four-point property as well. * Important spaces with the n-point property include, for any dimension, spaces with the following metrics: Euclidean, Jensen-Shannon, Triangular, and (a variant of) Cosine distances. * Important spaces which do not have the four-point property include those with the metrics: Manhattan, Chebyshev, and Levenshtein distances. * However, for any metric space (U,d), the space (U,d^α), 0<α≤1/2 does have the four-point property. In terms of practical impact on metric search, in <cit.> we show only how the four-point property can be used to improve standard hyperplane partitioning. We consider a situation where a subspace is divided according to which of two selected reference points p_1 and p_2 is the closer. When relying only on triangle inequality, that is in a metric space without the four-point property, then for a query q and a query threshold t, the subspace associated with p_1 can be excluded from the search only if d(q,p_1) - d(q,p_2) > 2t. As the region defined by this condition when projected onto the plane is a hyperbola (see Figure <ref>), we name this Hyperbolic Exclusion[In the literature, the Hyperbolic Exclusion is also referred to as Double-Pivot Distance Constraint <cit.>.]. If the space in question has the four-point property, however, weshow that, for the same subspaces, there is no requirement to search that associated with p_1 wheneverd(q,p_1)^2- d(q,p_2)^2/d(p_1,p_2) > 2t;this is a weaker condition and therefore allows, in general, more exclusion. We name this condition Hilbert Exclusion. A formula equivalent to |d(q,p_1)^2- d(q,p_2)^2|/2d(p_1,p_2)has been used in the context of metric search also in <cit.>, in order to estimate the distance between the point q and the hyperplane equidistant from p_1 and p_2. This formula was derived using the cosine law and was applied only with distances on metric space with the “semidefinite positive property" <cit.>, since this property allows defining a notion of “angle" in a generic metric space. To provide a bridge to our work, we observe that the semidefinite positive property is equivalent to the n-point property for a finite semimetric space (see Chapter IV, Section 43 of <cit.>). In this paper, we examine a more general consequence offour-point embeddable spaces and show some interim results including new best-performance search ofSISAP data sets. § TETRAHEDRAL PROJECTION ONTO A PLANE In a supermetric space, any two reference points p_1 and p_2, and query point q, and any solution to that query s where d(q,s) ≤ t, can all be embedded in 3D Euclidean space. As such, they can be used to form the vertices of a tetrahedron. It seems that, while simple metric search is based around theproperties of a triangle, there should be corresponding tetrahedral properties which give a new, stronger, set of guarantees.Assume that for some search context, points p_1, p_2 ∈ U are somehow selected and a data structure is builtfor a finite set S ⊂ U where, for s ∈ S, the three distances d(p_1,p_2), d(s,p_1) and d(s,p_2) are calculated during the build process and used to guide the structuring of the data. At query time, for a query q, the two distances d(q,p_1) and d(q,p_2) are calculated and may be used to make some deduction relating to this structure.This situation givesknowledge of two adjacent faces of the tetrahedron which can be formed in three dimensions. Five of the six edge lengths have been measured, and the final edge is upper-bounded by the value of t. Therefore, for a point s to be a solution to the query, it must be possible to form a tetrahedron with the five measured edge lengths, and a last edge of length t. Figure <ref> shows a situation where five edge lengths have been embedded in 3D space. The edge p_1p_2 is shared between the two facial triangles depicted.However the distance d(s,q) is not known, and therefore neither is the angle between these triangles. The observation which gives rise to the results presented here is that, if both triangles are now projected onto the same plane, which can be achievedby rotating one of them around the line p_1p_2 until it is coplanar with the other, then for any case where the final edge of thetetrahedron (qs) is less than the length t, then the length of this side in the resulting planar tetrahedron isupper bounded by t, as illustrated in Figure <ref>.Many such coplanar triangles can be depicted, representing many points in a single space, in a single scatter plot as in Figure <ref>. This shows a set of 500points, drawn from randomly generated 8-dimensional Euclidean space, and plotted with respect to their distances from two fixed reference points p_1 and p_2. The distance between the reference points is measured, and the reference points are plotted on the X-axis symmetrically either side of the origin. For each point in the rest of the set, the distances d(s,p_1) and d(s,p_2) are calculated, and used to plot the unique corresponding point in a triangle above the X-axis, according to these edge lengths. In this figure, in consideration with our observations over Figure<ref>, it can be seen that, if any twopointsare separated by less that some constant t in the original space, and thus also in the 3D embedding, then they are also within t of each other in this scatter plot.It is important to be aware, in this and the following figures, of the importance of the four-point property. The same diagram can of course be plotted for a simple metric space, but in thiscase nospatial relationship is implied between any two points plotted: no matter how close two points are in the plot, there is no implication for the distance between them in the original space. However ifthediagram is plotted for a metric with the four-point property, then the distance between any two points on the plane is a lower bound on their distance in the original space; two points that are further than t on the plot cannot be within t of each other in the original space. This observation leads to an arbitrarily large number of ways of partitioning the space and allowing these partitions to excluded based on a query position, and has many potential uses in metric indexing. §.§ Indexes Based on Tetrahedral/Planar Projection During construction of an index, the constructed 2D space can be arbitrarily partitioned according to any rule based on the geometry of this plane, calculated with respect to thedistances d(s_i,p_1), d( s_i,p_2) and d(p_1,p_2). At query time, if the query falls in any region of the plane that is further than the query threshold t from any such partition, points within that partition cannot contain any solution to the query. Since, as will be shown, different spaces give quite different distributions of points within the plane, build-time partitions can be chosen according to this distribution, rather than as a fixed attribute of an index mechanism. There is much potential for investigating partitions of this plane, and our work is ongoing. The simplest such mechanism to consider is the application of this conceptto normal hyperplane partitioning. Suppose that a data set S is simply divided according to which of the points p_1 and p_2 is the closer, which corresponds in the scatter diagram to a split over the Y axis. Then at query time, if the corresponding plot position for the query is further than t from the Y axis, no solutions can exist in the subset closer to the opposing reference point. Figure <ref> shows the same points, but now highlighted according to this distinction. Those drawn in solid, either side of the Y-axis, are guaranteed to be on the same side of the corresponding hyperplane partition in the original space; therefore, if they were query points, the opposing semi-space would not require to be searched. We refer to these points as “exclusive queries”.If the same diagram is drawn for a simple metric space, a query point can be used to exclude the opposing semi-space only according to acondition algebraically derived from triangle inequality: |d(q,p_1) - d(q,p_2)| > 2t,which describes a hyperbola with foci at the reference points and semi-major axis of the search threshold. For the same data and search threshold, the difference in exclusion capability is shown in Figure <ref>; of the 500 randomly selected queries, only 160fail to exclude the opposing semi-space, whereas with normal hyperbolic exclusion, the number is 421. The query threshold illustrated, 0.145, is chosen to retrieve around one millionth of the space and is not therefore artificially large.As stated, this particular situation has been extensively investigated and is fully reported in <cit.>. Here we will concentrate further onother properties of the planar projection, of which the derivation of Hilbert exclusion turns out to be a special case.§.§ Partitions of the 2D Plane For the purposes of this analysis only, for reasons of simplicity, we seek to divide a data set into precisely two partitions. This is without reference to details of any indexing structure which may use the concepts, although in all cases by implicationthere exists asimplebinary partition tree structure corresponding to the partitioning. In all cases the partition is defined in terms of the 2D plane onto which all points are projected as described above. §.§ Reference Point Separation An important observation is that the shape of the 2D “point cloud", upon which effective exclusion depends, is not greatly affected by the choice of reference points. In comparison with normal Hyperbolic exclusion this is a huge advantage. The hyperbola which bounds the effective queries, i.e. those which can be used to exclude the opposing semispace, is defined only by the (fixed) query radius, and the distance between the reference points, where the larger the separation of the reference points, the better the exclusion. In the extreme case where the separation is no larger thantwice the query radius, which can readily occur in high-dimensional space, it is impossible for any exclusions to be made. This effect can be ameliorated by choosing widely separated reference points, but in an unevenly distributed set this in itself can be dangerous: if one point chosen is an outlier, then the point cloud will lie close to the other point, andagain no exclusions will be made. Finding two reference points which are well separated, and where the rest of the points is evenly distributed between them, is of course an intractable task in general.Figures <ref> and <ref> show this effect. In these diagrams, the reference points have been selected as the furthest, and nearest, respectively out of 1,000 sample pairs of points drawn from the space. It can be seen that, when exclusion is based on tetrahedral properties allowed from the four-point property, the exclusive power remains fairly constant, as the size and shape of the point cloud is not greatly affected. However, when the hyperbolic condition is used, the exclusive power is hugely affected; in this case the query threshold is only slightly less than half the separation of the reference points, and the resulting hyperbola diverges so rapidly from the separating hyperplane that no exclusions are made from the sample queries.From Figure <ref> it should also be noted that, no matter how far the reference points are separated, the four-point property always gives a higher probability ofexclusions; in this case, although the separating lines do not appear visually to bevery different, the implied probability of exclusion in for the four-point property is 0.66, against 0.56.To allow most partition structures to perform well, a very large part of the build cost is typically spent in the selection of good reference points and this cost is largely avoidable with any such four-point strategy, as demonstrated experimentally in Sections <ref> and <ref>. §.§ Arbitrary Partitions Again we stress the fact that, given the strong lower bound condition on the projected 2D plane, we can choose arbitrary geometric partitions of this plane to structure the data.For randomly generated, evenly distributed points there seems to be little to choose. However it is often the case that “real world" data sets do not show the same properties as generated sets; in particular, they tend to be much less evenly distributed, with significant numbers of clusters and outliers. These factors can significantly affect the performance of indexing mechanisms.Figures <ref>, <ref>,<ref>and <ref> show a sample taken from the SISAP colors data set with Euclidean distance applied, showing eight different partitions. Eight different partitions of the plane have been arbitrarily selected and applied. The query threshold illustrated is 0.052 corresponding to a query returning 0.001% of the data.In all cases, it can be noted that the partitions are even, leading to balanced indexing structures. It is very likely that skewed partitions may perform better, an aspect we have not yet investigated. However one important balanced partition is illustrated on the left hand side of Figure <ref>, implying that a balanced hyperplane tree can be efficiently constructed.It can be seen that, in this case, partitioning the plane according to the height of individual points above the X-axis is the more effective strategy. The disadvantage with this is that a little more calculation is required to plot the height of the point, rather than its offset from the Y-axis; however this is a very minor effect when significantly more distance calculations can be avoided.Figures <ref> and <ref> illustrate more techical analyses of the point cloud, using Principle Component Analysis (PCA) and Linear Regression (LR) respectively. Either technique can be used along one of two axes in a two-dimensional space as illustrated. In Section <ref> we explain in the orthogonal linear regression technique in detail, and give experimental results showing its value as the best way to construct a balanced search tree over this data.Finally we give the illustrations in Figure <ref> to make the points that any partition of the plane can be used for this purpose. We have not yet found a compelling use for either partition, however this would depend on the nature of an individual non-uniform data set. §.§ BalanceAs already noted, any of the partitions shown above can be simply used to bisect the data and thus produced a balanced indexing structure. These examples are all defined using a single real value with respect to the planar geometry. This can be calculated for each object within the subset to be divided, and the median can be found very efficiently using the QuickMedianSort algorithm; for a random distribution of points, the practical cost of balancing a binary tree at construction timeappears similar to performing QuickSort once on all the data. While balanced structures are often slower than unbalanced ones for relatively small data sets, they become rapidly more desirable as the size of the data increases, and again more so if it is too large to fit in main memory and requires to be stored in backing store pages. The ability to balance the data without reducing the effectiveness of the exclusion mechanism therefore seems important. One further area of investigation, not yet performed, would be the effect of controlling the balance, which once again is arbitrarily possible simply by selecting different offset values. In general this will increase the probability of exclusion at cost of excluding smaller subsets of the data, and the effectiveness will depend on the individual distributions of the different strategies.§ THEIn this Section we revisit a key observation of Section<ref>, and in particular Section <ref>, where we pointed out that any partition of the two-dimensional projected plane may be used to form an indexing mechanism. Up to this point we have restricted the use to simple Hilbert partitioning, where the data is divided only according to the nearest reference point. Here, we demonstrate a more flexible approach.Figure <ref> shows a scatter plot resulting from an arbitrary choice of reference points for the SISAP colors data set. Although the pattern is not atypical, observation shows that the individual distribution shape is significantly affected by the choice of reference points and, more subtly, by the subset of data points that is to be stored at a given tree node; although a high-dimensional space implies that these would not necessarily have a strong regional identity, this factor does visibly affect the relative mean distances to the reference points.The partitions shown within the figure are based on the best-fit straight line which can be plotted through the points in two dimensions. This is parallel to the lines drawn in the right-hand figure. As this is calculated using the least-mean-squares algorithm, it is reasonable to assume that the perpendicular partition, shown in the left-hand diagram, will in general improve the spread of the data points and thus form a better partition for indexing [As shown above, PCA seems to give a better spread than LR; for the moment we have selected linear regression for the experiment due primarily to its simplicity of implementation, we continue to investigate alternative strategies.].To test this strategy, we define the (), which is a binary tree built recursively over a dataset S as follows. We select two reference points p_1, p_2 at each node. Each child node of the tree shares one reference points with its parents, as done in the Monotonous Bisector Tree <cit.>. We used the tetrahedral projection based on p_1 and p_2 to embed the data points onto a 2D plane, and we compute the best-fit line l through the projected points (or a subset of them) using a least squares minimization. Then, we rotate the 2D data points around the X-intercept of the line l, so that the new X-axis coincides with the line l, and we split the dataat the median X coordinate of the rotated space.Algorithm <ref> and <ref> give the simplest algorithms for constructing, and querying a balanced version of the.We compute the best fitting line l through the points {(x_i,y_i)}_i=1^N as the line y=mx +b that best fits the sample in the sense that the sum of the squared errors between the y_i and the line values mx_i+b is minimized. The fitting line is easily computed asy-y̅=m(x-x̅) where x̅=∑_i=1^N x_i/N,y̅=∑_i=1^N y_i/N, and m=∑_i=1^N(x_i-x̅)(y_i-y̅)∑_i=1^N(x_i-x̅)^2.Then, we rotate the data points by angle θ=arctan(m) around the X-intercept (h,0), where h=x̅-y̅/m:r_x = (x-h) cos(θ)-y sin(θ)r_y =(x-h) sin(θ)+y cos(θ). Experimental evaluation of the resulting search index was performed using exactly the same context as that described in Section <ref>, and all of the code used is available from the same repository. Figures <ref> and <ref> give results for the SISAP colors and nasa data sets respectively. For each data set, six different indexing structures were tested. A balanced monotone hyperplane tree,an unbalanced monotone hyperplane tree, and the Linear Regression Tree were each tested with two different reference point selection strategies. These are: “Rand" – random selection –and “Far" –in the monotone tree, one reference point is handed down from an ancestor, and the second point is simply the one from within the data subset used to construct that node that is the furthest distance from the ancestor node.The fair comparison is of the two balanced trees, and it can be seen that the Linear Regression Tree always outperforms the simple balanced tree. The unbalanced tree however is always the best performer over this data set. Reasons for this are not altogether clear. However we believe this is worth reporting for sake of further investigation: in this domain, successful analysis of these reasons should lead to the ability to mimic them and deterministically produce a tree with still better performance.In some of the further experiments performed in Section <ref>we find that the Linear Regression Tree performs best out of all the mechanisms tested. This seems to be for large data sets which have significant non-uniformity, searching with smaller thresholds. § HYPERPLANE PARTITION INDEXES AND THE FOUR-POINT PROPERTY Having established the full generality of indexing in the supermetric domain, we now return to the Hilbert Exclusion principle and investigate its application over a range of hyperplane indexing structures. This is important in the light of the preceding discussion as, having established that issues such as balance and separation of reference points have quite different consequences, we need to understand the best indexing structure for taking advantage of the increased tractability. It is certainly not reasonable to assumethat the best indexing structures for metric spaces will also the the best for supermetric spaces. The best recorded general performance for an exact-search partition-based indexing structure, before the identification of using thefour-point property within an exclusion mechanism, derives from the Distal Spatial Approximation Tree (DiSAT) <cit.>. This is therefore the obvious comparison to make between using normal metric properties and the four-pointproperty over a space which has both properties; it allows the exactly same data structure to be measured with the different exclusion algorithms and in this sense is a very fair comparison. This comparison has been made in <cit.> and a significant improvement shown for using the four-point property: for the SISAP benchmark data sets, at the lower thresholds, typically around half the number of distance calculations are required when the four-point property is used over the same search index.However, given the observations above on the different relative importance of the choice of reference point, it may be that the same data structure does not give the bestperformance when used for a supermetric space; the main differentiation between previous versions of the Spatial Approximation Tree (SAT) index and the DiSAT is the choice of widely separated reference points at higher levels of the tree, and it is therefore possible that different optimising factors will occur within a supermetric space.We therefore performed a thorough investigation on a number of different exact-search hyperplane tree structures, taking each possible orthogonal attribute separately and testing all possible combinations with both Hyperbolic and Hilbert Exclusion strategies to determine the best data structure for use in a supermetric space. §.§ Partition TreesThe basic structure of a partition tree is a recursively defined, n-ary tree. Each child of a parent node is governed by a single reference point, andevery element of thedata set contained below any parent node is associated with the child node whose reference point is closest to that element. The basic construction of a partition tree is given in Algorithm <ref>. Query of such a tree allows a number of exclusion possibilities. Most simply, the distance from the query to each reference point iscalculated; any partition may be excluded if this distance is greater than the cover radius (cr_i) added to the query threshold. In addition, for every partition, it may be excluded from the search based on the relative distances of the query to every other reference point, using the hyperplane exclusion principle. The basic query algorithm, not using the four-point property, is given as Algorithm <ref>. This can be changed to take advantage of the four-point property by the replacement of the inner partition test (lines ): d(q,p_i) - d(q,p_ j) > 2t is replaced byd(q,p_ i)^2 - d(q,p_ j)^2/d(p_ i,p_ j) > 2tThe extra required term, d(p_ i,p_ j) can be calculated at build time and stored for a small extra spacecost, explained in Section <ref>.The many different types of tree we tested are differentiated by how the reference set is chosen at each tree node. We tested a number of variants oftrees according to the following largelyorthogonal principles: Pure SAT property Each node of a purely-formed Spatial Approximation Tree has the property that no values from within the data set are closer to the “centre" node (the reference point in the parent node associated with each child node) than they are to any of the reference points one level down within the tree. This principle of construction allows further exclusion possibilities, as during a query the maximum distance between the query and any higher-level reference points may be passed recursively; if this distance is greater than max(d(q,p_j)), i ≠ j then it may be used to attempt to exclude N_i from the search. Any serial selection of reference points requires, as each node is constructed, that any point closer to the centre node than any previously selected reference point is added to the set of reference points. As pointed out by the authors of<cit.> the construction has very different properties depending on the order in which the contained set is considered. Two variants of such “pure" SATs were tested; for ,we considered the data set of inclusion in the reference set in order of distance from the centre node, and forwe considered them in reverse order. We also tried some hybrids but did not discover any interesting results, so these are not reported. For using the Hilbert Exclusion mechanism for trees with this pure SAT property, the distances to ancestor nodes can still be used, but (i) all distances need to be recursively passed, rather than just the maximum, and (ii) at build time, all distances from each p_i to all ancestors also require to be calculated and stored. This is because the Hilbert exclusion condition requires all three distances among any two reference points and the query to be known. We tested both construction and query for any extra cost associated with this extra information flow, and it was found to beinsignificant. SAT construction Faster versions of SAT as reported in <cit.> do not maintain the pure SAT property, but instead reuse the core reference point selection algorithm (traversing S according to an imposed order and adding reference points whenever they are closer to the centre node than any existing reference point), howeverthis process is terminated according to the number of reference points selected. This is because the pure SAT algorithm, applied to the distal ordering of values from the centre, leads to very wide, shallow trees which do not lead to good performance when using Hyperbolic exclusion. The extra exclusion possibilities from using the parent reference point distances are lost, but wider separation of reference points was found to lead to more exclusions. Therefore the extra factor of maximum branching factor is considered. We considered two;uses a fixed value of 4, found in <cit.> to give good performance, anduses a dynamic value selected according to the data size, chosen not to exceed the natural logarithm of the data size, thus reducing as the tree is descended. Choice of the centre point for the head node, for best performance, is described in <cit.> as being acheived by choice of an outlier, this giving the SAT_out class of algorithm ( and ); we confirmed this result and therefore reused this strategy for all of our experiments. Finally, the SAT_glob class of tree uses a single ordering for consideration of reference point selection, based on an ordering of the whole data set from the centre node of the entire tree, rather than the centre of each node. Non-SAT construction Finally, we considered a number of partition trees () where the choice of reference point was made independently of their distances from the parent reference point. Three arities were chosen; fixed and logarithmic as above, but also a binary version was constructed. Two strategies for reference point selection were used to fill these arities: random selection, and one usingthe FFT algorithm <cit.>. §.§ Experimental ProcedureThe above classification leads to the following set of tree structures used for experiments: – each of these should have a clear meaning given the above description. For each tree, both Hyperbolic and Hilbert exclusion mechanisms were tested, leading to a total of 24 different search indexes.Each was tested against the SISAP benchmark data sets colors and nasa <cit.> using Euclidean distance. For each test, three different query thresholds were used as is standard for the SISAP benchmark sets.10% of the data was randomly removed to act as a query set.Different distance metrics with the four-point property were also tested but showed no significantly different results, so we report only Euclidean. Actual threshold values used are as shown in Table <ref>. For sake of space, we give only the main results: for each index, and each data set, we give the number of distances per query at a single query threshold.We also tested query times; in all cases the number of distances was directly proportional to the query time, which is not surprising as all of the data sets used fit comfortably within the main memory and this is by far the dominant cost of the query. It is useful to confirm however that the extra administrative cost associated with the Hilbert exclusion is negligible with respect to distance costs. All tests were executed in Java, using the same Java abstract tree construction and query classes, specialisedonly according to the reference point selection strategy and query-time exclusion strategy. As a final semantic check, all results were cross-checked against a serial (exhaustive) search to ensureconsistency and therefore correctness. All code used is available in a publicrepository[https://bitbucket.org/richardconnor/metric-space-framework.git].For each test, multiple tree builds were performed and mean values are presented.For each build, the data was presented in randomised order, as the order of selection during tree build can have a significant serendipitous effect on performance. Tests were repeated until the standard error of the mean was ≤ 1%, which implies that all of the differences reported are statistically significant.Results are shown in Figure <ref> and<ref>. §.§ AnalysisThe most obvious conclusion is that the supermetric exclusion always gives better performance; while this is actually a guarantee as shown in <cit.> the interesting point is the magnitude of the improvement, which in some cases is quite startling.Although not quite the absolute best performance, the greatest improvement is in the pure SAT indexes, both classic and distal variants which in fact seem to give around the same performance. This is worthy of further study; as mentioned, the shape of these trees is very different, the classic SAT giving a relatively small branching factor against a very large branching factor at the higher levels of the distal SAT. The inventors of the distal SAT compromised the SAT property early on, presumably because performance was badly affectedby this property. Using the four-point property seems to overcome this. As these data sets are relatively small there is no real advantage to a shallow tree, but this may well be different with very large data.The lack of variance for thefour-point exclusion across all the different structures is also notable; this confirms our earlier hypothesis that the actual exclusion power of the Hilbert mechanism is much less affected by the choice of reference point, and certainly confirms that puttinghuge computational resources into building expensive data structures may be far less worthwhile in this context.Finally, we note the best performance data structure considered here, the log-sized hyperplane tree using the FFT algorithm to choose reference points. Paradoxically, this is one of the simplest, and fastest, structures to construct. It is likely that using a more sophisticated cluster-finding algorithm such as k-means or k-medoids may perform a little better, although at much higher tree build cost; given the rather small incremental improvement however of FFT over random, we are not convinced that this would be worthwhile in many cases.And as a last word: we note that thevalues of 1,704 distance measurements per query achieved over the SISAP colors data set, and 171 measurements per query over the nasa data set are, for the moment, new performance records against this benchmark.§ THECOST OF HILBERT EXCLUSION While it has been shown that Hilbert Exclusion performs better than Hyperbolic in run-time cost, there is an extra space cost involved as more information is required: the distance between the reference points is required for the query-time exclusion calculations, where it is not for the hyperbolic exclusion calculation. It is thus important to assess the extra overhead in time and space.In all cases, the choice of reference points is made during the building of the index structure; therefore it is possible for the distances to be calculated at build time and stored. Feasibly, they could instead be calculated at query time if the storage overhead was relatively great compared with the extra query-time cost; however here we show that it is not.In the case of binary trees, the extra space overhead is a single distance value, or 4 bytes[ as the value is only used for additive arithmetic and is not critical for correctness, single precision is sufficiently accurate ], per node. Even the leanest tree implementation will have a per-node space overhead much greater than this, although of course this depends on the language and implementation tactics used. A modern JVM has a minimum object overhead of 16 bytes, along with a further 32 bytes to store pointers to two subtrees even before any other node information is considered, such as object ids for the reference objects, cover radii, etc. Realistically the extra overhead is likely to be much less than 10%; given the further invariant that the number of internal tree nodes is less than one-half the number of data[ for a monotone tree;lessfor other types ], and each data object is likely to be much bigger than a tree node, the space overhead is minimal and unlikely to be significant in any realistic scenario.For general hyperplane trees with more than two reference points the overhead is potentiallygreater as all inter-reference point distances are required, giving a theoretical 𝒪(n^2) space cost. Pragmatically however the values of n involved are fairly small; furthermore as we are dealing with proper distances we only need store the upper triangular matrix, so the space overhead is n2 rather than n^2. For example, in <cit.> the authors observe that a DiSAT branching factor of 4 gives optimal performance in some contexts; here we can replace the 𝒪(n^2) observation with a constant value of 6, i.e. 24 bytes per tree node. In the case of a quadtree, the number of internal tree nodes will be less than approximately a third of the data size, giving a maximum overhead of less than 8 bytes per data objectFinally we consider the log-sized node strategy, where the number of reference points at each tree node is approximately the log of the volume of data stored below the node. Thisleads to much bigger overhead at the root of the tree; for example the root node of a tree for 10^10 data objects has 23 reference points, requiring 1KByte of overhead. However these trees are correspondingly shallower, and the node size decreases rapidly as the tree extends downwards. There is arecurrence relation to estimate the space overhead in bytes for a balanced tree: overhead (N)= 0,N ≤ 2 overhead (N)=|p|2× 4 + |p| ×overhead(N - |p||p| ) where |p|=⌊log N ⌋ When applied toany large size the overhead turns out to be only aroundone byte per data object. There isalso clearly an extra run-time cost in construction, from the measurement of extra distances. This is of much less concern; in any partition strategy, for every node built, there is a requirement to measure the distances between every data item below the level of that node against all of the reference points. Without further analysis it seems clear that the extra overhead of measuring the distances among the nodes is relatively trivial. Experimental evidence supports the notion thatthe extra overhead, in both time and space, is insignificant, independent of the strategy used. We have tried to detect significant differences in either construction time or space in the large experiments described in Section <ref>, but in all cases the differences have been hidden in the noise caused by the introduction of randomisation to the construction process. § SPACES WITH LARGER SCALE AND HIGHER DIMENSIONALITY In this section we investigate extending theuse of thesupermetric property to larger and higher-dimensional data sets. The purpose is to explorehow the behaviour of the mechanisms, which have been shown to give good results over (relatively) small benchmark sets,alters with sets that are inherently more difficult to index.We perform three types of test to demonstrate these behaviours: increasing dimensionality In these tests, we generate evenly-spaced points within generated Euclidean spaces of increasing dimension and test Hyperbolic and Hilbert exclusion mechanisms for query performance. These tests show the effect that increasing dimensionality has on the relative performance; pragmatically we show that at the point where increasing dimensionality starts to make search intractable, the use of Hilbert exclusion gives an extra 2-3 dimensions for the same level of performance. “real-world" high-dimensional data In these tests we use GIST representations of the MIR-Flickr <cit.> data set of one million images to perform a near-duplicate search; these are large data comprising 480 floating-point numbers, tested with various metrics. These tests show that the efficacy of Hilbert exclusion does not seem to be affected by the choice of metric. increasingly large data sets In these tests we use 80 dimensionalMPEG-7 Edge Histogram data taken from the CoPhIR <cit.> images set . Tests are made over increasingly large subsets (between 1 and 16 million images) to show how the mechanisms scale as the data size increases.§.§ Increasing DimensionsFor each dimension between 2 and 20 inclusive, evenly distributed Cartesian points were generated within the unit hypercube. For each dimension, a Euclidean space of one million data points was generated, and one thousand threshold queries were executed over each space. At each dimension a threshold was selected with a radius calculated to give one-millionth of the volume of the unit hypercube[ For dimension n, radius r_n = Γ(n2 + 1)/π^n2, where Γ is Euler's gamma function ].After first confirming that the log-sized hyperplane partition tree is still the most efficient index at all dimensions, measurements were made for four variants to establish the added value of the Hilbert exclusion mechanism. Trees were built using both random and FFT selection strategies for each node, and for both of these variants, querying was performed with, and without, the Hilbert exclusion mechanism. As previously noted, the same instance of the built data structure can be used in either way, as long as the metric has the four-point property. The figure recorded is thenumber of distance calculations required per query over the data set; at each dimension, each experiment was repeated until the standard error of the mean was less than 1%.Figure <ref> shows the outcomes. It can be seen that, across all dimensions, the FFT variant is better than a random choice of reference points for either exclusion mechanism, and more significantly for our purposes that the Hilbert exclusion variant is substantially better than the Hyperbolic. Most importantly perhaps is the observation that these two improvements are almost orthogonal, and between around 8 and 12 dimensions the result is a four-fold increase in performance. The tree(FFT pivot selection) using the Hilbert exclusion mechanism shows quite similar performance to the tree(random pivot selection) using the Hyperbolic exclusion mechanism; although subject to heuristics and uncertaintly, in our experiments the FFT-based choice of reference points gives abuild cost of around five times that of thehyperplane tree with randomly selected reference points, and indeed the latter is one of the cheapest indexing mechanisms to build; if build time is an important consideration, this could give the best compromise.An alternative view of the results is to consider where the different lines cross a given horizontal boundary in the chart. For example, if a particular situation indicates that accessing no more than 2.5% of the data is required to give sufficient performance, than this can be achieved with a data set whose dimensionality is around 13 using FFT and Hilbert exclusion, whereas only around 10 can be achieved, from the same indexing mechanism,without these.§.§ MirFlickr/GIST and Near-Duplicate DetectionIn these experiments we test a large data set for a real-world purpose, namely the detection of near-duplicate images. In previous work we have shown the use of the GIST characterisation gives the best tractable test for near-duplicate image detection within large sets of images <cit.>;these tests are inherently expensive because of the data size, and efficient similarity search isimportant to give tractability. Each GIST object[ using GIST parameters: 4 windows, 6 scales and 5 orientations per scale, taken from a monochrome 255 × 255 image with no border ]comprises 480 dimensions of floating point numbers, which usingIEEE single-precision format gives an object size of almost 2KBytes per object, i.e. just under 2GBytes per million images. While the intrinsic dimensionality of these spaces is relatively high – around 10-15 depending on the metric – the required search thresholds are quite low, therefore giving a nice example of spaces where metric search is particularly appropriate for the task in hand.We used the Mir-Flickr <cit.> set of one million images and generated GIST representations. We have previously demonstrated the use of this collection as a benchmark for near-duplicate image detection; the collection by chance contains around 2,000 clusters of near-duplicate images which we have identified, allowing both sensitivity and specificity to be accurately tested for different metrics and thresholds <cit.>. The GIST representations can be used for this purpose with any of Euclidean, Cosine, or Jensen-Shannon distances. A key aspect of such classification functions with very large collections is their specificity, which must be high to avoid very large numbers of false positives. All of these tests maintain high specificity up to a sensitivity of around 50%; to test the efficiency gains of the Hilbert Exclusion we therefore tested searches over the collection at five different thresholds, representing for each metric sensitivity of 10% to 50%, after which point afast drop-off in specificity occurs. Table <ref> gives, for each metric, the intrinsic dimensionality, the mean cost per distance measurement in milliseconds, and the thresholds used to search.For each of the three metrics, the first 1,000 images were used as queries against the remainder of the data. A log-sized hyperplane partition tree was constructed, with reference points chosen using the FFT technique. These trees were then tested with and without use of the Hilbert exclusion; in each test the number of distance calculations and query time were noted. These were found to be almost exactly directly proportional and so only the number of distance calculations are presented, given as the mean proportion of the total data size tested per query.Figure <ref> shows results from these experiments. It appears that the advantage given by using the Hilbert exclusion mechanism is relatively independent of the metric being considered. In all cases it is highly significant, giving a performance improvement of 2.5 to 3 times for all metrics, even at the top end of the thresholds tested.§.§ Increasing ScaleIn these experiments, we investigate howthe advantages shown by the Hilbert property are affected by the scale of the data. To measure this, we usedMPEG-7 Edge Histogram descriptors extracted from the CoPhIR <cit.> image data set, using the first 80 dimensionsof the raw data. We queried increasingly large subsets of the data, ranging from one million to sixteen million images, to test the scalability of the different search mechanisms. Results are reported for Euclidean distance; we repeated the tests using other metrics and found no interesting differences.We sampled 10^9 randomly selected distances to measure IDIM and to choose search thresholds; the IDIM of the data was measured as 7.5, and three thresholds were selected to return 10^-8, 10^-7 and10^-6 of the data per query[ 0.0196, 0.0834 and 0.1815 respectively ]; these thresholds are small but this is appropriate as data becomes larger.With relatively smaller threshold and larger data we did not make assumptions about which mechanisms of those tested earlier would perform best; we tried them all, and report here the most interesting representative results. As there is some anecdotal evidence that single-pivot strategies can be more effective that hyperplane partitions as thresholds decrease, we also included a vantage point tree <cit.> in the tests.For each search structure, we performed 1,000 queries selected randomly from a different part of the set and measured the number of distance calculations performed; we present these as a proportion of the data access per query.Figure <ref> shows the outcomes. We present results for: log-sized hyperplane trees, with Hilbert and Hyperbolic exclusion; monotone (binary, unbalanced) hyperplane trees, again with Hilbert and Hyperbolic exclusion; a balanced vantage point tree, and alinear regression tree. Reference points for the log-sized hyperplane trees are selected using FFT, and for the monotone binary trees by simply selecting the furthest object from the inherited reference point.For all mechanisms and thresholds, it can be seen that as the size of the dataset increases, the proportion of data accessed decreases. The rate of this decrease demonstrates scalability of the mechanism. The value of the Hilbert exclusion is very marked, especially with the larger threshold values. It is interesting to note that the vantage point tree performs very well with a very small threshold, but is relatively much worse as the thresholds get larger. Although a less marked effect, the log-sized hyperplane tree appears to scale slightly better than the binary version. Finally, we note the best overall performance achieved by the linear regression tree; this version is an early attempt at using the extra flexibility allowed by the stronger geometry of the supermetric space, and demands further research. § CONCLUSIONS We have presenteda novel observation based on the four-point property that is possessed by many useful distance metrics. We have shown how the property that any four points from the original space may be embedded inas a tetrahedron leads to further geometric guarantees, in particular we have shown a lower-bound distance that can be calculated from knowledge of the sides of two tetrahedral faces. We have shown a few examples of how metric indexes can be constructed from this property, and have achieved new best performances for Euclidean distance search over two of the SISAP benchmarks. Further we have demonstrated that the advantages shown over the relatively small and tractable benchmark sets extend to larger, less tractable spaces.There are some new areas of investigation opened up by this work. Further study of theuse of different partition strategies used to fit the reference points and data available at each node of an indexing structure should be worthwhile. Given the supermetric properties,much more information is available during tree construction than we have, so far, fully exploited. In particular, given an analytic expression for the discarding rule,a term for the distance between reference points, andvarious assumptions about the searching radius and the distance of the query to the reference pointsit should be possible to maximise the discarding power of the node. This would allow the construction of a controlled balancing which will outperform any randomly unbalanced index structure. We have not yet investigated the possibility of controlling the balance within n-ary partition trees, or applying domain-specific partition strategies to them, which seem to be the most promising avenues for achieving still better exact search performance.Finally, we are excited by the possibility of extending this work into higher dimensions. Inall but pathologically constructed cases, a space with the four-point property also has the so-called n-point property: that is, any n + 1 points may be isometrically embedded in n-dimensional Euclidean space. We are currently investigating various geometric guarantees that can be determined in arbitrarily high dimensions.§ ACKNOWLEDGEMENTSWe would like to thank the anonymous referees for helpful comments on an earlier version of this paper. We are particularly grateful to Dr. Fabrizio Falchi for his help in accessing the CoPhIR data set. Richard Connor would like to acknowledge support by the National Research Council of Italy (CNR) for a Short-term Mobility Fellowship (STM) in June 2015, which funded a stay at ISTI-CNR in Pisa during which this work was conceived. The work was also partially funded by Smart News, “Social sensing for breaking news", co-funded by the Tuscany region under the FAR-FAS 2014 program, CUP CIPE D58C15000270008. elsarticle-num | http://arxiv.org/abs/1707.08361v2 | {
"authors": [
"Richard Connor",
"Lucia Vadicamo",
"Franco Alberto Cardillo",
"Fausto Rabitti"
],
"categories": [
"cs.IR",
"H.3.3"
],
"primary_category": "cs.IR",
"published": "20170726103221",
"title": "Supermetric Search"
} |
Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ======================================================================================================================================== Historically, medical imaging repositories have been supported by indoor infrastructures. However, the amount of diagnostic imaging procedures has continuously increased over the last decades, imposing several challenges associated with the storage volume, data redundancy and availability. Cloud platforms are focused on delivering hardware and software services over the Internet, becoming an appealing solution for repository outsourcing. Although this option may bring financial and technological benefits, it also presents new challenges. In medical imaging scenarios, communication latency is a critical issue that still hinders the adoption of this paradigm. This paper proposes an intelligent Cloud storage gateway that optimizes data access times. This is achieved through a new cache architecture that combines static rules and pattern recognition for eviction and prefetching.The evaluation results, obtained through simulations over a real-world dataset, show that cache hit ratios can reach around 80%, leading reductions of image retrieval times by over 60%. The combined use of eviction and prefetching policies proposed can significantly reduce communication latency, even when using a small cache in comparison to the total size of the repository. Apart from the performance gains, the proposed system is capable of adjusting to specific workflows of different institutions.Keywords—Cloud, Medical imaging, Storage gateway, Data access latency, Pattern recognition, Machine learning.§ INTRODUCTION Medical imaging is a very important tool in medical practice, not only for diagnosis but also for patient management and treatment support <cit.>. It benefits from technological advances in several areas, including the creation of new imaging modalities and the implementation of the PACS (Picture Archiving and Communication System) concept <cit.>. PACS refers to systems that are responsible for the acquisition, management, storage, visualization and distribution of medical imaging data <cit.>. Nowadays, these systems proliferate in practically all healthcare institutions, and are also used to support distributed workflows <cit.>.The Cloud computing paradigm enables on-demand services, such as computing, storage and databases, and answers a current major problem in the medical field: the soaring volume of worldwide healthcare data that results in a Big Data problem <cit.>. Much of its interest resides in the fact that, with Cloud computing, computing resources are provided in an elastic way, supporting horizontal scalability, and development and maintenance of cloud software has become easier, more reliable, and safer. Based on this, a tremendous amount of ubiquitous computational power and an unprecedented number of Internet resources and services are used every day as regular commodities. This facility is also being explored for outsourcing of medical imaging services, with two main use cases <cit.>: * PACS archive outsourcing. In-house PACS solutions have high maintenance costs, infrastructure scalability is usually limited and over the years it easily becomes obsolete.* Inter-institutional workflows and sharing of medical imaging. For instance, the cloud is excellent for instantiating a teleradiology platform as a service. A major drawback associated with the migration of PACS services to the Cloud is access latency <cit.>, a particularly critical concern in medical imaging scenarios since remote access over the Internet is considerably slower than Intranet connections. Moreover, some studies can amount to a few gigabytes of data, which further exacerbates this issue <cit.>.The most common approach for reducing access times is based on the combination of local cache and prefetching mechanisms that attempt to anticipate the user requests. Nevertheless, their effectiveness depends on accurately predicting what data will be requested. Traditional approaches for cache and prefetching in the medical imaging scenario are based on static rules over specific parameters <cit.>. However, this strategy has innumerous problems and limitations, as discussed in section <ref>.This paper proposes an intelligent Cloud storage gateway for medical imaging repositories, focused on the reduction of communication latency. The proposed architecture is an improvement of a previous approach, having a combination of static rules with pattern recognition algorithms, enabling the system to adapt to user’s routines and behaviors, and is fully compliant with the Digital Imaging and Communications in Medicine (DICOM) standard.The remainder of this paper is structured as follows: section <ref> provides a brief overview of two main concepts (DICOM and PACS); section <ref> elaborates on the concepts of cache and prefetching, describing some approaches for both; in section <ref> the proposed architecture for the implemented system is explained; the experimental procedure followed for evaluating our proposal is detailed in section <ref>, and the results obtained are illustrated in section <ref>. Finally, section <ref> contains the concluding remarks regarding the developed work. §.§ Medical Imaging LaboratoriesMedical imaging processes are managed by systems called PACS. This kind of systems appeared in the early 1980’s as small systems composed mainly of an acquisition device, a visualization workstation, a small repository and a printer, having subsequently evolved to handle all digital medical imaging data produced in a healthcare institution.Figure <ref> shows a typical PACS instance that includes acquisition devices (i.e. the modalities), the repositories, PACS server, visualization workstations, printer, and the Radiology Information System (RIS). §.§.§ Digital Imaging and Communications in Medicine (DICOM) In the eighties, PACS were built in an ad-hoc fashion with proprietary communication protocols and file formats, among other aspects. For that reason, systems from distinct manufacturers, and in some cases even from the same manufacturer, were not interoperable, hindering the aggregation of all institutional devices into a single system capable of handling all medical imaging data. An international normalization effort to address these limitations resulted in the Digital Imaging and Communications in Medicine (DICOM) standard <cit.>. DICOM defines data structure formats and communication processes, and introduces the DICOM Information Model (DIM) <cit.> that outlines how relationships between real-world objects, such as studies and patients, must be represented. Moreover, it defines a set of network service commands for storing (C-Store), requesting (C-Get), querying (C-Find) and moving (C-Move) DICOM objects <cit.>.The DICOM network nodes are identified by their application entity title (AETitle) <cit.> and the communications are done as a three-part process: association negotiation, service request/response and, in the end, the association release.Because of its data encoding flexibility and the wide range of processes supported, DICOM was very well accepted in the medical imaging field. Nowadays, practically all devices follow this standard. §.§.§ PACS OutsourcingUsually, PACS are constrained to a single institution. Nevertheless, the Cloud and the proliferation of high-speed Internet connections created the means to broaden PACS horizons. For instance, it is now possible to deploy a Regional PACS over the Cloud or federated distributed facilities. In a previous work <cit.>, we described a federating systemfor two clinics (institution A and B in Figure <ref>). In this setting, the central PACS archive is hosted on a private Cloud located at institution A while institution B only has a gateway that communicates with the PACS server via Internet.The main concern with the solution depicted in Figure <ref> is that although some Internet connections can already provide an acceptable quality of service, these cannot compete with an Intranet based solution in terms of bandwidth and data transfer speeds. This constraint is hindering the adoption of PACS Cloud solutions by the institutions <cit.>. This issue could be minimized by endowing the gateway of institution B with a cache<cit.>, but nevertheless the question of how to populate and evict the cache in an effective way still remained. §.§ Cache and PrefetchingCache effectiveness depends on several factors, namely: (1) the probability of finding the needed data in the cache; (2) length of time needed to retrieve data from the cache; (3) delay introduced by the cache processes when it does not have the requested data and (4) overheads due to maintenance of cache consistency <cit.>. In this work we focused on the first aspect, namely on maximizing the likelihood of finding the needed data in the cache, trying to achieve better results than a previous architecture . Apart from the size of the cache, the two main contributing factors are the strategies for populating the cache and for selecting objects to be discarded when the cache is full. Cache population can be achieved by a passive mode, in which objects are stored in cache when they are first fetched from the source, or by prefetching, which consists on predicting future requests and retrieving the corresponding data before the requests actually occur. Passive cache population is most commonly used, and works well in scenarios that have high probability of repeating requests for a same object <cit.>. The cache eviction process can follow numerous approaches known as cache replacement policies, among which Least Recently Used (LRU) and Least Frequently Used (LFU) are common options. Some PACS archives are hierarchically organized in short-term, mid-term and long-term repositories (Figure <ref>). The long-term repository uses cheap and slow technologies to store all studies, while the mid-term and short-term repositories provide faster access times but only keep copies of some of the objects stored in the long-term repository <cit.>. These partial repositories are usually populated according to static rules associated with patient appointments, patient’s age, examination type, amongst other characteristics <cit.>.The main limitation of strategies based on static rules is that they have to be tailored for each institution, taking into account the workflow, software and user’s behavior. Furthermore, such rules have to accommodate all possible situations, which can lead to populating the cache with unnecessary data. For instance, Bui et al. describe a prefetching mechanism with 100% recall but only 50% precision <cit.>, which means that all data is in cache when needed but only 50% of the prefetched data is actually needed. Static rules have also the potential of producing perverse results in some situations. In a commercial solution, for example, we observed that a prefetching mechanism of a cache gateway associated to an outsourced PACS archive prefetches all results for all queries performed. So, the cache is populated with undesired studies when a user makes a bad query. Another reported case was related to the “poisoning” of cache population with massive number of studies requested by a user performing a non-standard task. For instance, an auditor requesting all CT (computed tomography) studies performed in the previous year could fill the cache with undesired studies, forcing all other requests to be served directly from the remote archive with consequent delay. Besides these considerations, traditional prefetching mechanisms can also overload the remote repository with requests and be stressing to the network.Pattern recognition and machine learning have been increasingly used for cache replacement policies as well as prefetching. For instance, Pal and Jain <cit.> proposed a prefetching mechanism for web browsing that uses Markov models to predict which pages the users will request next. On another reported case, Garcia et al. <cit.> used neural networks to predict the part of the map would be needed next, for map and navigation services. Hybrid approaches for cache replacement and prefetching have also been described, namely by combining traditional approaches, such as LRU and LFU, with machine learning algorithms. An example is the application of machine learning to enhance conventional cache replacement policies for web browsing, as described by Ali et al. <cit.>. However, reports about the exploration of hybrid approaches in medical imaging environments are not found in the literature.§ MATERIAL AND METHODS §.§ Proposed ArchitectureThis section proposes and describes an architecture of a new Cloud storage gateway (Figure 3) that supports cache replacement and prefetching for distributed medical imaging environments, aimed at minimizing the communication latency by learning the behavior of users. The solution was built to work with the gateway showed in the Figure 2.§.§.§ Sensors The proposed system performs predictions based on environmental conditions, and so it must be equipped with distinct types of sensors to capture those conditions.§.§.§ Message Sensor This sensor is the most important source for the new pattern recognition system because it allows capturing the messages interchanged between the local area network and the remote cloud archive. Since the proposed system has direct access to the network messages interchanged through the gateway, a set of listeners were integrated in the gateway, being awakened every time a new message is sent to (or received from) the repository.The listeners generate a event report every time a message is exchanged. A set of metadata is saved in a log file, including the following information elements:* time and date of message; * kind of request;* UID of the study requested or the query made;* requesting application entity;* destination application entity. The response messages contain additional information elements that are also registered in this log, such as the identifiers of the studies that match a query. All this information is then used in the pattern recognition system, as explained in <ref>. §.§.§ Study SensorIn some cases, the network data exchanged is not enough to assess the usage pattern. For that reason, a study sensor that can query the repository in order to extract characteristics regarding a specific study was also implemented.§.§.§ Network SensorIn order to optimize our system, a new sensor was developed from scratch, to allow assessing the network conditions before proceeding with prefetching. This component monitors the Cloud gateway communications, watching the network requests and responses. This way, this sensor is able to continuously assess if the network is overloaded or if it can support additional traffic generated by the prefetching process. If the network is stressed, prefetching usage could have a harmful effect on the performance of the system. This new sensor is essential for a correct deployment of the new mechanisms that are explored in this article, allowing a more robust and adequate performance of cache replacement.§.§.§ Labeller & Pattern Recognition ModuleThis module is responsible for detecting which usage pattern best fits the user’s behavior and results from the integration of a new Labeller module within an adapted version of the Pattern Recognition mechanism proposed in <cit.>, classifying user behavior to understand if it is relevant to prefetching mechanisms or an one-off event with little medical and health relevance. The key idea is to classify the behavior of the healthcare professionals based on the number and relationships of requested studies following each DICOM C-Find (i.e. query). To achieve this, healthcare professionals’ interactions were categorized into four distinct usage behavior patterns:* Patient revising (class 1):user is revising the studies of a single patient. This is potentially a usage pattern that demands fast access to images to preserve the quality of service;* Modality revising (class 2): user is revising the studies of a specific modality. This scenario is more critical for "heavier" modalities; * Inconsequent query (class 3): pattern representative of user error situations or queries that do not result in download of imaging data.* “Other” usage (class 4): pattern representative of usage scenarios not identified by this architecture, such as an auditor evaluating all images of a whole department in a certain time window. The functioning of this "module" proceeds as follows:firstly, it splits the events sensed by the Message Sensor by AETitle (i.e. host). After that, it pre-processes and performs feature extraction on that information. The result of this process is used in two distinct ways: (1) it is sent to a set of trained MultiLayer Perceptron (MLP) models so they can predict which usage pattern best fits the user’s behavior and (2) it is saved to a log file for training the models. Based on the evaluation performed in a previous work, the models are trained by incremental learning, leading to more representive and real-based results. For this, at the end of each day, each training instance (query event) is assigned to the corresponding usage pattern based on the consequent study requests.The features used by the models are of three kinds: (1) time features, e.g. the hour of the day, the day of the month, the month; (2) history features, e.g. number of patterns of each kind previous to this pattern, the last pattern, time since the last pattern; and (3) type of query, i.e. the parameters embedded in the query. §.§.§ Cache Replacement This module has two distinct agents: a cache manager agent, responsible for managing the imaging data stored in the cache system, and an eviction agent, responsible for dumping objects that are not necessary when the cache is full or has reached an occupation ratio that could hinder the storage of new objects. This eviction agent uses a LRU function, a broadly used solution in cache management which is relatively simple to implement and provides good performance <cit.>, hence being the golden standard and usually used in real-environment. The LRU function assigns a weight of 100 to the newest study in cache and 0 to the oldest one. The remaining studies are assigned a value that corresponds to the ratio between its distance to the oldest study and its distance to the newest study. The cache manager agent is important since it connects the repository and the database. The repository stores the image data as blobs of information, which are not processed and not searchable. A relational database is used to store information related to the repository objects. This way, it is possible to assess the amount of data stored, which images are stored, from what studies, from what patients, and also the amount of time that the images have been stored in the cache, allowing a better behavior of the eviction agent. §.§.§ Prefetching agent The prefetching agent has two distinct levels: short-term prefetching and long-term prefetching.The long-term mode is triggered using the information from Network Sensor. When the system has low usage level, for instance during nights and weekends, this type of prefetching is used. Nonetheless, each time a user requests a study, this module extracts some characteristics of the study, such as the modality and production date (last day, week, month or year). Every characteristic has a counter associated with, and when the long-term prefetching is triggered, it uses the counters to infer which subset of images will more likely be requested, for instance: CTs of the last two months. As such, if the cache has free-space and the network conditions allow, the prefetching agent requests all studies that match the most popular categories of images.The short-term mode uses the time between the query and the requests for prefetching the studies before being solicited by the user. The pattern recognition agent is used in this process. It is triggered when a query is made and predicts which usage pattern best fits the user’s behavior. After that, two parallel processes are executed:* The results of the user query are evaluated and the ones that match the usage pattern are selected for prefetching. Prefetching rules are applied in order to assess which ones have higher priority.* The prefetching agent instantly makes a query to the repository, based on the new outputs of the pattern recognition module. The query depends on the predicted usage pattern: if this is “patient revising” the query is performed by patient ID; if the usage pattern is “modality revising”, a query by modality with a time window of one month is performed. The two remaining usage patterns are not considered for prefetching. After that, all query results are evaluated by the prefetching rules and the ones with higher score are prefetched, if not already covered by process 1.This agent is essential to improve the system and have a more adequate and robust solution for the problem in question, as further demonstrated in results, when compared with the golden standard LRU.§.§.§ Prefetching Rule In order for the prefetching agent to know which studies should be fetched, a neural network function that learns with the AETitle history is provided. It uses a MLP neural network for each DICOM node, with the following inputs: the length of time since the study was produced, the body part, the modality, the patient gender, the patient age, the usage pattern and the institution that procuded the examination.The neural networks are trained each day (or week) using as training data the studies retrieved by the searches. These are labeled as positive instances if requested after the search, and as negative instances otherwise. The output of the function is a measure of the likelihood of that study being requested. §.§.§ DICOM Interface As previously described, some modules of this architecture need to communicate with third-party PACS equipment. For that reason, the system provides the DICOM interface module. This middleware converts module requests into DICOM requests and sends them to the destination. §.§ Experimental EvaluationTesting the proposed architecture in distinct healthcare institutions under different environment conditions (network, user schedules, number of workstations and so on) could be dangerous for the regular institutional processes, since some of the tests would overload the servers and network. For this reason we performed our tests through simulations, under different conditions, over a real-world dataset. Each distinct scenario was simulated ten times to minimize the impact of random initialization of some parameters of the system, such as the MLPs. For each condition, two metrics were used to analyze system performance:* Hit Ratio, calculated by dividing the number of times a requested object was stored in the cache by the number of object requests.* Retrieval time per image, calculated by dividing the total time needed to retrieve a requested study by the number of object requests. §.§.§ Real-World Dataset The real-world dataset is composed of two parts: (1) an XML file containing information about the messages exchanged in the network and (2) an index containing anonymized information about the studies stored in the institutions.The XML file contains data for 5186 DICOM requests: C-Move and C-Find. These requests are from the workstations to the PACS server, during a 3-month period. Nevertheless, the simulation needed more information, including: (1) the size of the studies; (2) the number of files; (3) the results retrieved for a query; (4) the anonymized patient data; and (5) the characteristics of the study. For that, we used a replica of the repository database, with sensitive data removed by applying a one-way function (i.e. hash) to some of the fields, for instance: patient name. The same function was also applied to the XML file, in order to hide some parameters of the queries that contain private data. This strategy allowed us to reproduce the results retrieved by a user’s query, without having access to the actual raw data.This dataset was extracted from the Cloud gateway system described in section <ref>, which has three remote workstations on Institution B. The shared repository holds 2 terabytes of medical imaging studies. For the validation experiments, the static rules already used in this real-world gateway wereimported and applied in our system. §.§.§ Test conditions Tests were executed simulating distinct situations, including cache size. For that, and considering the size of the dataset, cache sizes of 2.5, 10, 20, 50 and 100 gigabytes were used. For each cache size, the system was tested with the following configurations:* Configuration 1: LRU is selected as eviction policy, and no prefetching policy is used, as explained in <ref>, representing the golden standard in the area. * Configuration 2: LRU is selected as eviction policy, and both short and long-term prefetching modes are used.Additionally, all configurations have a passive cache population mode, which means that all studies that pass through the gateway are also stored in the cache. §.§ Search for similar or related works Since this a area of growing interest, an exhaustive search of material and related works was performed. Although several works are published in the area of cloud storage in health/medical environment, none of the literature in the forums (Google Scholar, Web of Science, IEEE Xplore Digital Library) compared the hit ratio with a growing cache size and retrieval time with a growing cache size, resourcing in other measurements that, in our opinion, are less relevant to this area and the objective of this work. So LRU was selected to compare to our system, since it is considered the golden standard in the area. Furthermore, for a more adequate and fair comparison, LRU was performed with the pipeline of our system, with modifications to improve its results. § RESULTS & DISCUSSIONNumerous tests were carried out to evaluate the effectiveness of the proposal. Configuration 1 was used as the reference for comparison with the proposed system, since LRU is a widely used eviction approach which provides good results with relatively simple implementation <cit.> and as been the golden standard in this area. In addition to LRU, our system uses also short and long-term prefetching modes, needing for that a Network Sensor, as previously explained. This configuration have lower computational cost when compared with more complex configurations, allowing its application in different systems with low impact in their performance.While there is no rule of thumb for defining the necessary cache size for a hospital information system, and literature supporting appropriate cache sizing procedure is scarce, some literature recommends that such storage component should cover actual imaging of 1 month up to 3 months <cit.>, which can reach several hundred gigabytes of data <cit.>. Despite being desirable to have a large cache, this brings significant costs to the system <cit.>, thus, adequate cache size should be chosen so that its size is "optimized" regarding the specific workflow of the system where it will be implemented (e.g. in a RIS) <cit.>.So, we opted to analyse the performance of the proposed system with a range of smaller cache sizes, which go up to a maximum of 100 gigabytes. The results obtained for hit ratio and retrieval time per image are shown in Figure <ref> and Figure <ref>, respectively. Cache sizes are presented as percentages, representing the corresponding fraction of the dataset instead of the actual value in gigabytes.Analysing hit ratio, it is possible to observe in Figure <ref> that the proposed system exhibits higher hit ratio than the "baseline" configuration for every cache size tested, with the maximum value being a hit ratio of approximately 81% for a cache size of 5% (equivalent to 100 gigabytes). In what concerns retrieval time per image, time reduces with the increase in cache size, as expected. Moreover, the proposed system has lower retrieval times when compared to the base system, for every cache size considered in this work.These results show that the proposed implementation improves system performance in every tested scenario, when compared to configuration 1. In fact, with a cache size of 100 gigabytes, the proposed system required on average 73% less time to retrieve each image, when compared to configuration 1. This shows that using short and long-term prefetching modes can considerable decrease the impact of communication latency. Furthermore, a hit ratio above 80% is very significant taking into account that 100 gigabytes represent only 5% of the total data stored in the main repository that is remotely located.Moreover, when comparing both configurations in the larger cache spectrum, we can see that our system achieves a higher hit ratio and 33% lower retrieval time using a cache size of 2.5%, than the baseline configuration with a cache with double the size. These results show that our system proposal brings considerable performance benefits even with a smaller cache.Another interesting aspect to take into account is the fact that static rules, which were imported from the real-world validation scenario, apply specifically to the more common cases (classes 1 & 2, described in <ref>) that cover, in the dataset used, approximately 80% of the user requests. While these static rules allow a more efficient system response in typical requests, the remaining 20% are not contemplated. Due to the adaptive nature of our architecture, distinct behavior patterns are correctly detected and classified, enabling the system not only to match and even improve the performance for typical requests, but also to greatly improve performance for the remaining requests, which would normally be processed with a basic configuration similar to that of configuration 1.It is important to refer that this work focused mainly on the small-cache regime, where hit ratios vary more significantly depending on the caching policies that are used. The small-cache regime is of particular interest, since it is a regime where the use of eviction policies coupled with other policies, namely prefetching policies, produces significantly better performances compared to the sole use of eviction policies <cit.>. As aforementioned, this was effectively observed, as configuration 2 outperformed configuration 1 in every tested scenario.§ CONCLUSIONSIn this paper, we propose an intelligent Cloud storage gateway that supports prefetching and eviction policies, aiming to reduce the communication latency when accessing remote medical imaging repositories. This scenario is particularly important due to the current trend for outsourcing PACS archives to the Cloud.The proposed architecture uses a set of prefetching and eviction policies. In what concerns eviction, LRU was the selected policy. Regarding prefetching, two prefetching modes were used: long-term and short-term prefetching. The first is responsible for fetching objects that will probably be requested in the next day or week, whereas the latter is for more immediate needs, i.e. for the next minutes.The system was subjected to exhaustive tests over a real-world dataset, leading to observed reduction of image retrieval times close to, or even over, 60% for the larger cache sizes. The results obtained show that the combined use of eviction and prefetching policies proposed in this paper can significantly reduce communication latency, even with a considerably reduced cache in comparison to the total size of the main repository (small-cache regime).The hybrid solution herein proposed yields a system capable of adjusting to the distinct, specific workflows of different institutions, whilst offering significant improvements in system performance, namely regarding hit ratio and retrieval time metrics.§ COMPLIANCE WITH ETHICAL STANDARDS Funding This work has received support from theERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme, and by National Funds through the FCT – Fundação para a Ciência e a Tecnologia within project PTDC/EEI-ESS/6815/2014; POCI-01-0145-FEDER-016694. Sérgio Matos is funded under the FCT Investigator programme.Conflict of InterestAll authors declare that there are no conflicts of interest in this work.Ethical ApprovalThis article does not contain any studies with human participants or animals performed by any of the authors.ieeetr | http://arxiv.org/abs/1708.06334v1 | {
"authors": [
"Carlos Viana-Ferreira",
"António Guerra",
"João F. Silva",
"Sérgio Matos",
"Carlos Costa"
],
"categories": [
"cs.DC"
],
"primary_category": "cs.DC",
"published": "20170727111044",
"title": "An Intelligent Cloud Storage Gateway for Medical Imaging"
} |
#1 1 0Sharpening Jensen's Inequality J. G. Liao and Arthur BergDivision of Biostatistics and BioinformaticsPenn State University College of Medicine December 30, 2023 ================================================================================================================================= 1 TitleThis paper proposes a new sharpened version of the Jensen's inequality. The proposed new bound is simple and insightful, is broadly applicable by imposing minimum assumptions, and provides fairly accurate result in spite of its simple form. Applications to the moment generating function, power mean inequalities, and Rao-Blackwell estimation are presented. This presentation can be incorporated in any calculus-based statistical course.Keywords:Jensen gap, Power mean inequality, Rao-Blackwell Estimator, Taylor series 1.45 § INTRODUCTIONJensen's inequality is a fundamental inequality in mathematics and it underlies many important statistical proofs and concepts. Some standard applications include derivation of the arithmetic-geometric mean inequality, non-negativity of Kullback and Leibler divergence, and the convergence property of the expectation-maximization algorithm <cit.>. Jensen's inequality is covered in all major statistical textbooks such as <cit.> and <cit.> as a basic mathematical tool for statistics.Let X be a random variable with finite expectation and let φ(x) be a convex function, then Jensen's inequality <cit.> establishes 𝔼[φ(X)] - φ(𝔼[X])≥ 0.This inequality, however, is not sharp unless var(X)=0 or φ(x) is a linear function of x.Therefore, there is substantial room for advancement. This paper proposes a new sharper bound for the Jensen gap [φ(X)]-φ([X]). Some other improvements of Jensen's inequality have been developed recently; see for example <cit.>, <cit.>; <cit.> and references cited therein. Our proposed bound, however, has the following advantages. First, it has a simple, easy to use, and insightful form in terms of the second derivative φ”(x) and var(X).At the same time, it gives fairly accurate results in the several examples below. Many previously published improvements, however, are much more complicated in form, much more involved to use, and can even be more difficult to compute than E[φ(X)] itself as discussed in <cit.>. Second, our method requires only the existence of φ”(x) and is therefore broadly applicable. In contrast, some other methods require φ(x) to admit a power series representation with positive coefficients <cit.> or require φ(x) to be super-quadratic <cit.>. Third, we provide both a lower bound and an upper bound in a single formula. We have incorporated the materials in this paper in our classroom teaching. With only slightly increased technical level and lecture time, we are able to present a much sharper version of the Jensen's inequality that significantly enhances students' understanding of the underlying concepts. § MAIN RESULT Let X be a one-dimensional random variable with mean μ, and P(X∈(a,b))=1, where -∞≤ a<b≤∞.Let φ(x) is a twice differentiable function on (a,b), and define functionh(x;ν)≜φ(x)-φ(ν)/(x-ν)^2-φ '(ν)/x-ν.Thenx∈(a,b)inf{h(x;μ)}var(X) ≤ E[φ(X)]-φ(E[X])≤sup_x∈(a,b){h(x;μ)}var(X).Let F(x) be the cumulative distribution function of X. Applying Taylor's theorem to φ(x) about μ with a mean-value form of the remainder givesφ(x)=φ(μ)+φ'(μ)(x-μ)+φ”(g(x))/2(x-μ)^2,where g(x) is between x and μ.Explicitly solving for φ”(g(x))/2 gives φ”(g(x))/2=h(x;μ) as defined above.Therefore𝔼[φ(X)] - φ(𝔼[X])=∫_a^b {φ(x)-φ(μ)} dF(x)=∫_a^b {φ'(μ)(x-μ)+h(x;μ)(x-μ)^2} dF(x)=∫_a^b h(x;μ)(x-μ)^2 dF(x),and the result follows because inf_x∈(a,b) h(x;μ)≤ h(x;μ)≤sup_x∈(a,b) h(x;μ).Theorem 1 also holds when inf h(x;μ) is replaced by infφ”(x)/2 and sup h(x;μ) replaced by supφ”(x)/2 sinceinfφ”(x)/2≤inf h(x;μ) andsupφ”(x)/2≥sup h(x;μ). These less tight bounds are implied in the economics working paper <cit.>.Our lower and upper bounds have the general form J·var(X), where J depends on φ.Similar forms of bounds are presented in <cit.>, but our J in Theorem 1 is much simpler and applies to a wider class of φ.Inequality (<ref>) implies Jensen's inequality when φ”(x)≥ 0. Note also that Jensen's inequality is sharp when φ(x) is linear, whereas inequality (<ref>) is sharp when φ(x) is a quadratic function of x.In some applications the moments of X present in (<ref>) are unknown, although a random sample x_1,…,x_n from the underlying distribution F is available.A version of Theorem 1 suitable for this situation is given in the following corollary. Let x_1,…,x_n be any n datapoints in (-∞,∞), and letx̅=1/n∑_i=1^n x_i,φ_x=1/n∑_i=1^nφ(x_i), S^2=1/n∑_i=1^n(x_i-x̅)^2.Theninf_x∈[a,b]h(x;x̅)S^2≤φ_x-φ(x̅)≤sup_x∈[a,b]h(x;x̅)S^2,where a=min{x_1,…,x_n} and b=max{x_1,…,x_n}. Consider the discrete random variable X with probability distribution P(X=x_i)=1/n, i=1,…,n.We have E[X]=x̅, E[φ(X)]=φ_x, and var(X)=S^2.Then the corollary follows from application of Theorem 1. If φ '(x) is convex, then h(x;μ) is monotonically increasing in x, and if φ '(x) is concave, then h(x;μ) is monotonically decreasing in x.We prove that h'(x;μ)≥0 when φ'(x) is convex. The analogous result for concave φ'(x) follows similarly.Note that 𝑑ℎ(x;μ)/𝑑𝑥=φ '(x)+φ '(μ)/2-φ(x)-φ(μ)/x-μ/1/2(x-μ)^2,so it suffices to prove φ '(x)+φ '(μ)/2≥φ(x)-φ(μ)/x-μ.Without loss of generality we assume x>μ.Convexity of φ'(x) gives φ '(y)≤φ'(μ)+φ'(x )-φ'(μ)/x -μ(y-μ)for ally∈(μ,x).Therefore we have φ(x)-φ(μ) =∫ _μ^xφ '(y)𝑑𝑦≤∫ _μ^x{φ'(μ)+φ'(x )-φ'(μ)/x -μ(y-μ) }𝑑𝑦=φ '(x)+φ '(μ)/2(x-μ).and the result follows. Lemma 1 makes Theorem 1 easy to use as the follow results hold:inf h(x;μ)= lim_x→ ah(x;μ)sup h(x;μ)= lim_x→ bh(x;μ) ,when φ'(x)is convexinf h(x;μ)= lim_x→ bh(x;μ)sup h(x;μ)= lim_x→ ah(x;μ) ,when φ'(x)is concave.Note the limits of h(x;μ) can be either finite or infinite. The proof of Lemma 1 borrows ideas from <cit.>.Examples of functions φ(x) for which φ' is convex include φ(x)=exp(x) and φ(x)=x^p for p≥2 or p∈(0,1]. Examples of functions φ(x) for which φ' is concave include φ(x)=-log x and φ(x)=x^p for p<0 or p∈[1,2].§ EXAMPLES[Moment Generating Function] For any random variable X supported on (a,b) with a finite variance, we can bound the moment generating function E[e^tX] using Theorem 1 to get inf_x∈(a,b){h(x;μ)}var(X)≤ E[e^tX]-e^tE[X]≤sup_x∈(a,b){h(x;μ)}var(X),where h(x;μ)=e^tx-e^tμ/(x-μ)^2-te^tμ/x-μ.For t>0 and (a,b)=(-∞,∞), we have inf h(x;μ)=lim_x→-∞h(x;μ)=0andsup h(x;μ)=lim_x→∞h(x;μ)=∞.So Theorem 1 provides no improvement over Jensen's inequality.However, on a finite domain such as a non-negative random variable with (a,b)=(0,∞), a significant improvement in the lower bound is possible becauseinf h(x;μ)=h(0;μ)=1-e^tμ+tμ e^tμ/μ^2>0.Similar results hold for t<0.We apply this to an example from <cit.>, where X is an exponential random variable with mean 1 and φ(x)=e^tx with t=1/2. Here the actual Jensen's gap is [e^tX]-e^t[X]= 2-√(e)≈.351.Since var(X)=1, we have.176≈ h(0;μ)≤[e^tX]-e^t[X]≤lim_x→∞h(x;μ)=∞.The less sharp lower bound using infφ”(x)/2 is 0.125. Utilizing elaborate approximations and numerical optimizations <cit.> yielded a more accurate lower bound of 0.271.[Arithmetic vs Geometric Mean] Let X be a positive random variable on interval (a,b) with mean μ. Note that -log(x) is convex whose derivative is concave.Applying Theorem 1 and Lemma 1 leads to lim_x→ bh(x;μ) var(X) ≤ -E{log(X)} + logμ≤lim_x→ ah(x;μ)var(X),whereh(x;μ)=-log x+logμ/(x-μ)^2+1/μ(x-μ). Now consider a sample of n positive data points x_1,…,x_n.Let x̅ be the arithmetic mean and x̅_g=(x_1x_2⋯ x_n)^1/n be the geometric mean.Applying Corollary <ref> givesexp{S^2 h(b;x̅)}≤x̅/x̅_g≤exp{S^2 h(a;x̅)},where a, b, S^2 are as defined in Corollary <ref>.To give some numerical results, we generated 100 random numbers from uniform distribution on [10,100].For these 100 numbers, the arithmetic mean x̅ is 54.830 and the geometric mean x̅_g is 47.509.The above inequality becomes1.075≤x̅/(x_1x_2⋯ x_n)^1/n = 1.154≤ 1.331,which are fairly tight bounds. Replacing h(x_n;x̅) by φ”(x_n)/2 and h(x_1;x̅) by φ”(x_1)/2 leads to a less accurate lower bound 1.0339 and upper bound 21.698.[Power Mean] Let X be a positive random variable on a positive interval (a, b) with mean μ. For any real number s≠0, define the power mean as M_s(X)=(E X^s)^1/sJensen's inequality establishes that M_s(X) is an increasing function of s. We now give a sharper inequality by applying Theorem 1. Let r≠0, Y=x^r, μ_y=EY, p=s/r and φ(y)=y^p.Note that E X^s = E{φ(Y)}.Applying Theorem 1 leads to inf h(y;μ_y)var(Y)≤ E[X^s] - (E X^r)^p ≤sup h(y;μ_y) var(Y),whereh(y;μ_y)=y^p-μ_y^p/(y-μ_y)^2-p μ_y^p-1/y-μ_y.To apply Lemma 1, note that φ'(y) is convex for p≥ 2 or p∈(0,1] and is concave for p<0 or p∈[1,2] as noted in Section 2.Applying the above result to the case of r=1 and s=-1, we have Y=X, p=-1. Therefore((EX)^-1 +lim_y→ ah(y;μ_y)var(X))^-1≤(E X^-1)^-1≤((EX)^-1 + lim_y→ bh(y;μ_y)var(X))^-1. For the same sequence x_1,…, x_n generated in Example 2, we have x̅_harmonic=39.113.Applying Corollary <ref> leads to 25.337≤x̅_harmonic=39.113≤ 48.905.Note that the upper bound 48.905 is much smaller than the arithmetic mean x̅=54.830 by the Jensen's inequality.Replacing h(b;x̅) by φ”(b)/2 and h(a;x̅) by φ”(a)/2 leads to a less accurate lower bound 0.8298 and 51.0839.In a recent article published in the American Statistician, <cit.> revisited Kolmogorov's formulation of generalized mean as E_φ(X)=φ^-1(E[φ(X)]),where φ is a continuous monotone function with inverse φ^-1. The Example 2 corresponds to φ(x)=-log(x) and Example 3 corresponds to φ(x)=x^s.We can also apply Theorem 1 to bound φ^-1(E φ(X)) for a more general function φ(x).[Rao-Blackwell Estimator] Rao-Blackwell theorem (Theorem 7.3.17 in Casella and Berger, 2002; Theorem 10.42 in Wasserman, 2013) is a basic result in statistical estimation. Letθ̂ be an estimator of θ,L(θ, θ̂) be a loss function convex in θ̂, and T a sufficient statistic. Then the Rao-Blackwell estimator, θ̂^*=E[θ̂| T], satisifies the following inequality in risk function E[L(θ, θ̂)]≥ E[L(θ, θ̂^*)].We can improve this inequality by applying Theorem 1 to φ(θ̂)=L(θ, θ̂) with respect to the conditional distribution of θ̂ given T:E[L(θ, θ̂)| T] - L(θ, θ̂^*)≥inf_x∈(a,b) h(x;θ̂^*) var(θ̂| T),where function h is defined as in Theorem 1 for φ(θ̂) and P(θ̂∈(a,b)| T)=1. Further taking expectations over T givesE[L(θ, θ̂)] - E[L(θ, θ̂^*)]≥ E[inf_x∈(a,b) h(x;θ̂^*)var(θ̂| T)].In particular for square-error loss, L(θ, θ̂)=(θ̂-θ)^2, we haveE[(θ-θ̂)^2] - E[(θ-θ̂^*)^2] = E[var(θ̂| T)].Using the original Jensen's inequality only establishes the cruder inequality in Equation (<ref>). § IMPROVED BOUNDS BY PARTITIONINGAs discussed in Example 1 above, Theorem 1 does not improve on Jensen's inequality if inf h(x;μ) = 0.In such cases, we can often sharpen the bounds by partitioning the domain (a,b) following an approach used in <cit.>. Let a=x_0<x_1<⋯<x_m=b,I_j=[x_j-1,x_j), η_j=P(X∈ I_j), and μ_j=E(X| X∈ I_j). It follows from the law of total expectation thatE[φ(X)] =∑_j=1^m η_j E[φ(X) | X∈ I_j]=∑_j=1^m η_j φ(μ_j)+ ∑_j=1^m η_j (E[φ(X) | X∈ I_j]-φ(μ_j)).Let Y be a discrete random variable with distribution P(Y=μ_j)=η_j, j=1,2,…,m. It is easy to see that EY = EX.It follows by Theorem 1 that∑_j=1^mη_jφ(μ_j) = E[φ(Y)]≥φ(EY)+inf_y∈[μ_1,μ_m] h(y;μ_y)var(Y).We can also apply Theorem 1 to each E[φ(X | X∈ I_j)]-φ(μ_j) term: E[φ(X | X∈ I_j)]-φ(μ_j)≥inf_x∈ I_j h(x;μ_j)var(X| X∈ I_j).Combining the above two equations, we haveE[φ(X)]- φ(EX) ≥inf_y∈[μ_1,μ_m] h(y;μ_y)var(Y) + ∑_j=1^m η_j inf_x∈ I_j h(x;μ_j)var(X| X∈ I_j).Replacing inf by sup in the righthand side gives the upper bound. The Jensen gap on the left side of (<ref>) is positive if any of the m+1 terms on the right is positive. In particular, the Jensen gap is positive if there exists an interval I⊂(a,b) that satisfies inf_x∈ Iφ”(x)>0, P(X∈ I)>0 and var(X| X∈ I)>0.Note that a finer partition does not necessarily lead to a sharper lower bound in (<ref>). The focus of the partition should therefore be on isolating the part of interval (a,b) in which φ”(x) is close to 0. Consider example X∼N(μ ,σ ^2) with μ=0 and σ=1 and φ(x)=e^x. We divide (-∞,∞) into three intervals with equal probabilities. This givesThe actual Jensen gap is e^μ+σ/2-e^μ=0.649.The lower bound from (<ref>)is 0.409, which is a huge improvement over Jensen's bound of 0.The upper bound ∞, however, provides no improvement over Theorem 1.To summarize, this paper proposes a new sharpened version of the Jensen's inequality. The proposed bound is simple and insightful, is broadly applicable by imposing minimum assumptions on φ(x), and provides fairly accurate result in spite of its simple form. It can be incorporated in any calculus-based statistical course.chicago | http://arxiv.org/abs/1707.08644v2 | {
"authors": [
"J. G. Liao",
"Arthur Berg"
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"categories": [
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"primary_category": "math.ST",
"published": "20170726211341",
"title": "Sharpening Jensen's Inequality"
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SF2A 2017Secular evolutionObservatoire de Paris, LERMA, College de France, CNRS, PSL, Sorbonne University UPMC, F-75014, Paris, France Secular evolution of Milky Way-type galaxies F. Combes Received: December 30, 2023/ Accepted: date ===============================================The internal evolution of disk galaxies like the Milky Way are driven by non-axisymmetries (bars) and the implied angular momentum transfer of the matter; baryons are essentially driven inwards to build a more concentrated disk. This mass concentration may lead to the decoupling of a secondary bar, since theorbit precessing frequency is then much enhanced. Vertical resonances with the bar will form a box/peanut bulge in a Gyr time-scale. Gas flows due to gravity torques can lead to a young nuclear disk forming stars, revealed by a σ-drop in velocity dispersion. These gas flows moderated by feedbackproduce intermittent accretion of the super-massive black hole, and cycles of AGN activity. The fountain effect due to nuclear star formation may lead to inclined, and even polarnuclear disks.The Galaxy, Secular evolution, bulge, disk, gas flowsHow do we define secular evolution? This is slow and internal evolution of a galaxy, which can be fueled by long-term gas accretion from cosmic filaments, as opposed to violent evolution in galaxy mergers, or through interactions with a galaxy cluster. At least three types of galaxy formation and evolution can be considered:(i) the monolithic scenario, in which the gas collapses and forms stars in a time shorter than the time-scale for clouds to collide and flatten into a disk. This forms a spheroidal galaxy, or a bulge that can later accrete gas to form a disk. (ii) the hierarchical scenario in whichthe gas flattens in disks before forming stars, leading to disky galaxies, and their interaction/merger with random angular momentum lead to the formation of spheroids. (iii) the third possibility is just a branching of (ii) in low-density environments, when mergers are rare, and the galaxy disks evolve internally, and form boxy bulges from their own disk stars.§ TWO KINDS OF BULGES, CLASSICAL AND PSEUDO (BOX/PEANUT)Classical bulges are generally the result of major mergers, with unaligned spins: the remnant is not flattened and has little rotation. Its light profile has a Sersic index n=4 or higher.In minor mergers, disks are more easily conserved, while the classical bulge grows.During secular evolution,bars and vertical resonances elevate stars in the inner parts into a pseudo-bulge: a component intermediate between a spheroid and a disk (Combes & Sanders 1981). It is flattened, rotating, and has an exponential lightprofile (n∼1). They are more frequent in late-type galaxies.Clumpy galaxies at high redshift can also form a classical bulge,throughdynamical friction of the massive clumps againt dark matter. The formation of classical bulges is favored, and this makes even moredifficult to form bulgeless galaxies. Observations tell us that the majority oflate-type galaxies have no or little bulge today (Kormendy & Fisher 2008, Weinzirl et al 2009).The fraction of pseudo-bulges has been quantified recently by Fisher & Drory (2016), as a function of stellar mass: classical bulges begin to dominate only for stellar masses larger than 5 10^10 M_⊙. The impact of environment is important: there exists half less pseudo-bulges in centrals with respect to satellites and field galaxies (Mishra et al. 2017).From HST images at high redshift (Goods-South) it was possible to decompose galaxies in disks and bulges, and distinguish pseudo and classicals (Sachdeva et al. 2017). Although pseudo bulges have masses about half that of classicals, both bulges double in mass since z∼1: the mass fraction increases from 10 to 26% for the pseudo, and from 21 to 52% for the classicals. This points towardssecular evolution with at most minor mergers.It might not be as easy to separate the formation of the two kinds of bulges, since the dynamical evolution implies an angular momentum transfer with the bar, ending with a spin-up of the classical bulge (Saha et al. 2012).This is particularly important for low-mass classicals, but also for higher masses (Saha et al. 2016). §ANGULAR MOMENTUM TRANSFER Cold disks form by transferring angular momentum (AM) outwards. Bars, as negative AM waves, are amplified in the process. Stars exchange AM only at resonances (unless the potential is varying). The stars emit AM at ILR,absorb at CR and OLR (Lynden-Bell & Kalnajs 1972). The AM is also absorbed by the dark matter halo (Athanassoula 2002).During bar growth, more and more particles are trapped along the bar, and their orbits are more elongated; thebar pattern speed Ω_b slows down, and thecorotationmoves outwards, the bar is longer in the disk.When gas is present in the disk, it feels the gravity torques from the bar. It is driven inwards from CR, gives its AM to the bar, which weakens the bar (Bournaud & Combes 2002). When disks are refilled with gas and become more massive, they can reform a bar with higherΩ_b (shorter bars). §.§ Decoupling of a secondary barWhen the bar has slown down and there exist two ILR, the orbits become perpendicular (x2) to the bar in between the two ILRand do not sustain the bar anymore. This produces the decoupling of a new faster bar inside the oILR ring. The z-resonance and formation of the peanut also contribute to weaken the inner primary bar.The gravity torques of the second bar drive the gas to the center, forming a cool nuclear disk with young stars, which might be observed through a σ-drop, i.e. a dip in the stellar velocity dispersion (Emsellem et al. 2001, Wozniak et al 2003).New simulations of σ-drops have been done recently (Portaluri et al. 2017, Di Matteo et al. 2017) including spectral synthesis modelling, and chemical tagging. They show clearly that the drop is seen in luminosity-weightedimages.Embedded bars are observed in about 30% of all barred galaxies. It is also possible to form long-lived two-bar galaxies, with no resonance in common,no mode coupling (Wozniak 2015). The star formation in the gas stabilises the nuclear bar. Also the nuclear bar could form first, in an inside-out two bar formation scenario (Du et al. 2015), in clumpy high-z galaxies.§.§ Bar gravity torquesThere exists a weak correlation between bars andAGN (Schawinski et al. 2010,Cardamone et al. 2011) and certainly bars help to drive gas to the nuclear region by their gravity torques. But the situation is complex, and depends on the various time-scales.In a survey of 20 nearby Seyferts, we have been able to compute the torques exerted by bars on the gas distribution, obtaining the gravitational potential from HSTred images (NUGA project). At the scale of 10-100 pc, at which the molecular gas maps were obtained, the statistics of fueling are not high: only 35% of negative torques were measured in the center(e.g. Garcia-Burillo & Combes 2012). The rest of the times, the torques are positive, and the gas is stalled in resonant rings. The fueling phases are short, a few 10^7 yrs, may be due to feedback. There is also star formation fueled by the torques, always associated to AGN activity, but with longer time-scales.Embedded non-axisymmetries will occur at smaller-scales to control gas accretion. Zoomed simulations of gas accretion onto a central black hole have revealed a cascade of m=2, m=1 perturbations (Hopkins & Quataert 2011), providing an intermittent inflow rate. When the gas fraction is high, the nuclear disk is unstable against warps, bending, and forms clumps, sensitive to dynamical friction, which will drive gas inwards. High resolution (22pc) ALMA observations have been able tomeasure gravity torques even further towards the center. In the barred spiral NGC 1433, a second resonant ring has been discovered at 200 pc, at the ILR of the nuclear bar (Combes et al. 2013). But the torques are positive inside, and the gas is piling up at this second ring (Smajic et al. 2014). There is also a molecular outflow on the minor axis,due to the AGN feedback. This is a weak outflow of 100 km/s in velocity, and dragging 7% of the molecular gas mass.The case of the nearby Seyfert type 1 galaxy NGC1566 is different: trailing nuclear spiral arms have been discovered within 100pc around the black hole,torques are negative, fueling the nucleus (cf Figure <ref>). This is due to the gravitational impact of the black hole, since the gas enters its radius of influence.In a high-resolution simulation meant to approach the MW, peculiar hydrodynamical processes have been revealed in the central 200 pc region (Renaud et al. 2016, Emsellem et al. 2015). When the gas has sufficiently concentrated to the center, it becomes unstable, fragments and forms stars. Through strong supernovae feedback and its fountain effect, the gas is projected above the plane, and falls back to settle in a polar disk around the black hole (cf Figure <ref>). § SUMMARYSecular evolution is the dominant scenario in Milky-Way type galaxiesin the second part of the Universe age. The stellar bar favors the mass concentration, and through vertical resonance, elevates starsto form a box/peanut bulge. Primary bars drive gas from 10 kpc-scaleto R ∼100 pc, then nuclear bars continue from 100 pc to 10 pc. Young nuclear disks arerevealed byσ-drops in their velocity dispersion.The mass concentration, and inward gas flows fuel nuclear starbursts and AGN. At scales ∼1-10 pc, viscous turbulence, clumps, disk warps and bending, take over to fuel the super-massive black hole. The process is intermittent, moderated by feedback and gas outflows.[Atha02]Atha02 Athanassoula, E.: 2002, ApJ, 569, L83 [Bou02]Bou02 Bournaud, F., Combes, F.: 2002, A&A392, 83 [Car11]Car11 Cardamone, C. N., Schawinski, K., Masters, K., Lintott, C., Fortson, L.: 2011, AAS 21820603 [Com81]Com81 Combes, F., Sanders, R. H.: 1981, A&A 96, 164 [Com13]Com13 Combes, F., Garcia-Burillo, S., Casasola, V. etal.: 2013 A&A, 558, A124[Com14]Com14 Combes, F., Garcia-Burillo, S., Casasola, V. etal.: 2014 A&A, 565, A97[Dim17]Dim17 Di Matteo, P et al.: 2017, A&A in prep [Du15]Du15 Du, M., Shen, J., Debattista, V. P.: 2015, ApJ804, 139 [Em01]Em01 Emsellem, E., Greusard, D., Combes, F. et al.: 2001, A&A368, 52 [Em15]Em15 Emsellem, E., Renaud, F., Bournaud, F., Elmegreen, B., Combes, F., Gabor, J. M.: 2015, MNRAS 446, 2468 [Fis16]Fis16 Fisher, D. B., Drory, N.: 2016,Astr. Space Science, 418, 41 [Gar12]Gar12 Garcia-Burillo, S. Combes, F.: 2012, JPhCS 372, a2050 [Hop11]Hop11 Hopkins, P. F., Quataert, E.: 2011, MNRAS415, 1027 [Kor08]Kor08 Kormendy, J., Fisher, D. B.: 2008, ASP Conf 396, 297 [Lyn72]Lyn72 Lynden-Bell, D., Kalnajs, A. J.; 1972, MNRAS 157, 1 [Mis17]Mis17 Mishra, P. K., Wadadekar, Y., Barway, S.: 2017, MNRAS,467, 2384 [Por17]Por17 Portaluri, E., Debattista, V. P., Fabricius, M. et al.: 2017, MNRAS 467, 1008 [Ren16]Ren16 Renaud, F., Bournaud, F., Emsellem, E., et al.: 2016, MNRAS 463, 251 [Sac17]Sac17 Sachdeva, S., Saha, K., Singh, H. P.: 2017, ApJ840, 79 [Sah12]Sah12 Saha, K., Martinez-Valpuesta, I., Gerhard, O.:2012, MNRAS, 421, 333 [Sah16]Sah16 Saha, K., Gerhard, O., Martinez-Valpuesta, I.: 2016, A&A588, A42 [Sch10]Sch10 Schawinski, K., Urry, C. M., Virani, S. et al.: 2010, ApJ711, 284 [Sma14]Sma14 Smajic, S., Moser, L., Eckart, A. et al.: 2014, A&A567, A119 [Wei09]Wei09 Weinzirl, T., Jogee, S., Khochfar, S., Burkert, A., Kormendy, J.: 2009, ApJ 696, 411 [Woz03]Woz03 Wozniak, H., Combes, F., Emsellem, E., Friedli, D.: 2003, A&A409, 469 [Woz15]Woz15 Wozniak, H.: 2015, A&A575, A7 | http://arxiv.org/abs/1707.08733v1 | {
"authors": [
"F. Combes"
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"published": "20170727073953",
"title": "Secular evolution of Milky Way-type galaxies"
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An Invitation to Polynomiography via Exponential Series Bahman Kalantari Department of Computer Science, Rutgers University, [email protected] ========================================================================================================The subject of Polynomiography deals with algorithmic visualization of polynomial equations, having many applications in STEM and art, see <cit.>-<cit.>. Here we consider the polynomiography of the partial sums of the exponential series.While the exponential function is taught instandard calculus courses, it is unlikely that properties of zeros of its partial sums are considered in such courses, let alone their visualization as science or art. The MonthlyarticleZemyan <cit.> discusses some mathematical properties of these zeros.Here we exhibit some fractal and non-fractal polynomiographs of the partial sums while also presenting a brief introduction of the underlying concepts.Polynomiography establishes a different kind of appreciation of the significance of polynomials in STEM, as well as in art. It helps in the teaching of various topics at diverse levels. It also leads to new discoveries on polynomials and inspires new applications.We also present a link for the educator to get access to a demo polynomiography software together with a module that helps teach basic topics to middle and high school students, as well as undergraduates.Keywords: Exponential Function, Complex Polynomial, Iterative Methods, Polynomiography. empty § INTRODUCTIONEver since introducing the term polynomiography for the visualization of polynomial equations via iteration functions, when encountering certain polynomials I have found ittemptingtoconsider the shape of their polynomiographs in the complex plane. The word polynomiography is a combination of the term polynomial, first used in the 17th century, see Barbeau <cit.>, and the suffix graphy. Polynomiography grew out of my research into the subject of polynomial root-finding, an ancient and historic subject that continues to grow and finds new applications with every generation of mathematicians and scientists. We cancreateliterally hundreds of polynomiographs for a single polynomial equation, even when restricted to the same portion of the complex plane.One familiar with the term fractal, coined by Mandelbrot many years earlier, might think polynomiography is just another name for fractal images. This however is not validand underestimates the significance of the theory and algorithms that have led to polynomiography. One reason to call an image a polynomiograph rather than a fractal is because the image may exhibit no fractal behavior no matter in what part of the Euclidean plane it is generated.It would thus not make sense to call it a fractal.After all, fractal properties that may be inherent in some iterations may not be present everywhere in an image and an iterative method may be well behaved in certain areas of the Euclidean plane, or exhibit nofractal pattern anywhere.Consider for instance the polynomiograph of the polynomial 1+z+0.5z^2, the quadratic partial sum of the exponential function under the iterations of Newton's method,shown in Figure <ref>, top-left.Given a particular point z_0, the iterations of Newton's method generate a sequence z_0, z_1, z_2, …, that may or may not converge to a root. The sequence is called the orbit of z_0.The basin of attraction of a root is the set of all points in the plane whose orbit converges to that root. In the figure the upper half represents thebasin of attraction of the one root and the lower half the basin of attraction of the other root.The points on the perpendicular bisector of the line connecting the roots, the x-axis,does not belong to either one of the two basins. There is no fractal property in this image,no broken lines, no self-similarity.There is chaos corresponding to orbits of the points on the bisecting line, however this set is not a fractal set.The basins of attraction form the Fatou set, and the bisecting line the Julia set. This is an example of a non-fractal polynomiograph.On the other hand, for a polynomial of degree three with distinct roots, the corresponding Newton's polynomiograph is fractal if it contains a portion of the Julia set.TheJulia set is the boundary of each basin of attraction of a root and is fractal but the Fatou set may contain more than the union of the basins of attractions.There are deep results on the Julia and Fatou sets of polynomials under the iterations of Newton's method and other iteration functions, including their own iterations.For instance,the animation <cit.> givesa 3D depiction of the dynamics of the Fatou sets for the polynomial z^3-1. About fractals and the amazing mathematical properties of iterations of rational functions one can consult Beardon <cit.>, Devaney <cit.>, Mandelbrot <cit.>, Milnor <cit.>, and Kalantari <cit.> with emphasis on iteration functions for polynomial root-finding. A polynomiograph may exhibit fractal features, in which case one can refer to it as a fractal polynomiograph. Even if a polynomiograph exhibits fractal behavior it is more informative to refer to it as such, rather than plain fractal. Indeed the word fractal is used in very general terms. It may refer to many different types of objects, such as 2D images coming from iterations of all kinds, 3D fractals andeven objects such as trees, clouds, mountains,nature and the universe.After all, just because we may refer to an image or an object as fractal it does not imply that we understand all its properties.In <cit.> and several other articlesI have described reasons in support of the definition of the term.While originally polynomiographywas to represent as algorithmic visualization of a polynomial equation via a specific family of iteration functions, called the basic family to be discussed later in the article, after many more years of experiences I would likepolynomiography to refer to visualization of polynomials in broader terms,allowing the possibility of other iterative methods, even a mixture of iterative methods,even 3D visualizations or visualizations that pertain to the zeros of complex polynomials, not necessarily via iterations.Based on many educational experiences, including those of educators and students who have come to experiment with polynomiography software,there is convincing evidence that the imagesconvey meaningful mathematical attributes of polynomialsand algorithmic properties that make the images interesting beyond their aesthetic beauty as art, especially to the youth and students.There have been attempts by educators to popularize fractals in education and to teach some basic properties, for instance at high schools.However, introducing fractals in a very general setting could be confusing to the youth. After all, the underlying mathematics of fractals and iterations is sophisticated.On the other hand,sincesolving quadratic equations is common knowledge in middle and high school, students can connect to polynomiography in an easy fashion, turning polynomials of any degree into fun objects to deal with.Studying the underlying theory of polynomiography also makes it possible to teach and learn about fractals. Polynomiography helps motivate the teaching of fractals and related material at the K-12 level and beyond in a constructive manner, alsoconnecting geometry and algebra. The present article is an attempt to demonstrate the beauty of a well known class of polynomials, seldom considered as complex polynomials in the manner presented here.What makes visualization of a polynomial equation interesting, even when all its coefficients are real numbers, is to view its domain not as the real line, but the Euclidean plane. The simplest case of polynomiography is the visualization of the basins of attraction of Newton's method when applied to a quadratic equation, e.g. z^2-1, historicallyconsidered by Cayley <cit.> in 1879. Except for a shift, its polynomiograph is identical with the one shown in Figure <ref>, top-left image. It is not difficult to mathematically prove this property of the basins of attraction without computer visualization. On the other hand, the analysis of basins of attraction of z^3-1under Newton's methodis very complex.It is well known that the resulting image is fractal, afractal polynomiograph.With the advent of computers it became more plausible to understand the shape of basins of attraction and apparently the first person who tried this and saw the surprising fractal behavior was the mathematician John Hubbard, see Gleick <cit.>.While polynomiographs of Newton's method for quadratic are not fractal, one can easily modify Newton's method to get fractal polynomiographs.This modified method is a parametrized Newton's method, described in more generality in the next section.The subject of fractals is significant, vastly rich and very beautiful. Mandelbrot not only coined the term fractal but undoubtedly played an enormous role in bringing them into view and this in turn has resulted in many theoretical advancements and visualizations, including new kinds of algorithmic mathematical art.Polynomiography does overlap with fractals in many way. However, it is not a subset of fractals, not in theory, nor in practice, nor in terms of its images as art or otherwise. I believe that polynomiography can play an important role in the teaching of fractals and dynamical systems at various levels. To support this point, in numerous personal experiences that include presentationsto hundreds of middle and high school students,lectures during a first-year seminars or formal courses,only a very small percentage of students have ever heard of the term “fractal.”Even those who had familiarity with fractals could only identify them as visual images that represent self-similarity.Even at universities, topics on fractals and dynamical systems are only offered as graduate level courses.For those who wish to teach or learn basic concepts from the theory of fractals and dynamical systems polynomiography can provide apowerful bridgeinto these subjects areas, as well as many others. Polnomiography appeals to students because they can connect it with a task they have leaned early on, namely solving a polynomial equation. This in particular makes polynomiography effective for introducing it to K-16 students at elementary or advanced levels.As an example consider introducing the Mandelbrot set to middle and high students beyond just showing the aestheticbeauty of the set. We must first introduce them to Julia sets resulting fromthe iterations of a quadratic function.However, these iterations attempt to find fixed points but not roots. The notion of fixed points, while implicit in Newton's method, typically is not taught in K-12, and hence is unfamiliar to students.In order to popularize fractals, firstthe notion of fixed points and fixed point iterations must be introduced. Having introduced these, then we can considerthe task of approximating the roots of a polynomial p(z) as that of finding the fixed points of the polynomial, q(z)=p(z)+z. A fixed point θ of q(z) isattractive, repulsive and indifferent if the modulus of p'(θ) isless than one, larger than one, or equal to one.For a complex number z=x+i y, i=√(-1),its modulus is |z|= √(x^2+y^2).The fixed point iteration refers to computing z_k=q(z_k-1) for k=1, 2, …,where z_0 is a starting seed. If a fixed point is attractive, the iterations are guaranteed to converge to it, provided z_0 is close enough. That iterations are necessary to approximate the roots even of a quadratic equation is clear to a middle schooler who knows the quadratic formula fails to provide a numeric decimal value to the solution of x^2-2.Students can experience the behavior of iterations of quadratic functions via polynomiography. These experiences will demonstrate that not both fixed points of a quadratic can be attractive.However,Newton's method will never fail to approximate the roots because both fixed points are attractive with respect to Newton's function.While Newton's method doesn't result in fractal polynomiographs forany quadratic, fractal Julia sets result in polynomiography of a quadratic corresponding to a parametrized Newton's method, see Figure <ref>. By introduction ofdifferent values for the parameter, students quickly learn the notion of attractive and repulsive fixed points andappreciate Newton's method for numerically solving a quadratic equation. Students can also be introduced to the notion of open and closed sets, as well as Fatou and Julia sets.By considering polynomiography students and teachers quickly discover the vastness of the world of polynomials and they discover new applications of them, distinct from the traditional applications considered in standard textbooks.Among these applications one can include educational lesson plans, visual cryptography, art and much more. Inmy personal experiences in the teaching of the subject of polynomiography I am often delighted to find many creative applications that students are able to think of, including those that I did not even imagine.Why limit the applications of polynomials to the standard ones discussed in typical textbooks on algebra, calculus, or numerical analysis? Why not think of the shape of zeros of a polynomial, even if these do not come up in ordinary application? Once we think of the zeros of a polynomial, its polynomiographybecomes a relevant matter of curiosity, leading to new images, new discoveries, new applications, new questions, and new art. Polynomiography may be amouthful of a word, however it is a meaningful one. Students quickly accept it.In order to introduce polynomiography we need to consider polynomial equations over the complex plane. Since everyone is already familiar with the Cartesian coordinate system in the plane, it is easy to describea polynomial equation as a way to encrypt a bunch of points in the Euclidean plane. We think of the points as complex numbers. This allows for turning points in the Euclidean plane into objects that inherit the four elementary operations on real numbers. For a middle or high school student,learning about elementary operations on complex numbers is a matter of minutes rather than hours. Once these operations are understood, a polynomial equation together with the fundamental theorem of algebra is nothing more than a way to encrypt points.Solving a polynomial equation is a game of hide-and-seek.For a fun introduction to the fundamental theorem of algebra, see Kalantari and Torrence <cit.>.In this article I present some polynomiography for the partial sums of the exponential series, familiar to every student who has come across calculus. The exponential function is consideredby some mathematicians to be the most important function in mathematics. Polynomiography for the partial sums of some analytic functions such as sine and cosine is already considered in <cit.>.In fact we can do polynomiography for functions that are not polynomial, and witness a visual convergence in the sense of polynomiography. The irony is that the exponential function itself has no zeros.At the end for the educator I will provide links to demo polynomiography software and a teaching module. § THE N-TH PARTIAL SUM OF EXPONENTIAL SERIES AND THE BASIC FAMILY The exponential function and itsn-th partial sum polynomial are, respectivelyexp(z)= ∑_k=0^∞z^k/k!,P_n(z)= ∑_k=0^n z^k/k!=1+z+z^2/2!+⋯ + z^n/n!.The shape of zeros of these polynomials has been studied, see Zemyan <cit.> for a wonderful review. Many results are known, for instance, bounds on the zeros of P_n(z). Some conjectures are raised on the zeros, such as the convexity of the roots are described in <cit.>. Undoubtedly meany research questions can be stated.The polynomiographs of the partial sums to be exhibited here are generated via the basic family of iteration functions. For a given arbitrary polynomial p(z), the basic family is an infinitecollection of iteration functions. To define the basic family even in more generally, given a complex number α satisfying| 1- α| < 1, theparametrized basic family is:B_m, α(z)=z- α p(z) D_m-2(z)/D_m-1(z),m=2,3, …whereD_0(z)=1,D_k(z)=0 for k <0, and, D_m(z) satisfies the recurrenceD_m(z)= ∑_i=1^n (-1)^i-1p(z)^i-1p^(i)(z)/i!D_m-i(z).The range for α guarantees that each root of p(z) remains an attractive fixed point of B_m, α(z), see <cit.>.When α=1 we denote the family by B_m(z). Each member is capable of generating different polynomiography of the same polynomial.The first two members are, the B_2(z), Newton, and the B_3(z), Halley, iteration functions.This family and its variations are extensively studied in <cit.> establishingmany fundamental properties of why they are probably the most important family of iteration functions for polynomial root-finding. In particular, for each fixed m≥ 2, there exists a disc centered at a root θ such that for any z_0 in this disc the sequence of fixed point iteration z_k+1=B_m(z_k), k=0,1,…, is well-defined and converges to θ. Whenθ is a simple root, i.e. p(θ)=0, p'(θ) ≠0,the order of convergence is m. Variations of the basic family,other than the parameterized version, are described in <cit.>.In contrast to using individual members of the basic family, there is a collective application, using thebasic sequence, {B_m(w), m=2, …}, where w is some fixed complex number, see <cit.>. To describe the convergence of the basic sequence we need to define the notion of the Voronoi diagram of a set of points in the Euclidean plane.Given a set of points θ_1, …,θ_n in the Euclidean plane, theVoronoi cell of a particular point θ_i is the set of all points in the plane that are closer to θ_i than to any other θ_j.If w lies in the Voronoi cell of a particular root θ of p(z), then it can be shown that the sequence B_m(w) converges to θ. For pointwise convergencesee <cit.>, and for a proof of a stronger property, uniform convergence of the basic family, see <cit.>. Based on this convergence property we can produce non-fractal polynomiographs that are very different from the usual fractal ones. § POLYNOMIOGRAPHS FOR THE PARTIAL SUM Here we describe several polynomiographies for the first few partial sums based on the basic family. It is easy to show that the roots of P_n(z) are simple, i.e. P_n(z) and its derivative P'_n(z) have no common zeros.Also, it can be shown that the modulus of any root θ a root of P_n(z) satisfies 0 <|θ | < n.The polynomiography of P_1(z)=1+z is quite simple.Any member of the basic family will converge to the root in one iteration. Polynomiographs of P_n(z) under Newton's method for n=2, …, 10 are depicted in Figure <ref>,showing them in increasing order from left to right and top to bottom.The shape of the zeros form a convex shape,reminiscent ofa parabola of the form x=y^2. This can be seen in the polynomiographs.The norm of the roots goes to infinity as n does. Figure <ref> shows the polynomiography of P_n(z), n=2, …, 7 under the point-wise convergence.These are not fractal images.Figure <ref> shows the polynomiography of P_n(z), n=2, …, 7 under parametrized Newton's method all for a particular value of α.§ POLYNOMIOGRAPHS OF SZEGÖ PARTIAL SUMSThe norm of the roots of P_n(z) get large as n does. Szegö partial sums areS_n(z)=P_n(nz)= ∑_k=0^n (nz)^k/k!=1+nz+(nz)^2/2!+⋯ + (nz)^n/n!.Many interesting properties of this polynomial are known, see Pólya and G. Szegö <cit.> and Zemyan <cit.>. If θ is a root of P_n(z) then θ/n is a root of S_n(z). Thus all the roots of S_n(z) are inside the disc of radius one, centered at the origin.Polynomiographs of S_n(z) for small n look like scaled version of those of P_n(z). However, as n goes to infinity the zeros of S_n(z) bend, forming an almond-shape inside the unit disc, see Zemyan <cit.>. Figure <ref> shows the polynomiography of S_n(z) × (z^n-1) under point-wise convergence of the basic family.The reason for multiplying by the roots of unity z^n-1 is two-fold: To show rootslie inside the unit disc andthat under multiplication of polynomials we can generate interesting polynomiographs as science and as art. § CONCLUDING REMARKS In this article I have demonstrated polynomiography for the partial sums of the exponential series.One can appreciate the images as art, but also as a way to get interested in learning or teaching root-finding algorithms.What is intriguing about polynomiography software is that in the course of generating images we can learn about the shape of the zeros and get introduced to many other concepts in math and related areas.Polynomiography is a medium for STEM, a bridge to learning or teaching about different subject areas, and making artistic images by considering variations of polynomials,root-finding algorithms, coloring techniques, and operations such as multiplication of polynomials, scaling their zeros, compositions and more. Indeed we could make many artistic images based on the partial sums alone. Interested educators can get a student module, see <cit.>, as well as a link to a free demo polynomiography software upon registration at<http://www.comap.com/Free/VCTAL/>.See also, Choate <cit.> for lesson plans and a short manual for the demo software.99Ander C. Anderberg, J. Choate and B. Kalantari, “Computational Thinking Module, Polynomiography: Visual Displays of Solutions of Polynomial Equations”, (2016), <http://www.comap.com/Free/VCTAL/PDF/Polynomiography_SE.pdf>.Bar E. J. Barbeau, Polynomials, Springer, 2003.Bea A. F. Beardon, Iteration of Rational functions: Complex Analytic Dynamical Systems, Springer-Verlag, New York, 1991.cay A. Cayley, “The Newton-Fourier imaginary problem”,American Journal of Mathematics, 2 (1879), pp. 97.jon J. Choate, “Polynomiography”,Geometer's Corner, Consortium for Mathematics and Its Applications,105 (2013) pp. 1-3. Dev R. L. Devaney, A First Course in Chaotic Dynamic System Theory and EXPERIMENT (ABP), 1992.Gl J. Gleick, Chaos: Making of A Science, Harmondswoth: Pengin Books, 1988. Kal04 B. Kalantari, “Polynomiography and applications in art, education, and science”,Computers & Graphics, 28 (2004), pp. 417–430.kalsig04 B. Kalantari, “A new visual art medium: Polynomiography”, ACM SIGGRAPH Computer Graphics Quarterly, 38 (2004), pp. 21–23.kal2005c B. Kalantari, “Polynomiography: From the Fundamental Theorem of Algebra to Art”, Leonardo, 38 (2005), pp. 233–238.Kalbook B. Kalantari, Polynomial Root-Finding and Polynomiography, World Scientific,Hackensack, NJ, 2008. KalDCG B. Kalantari,“Polynomial root-finding methods whose basins of attraction approximate Voronoi diagram”,Discrete & Computational Geometry,46 (2011), pp. 187–203.KT B. Kalantari and B. Torrence, “The Fundamental Theorem of Algebra for Artists,” Math Horizons,20(2013), pp. 26–29.KS B. Kalantari and A. Sinclair, “The Rise of Polynomials”, (2008) <https://www.youtube.com/watch?v=kMP0vclKlDA>.Man B. B. Mandelbrot, Fractal Geometry of Nature, W. F. Freeman, New York, 1993. Milnor J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vol 160, 3rd en. Princeton University Press, New Jersey, 2006. PolG. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Erster Band, Springer-Verlag, Berlin, 1964.ZemyanS. M. Zemyan, “On the Zeroes of the Nth Partial Sum of the Exponential Series”,The American Mathematical Monthly, 112 (2005), pp. 891–909. | http://arxiv.org/abs/1707.09417v1 | {
"authors": [
"Bahman Kalantari"
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"published": "20170726172556",
"title": "An Invitation to Polynomiography via Exponential Series"
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Fabrizio Lucarelli [email protected] ASI Science Data Center (ASDC), Via del Politecnico snc, I-00133 Roma, Italy INAF–OAR, via Frascati 33, I-00078 Monte Porzio Catone (Roma), Italy ASI Science Data Center (ASDC), Via del Politecnico snc, I-00133 Roma, Italy INAF–OAR, via Frascati 33, I-00078 Monte Porzio Catone (Roma), Italy ASI Science Data Center (ASDC), Via del Politecnico snc, I-00133 Roma, Italy INAF–OAR, via Frascati 33, I-00078 Monte Porzio Catone (Roma), Italy Agenzia Spaziale Italiana (ASI), Via del Politecnico snc, I-00133 Roma, Italy INAF/IAPS–Roma, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy Univ. “Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Roma, Italy Gran Sasso Science Institute, viale Francesco Crispi 7, I-67100 L`Aquila, Italy INAF/IASF–Bologna, Via Gobetti 101, I-40129 Bologna, Italy INAF/IASF–Milano, via E.Bassini 15, I-20133 Milano, Italy ASI Science Data Center (ASDC), Via del Politecnico snc, I-00133 Roma, Italy INAF–OAR, via Frascati 33, I-00078 Monte Porzio Catone (Roma), Italy INAF/IASF–Milano, via E.Bassini 15, I-20133 Milano, Italy INFN–Pavia, Via Bassi 6, I-27100 Pavia, Italy University of Witwatersrand, Johannesburg, South Africa INAF–OAR, via Frascati 33, I-00078 Monte Porzio Catone (Roma), Italy Dip. di Fisica, Universita’ di Trieste and INFN, Via Valerio 2, I-34127 Trieste, Italy INAF/IASF–Milano, via E.Bassini 15, I-20133 Milano, Italy INFN–Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy INAF/IAPS–Roma, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy INAF/IAPS–Roma, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy INAF – Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047 Selargius (CA), Italy INAF – Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047 Selargius (CA), Italy INFN–Pavia, Via Bassi 6, I-27100 Pavia, Italy INAF – Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047 Selargius (CA), Italy INAF Oss. Astron. di Brera, Via E. Bianchi 46, I-23807 Merate (LC), ItalyOn July 31st, 2016, the ICECUBE collaboration reported the detection of a high-energy starting event induced by an astrophysical neutrino.We report here about the search for a gamma-ray counterpart of theICECUBE-160731 event made with thesatellite.No detection was found spanning the time interval of ± 1 ks around theneutrino event time T_0 using the“burst search” system.Looking for a possible gamma-ray precursor in the results ofthe AGILE-GRID automatic Quick Look procedure over predefined48-hours time-bins, we found an excess above 100 MeV betweenone and two days before T_0, positionally consistent with theICECUBE error circle, having a post-trial significanceof about 4σ. A refined data analysis of this excess confirms a-posteriori theautomatic detection. The newtransient source, namedAGL J1418+0008, thus stands as possible ICECUBE-160731 gamma-ray precursor.No other space missions nor ground observatories have reported anydetection of transient emission consistent with the ICECUBE event. We show that Fermi-LAT had a low exposure of the ICECUBE regionduring thetransient. Based on an extensive search for cataloged sources withinthe error regions of ICECUBE-160731 and AGL J1418+0008, we finda possible common counterpart showing some of the key features associatedto the high-energy peaked BL Lac (HBL) class of blazars. Furtherinvestigations on the nature of this source using dedicatedSWIFT ToO data are presented.§ INTRODUCTION Neutrino astronomy by under-water and under-ice Cherenkov detectors hasentered a new era since the completion of the ICECUBE and ANTAREStelescopes <cit.>and the subsequent first clear detection of a diffuse backgroundof Very High Energy (VHE) extra-terrestrialneutrinos <cit.>.No significant clustering of neutrinos above background expectation has beenobserved yet <cit.>, although the ICECUBEapparatus might reach the sensitivityor accumulate enough statistics to unambiguously detect anisotropyor clustering of events within a few more years of observations.Emission of TeV-PeV neutrinos might be due to exceptionally energetictransient phenomena like flaring activities from Active Galactic Nuclei (AGNs),Gamma-Ray Bursts (GRBs) or Supernovae explosions <cit.>.A direct correlation between gamma-rays and neutrinos from astrophysicalsources is expected whenever hadronic emission mechanisms are at work.In particular, several theoretical works assume that neutrinos productionoccurs in astrophysical beam dumps, where cosmic rays accelerated in regionsof high magnetic fields near black holes or neutron stars interact via proton-proton (pp) or proton-photon (pγ) collisions with the matter or the radiationfield surrounding the central engine or in a jet of plasma ejected from it,giving raise also to gamma-rays emission (see <cit.>for a review).Supernovae remnants (SNRs) expanding in dense molecular clouds andmicroquasars in our Galaxy as well as AGNs of the blazars categoryare the main neutrino source candidates up to PeVenergies <cit.>.Besides the identification of the pion excess in observations of SNRs interacting with molecularclouds <cit.>,detection and identification of a clear neutrino point-likesource would represent the evidence of proton and hadronacceleration processes, resolving as well the long-lasting problem of thecosmic ray origin (at least up to multi-PeV energies).Since April 2016, the ICECUBE experiment alerts almostin real time the astronomical community whenever an extremelyhigh-energy single-track neutrino event (with energy in the sub-PeV to PeV range) isrecorded. The communication is sent through the ICECUBE_HESE(a single high-energy starting ICECUBE neutrino) and the ICECUBE_EHE(extremely high-energy ICECUBE neutrino) GCN/AMON noticessystem <cit.> a few seconds after the eventtrigger. The instant notice provides a first determinationof the statistical relevance of the event and thereconstructed neutrino arrival direction, projected onto the sky, withits 90% and 50% containment radius (c.r)[For ICECUBE_EHE notices,only source errors at 50% c.r. are given.].On July 31st, 2016, the ICECUBE Collaboration reported a HESE GCN/AMONnotice[http://gcn.gsfc.nasa.gov/notices_amon/6888376_128290.amon]announcing the detection of a high-energy neutrino-inducedtrack-like event at time T_0 = 01:55:04.00 UT (MJD=57600.07990741). The event was also classified as EHE event, possibly having anenergy higher than several hundred TeV[As quoted in the ICECUBE_EHE event information web pagehttps://gcn.gsfc.nasa.gov/amon_ehe_events.html] and a signalness[ Probability that the neutrino event is of astrophysical origin.]of ∼ 0.85. This neutrino detection triggered a broad-bandfollow-up by several space and ground-based instruments, searchingfor an electromagnetic (e.m.) counterpart to be associated to the neutrino emission.In what follows, we report about the search for a counterpart ofthe ICECUBE-160731 neutrino event made using the data of theAGILE satellite. The paper is organized as follows: in Section 2, wedescribe the main AGILE instrumental characteristics and its uniquecapabilities for the search of gamma-ray counterparts to such triggered eventsof very short duration. In Section 3, we present the resultsof the AGILE observations, both near the prompt neutrinoevent time T_0 and in archival data. In Sections 4, we report about themulti-wavelength (MWL) follow-up and in Section 5 we search for a possiblee.m. counterpart candidate using the cross-catalog search tools availablefrom the ASI Science Data Center (ASDC)[http://www.asdc.asi.it].§AS DETECTOR OF TRANSIENT GAMMA-RAY SOURCES The satellite AGILE <cit.>,launched on 2007, has just completed its tenth year of operations in orbit.The main on-board instrument is the imaging detector (GRID)sensitive to gamma-rays in the energy range 30 MeV–50 GeV, composedby the Silicon Tracker, the Mini-Calorimeter (MCAL) and the anti-coincidence (AC) system for the particle background rejection. The co-axial X-ray (20-60 KeV) detector Super- completesthe satellite scientific payload.Since Nov. 2009, AGILE is operated in the so calledspinning observation mode, in which the satellite rotatesaround the Sun-satellite versor. In this operation mode, the AGILE imager approximately observes the whole sky every day, with asensitivity (at 5σ detection level) to fluxes above 100 MeVof the order of (3 ÷ 4) × 10^-6 ph cm^-2 s^-1. As already demonstrated in the recent follow-up of the gravitational-waveevent GW150914 <cit.> and in dozensof Astronomer's Telegrams (ATel) and GCN circulars,AGILE is a very suitable instrument to perform searchesfor short transient sources and counterparts tomulti-messenger transient events like the neutrino eventobserved on July 31st, 2016.The main characteristics that makein spinning mode an important instrument for follow-up observations of multi-messenger counterparts are: * a very large field of view (FoV) of 2.5 sr for the -GRID; * best sensitivity to fluxes above 30 MeV of the orderof (2 ÷ 3) × 10^-4 ph cm^-2 s^-1 for typical single-passintegrations of 100 s;* a coverage of 80% of the whole sky every 7 minutes; * a gamma-ray exposure of ∼2 minutes of any field in theaccessible sky every 7 minutes;* between 150-200 passes every day for any region in the accessible sky. * sub-milliseconds trigger for very fast events. Despite the small size (approximately a cube of side ∼ 60 cm),the -GRID achieves an effective area of the order of 500 cm^2 between200 MeV and 10 GeV for on-axis gamma-rays, and an angular resolution(FWHM) of the order of 4^∘ at 100 MeV, decreasingbelow 1^∘ above 1 GeV <cit.>.A very fast ground segment alert system allows the AGILE Teamto perform the full -GRID data reduction and the preliminaryQuick Look (QL) scientific analysis only 25/30 minutesafter the telemetry download from the spacecraft <cit.>. The AGILE QL on-ground system implements two differentkinds of automatic analysis: * A “burst search” system, involving both GRID and MCAL instruments,is used to look for transients and GRB-like phenomenaon timescales ranging from a few seconds to tens of seconds[Aspecial sub-millisecond search for transient events detected by MCAL is operational on board <cit.>.]. The burst search systemruns on predefined time windows of 100 seconds, and it may be also triggeredby external GCN notices <cit.>.* A “standard” -GRID QL analysis, based on a Maximum Likelihood(ML) algorithm <cit.>, is used to detectgamma-ray transients above 100 MeV on timescales of 1-2 days <cit.>.This automatic procedure routinely runs over predefined 48-hours time-bins.Given theeffective area and sensitivity, thesecollecting time intervals are the most appropriate to accumulateenough statistics and to maximize the signal-to-noise ratio in both cases. §INVESTIGATIONS OF ICECUBE-160731 The ICECUBE-160731 best-fit reconstructed neutrino arrival direction inequatorial coordinates is (from Rev. #1 of the GCN notice):R.A.,Dec (J2000)=(214.5440, -0.3347) +/- 0.75 [deg](90% statistical plus systematic c.r.), corresponding toGalactic coordinates: l,b=(343.68, 55.52) [deg]. In the next sections, details of the automatic and refineddataanalysis of the ICECUBE-160731 event are reported. §.§ Prompt event The search for a GRB-like prompt event on short time-scales ranging froma few to tens of seconds connected to the ICECUBE neutrino emission was performed with theburst search system. The system was triggered by the firstICECUBE GCN/AMON notice reported a few tens of seconds after T_0.The automatic procedure searches for prompt emissionon predefined 100 s time-interval bins ranging fromT_0-1000 to T_0+1000 s. On these short timescales, themethod of the ML is not applicable, and an aperture photometryis applied. The significance of the signal with respect to thebackground is calculated using the Li&Ma formula <cit.>.Near T_0, the reconstructed neutrino-source position was ingood visibility for the -GRID FoV, neither occulted bythe Earth nor by the exclusion regions around the Sun and anti-Sunpositions (see Fig. <ref>). No significantdetection was found in the GRID data from the event positionin any of the 100 s time-bins scanned. The 3σ Upper Limit (UL)for the emission in the range 30 MeV–50 GeV estimated in the100 s time-bin with the highest exposure on the event position is:5.7 × 10^-4 ph cm^-2 s^-1.Moreover, using the data of the -MCAL and the AC scientificratemeters, we have searched for burst-like events in the energy range of0.4 – 100 MeV and 70 keV – tens of MeV, respectively. No significantevent has been detected in neither of the two detectors.§.§ Search for precursor and delayed emissionSince the astrophysics and the time scales of the phenomenarelated to the emission of these extremely high-energy neutrinos arestill uncertain, besides the investigations near T_0we also explored the AGILE-GRID data taken few days before andafter T_0, searching for a possible precursor ordelayed emission on longer (daily) time-scales possibly connectedto the neutrino event. Interestingly, a excess above 100 MeV with apre-trial ML significance of 4.1σ compatiblewith the ICECUBE error circle appeared in the results ofthe -GRID automatic QL procedure between one andtwo days before T_0. This detection was reported inthe ATel #9295 <cit.>.The automatic AGILE QL procedureruns on predefined 2-days integration time since Nov. 2009,the starting of the spinning observation mode.The AGILE source ML detection methodderives, for each candidate source, the best parameter estimatesof source significance, flux, and source location.The ML statistical technique, used since the analysisof EGRET data <cit.> and adaptedto thedata analysis <cit.>, comparesmeasured counts in each pixel with the predicted counts derivedfrom the diffuse model to find statistically significantexcesses consistent with the instrument point spread function.AnQL “detection” is in generaldefined by the condition √(TS)≥4, where TS is the TestStatistic of the ML method defined as-2 log(ℒ_0/ℒ_1), where ℒ_0/ℒ_1is the ratio between the maximum likelihood of the null hypothesis overthe point-like source hypothesis, given the diffusebackgroundmodel <cit.>. This threshold has been calibratedover various timescales and different background conditions(e.g., on or outside the Galactic plane) <cit.>. To evaluate the post-trial significance of the automaticQL detection mentioned above, we used the probability distribution of the ML TestStatistic (TS) computed in <cit.>. The probability of havingat least one detection due to a background fluctuationfor any position within the predefined Region of Interest(ROI) of 10^∘ radius used in the ML fitting procedure with a significance√(TS)≥ h, in N independent trials,is given by P_1(N)=1-(1-p)^N, where p isthe p-value (that is, the probability of finding a false positive detectionin a single observation) corresponding to h. The p-value for adetection with √(TS)≥ 4.1 outside the Galactic plane[ As expected by the Wilks' theorem <cit.>, the TS valuesfollow in this case the 1/2χ^2 distribution with onedegree of freedom.] is 3.8× 10^-5.By considering all the generated maps having enough exposure inspatial coincidence with the neutrino error circle (amounting to 226since the beginning of the spinning observation mode),the probability of having one detection by chance in N=226 trialsis P_1(226)=8.5 × 10^-3. The chance probability of the detection becomes at least two orders of magnitudes lowerif we consider the probability P_2 of spatialcoincidence of the -GRID excess with the ICECUBE error regionwithin the 10^∘ radius ROI. The combined post-trial probability becomes thenP_1 × P_2 ∼ 8.5 × 10^-5, which corresponds toa 3.9σ post-trial significance.A refined analysis has been performed both to confirm theautomatic QL result (applying more stringent cuts to further reduce the backgroundcontamination from albedo events) and to find a bettertemporal characterization of the gamma-ray transient positionallyconsistent with the ICECUBE-160731 position.In the refined GRID data analysis, we created alightcurve symmetric with respect to T0, usinga time-bin of 24 hours, which is the minimum integrationtime needed by the GRID to detect amedium/high flaring gamma-ray source above 100 MeV with enoughstatistics[Only in someexceptional bright flares the integration time-bin may be reducedbelow 24 hours (see, e.g., <cit.>).].A search for emission above 100 MeV using theMLaround the ICECUBE position has thus been performed overthe time interval (T_0-4; T_0+4) days.Exposure, counts and diffuse emission maps of each time-bin were generatedusing the officialscientific analysis software (release: BUILD 21;response matrices: I0023) [http://agile.asdc.asi.it/public/AGILE_SW_5.0_SourceCode/] <cit.>, applying a cut of 90^∘ on the albedo events rejection parameterand taking an AGILE-GRID FoV radius of 50^∘. In comparison,the predefined QL maps are generated with a looser albedo cut of 80^∘and a larger acceptance FoV radius of 60^∘. GRID data acquisition during the passage over the South Atlantic Anomaly (SAA)is suspended. Each time-bin of the lightcurve has beenanalyzed by means of the ML algorithm assuming a gamma-ray sourceat the ICECUBE position. Figure <ref> shows the resulting light-curve,where for each bin, the ML gamma-ray flux estimate above 100 MeV or the 95% C.L. UL atthe input ICECUBE-160731 position is shown.A gamma-ray excess above 100 MeV with a ML significanceof 4.1σ is detected in the bin centered one day and a halfbefore the T_0 (from MJD=57598.07991 to MJD=57599.07991), confirming theautomatic QL detection <cit.>.The candidate gamma-ray precursor has an estimated flux of F(E>100 MeV)=(3.0 ± 1.2) × 10^-6ph cm^-2s^-1with centroid Galactic coordinatesl,b=(344.01, 56.03)± 1.0 [deg] (95% stat. c.l.) ± 0.1 [deg] (syst.),compatible with the ICECUBE-160731 position.Thea-posteriori refined analysis on a 24-hoursbasis shows that the excess is particularly short in time, mostlyconcentrated between July 29th and 30th, 2016. By examining the arrival times of the gamma event file, we found aa clusterization of five counts in less than 7 hours around (T_0-1) daywithin 1.5 degrees from the ICECUBE centroid.In particular, on the 24-hours integrationfrom MJD 57598.25 to 57599.25 ((T_0-1.8; T_0-0.8) days), which fullycontains the event clusterization, we obtained a ML significanceof the peak emission of 4.9σ at the Galactic centroid coordinates:l,b=(344.26, 55.86) ± 0.8 [deg] (95% stat. c.l.) ± 0.1 [deg] (syst.),with a flux F(E>100 MeV)=(3.5 ± 1.3) × 10^-6ph cm^-2s^-1. The newtransient, named AGL J1418+0008,positionally consistent with the ICECUBE-160731 error circle mightthen be a possible precursor of the neutrino event.Figure <ref> shows the -GRID intensity map centeredat the ICECUBE-160731 position, in the 24-hours time intervalcorrespondent to the peak significance. The white region definesthe 95% C.L. ellipse contour of the -GRID detection AGL J1418+0008,which is well compatible with the ICECUBE-16073190% c.r. error circle (black circle). Figure <ref>shows also the position of the known sources from the 5th edition ofthe BZCAT and FERMI-LAT 3FGL catalogs <cit.>.None of these known sources lies within theor ICECUBE error circles. A further search in the Second and Third FERMI-LAT high-energysources Catalogs (2FHL and 3FHL, <cit.>)does not show again any possible association with known counterparts.The closest 3FHL source is 3FHL J1418.4-0233 (associated to theBL Lac blazar 5BZB J1418-0233 <cit.>), which is morethan 2^∘ away from the neutrino position. §.§ Search for gamma-ray emission inarchival dataThe whole public -GRID archival data from Dic. 2007 up to Nov. 2016 have been investigated in order to search for other possibleprevious and later transient episodes around theICECUBE-160731 position. This long time-scale search has been performed by usingthe -LV3 online tool <cit.> accessible fromthe ASDC Multi-Mission Archive (MMIA) web pages[URL:http://www.asdc.asi.it/mmia/index.php?mission=agilelv3mmia]. This tool allows fast online interactive analysis basedon the Level-3 (LV3) -GRID archive of pre-computed counts,exposure and diffuse background emission maps.The search for transient emission above 100 MeV on 2-dayintegration times did not show any other significantdetection but the one compatible with theQL resultbetween one and two days before T_0 (over a total of 271 analyzed maps).We finally performed a ML analysis centered on the ICECUBEposition using the LV3 pre-computed maps for the whole AGILEobserving time (9 years). We obtained an UL of3.5 × 10^-8ph cm^-2 s^-1(E>100 MeV, for a 95% C.L.). § MULTI-WAVELENGTH FOLLOW-UP OF ICECUBE-160731 The ICECUBE-160731 detection triggered a thorough campaign ofMWL follow-up observations. These observations covered a largepart of the entire e.m. spectrum, from the optical band(Global MASTER net, iPTF P48, LCOGT) to the VHE gamma-rays(HAWC, MAGIC, HESS, ...).Very few observatories and space missions were observingthe neutrino event position to T_0. Apart from and facilities like HAWC, ANTARES and FERMI-LAT, which haveaccess to a large part of the sky almost the whole day, all theothers had to re-point to the ICECUBE positiona few minutes or even hours after T_0. In this section, wewill summarize the most interesting results of the MWLfollow-up, reminding the reader to theAppendix <ref> for a summary ofall other observations published in ATel andGCNs in the hours or the days after the event.In the X-ray band, SWIFT observed the ICECUBE-160731 error circleregion starting approximately from (T_0 + 1) hrs till(T_0 + 12) hrs <cit.>.The XRT instrument on-board of the SWIFT satellite detected sixsources in the 0.3-10 keV band. Figure <ref> shows a zoomof the -GRID intensity map over the integration ofthepeak detection, with the location of thesix SWIFT-XRT sources, numbered 1 to 6 (blue crosses inFig. <ref>). After the revision of thebest-fit neutrino arrival direction and its error radius, threeof the detected XRT sources eventually lay outside the revised ICECUBE-160731 error circle. Only sources #5 and#6 are still compatible with the neutrino position(and within the AGILE ellipse contour), while source#2 remains just on the border.In the optical region, the Global MASTER Optical Networkperformed a search for optical transients in the time interval(T_0 + 17; T_0 + 21) hrs <cit.>. They only detected a point-like event, classified asMASTER OT J142038.73-002500.1, that might have been inducedby particle crossing the CCD, and the bright NGC 5584 galaxy(which, anyhow, is already outside the revised error circle)(yellow boxes in Fig. <ref>).Rapid follow-up observations in the Optical/IR band, startedonly 3.5 hours after T_0, were performedby the Palomar 48-inch telescope (iPTF P48) <cit.>.They detected two optical transient candidates at 1.1 and 2.0^∘from the initial neutrino candidate position (magentadiamonds in Fig. <ref>).In the gamma-ray band, FERMI-GBM could not observe the regionat T_0 since the position was occulted by the Earth <cit.>while FERMI-LAT reported only flux ULs (95% C.L.) above100 MeV of 10^-7 ph cm^-2 s^-1in 2.25 days of exposure starting from a 2016-07-31 00:00 UTC, andof 0.6 × 10^-7 ph cm^-2 s^-1 in8.25 days of exposure starting from 2016-07-25 at00:00 UTC <cit.>.As shown in Appendix <ref>, the non-detection ofany precursor by Fermi-LAT might be due to a low exposure ofthe ICECUBE region during thetransient.At the time of the neutrino event T_0, the INTEGRAL satellite,which also has the capability to cover almost the wholesky <cit.>, was not observing becauseit was close to perigee inside the Earth radiation belts.The ICECUBE region was also observed in the VHE band by severalexperiments (see Appendix <ref> for details).Apart from HAWC, that has a 24-hours duty cycle, all the otherscould re-point to the ICECUBE position hours later than T_0,reporting only flux ULs above different energy thresholds. On asearch for steady source using archival data, the HAWC Collaboration reported about a location with a pre-trial significance of3.57σ at R.A.,Dec (J2000)=(216.43, 0.15) [deg] <cit.>(shown as cyan cross in Fig. <ref>), although morethan 2^∘ away from the neutrino error circle. Consideringthe number of trials quoted in the HAWC GCN, this is not a significant detection. § POSSIBLE NEUTRINO-EMITTER E.M. SOURCES IN THE ICECUBE-160731ANDAGL J1418+0008 ERROR REGIONS In what follows, we will furtherinvestigate whether some of the steady/transient sources found duringthe MWL follow-up are good candidates as the ICECUBE-160731 emitter.In particular, we decided to review only the e.m. sourcesstill within the revised ICECUBE error region plus theclosest optical transient detected by iPTF48 (named iPTF16elf,<cit.>) (see Fig. <ref>).Table <ref> shows the main characteristicsof the five e.m. sources satisfying the chosen selection criteria.The table also shows the most likely known association as reportedfrom each of the ATel announcing the detection obtained during the follow-up.To find some of the key features of oneof the most promising neutrino-emitter candidates, the High-energypeaked BL Lac (HBL) types of AGNs <cit.>,we reviewed the initial counterpart association and, moreover, weinvestigated the broad-band spectral properties of each object.The first two SWIFT-XRT sources detected during thefollow-up, #2 and #5 <cit.>, are consistentwith the position of two known quasars: source #2 is 9.12" from2QZ J141936.0-010841[Also known as [VV2010] J141936.0-010840(VV2010 Cat., <cit.>) and SDSS J141935.99-010840.2(SDSS Cat. – Release #7, <cit.>).](2QZ Cat, <cit.>) while source #5 is 4.5" from2QZ J141949.8-000644[Also known as [VV2010]J141949.9-000644 <cit.>, 2MASS J14194982-0006432(2MASS Cat., <cit.>), and SDSS J141949.83-000643.7 <cit.>.].By looking to their spectral energy distributions (SEDs),built using both the XRT detections and MWL archival data,neither of the two quasars shows hints of high-peaked synchrotron emission, which isone of the key feature used to identify a HBL type of AGN. Moreover, theycompletely lack radio emission, which leads us to conclude that they might beradio-quiet quasars and we can discard them aspossible emitter of the ICECUBE-160731 neutrino.XRT source #6 is ∼ 2.5" from 2MASS J14182661+0014283, a knownG-type star, and it thus can as well be excluded aspossible source candidate of the neutrino emission.Concerning the two optical transient candidates OTJ142038.73-002500.1 and iPTF16elf, they are both positionallyconsistent with two galaxies (respectively, SDSS J142041.62-002413.1(z=0.054) and Z 18-88 (z=0.038)), which form part of a cluster. Forboth, there are no evident indications of blazar features in theirrespective SEDs.Besides the review of five e.m. candidatesfound during the ICECUBE-160731 MWL follow-up, we searched for otherpossible counterparts within the ICECUBE 90% error circleby exploring the ASDC resident and external catalogsusing the online ASDC SkyExplorer tool[https://tools.asdc.asi.it].In particular, we focused our search to known radio and X-ray sourceswhich might show the typical characteristics of HBL/HSP AGNblazars <cit.>: low radio fluxes and low IR-radiospectrum slopes; high X-ray-to-radio flux ratios; ν synchrotron peaks above 10^15 Hz. A query of 50 arcmin around the ICECUBE-160731 centroid Galactic coordinatesl,b=(343.68, 55.52 deg) selecting, among others, radio and X-ray sources fromthe FIRST <cit.> and the RASSCatalogs <cit.>,returns several objects (see Figure <ref>).Following the search criteria defined above, oneof the most interesting object resulting from the query is a RASS sourceappearing at ∼19 arcmin from the center, 1RXS J141658.0-001449,with position and related uncertainty R.A.,Dec (J2000)=(14^h16^m58^s.0,-00^∘14'49") ± 25", (indicated by the dashed circle inFig. <ref>). This catalogedX-ray source is the only one in the field showing a FIRST weakradio source (F=1.99 mJy; R.A.,Dec (J2000)=(14^h16^m58^s.27,-00^∘14'44.87")) within its error circle.A further search in the ASDC optical catalogs founda faint galaxy, SDSS J141658.90-001442.5 (mv ∼23), at 9.6 arcsecfrom the FIRST source (14.8 arcsec from the RASS source).Assuming the radio/optical/X-ray emission comes from the same galaxy, we haveproduced the SED shown in Figure <ref>.The high value of the ratio between the 1RXS J141658.0-001449flux density in the 0.1-2.4 keV band and the FIRST radio source ν F_ν valueat 1.4 GHz (respectively, black and red points in Fig. <ref>)might hint to a non-thermal synchrotron emission peaking above 10^15 Hz, typical ofa HBL AGN blazar. Considering these types of e.m. sources as the mostlikely neutrino-emitters, the X-ray source 1RXS J141658.0-001449(and the plausible host galaxy SDSS J141658.90-001442.5) appears as oneof the candidate as origin of the ICECUBE-160731 event. This source was not in the field covered by the July 31st, 2016, SWIFTseries of ToO observations <cit.>. Interestingly, thesource lies also within the 95% error ellipse contour of the AGILE detectionoccurred before the neutrino event time T_0 (see Fig. <ref>). §.§ SWIFT ToO data on the 1RXS J141658.0-001449 fieldTo better estimate the position and the spectrum of the RASS1RXS J141658.0-001449 source (which was not in the field covered bythe first SWIFT series of ToO observations <cit.>)and determine a stronger spatial correlation with the radio andoptical sources described above, a new SWIFT ToO has been submittedand executed in December 2016, almost six months later thanthe ICECUBE-160731 neutrino detection.The data were collected in five distinct ∼1 ks exposures centered on the1RXS J141658.0 source position between 2016-12-11 00:32:59 UT and 2016-12-15 07:07:53 UT and are entirely in Photon Counting (PC) mode[ Correspondent SWIFT OBSERVATION IDs: from 00034815001 to 00034815005.].Figure <ref> shows the (smoothed) cumulative XRT count map in the 0.3-10 keVenergy range, with an overall exposure of 4.9 ks. The positionof the 1RXS J141658.0 source (with its quoted error circle) issuperimposed to the map (white circle near the map center). No apparentX-ray excess is visible at the 1RXS J141658.0 position.Using the XIMAGE sosta algorithm, we derive a 3σ UL of3.1 × 10^-3cts s^-1 in the XRT energyband on the 1RXS J141658.0 position. Assuming a source with apower-law photon index of 1.7, we evaluated an upper limit of4.6 × 10^-3cts s^-1 in the ROSAT PSPC band.This value is well below the count rate of (2.19 ± 1.04) × 10^-2quoted for 1RXS J141658.0-001449 in the RASS-FSC Catalog.This might indicate an intrinsic variability of the source,which was significant only during the RASS observation.It should be noted that this source does not appear anymore in the secondROSAT all-sky survey (2RXS) Catalog <cit.>, anextended and revised version of the 1RXS Catalog thatcontains a significant reduced number of low reliability sources.Applying the XIMAGE detect algorithm on the overall 5 ks XRT count map,weighted by the correspondent sum of each single XRT exposure, five(uncataloged) X-ray field sources are detected within the FoV (see Fig. <ref>).Table <ref> reports count rates, source coordinates, SNR ratio andprobability to be a background fluctuation for all the five detections.Studies of the characteristics of the five field sources is ongoing. § DISCUSSION AND CONCLUSIONS We reported the results ofobservations of the ICECUBE-160731neutrino event error region. These observations covered the event skylocation at the event time T_0 and also allowed us to search fore.m. counterparts before and after the event.The analysis of the -GRID data in the time windowT_0 ± 1 ks with theburst search system has not shown any significant gamma-ray excess above30 MeV from the neutrino position. Moreover, no burst-likeevents using the -MCAL and the AC ratemeters around T_0have been detected.Instead, an automatic detection above 100 MeV,compatible with the ICECUBE position, appeared fromtheQL procedure on a predefined48-hours interval centered around one day and a half before T_0.Considering all the number of trials performed by theQLsystem and the chance probability to have a excess in coincidencewith the neutrino position, the automatic detection reaches acombined post-trial significance of about 4σ.A refined data analysis confirms the QL detection alreadyreported in the ATel #9295 <cit.>.This new -GRID transient, named AGL J1418+0008, israther concentrated in time, showing a clusterization ofevents around (T_0-1) days, and reaching a peak ML significance of4.9σ on the 24-hours integration covering theinterval (T_0-1.8; T_0-0.8) days. AGL J1418+0008 thus stands aspossible ICECUBE-160731 gamma-ray precursor.No other space missions or observatories have reported anyclear indication of a transient e.m. emission consistentwith the neutrino position and time T_0. This non-detectionof an e.m. counterpart in any of the wavelengths covered by the ICECUBE-160731follow-up does not exclude the possibility of a bright rapid gamma-rayflare precursor just before the neutrino detection.Most of the instruments involved in the e.m. follow-up, in fact,could re-point their instruments only hours or even a day after T_0,and might have missed the flaring episode seen byat E>100 MeV.As said in the MLW follow-up summary, FERMI-LAT did not report any evidenceof a precursor above 100 MeV. As we show in Appendix <ref>, thismight be due to a very high FERMI-LAT observing angle and a very low exposureof the ICECUBE region with respect to theobservations.Given the high Galactic latitude of the ICECUBE neutrino arrivaldirection (b=55.52 [deg]), we do expect an extra-galactic origin of thisevent. Indeed, several authors (i.e., <cit.>)assume that blazar AGNs are the main VHE neutrino-emitter candidatesand the only sources able to explain the common origin ofthe diffuse neutrino background seen by ICECUBE, the extra-galacticcosmic-ray component and the isotropic diffuse gamma-ray backgroundobserved by FERMI <cit.>. <cit.>found for the first time a significant probability that one ofthe ICECUBE PeV event was spatially and temporally coincidentwith a major gamma-ray outburst of the Flat Spectrum Radio Quasar (FSRQ)PKS B1424-418. Considering that there is a substantial fraction of the blazarpopulation not resolved yet, Kadler et al. estimatethat around 30% of the detected multi-TeV/PeV neutrinos will not beassociated with any known blazar, like appears to be the case of the ICECUBE-160731 event.Recently, <cit.> found that a significantcorrelation between known HBL blazars, ICECUBE neutrinos andUHECRs detected by Auger and the Telescope Array (TA) exists.We thus searched for a HBL candidate counterpart inside the common ICECUBE and AGL J1418+0008 error circles and found a possible HBL source,the Sloan faint galaxy SDSS J141658.90-001442.5, which appearswithin the positional error of the RASS source 1RXS J141658.0-001449 and closeto a FIRST 2 mJy radio source. The ICECUBE-160731 SWIFT follow-up,although rapid, did not cover the field around this possiblee.m. candidate. A new SWIFT ToO then has been submitted in orderto characterize better this RASS-FSC source. Unfortunately, theToO was performed about 6 months after the neutrino event, andthe analysis of the XRT data from the almost 5 ks exposuredid not reveal any significant X-ray emission at the 1RXS J141658.0 position,providing a 3σ UL of 3.1 × 10^-3 cts s^-1 in the0.3–10 keV band. We then cannot confirm at the moment our hypothesisabout the HBL nature of this source that, anyhow, mighthave been detected during the ROSAT survey because in an intrinsicX-ray high-state.Other possible PeV neutrino-emitters have been proposed, like Starburst galaxies,giant radio galaxies with misaligned jets, gamma-ray bursts (GRBs)(see <cit.> for a review).<cit.>, for example,correlate another recent ICECUBE HESE neutrino event (ICECUBE-160814)with an optical transient occurred almost ten daysafter the event time. They postulate the possibility that the neutrino emitter might be an ejecting white dwarf in a binary system. This isan intriguing possibility, although the power budget available in these systems(optical companion plus compact object) could not be sufficient toaccelerate protons up to multi-PeV energies in order to produce sub-PeV/PeVneutrinos from pp collisions.Eventually, none of the other e.m. sources proposed up to now asneutrino-emitter candidates are able to explain the bulk ofmulti-wavelength/multi-messenger (neutrinos plus cosmic rays)observational data like the HBL/HSP class of blazars <cit.>. Indeed, the probability to find a blazar of this class ina 1^∘ radius sky-area like the ICECUBE-160731 errorcircle is quite low. Assuming, in fact, an HSPdensity of the order of 5 × 10^-2 deg^-2 from the 2WHSPcatalog <cit.>, there are approximately5 HSP/HBL AGNs every 100 squared degrees of sky. Thus, theprobability to find one of these objects within the roughly 3squared degrees covered by the ICECUBE error circle is of about 0.15%.In the specific case of the ICECUBE-160731 neutrino, for example,we have not found yet any other potential HBL candidatebut the one not confirmed with the dedicated SWIFT ToO observations.Moreover, thetransient, not confirmed by FERMI (although caused by a poor FERMI-LAT visibility just before T_0)might indicate a possible soft source, in disagreementwith the hard-spectrum features expected for the HBLs. Nevertheless, the HBL scenario can still hold if we assumea lepto-hadronic process occurring within the blazarjet <cit.>, where the bulk ofbroad-band e.m. emission is due to synchrotron and Inverse Comptonleptonic processes, while protons would be mainlyresponsible for the neutrino flux (from the decay ofcharged pions produced by photo-meson production on the softphotons field within the jet). In this case, <cit.>foresee that a soft component, peaking at MeV/GeV energies, wouldbe expected from re-processing of VHE photons from the decay of π^0'soriginated in the pγ collisions within the jet. The observation of the transient AGL J1418+0008, compatible withthe neutrino position and very close in time to the event T_0,if associated with the ICECUBE event, could be thenexplained by such hadronic mechanism.To conclude, there is also the possibility that the source of the ICECUBE-160731neutrino event might be either a different AGN type or a differentclass of source, even though we cannot exclude at the moment amoderately bright HBL not yet identified.We would like to thank Paolo Giommi and Matteo Perri, for many fruitful discussionsand the valuable help with the analysis of the SWIFT ToO, and Paolo Liparifor the very useful comments about the paper. We also thank the Swift Team for makingthe SWIFT ToO observations possible, in particular M. H. Siegelas the Swift Observatory Duty Scientist. Lastly, we thank the anonymousreferees for their valuable comments that helped to improve our paper. AGILE is an ASI space mission with programmatic support from INAF and INFN. Weacknowledge partial support through the ASI grant no. I/028/12/0. Part of thiswork is based on archival data, software or online services provided bythe ASI SCIENCE DATA CENTER (ASDC). It is also based on data and/or softwareprovided by the High Energy Astrophysics Science Archive ResearchCenter (HEASARC), which is a service of the Astrophysics Science Divisionat NASA/GSFC and the High Energy Astrophysics Division of theSmithsonian Astrophysical Observatory. This research has also made useof the SIMBAD database and the VizieR catalog access tool, operated atCDS, Strasbourg, France. scientific analysis software (BUILD 21; <cit.>), XIMAGE§ COMPARISON BETWEEN AGILE AND FERMI-LAT DATA DURINGTHE ICECUBE-160731 EVENTIn this Appendix, we verify that the FERMI-LAT non-detection of the possible precursor of the neutrino 160731 event might be dueto a poor exposure and non optimal viewing angle of the ICECUBEerror circle.We have compared the FERMI-LAT attitude data with theonesduring the time interval (T_0-2;T_0) days (MJD 57598.07991÷57600.07991)and found that FERMI-LAT observed the ICECUBE error circle at an off-axis anglelower than 50^∘ only for a 3.9% of its total exposure time, while for the exposure time below the same off-axis angle amounted to 27.4% of the total (seeFigure <ref>)[At high values ofthe off-axis angle (>50^∘), the Fermi/LAT sensitivity is up to 50% lowerthan the nominal on-axis value.].Further investigations of the FERMI spacecraft data show also severalperiods of not data-taking during the same time-interval (amounting to ∼15% of the total observation time),particularly near (T_0-1) days (as it is possible to seefrom Fig. <ref>), where AGILE found a clusterization of gamma-like events compatible with the ICECUBE error circle.To prove that during this period theand FERMI-LAT exposureson the ICECUBE region were at least comparable, we have evaluatedthe exposures for both instruments on time intervals of 24, 12 and 6 hourscentered at (T_0-1) days (MJD=57599.07991), where thedetectionreached its peak significance.We downloaded Pass8 data[From the FERMI data ASDC mirror(https://tools.asdc.asi.it).] aroundthe position of ICECUBE-160731 and, using version v10r0p5 of the Fermi Science Toolsprovided by the Fermi satellite team[http://fermi.gsfc.nasa.gov] and the instrument response function P8R2_SOURCE_V6,we calculated the mean exposure values on the neutrino error circleon those different integration times. We selected Pass8 FRONT and BACK source classevents and, in order to be comparable with thespectral sensitivity(optimized to the observation of soft sources with typicalspectral indexes of 2÷2.1), we limited the event energiesbetween 0.1 and 10 GeV.Table <ref> shows the values of the FERMI-LAT andexposureson the different time intervals chosen and for a maximum off-axisangle between source and FoV center of 50^∘.The LAT exposure on the 24-hours interval MJD 57598.25 ÷ 57599.25becomes comparable with theexposure of 3.7 × 10^6cm^2 sobtained under the same maximum viewing angle and thesame integration time. On the shorter intervals of 12 and 6 hoursaround (T_0-1) days, theexposure becomes even largerthan the FERMI one. Assuming, thus, a very short flare, asthedetection indicates, it might imply the possibility thatFERMI, given the very low exposure and the large viewing angleof the ICECUBE-160731 position during this period, lost most ofthe transient episode. Differences in the event classificationalgorithms between the two instruments can also bring to adetection/non-detection in such cases of short transientsat the level of 4σ above the background. § SUMMARY OF THE ICECUBE-160731 MWL FOLLOW-UP|l|c|c|c|p2in| Summary of the MWL follow-up of the ICECUBE-160731 event 700pt Mission/Observatory ATelGCN Circular Observation/integration timeComments(Energy band) # # [UTC] HAWC (TeV gamma-rays) - 19743 2016-07-30 21:28:57 – 2016-07-31 02:59:15 No detection around neutrino event time T_0 (most significant location (1.12σ) at R.A., Dec (J2000) = 214.67, 1.04 deg). From archival data, a pre-trial 3.57σ detection from R.A., Dec (J2000) = 216.43, 0.15 deg is reported. SWIFT (X-ray, Optical/UV) 9294 19747 2016-07-31 03:00:46 – 2016-07-31 14:51:52 Six known or cataloged X-ray sources detected (0.3-10 keV) but no transient events. No transient sources detected in the simultaneous UVOT data. AGILE (Gamma-rays) 9295 - 2016-07-29 02:00 – 2016-07-31 02:00 2016-07-28 08:00 – 2016-07-30 08:00 >4σ pre-trials detection on the interval 2016-07-28/2016-07-30 (08:00) UT. Global MASTER net (Optical) 9298 19748 From 2016-07-31 19:23:17 on No optical transients detected inside 2 square degrees around center of ICECUBE-160731 Rev. #0 error circle. Detected one likely particle CCD event (OT J142038.73-002500.1) and the NGC 5584 galaxy. FACT (TeV gamma-rays) - 19752 2016-07-31 21:42 – 2016-07-31 22:25 No detection. HESS (TeV gamma-rays) 9301 - 2016-07-31/08-01 (1 hr) 2016-08-01/02 (1 hr) No detection. FERMI-LAT (Gamma-rays) 9303 - 2.25 days from 2016-07-31 00:00 8.25 days from 2016-07-25 00:00 No detection above 100 MeV. FERMI-GBM (X-ray/Gamma-rays) - 19758 Neutrino event trigger time (T_0). Position occulted by Earth at T_0. Flux U.L. at 3σ level (12-100 keV) on the interval July 30th-Aug. 1st. iPTF P48 (Optical/IR) - 19760 From 2016-07-31 05:22 on. No optical transients detected close to the ICECUBE updated error circle. Two optical transients candidate (iPTF16elf and iPTF16elg) detected at 1.1 and 2.0 deg from the neutrino candidate position, both consistent with known galaxies. MAXI/GSC (X-ray) 9313 -At 2016-07-31 02:32. From 2016-07-20 to 2016-08-03. No detection on the 2-20 keV band within the ICECUBE error circle, neither near T_0 nor in the period July 20th – Aug. 3rd. 3σ U.L. are provided. MAGIC (TeV gamma-rays) 9315 - 2016-07-31 21:25 – 2016-07-31 22:47 No detection above 600 GeV.ANTARES (TeV/PeV neutrinos) 9324 19772 T_0 ± 1 hr T_0 ± 1 day No up-going muon neutrino candidate events recorded within three degrees of the ICECUBE event coordinates. 90% U.L. on the fluence from a point-like source are reported. Konus-Wind (X-ray/Gamma-rays) - 19777T_0 ± 1000s From 5 days before to 1 day after T_0.No triggered events detected. 90% C.L. upper limits are reported on the 20-1200 keV fluence for typical short and long GRB spectra. LCOGT (Optical) 9327 - From 2016-07-31 23:04:41 till 2016-08-03 18:29:11. No detection of new optical sources down to 3σ limiting magnitudes >19.aasjournal | http://arxiv.org/abs/1707.08599v1 | {
"authors": [
"F. Lucarelli",
"C. Pittori",
"F. Verrecchia",
"I. Donnarumma",
"M. Tavani",
"A. Bulgarelli",
"A. Giuliani",
"L. A. Antonelli",
"P. Caraveo",
"P. W. Cattaneo",
"S. Colafrancesco",
"F. Longo",
"S. Mereghetti",
"A. Morselli",
"L. Pacciani",
"G. Piano",
"A. Pellizzoni",
"M. Pilia",
"A. Rappoldi",
"A. Trois",
"S. Vercellone"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726182417",
"title": "AGILE detection of a candidate gamma-ray precursor to the ICECUBE-160731 neutrino event"
} |
A new single-particle basis for nuclear many-body calculations. G. Puddu E-mail: [email protected] Dipartimento di Fisica dell'Universita' di Milano, Via Celoria 16, I-20133 Milano, Italy Received: December 30, 2023/ Accepted: date =============================================================================================================================================================== Recent years witnessed a growing interest in non-standard epistemic logics of knowing whether, knowing how, knowing what, knowing why and so on. The new epistemic modalities introduced in those logics all share, in their semantics, the general schema of ∃ x ϕ, e.g., knowing how to achieve ϕ roughly means that there exists a way such that you know that it is a way to ensure that ϕ. Moreover, the resulting logics aredecidable. Inspired by those particular logics, in this work, we propose a very general and powerful framework based on quantifier-free predicate language extended by a new modality ^x, which packs exactly ∃ x together. We show that the resulting language, though much more expressive, shares many good properties of the basic propositional modal logic over arbitrary models, such as finite-tree-model property and van Benthem-like characterization w.r.t. first-order modal logic. We axiomatize the logic over S5 frames with intuitive axioms to capture the interaction between ^x and know-that operator in an epistemic setting.§ INTRODUCTIONStandard epistemic logic studies valid reasoning patterns about knowing that. However, in natural language, knowledge is also expressed by knowing whether, knowing how, knowing what, knowing why and so on. Recent years witnessed a growing interest in the non-standard epistemic logics of such expressions (cf. e.g., <cit.> and the survey <cit.>).[This is not meant to be an exhaustive list, e.g., see also <cit.> about logics of knowing how in the setting of ATL.] In this line of work, various new modalities of know-wh are introduced,[Know-wh means verb know followed by a wh-word.] all of which share the general de re schema ∃ x ϕ(x) in their semantics, e.g, “knowing how to achieve ϕ” roughly means that there exists a way such that you know that it is a way to ensure that ϕ <cit.>; “knowing why ϕ” means that there exists an explanation such that you know that it is an explanation to the fact ϕ <cit.>. Actually, in the early days of epistemic logic, Hintikka already used such formulations to handle knowing who <cit.> (cf. the survey <cit.> for a detailed discussion on Hintikka's early contributions). Such interpretations aregrounded also in philosophy and linguistics (cf. e.g., <cit.>). Such a semantic schema is based on the so-called mention-some interpretation to the wh-questions embedded in those knowledge expressions <cit.>. There is also a mention-all interpretation <cit.>, which makes sense in many other situations, e.g., “knowing who came to the party”means, under an exhaustive reading, that for each relevant person, you know whether (s)he came to the party or not, which can be summarized as ∀ x (ϕ(x)ϕ(x)). There are degenerated cases when the two interpretations coincide, e.g., “knowing [what] the value of c [is]” means, under the interpretation of mention-some, that there exists a value such that you know that it is the value of c, which is equivalent to the mention-all interpretation: for any value, you know whether it is the value of c, given there is one and only one real value of c.Given the experiences in dealing with those particular cases of know-wh, it is the time to lay out a general backgroundframework for the shared “logical core” of thoselogics.[Note that this does not imply that we can simply use a single framework to cover all of those particular cases, since the details of the semantics in each setting matter a lot in deciding the characteristic axioms and rules in each different setting. Moreover, for example, in the setting of knowing how logics, we need to quantify over second-order objects (plans or strategies). ] This paper is the initial step towards this purpose by extending thepredicate language with the mention-some operator ^x, which is essentially a package of ∃ x. With the variable in place, we can say much more than those existing logics of know-wh, e.g., “I know a theorem of which I do not know anyproof” ^y Prove(y, x), or, in a multi-agent setting, “i knows a country which j knows its capital”: ^x_i^y_j Capital(y, x). Actually, when x does not appear in ϕ, ^xϕ is equivalent to ϕ, thus our language is indeed an extension of the standard modal language. Moreover, we will show that in the epistemic setting, this mention-some operator can also express mention-all. By having the predicate symbols in the language, we can also talk about the content of knowledge, which may be useful to bridge epistemic logic and knowledge representation. We alsobelieve that the new modality can be interpreted meaningfully not only in the epistemic context.From a technical point of view, what we discovered is a well-behaved yet powerful fragment of first-order modal logic (over arbitrary models), as the following maintechnical contributions of this paper demonstrate:* We propose a novel notion of bisimulation which can characterize the expressive power of our language within first-order modal logic precisely, over arbitrary models. * Over arbitrary models, satisfiability of the equality-free fragment is not only decidable but also -complete, just as the complexity of basic propositional modal logic, demonstrated by a tableau-like method to show some strong finite-tree-model property.* We give a sound and complete proof system to our logic over epistemic (S5) models with the equivalence relation. However, we show that over S5 models, our logic is undecidable. Due to historical and technical reasons, first-order modal logic (FOML), in particular in the epistemic setting, has not beenthoroughly studied as its propositional brother (cf. e.g., <cit.>). Decidable fragments of first-order modal logic are usually obtained by restricting the occurrences of variables (particularly in the scope of ) (cf. e.g., <cit.>). We hope our framework and techniques can pave a new way to some interesting fragments of first-order modal logic, using new modalities to pack quantifiers and standard modalities together, which also reflects the “secret of success” of basic propositional modal logic as a nicely balanced logic between expressivity and complexity. In the rest of this paper, we introducethe language and semantics of our framework in Section <ref>, study its expressivity over arbitrary models in Section <ref>, prove the complexity of the equality-free fragment in Section <ref>, give the proof systems in the epistemic setting in Section <ref>, and prove their completeness and undecidability in Section <ref>. § SYNTAX AND SEMANTICSThroughout this paper we assume a fixed countably infinite set of variables , and a fixed set of predicate symbols . Furthermore, we assume that each predicate symbol is associated with a unique non-negative integer called its arity. We use x⃗ to denote a finite sequence of (distinct) variables in , in the order of a fix enumeration of . By abusing the notation, we also view x⃗ as a set of variables in x⃗ occasionally. In this paper, for the brevity of presentation, we focus on the following unimodal language, but the results and techniques can be generalized to the polymodal language.[We leave the discussion about the extension with constants or function symbols to the full version of the paper.] Given and ,ϕ::= x≈ y | Px⃗|ϕ| (ϕϕ)|ϕwhere x, y∈, P∈. We call theequality-free fragment(modal logic of mention-some). ϕ can be read as knowing some x such that ϕ(x) in the epistemic context. We use the usual abbreviations ⊤, , , →, and writefor , i.e., the dual of . We define the free and bound occurrences of variables as in first-order logic by viewing ^x as a quantifier. We call x is a free variable in ϕ (x∈ FV(ϕ)), if there is a free occurrence of x in ϕ. We write ϕ(x⃗) if all the free variables in ϕ are included in x⃗. Given anformula ϕ and x, y∈, we write ϕ[y x] for the formula obtained by replacing every free occurrence of x by y. To simplify the discussion, we do not include constant symbols and function symbols in the language and leave them to a future occasion.[Note that constant and functions can be coded usingand ≈ in full(cf. e.g., <cit.>). However, our language is a fragment of .]. As for the semantics, to be general enough, following <cit.> we use the first-order Kripke model with an increasing domain, and flesh out the intuitive idea of mention-some discussed in the introduction. An (increasing domain) modelforis a tuple W, D, δ, R, ρ where:W is a non-empty set.D is a non-empty set.R⊆ W× W is a binary relation over W.δ:W→ 2^D assigns to each w∈ W a non-empty local domain s.t. wRv implies δ(w)⊆δ(v) for any w,v∈ W. We also write D_w for δ(w).ρ:× W→⋃_n∈ω2^D^n such that ρ assigns each n-ary predicate on each world an n-ary relation on D.[Following <cit.>, we do not require that the interpretation of P at a world is based on the local domain. Actually, as we will see later in Corollary <ref>, this seemingly `counterintuitive' generalization does not affect the satisfiability or validity: each satisfiable formula ϕ is satisfiable in a model where ρ(P,w) is based on objects in D_w (cf. also <cit.>). ] A constant domain model is a model such that D_w=D for any w∈ W. A finite model is a model with both a finite W and a finite D. Given a model , we denote its components as W^, D^, δ^, R^, and ρ^. To interpret free variables, we also need a variable assignment σ: → D.The formulas are interpreted on models with variable assignments.[, w, σ x≈ y⇔σ(x)=σ(y); , w, σ P(x_1⋯ x_n)⇔ (σ(x_1), ⋯, σ(x_n))∈ρ(P,w);, w, σϕ⇔ , w, σ⊭ϕ;, w, σ (ϕψ)⇔, w, σϕ and , w, σψ;, w, σϕ⇔ there exists an a∈δ(w) such that;, v, σ[x↦ a]ϕ for all v s.t. wRv;]where σ[x↦ a] denotes another assignment just like σ except mapping x to a. As an intuitive definition, following <cit.>, we say ϕ is valid, if ϕ is true on any , w w.r.t. any σ such that σ(x)∈δ^(w) for all x∈. Correspondingly, ϕ is satisfiable if ϕ is not valid, i.e., ,w,σϕ for some , w and σ such that σ(x)∈δ^(w) for all x∈.It is not hard to see that if σ(x)=σ'(x) for all the free x in ϕ, , w, σϕ,w, σ'ϕ. In this light, wewrite , wϕ [a⃗] to denote ,w,σϕ(x⃗) for any σ such that σ assigns free variables x⃗ of ϕ the corresponding objects in a⃗ given |x⃗|=|a⃗|, where |·| denotes the length. For comparison, the standard semantics foris defined as:[, w, σϕ⇔ for all v such that wRv, v, σϕ;]Truth conditions of ϕ and ϕ can then be definedusing ,:[, w, σϕ⇔ there exists an a∈δ(w) such that , w, σ[x↦ a]ϕ;, w, σϕ⇔for all a∈δ(w), , w, σ[x↦ a]ϕ;]It is now clear that ϕ is equivalent to ϕ wherex∉FV(ϕ). Thereforecan be viewed as an extension of the basic propositional modal language. Therefore, in the context of , ϕ can be viewed as an abbreviation. It also becomes evident that ϕ and ϕ are essentially ∃ x ϕ and ∀ xϕ inrespectively. In the following we study the expressivity ofin comparison with . § EXPRESSIVITY Note that our semantics forhas the ∃∀ pattern which is similar to the neighbourhood semantics for modal logic <cit.>. Moreover,can be viewed as a fragment of the corresponding . Indeed, inspired by the world-object bisimulaion for<cit.> and bisimulation for monotonic neighbourhood modal logic <cit.>, we propose a novel notion of bisimulation for . Before the formal definition, given a model , let D_^* be the set of (possibly empty) finite sequence of objects in D^. A partial isomorphism between a⃗∈D^*_ and b⃗∈D^*_ such that |a|=|b| is an isomorphism mapping a_i to b_i w.r.t. relevant interpretations of predicates (cf. e.g, <cit.>). It is partial since it is not about all the objects in D^ and D^.Given two modelsand , the relation Z⊆ (W^× D_^*)× (W^× D_^*) is call an ∃-bisimulation, if for every ((w,a⃗), (v, b⃗))∈ Z such that |a⃗|=|b⃗| the following holds (for brevity, comma in (w,a⃗) is omitted): PISO a⃗ and b⃗ form a partial isomorphism w.r.t. identity and interpretations of predicates at w and v respectively. Zig For any c∈ D^_w, there is a d∈ D^_v such that for any v'∈ W^ if vR v' then there exists w' in W^ such that wR w' and w'a⃗c Z v'b⃗d. Zag For any d∈ D^_v, there is a c∈ D^_w such that for any w'∈ W^ if wR w' then there exists v' in W^ such that vR v' and w'a⃗c Z v'b⃗d. We say , wa⃗ and ,vb⃗ are -bisimilar (, wa⃗_,vb⃗) if |a|=|b| and there is an ∃-bisimulation linking wa⃗ and vb⃗. In particular, we say ,w and ,v are -bisimilar if , w_,v, i.e., when |a⃗|=|b⃗|=0.It is not hard to show that _ is indeed an equivalence relation.Note that our bisimulation notion is much weaker than isomorphism, in particular the domains of the two bisimilar models do not necessarily have the same cardinality.Consider the constant domain models ,:@R-20pt: w[r][dr]v: Pa :s[r][dr]t: Pc u: Pbrwhere D^={a,b}, D^={c}, ρ^(P,w)=ρ^(P,s)=ρ^(P,r)=∅, ρ^(P,v)={a}, ρ^(P,u)={b}, ρ^(P,t)={c}. Suppose P is the only predicate, we can show that ,w_,s by an ∃-bisimulation Z (pay attention to the switch of the two models in the second half of the definitions of Zig and Zag): {(w,s), (va, tc), (ub, tc), (vb, rc), (ua, rc) }Also note that Zig and Zag hold trivially for wa⃗ and vb⃗ if w and v do not have any successor, based on the fact that local domains are non-empty by definition. We write , wa⃗≡_, vb⃗ if |a⃗|=|b⃗| and for all anyformula ϕ(x⃗) such that |x⃗|=|a⃗|: , wϕ [a⃗] , vϕ[b⃗]. We can show thatis invariant under -bisimilarity., wa⃗_,vb⃗ then , wa⃗≡_, vb⃗.It suffices to prove that if Z is an ∃-bisimulation linking wa⃗ and vb⃗ such that |a⃗|=|b⃗|,then , wϕ [a⃗] , vϕ[b⃗].Given a bisimulation Z, we do induction on the structure of ϕ. Supposing ϕ is an atomic formula, due to PISO, ,wϕ [a⃗] iff ,vϕ [b⃗] for any atomic formula ϕ (x⃗). Suppose ϕ is in the shape of ψ(x⃗) and,wϕ [a⃗]. By the semantics there is a c∈ D^_w such that ,w'ψ [a⃗c] for all the w' such that wRw'. According to Zig, there exists a d∈ D^_v such that the second half of the condition holds. We claim that,v'ψ [b⃗d] for any v' such that vRv'. Suppose not, then there is a v' such that ,v'⊭ψ [b⃗d] and vRv'. Then according to Zig, there is w' such that wRw' andw'a⃗c Zv'b⃗d. By IH, ,w'⊭ψ [a⃗c]. Contradiction. , w_,v then for any closed -formula ϕ:, wϕ, vϕ.By the above invariance results, we can show that many naturalcombinations of quantifiers and modalities are not expressible in . ∃ x Px, ∃ xPx and ∃ x Px are not expressible in .In this proof, we again consider constant domain models.For ∃ x Px, consider the bisimilar models in Example <ref>. ∃ x Px holds on ,w but not on ,s, thus it is not expressible in .For ∃ xPx and ∃ x Px, consider:@R-20pt: w[r][dr]v: Pa :s[r]t uwhere D^={a,b}, D^={c} as before, ρ^(P,w)=ρ^(P,u)=ρ^(P,t)=ρ^(P,s)=∅, ρ^(P,v)={a}. Clearly, ∃ x Px and ∃ x Px are true at , w but false at , s. However, we can show that ,w_,s by an ∃-bisimulation Z: {(w,s), (ua, tc), (vb, tc), (ub, tc)}.To precisely characterizewithin the corresponding , we need a notion of saturation. In the following, we write Γ(x⃗) if all the free variables in the set of -formulas Γ are included inx⃗. Inspired by <cit.>, we can generalize the concept of m-saturation for propositional modal logic (cf. <cit.>) as follows:A modelis said to be -saturated, if for any w∈ W^, and any finite sequence a⃗∈D^*_, the following two conditions are satisfied:∃-type If for each finite subset Δ of a set Γ (y⃗x) where |y⃗|=|a⃗|, , w⋀Δ [a⃗],[Here we require y⃗ are assigned a⃗ correspondingly. This is only to avoid introducing extended language with constants for a⃗ as in standard model theory, since we did not define interpretation for constant symbols. Similarly below.] then there is a c∈ D^_w such that , wϕ[a⃗c] for all ϕ∈Γ, where x is assigned c.-type If for each finite subset Δ of Γ(x⃗) such that |x⃗|=|a⃗|, , w⋀Δ [a⃗], then , wϕ[a⃗] for each ϕ∈Γ. Note that in the above definition, to simplify the presentation, we use , which are expressible in . -type condition is essentially the m-saturation adapted with variable assignments.Recall that a finite model has a finite domain and finitely many worlds. It can be verified that:[Due to limited space, we omit some proofs and leave them to the extended version of this conference paper.]Every finite model is -saturated. We can obtain the Hennessy-Milner-type theorem (cf. <cit.>) to establish the equality between -bisimilarity and -equivalence.For -saturated models , and |a⃗|=|b⃗|:, wa⃗_,vb⃗,w a⃗≡_,vb⃗ Due to Theorem <ref>, we only need to show the right-to-left direction. We define Z={(wa⃗, vb⃗)| w∈ W^, v∈ W^, a⃗∈ D_^*, b⃗∈ D_^*, |a⃗|=|b⃗|, , wa⃗≡_, vb⃗}. We need to show that Z is an ∃-bisimulation. PISO is straightforward as a⃗ and b⃗ are finite and partial isomorphism between them can be expressed by atomic formulas. We only show Zig since Zag is similar. Assuming wa⃗Z vb⃗ and there is a c∈ D^_w. Let Γ={ϕ(y⃗x)| wϕ [a⃗c], |y⃗|=|a⃗|}. Now for any finite set Δ⊆Γ we have w⋀Δ[a⃗]. Sincewa⃗≡_vb⃗, v⋀Δ[b⃗]. Now by ∃-type condition, we know there is a d∈ D^_v such that vϕ [b⃗d] for all ϕ∈Γ (⋆). Now take an arbitrary v' such that vR v', we show there is a w' such that wRw' and w'a⃗c≡_v'b⃗d. Let Σ={ϕ(x⃗)|,v'ϕ [b⃗d], |x⃗|=|b⃗d|}. It is clear that for each finite set Δ⊆Σ, v⋀Δ[b⃗d], namely v⋀Δ[b⃗d]. We can show w⋀Δ[a⃗c]. Suppose not, then w⋀Δ[a⃗c], thus ⋀Δ∈Γ. By (⋆), v⋀Δ[b⃗d], which is in contradiction with v⋀Δ[b⃗d]. Now it is clear that w⋀Δ[a⃗c] for each finite Δ⊆Σ. By the -type condition of -saturation, there is a successor w' of w such that w'Σ[a⃗c]. Therefore w'a⃗c≡_ v'b⃗d, namely (w'a⃗c, v'b⃗d)∈ Z. This completes the proof for Zig.For finite models: , wa⃗_,vb⃗,w a⃗≡_,vb⃗. For -saturated models: , w_,v , wand ,v satisfies the same closed -formulas (sentences). Now we can characterizein the corresponding first-order language. Given , as before,the correspondinglanguage is defined as follows: ϕ::= x≈ y | Px⃗|ϕ| (ϕϕ)|∀ xϕ|ϕThe corresponding two-sorted first-order languageis: ϕ::= x≈ y | Q ux⃗| Ruv | Eux |ϕ| (ϕϕ)|∀ xϕ|∀ uϕwhere x, y∈ and u,v∈ which is a collection of world variables disjoint from ; Q∈ andis the smallest collection of predicate symbols such that for each n-ary P∈ there is a unique Q_P∈ with the arityn+1. E is a new predicate symbol to say whichobject exists on which world.It is trivial to define a translation r fromto the correspondingby setting r(ϕ)=∃ xr(ϕ). It is clear that this translation is truth preserving. To translateinto the formulas in the corresponding(with u,v as the only world variables), we define the following t_u inspired by <cit.>: [t_u(x≈ y)= x≈ yt_u(Px⃗)= Q_P(u, x⃗); t_u(ψ)= t_u(ψ)t_u(ϕψ)=t_u(ϕ) t_u(ψ).; t_u(ψ)=∃ x (Eux ∀ v (Ruv → t_v(ψ));]t_v is defined symmetrically, by swapping u and v.In this way, every -formula is translated into a two-world-variable formula ofwith one free variable, similar to the standard translation ofbasic modal language to first-order language. We can also view a model foras a model forby turning ρ(w, P) into the interpretation of the corresponding Q_P in the most natural way. It is not hard to show: For anyformula ϕ: ,w, σϕ, σ' ⊩ t_u(ϕ)where σ' is σ extended by u↦ w, ⊩ is the standard semantics for(cf. e.g. <cit.>).It follows from the compactness ofthat:[ inherits the compactness from first-order logic since there is a truth preserving translation fromtoby using two new unary predicates S_1, S_2 for the two sorts, together with a single unrestricted quantifiers to mimic the two-sorted quantifiers. Note that we can also express the constraints on the sorts inby θ(S_1,S_2)=∀ x ((S_1 x S_2x)(S_1x S_2x)).]is compact. Based on Theorem <ref> and the compactness of , it is relatively routine to prove the van-Benthem-like characterization theorem using the strategy in <cit.>, which makes use of ω-saturation in first-order model theory.We omit the proof due to limited space. A -formula ϕ(x⃗u) is equivalent to an -formula (over -models) iff it is invariant under -bisimilarity.Since anyformula can also be viewed as aformula ϕ(x⃗u), via a natural translation like t_u (cf. <cit.>), it follows that: A -formula is equivalent to an -formula iff it is invariant under -bisimilarity. § SATISFIABILITY OF Note that by t_u we can translate anformula to an equivalentformula, and eventually to aformula with one free world variable u (see Footnote <ref>). For example Px becomesϕ(u)=∃ x (S_1xS_2uEux∀ v ((S_2vRuv)→ Q_Pvx))in the correspondinglanguage. Note that since we only consider increasing domain models, ϕ is equivalent to ϕ(u)=∃ x (S_1xS_2uEux∀ v ((S_2vRuvEvx) → Q_Pvx))which is in the (loosely) guarded fragment of first-order logic known to be decidable <cit.>. However, it does not directly imply that our logic is decidable, since, for example, to capture the increasing domain model we do need the following first-order constraint: χ=∀ u∀ v ∀ x((S_2u S_2v S_1x Eux Ru v)→ Evx)χ can be viewed as some form of transitivity, and it is not in the guarded fragment, since v and x are free in the consequent but they do not co-exist in any of the atomic guards in the antecedent. On the other hand, such a transitivity-like constrained predicate E only appears in the guards of the translations of -formulas, which may be handled by the techniques in <cit.> for the guarded fragment with transitive guards known to be decidable. We leave the detailed discussion connectingto guarded fragments to a future occasion.In this section, we give an intuitive tableau-like method to decide whether an equality-free -formula is satisfiable, inspired by thesatisfiability games for various temporal logics <cit.>.[See <cit.> for tableau methods for first-order modal logic in general.] For the ease of presentation, we consider the following equivalent language ofin positive normal form (PNF), where negations only appear with atomic formulas:[PNF is often used in automata-theoretical methods to satisfiability problems of logics in computer science (cf. e.g., <cit.>).]ϕ::=Px⃗| Px⃗| (ϕϕ) | (ϕϕ)|ϕ|ϕNote that for an arbitrary -formula ϕ, we can rewrite it into an equivalent formula in PNF by the following rewriting rules and the replacement of equals: r( (ϕψ))=r(ϕ) r(ψ) r(ϕ)= r(ϕ)r(ϕ)=r(ϕ)Note that |r(ϕ)|≤ 2|ϕ|. In the following, we will focus on the formulas that are clean in the sense that no variable occurs both free and bound,[ In particular, iforappears in the formula then x does not have any free occurrence, even whenanddo not bind any occurrence of x, e.g., ( Py)Px is not clean.] and no two distinct occurrences of modalities bind the same variable. As in first-order logic, it is easy to show that anyformula can be relettered into an equivalent clean formula with the same length, by renaming the bounded variables.We define the following tableau rules for allformaulas in PNF.Tableau rules w: ϕ_1ϕ_2,Γ, σw:ϕ_1,Γ, σ| w:ϕ_2,Γ, σ ()w: ϕ_1ϕ_2,Γ, σw:ϕ_1,ϕ_2,Γ,σ()Given n≥ 0, m≥ 1:w:ϕ_1, …, ϕ_n,ψ_1,…, ψ_m, l_1… l_k, σ{(wv^y_y_i: {ϕ_j| 1≤ j≤ n}, ψ_i[y y_i], σ') | y∈ Dom(σ'), i∈ [1,m]} ()Given n≥ 1, k≥ 0:w:ϕ_1, …, ϕ_n, l_1… l_k, σw: l_1… l_k, σ ()where σ'=σ∪{(x_j, x_j)| j∈ [1,n] } and l_k∈ lit (the literals).Those rules are used to generate tree-like structure, where the nodes are triples (w,Γ,σ) in which w is some name to denote a world, Γ is a finite set of -formulas, and σ is a partial function fromtoas an assignment. Note that Rule () is essentially a choice: given the numerator, you can only select one of the denominator to continue. On the other hand, the rule () is a branching one, which generates all the nodes in the denominator set (even when n=0 i.e., the -part is empty). It is a generalized version of the corresponding rule for basic modal logic (cf. <cit.>): w: ϕ,ψ, l_1… l_lv: ϕ, ψThe idea is that if there is a diamond formula then we need to generate a successor while keeping the information given by -formulas. Here the complication is to manage the variable assignment properly. Note that if there are merely -formulas without any -formula, then we do not need to generate any successor, as captured by the rule . A tableau starting from (w:Γ, σ) is a tree where the successors of a node are generated by applying the rules,until no rule is applicable. Recall that you can only select one of the denominators to continue when applying (). Below is an example of a tableau where the (partial) function σ is represented by a set of ordered pairs:[w: {(Px Qx) Qy Pz, {(z,z)} ()× 2 [w: {(Px Qx),Qy,Pz}{(z,z)} () [wv^x_y: {Px Qx,Qx}, {(x,x), (z,z)} () [wv^x_y: {Px,Qx}, {(x,x), (z,z)}] ] [wv^z_y: {Px Qx,Qz}, {(x,x), (z,z)}[wv^z_y: {Qx,Qz}, {(x,x), (z,z)}] ] ] ]Intuitively, () and () will `decompose' the formula untilis applicable. It is not hard to see that all the leaf nodes are in the shape of (w: l_1… l_k, σ). Moreover, any tableau starting from (w: Γ, σ), where Γ and σ are finite, is a finite tree, both in depth and width, since we only generate simpler formulas by the rules and the domain of the assignment is always finite. A tableau is called open if all its branches do not contain contradictions of literals at the same world, i.e. no Px⃗ and Px appear together.Given a tableau , we say a node (w: Γ, σ) is a branching node if it is branching due to the application of . We call (w: Γ, σ) the last node of w, if it is a leaf node or a branching node. Clearly, given a w appearing in a tableau , the last node of w always uniquely exists, since we always only select one of the denominators for the rule (). We denote the last node of w in a givenas t_w. LetDom(t_w)={[ Dom(σ') if t_w is branching;Dom(σ) otherwise ].where σ is the assignment in t_w and σ' is the assignment defined as inw.r.t. t_w. Actually, the open tableaux are pseudo models. For any clean -formula θ in PNF, the following are equivalent:* There is an open tableau from(r: {θ}, σ_r={(x, x)| xis free in θ}∪{(z,z)})where z∈ and it does not appear in θ.* θ is satisfiable in an increasing domain model. Before the main proof, we need the following handy observations about any tableaustarting from (r, {θ}, σ_r) where θ is clean. For any node (v: Γ, σ) in , we claim: (1) If() occurs in Γ, then it only occurs once and there is no() occurring in Γ. (2) For all x∈ Dom(σ),anddo not appear in Γ, thus all the occurrences of x in Γ are free.(3) All the free variables x in Γ are in Dom(σ). (4) If ϕ, ψ∈Γ then y is not a free variable in ϕ.(5) If ϕ, ψ∈Γ then y is not a free variable in ϕ and x is not a free variable in ψ. (6) For any x in Dom(σ), σ(x)=x.(1): We prove it by induction on the structure offrom the root: it is true for the clean formula θ by definition, and all the rules preserve this property since they never add any new occurrences of modalities. (2): Again, we prove it by induction from the root: at the root the claim is true by the definition of σ_r and the cleanness of θ (cf. also Footnote <ref>). Moreover, all the rules preserve this property. In particular, for the rule (), by induction hypothesis, for any variable x∈ Dom(σ), it only occurs free in the formulas of the numerator. Thus the occurrences of x∈ Dom(σ), if any, are also free in any ϕ_j and ψ_i[y y_i] which have less modalities to bind than the numerator. Now for x∈ Dom(σ')∖ Dom(σ), i.e.,appears in the numerator, the statement also holds by Claim (1), since there is only one occurrence of any modality for such an x in the numerator.(3): Again, we can show by induction: Dom(σ_r) has all the free variables in θ, and all the rules preserve this property. For the rule (), we need to check the free variables in those ϕ_j and ψ_i. Note that the only possible extra free variable in ϕ_j but not in ^x_jϕ_j is x_j but it is already included in Dom(σ'). The only possible extra free variable in ψ_i[y y_i] but not in ^y_iψ_i is y which is also already in Dom(σ').(4): Towards contradiction, suppose ϕ, ψ∈Γ, but y is a free variable in ϕ. According to Claim (3) y∈ Dom(σ), then by Claim (2),does not appear in Γ, which is in contradiction with ψ∈Γ. (5): Similarly to (4).(6): Obvious, by definition. Now we are ready to prove the main theorem. From top to bottom: Given an open tableaufrom the root node (r:ϕ, σ_r), we define =W, D, δ, R, ρ where:* W={w| (w, Γ, σ)appears infor some Γ and σ} * w R v iff v=wv' for some v'.* δ(w)=Dom(t_w)* D=⋃_w∈ Wδ(w) * x⃗∈ρ(w, P) iff the atomic formula Px⃗ appears in t_w. where t_w is the last node of w in , as defined before.According to the rule () and the definition of δ,is indeed an increasing domain model.[Note that it does not mean D_w is everywhere the same due to branching nodes in .] Since (z,z)∈ Dom(σ_r), D_w is not empty for any w∈ W. Moreover ρ is well-defined due to the openness of . Note that due to Claim (3), if the atomic formula Px⃗ appears in t_w then x⃗⊆ D_w. We will show that ,r is indeed a model of θ w.r.t. σ_r.From Claim (3), if (w: Γ, σ) appears in , then all the free variables are in Dom(σ)⊆ D_w=Dom(t_w), thus it makes sense to ask whether , w, σΓ (the assignment to the bound variables are irrelevant for the truth of formulas in Γ). To prove that , w, σΓ for all nodes (w: Γ, σ) in , we do induction on the nodes ofin a bottom-up fashion from leaf nodes, by following the the rules conversely.Leaf For leaf nodes (w: l_1, …, l_k, σ), by definition of ρ, the statement holds based on the fact that σ(x)=x for all the free variable x in those literals by Claims (3), (6).Supposing , w, σ l_1… l_k, then it is clear that , w, σϕ_1 …ϕ_n l_1… l_k since there is no outgoing transition from w, and D_w is not empty.,Obvious.Suppose , wv_y_i^y, σ'ψ_i[y y_i]⋀_1^nϕ_j for every y∈ Dom(σ') and i∈ [1, m], and (w: Γ, σ) is the branching predecessor whereΓ={^x_jϕ_j| j∈[1,n]}∪{^y_iψ_i| i∈[1,m]}∪{l_h| h∈[1,k]}.Note that D_w=Dom(t_w)=Dom(σ'). We need to show that , w, σΓ. The l_h part is as in the case of the leaf nodes. For ^x_jϕ_j, let x_j∈ Dom(σ')=D_w be the witness, we need to show that at all the successors wv_y_i^y of w, , wv_y_i^y, σ[x_j↦ x_j]ϕ_j . Now by Claim (4), all x_k such that k≠j are not free in ϕ_j. From the induction hypothesis (IH) that , wv_y_i^y, σ'ϕ_j for each wv_y_i^y, and the fact that σ[x_j↦ x_j] and σ' agree on the free variables in ϕ_j, we know , wv_y_i^y, σ[x_j↦ x_j]ϕ_j for each wv_y_i^y. Thus , w, σ^x_jϕ_j.As for each ^y_iψ_i, we show that for each y∈ D_w= Dom(σ'), , wv^y_y_i, σ[y_i↦ y]ψ_i. Note that y might be not in Dom(σ). By the IH, , wv_y_i^y, σ'ψ_i[y y_i]. Since σ' assigns y to y, we just need to show that σ[y_i↦ y] and σ' agree on all the free variables in ψ_i[y y_i] except y. Note that y_i is not a free variable in ψ_i[y y_i], therefore the only possibledifferences between σ[y_i↦ y] and σ' are about those x_j wherex_j≠y. By Claim (5), we know that x_j is not a free variable in ψ_i[y y_i] if x_j≠ y. Therefore , wv^y_y_i, σ[y_i↦ y]ψ_i for each y∈ D_w, thus , w,σ^y_iψ_i for each i. It follows that ,r,σ_rθ.Now from bottom to top: We just need to show that the rule applications preserve the satisfiability of the formula set. Note that for () it suffices to show one outcome node is still satisfiable. In this way, there is an open tableau since if the formula sets at the leaf nodes and branching nodes are satisfiable,then there is no contradiction among the literals. It is obvious that () and () preserves satisfiability, and one of the denominator ofpreserves it too. We now show thatalso does so. Supposing Γ={ϕ_1, …, ϕ_n,ψ_1,…, ψ_m, l⃗}, in some branching node (w: Γ, σ), is satisfiable, then there is a model , w and an assignment η such that η(x)∈ D_w for all x∈ and:, w, η{ϕ_1, …, ϕ_n,ψ_1,…, ψ_m} (∘)By the semantics, we know there are a_1, …, a_n∈ D_w, such that for all the successors v of w: , v, η[x_j↦ a_j] ϕ_j. Due to Claim (4), each x_j is not free in ϕ_k for k≠j thus we can safely obtain for all successor v of w:, v, η[x⃗↦a⃗] {ϕ_1, …, ϕ_n}.(A) Now also by (∘) and the semantics for ^y_i, for each ψ_i and each b∈ D_w, there is a successor v^b_i of w such that , v^b_i,η[y_i↦ b]ψ_i(B)From Claim (5), y_i is not one of x_j. Now from(A) and (B), we have for each ψ_i and each b∈ D_w, there is a successor v^b_i of w such that: , v^b_i,η[x⃗↦a⃗, y_i↦ b]{ϕ_1, …, ϕ_n, ψ_i}(⋆) Now consider any y∈ Dom(σ') as in the denominator of the rule , we need to show {ϕ_1, …, ϕ_n, ψ_i[y y_i]} is satisfiable. Suppose that y∈ Dom(σ') andη(y)=b∈ D_w. By Claim (2), y is free in ψ_i[y y_i], thus by (⋆):, v^b_i, η[x⃗↦a⃗]{ϕ_1, …, ϕ_n, ψ_i[y y_i]} Therefore, {ϕ_1, …, ϕ_n, ψ_i[y y_i]} is satisfiable for each y∈ Dom(σ'). This completes the whole proof.A strong finite tree property then follows: [Interested readers may go back to Footnote <ref>.]If an -formula ϕ (with length n) is satisfiable, then it is satisfiable in a finite tree model ,w, w.r.t. an assignment σ such that |W^|≤ n^4n, |D^|≤ n, ρ(P, w)⊆ D_w^k (if P is k-ary), the depth of the tree is bounded by 2|ϕ|, and σ(x)∈ D^_w for each free variable x in ϕ.The upper bound for the depth of the tree comes from the bound on the length of the PNF of ϕ. The (very loose) upper bound on the size of W^ comes from the fact that each node in the tableau may contain up to n modalities and each modality may have n-successors (due to the size of the domain), and the depth of the tableau is up to 2n. As in normal modal logic <cit.>, we can force a binary branching tree by an -formula to get an exponential-sized model.Satisfiability problem of -formulas is -complete. (Sketch) Note that standard modal logic formulas can be translated into ourby using 0-ary predicate P for each propositional letter. For example, p can be translated as ^y P. The -hardness then follows since satisfiability for basic normal modal logic is -complete <cit.>. For the upper bound, note that to rewrite a formula into PNF and to reletter the formula into a clean shape are efficient in space, and the length of the resulting formula is still linear in the length of the original formula. From then on, a standard -algorihm traversing a tree structure including all the the possible -branches suffices, as in standard modal logic, with some care about efficiently encoding the descriptions of each node and the result of the consistent checking.Our notion of the tableaux is closely related to two-player satisfiability games and alternating-tree automata <cit.>, which will give us the algorithmic tools forin the future.We conjecture that a similar tableau method would work for , with more careful assignment management by selecting representatives of provably equivalent variables w.r.t. ≈ as the local domain. We leave it to the extended version of the paper. § A NEW EPISTEMIC LOGIC In this session, we go back to the motivating epistemic setting, and give a complete axiomatization of our logic over epistemic models.§.§ Epistemic languageTo ease the presentation of the proof system,we also include the standard modalityas a primitive modality in the following language :[In the axiomatization we will make use of . If we do not introduce them explicitly then every time we need to use ϕ where x is not free in ϕ.]ϕ::= x≈ y | Px⃗|ϕ| (ϕϕ)|ϕ|ϕRecall thatis equally expressive as . In the epistemic setting, the intended reading of ϕ is i knows that ϕ, and the intended reading of ϕ is that i knows something such that ϕ. The semantics is as before, but we have two extra conditions on the model:* Each R is an equivalence relation as in (idealized) standard epistemic logic S5. We also write ∼ for R. * For all w∈ W, D_w=D.Note that for an increasing domain model , if R is an equivalence relation, then for any two worlds w,v such that wRv, we have D_w=D_v. Therefore, we can simply assume that there is a constant domain over the whole model.[Without the condition of constant domain, different partitions w.r.t. ∼ may still have different local domains.] We also call such models S5-models, following the terminology in epistemic logic. ϕ∈ is valid if for any pointed S5-model ,w, any assignment σ, ,w,σϕ. Note that in constant domain models, we no longer need to give the restriction on the assignment: σ(x)∈ D=D_w for all assignment σ, all w∈ W_ and all x∈.over S5-models is a quite powerful language. As we mentioned, the semantics foris in line with the mention-some reading. We can also introduce a modal operatorbased on mention-all semantics as below: [ , w, σϕ ⇔ for each d∈ D, either, w, σ[x↦ d]ϕor, w, σ[x↦ d]ϕ; ]Intuitively, ϕ means for all objects in the constant domain D, the agent knows whether ϕ, e.g., I know who came to the party means that for each person in concern, I know whether (s)he came or not. In terms of , ϕ is essentially ∀ x (ϕϕ). Note thatandnot only differ only in the quantifiers. On the other hand, the natural ∀-version ofis defined:[, w, σ^∀ xϕ⇔ for each d∈ D, , w, σ[x↦ d]ϕ;], w, σ^∀ xϕ only says that of each object, the agent knows that it satisfies property ϕ(x), which differs from the semantics of the mention-all operator.There is also a very natural generalization of ourmention-some operator with multiple variables: ^x⃗: [ , w, σ^x_1⋯ x_nϕ⇔ there exist d_1,⋯, d_n∈ D such that , t, σ[x⃗↦d⃗]ϕ;],^∀ x, and ^x⃗ can all be defined inover S5-models:The following equivalences are valid for any -formula ϕ, based on the fact that ϕϕ and ϕϕ are valid on S5 models:* ϕ (^x (ϕϕ))* ^∀ xϕϕ * ^x⃗ϕ^x_1⋯^x_nϕ§.§ Proof systemIn the rest of this section we propose a Hilbert-style proof system for the equality-freeand then extend it to a proof system for . First note that the following “K axiom” foris not valid on S5-models, since the witnesses object for x in (ϕ→ψ) and ϕ may be different: (ϕ→ψ)→ (ϕ→ψ)Therefore the modal logic ofis not normal, asexpected. We propose the following proof system: 2cSystem Axioms all axioms of propositional logic (ϕ→ψ)→ (ϕ→ψ) ϕ→ϕ ϕ→ϕ ϕ→ϕ (ϕ[y x]) →ϕ (if ϕ[y x] is admissible) ϕ→ϕ (if x∉FV(ϕ)) ϕ→ϕ⊤ Rules: φ,φ→ψψ ⊢φ→ψ⊢φ→ψwhere ϕ[y x] is admissible if no free occurrence of x in ϕ is in the scope of a modality binding y. The axioms are quite intuitive as they are, but they can be understood even more easily if we put them in some context to give concrete intuitive readings. For example, supposing we use ϕ(x) to express that the agent knows how to achieveϕ, then ,express the (idealised) introspections of such goal-directed know-how: if you know how then you know you know how, and if you do not know how then you know you do not;says that if you know that a particular way can achieve ϕ for sure, then you know how to achieve ϕ;trivializes know-how to know-that, if x is irrelevant to ϕ;says that if you know how to achieve ϕ then you know how to ensure that you know ϕ;says we know tautologies;[This is a technical axiom to recover the necessitation rule for K from . Alternatively, we can also just include the necessitation rule forinstead of .] andis the monotonicity rule for know-how.Actually, all the above axioms have incarnations in the logic system 𝕊𝕂ℍ of know-how proposed in <cit.>. Modulo the replacement of the know-how operatorby , all the axioms and rules in 𝕊𝕂ℍ are derivable in , except the composition axiom 𝙺𝚑𝙺𝚑𝚝𝚘𝙺𝚑 (ϕ→ϕ), which indeed captures the characteristic feature about know-how: the witness strategies can be composed.It is routine to verify the soundness, assuming the familiarity of first-order modal logic over S5-models. is sound over S5 models.is very powerful, as demonstrated by the following proposition: many usual “suspects” can be derived. The following theorems are derivable and the rules are admissible in :ϕ→ϕϕϕ (x∉FV(ϕ))⊢φ⊢φ⊢φ⊢φ⊢φψ⊢χ(ψ)χ(ϕ)⊢φ→ψ⊢φ→ψ (x∉FV(ψ)) ⊤ ϕϕ[y x] (yis not in ϕ)) ϕ→ϕ ϕ→ϕ(Sketch)is based onand . ϕ→ϕ is a special case ofwhen y=x. Together withwe have . Now fromandwe have . Then based onand , andfollow from . The rule of Replacement of Equals () can be proved inductively based on .is based on , ,and .The standard introspection axioms ofare special instances of inspection axiomsandbased onand .[ can also be derived fromandin the systemlike in the case of propositional S5 system.] Let us now look at , which is the renaming axiom for bound variables. Right-to-left can be derived by starting with : ⊢(ϕ[y x])→ϕ, then since y does not appear in ϕ, we have ⊢(ϕ[y x])→ϕ by . For the other direction, note that ϕ=(ϕ[y x])[x y] if y is not in ϕ. Then since (ϕ[y x])[x y] is admissible, by , ⊢(ϕ[y x])[x y]→ϕ[y x], namely ⊢ϕ→ϕ[y x]. Then by , ⊢ϕ→ϕ[y x].Now we know that the S5 system ofis a subsystem of . We may also view ,as analogues of the following axiom and rule respectively in first-order logic: ∀ x ϕ→ϕ[y x] ⊢ϕ→ψ⊢ϕ→∀ xψ(x∉FV(ϕ)) is the rule of replacement andallows us to reletter the bound variables, as in . Moreover, the Barcan formula (∀ xϕ→∀ xϕ), over S5-models, can be expressed as:ϕ→ϕwhich is also derivable indue to .The following provable formula plays a role in the later completeness proof.⊢_ (ϕ→ψ) →(ϕ→ψ). Consider the contrapositive of the formula to be proved: (ϕψ)→ (ϕψ). It is routine to show that ⊢(ϕψ) → (ϕψ). Now by, , and , ⊢ϕϕ and ⊢ψψ. Therefore, by , ⊢(ϕψ)→ (ϕψ).In the following we define the Hilbert systemforby extendingwith the following two extra axiom schemata: Axioms x≈ xx≈ y → (ϕ→ψ) if ϕ and ψ only differ in thatsome free occurrences of x in one formulaare replaced by free occurrences of y in another. is sound. It is routine to show: The following are provable in : x≈ y→ y ≈ x x ≈ y y≈ z → x≈ zx≈ y→ (x≈ y)x≉y→ (x≉y) andare due to . To derive , we first have ⊢ (x≈ x) byand . Bywe have ⊢ x≈ y→ ( (x≈ y)→ (x≈ x)). Thereforeis provable.For , fromwe have ⊢ x≉y→ x≉y, thus ⊢ x≉y→x≉y byand . Note that ϕ→ϕ is derivable byand . Therefore by taking ϕ=x≉y and usingwe have ⊢. § COMPLETENESS Let us first focus on the completeness ofwithout axioms for equalities.We first extend the language ofwith countably infinitelymany new variable symbols. Call the new language ^+ and the variable set ^+. In the following, we say a set of formulas is ^+-consistent if it is -consistent w.r.t. the extended language ^+. A set of ^+ formulas has ∃-property if for each ϕ∈^+ it contains a “witness” formula ϕ→ϕ[y x] for some y∈^+ where ϕ[y x] is admissible.The canonical model is a tuple W^c, D^c, ∼^c, ρ^c where:* W^c is the set of maximal ^+-consistent setswith ∃-property,* D^c=^+,* s∼^c t iff (s)⊆ t where (s):={ϕ|ϕ∈ s},* x⃗∈ρ^c(P,s) iff Px⃗∈ s. It is routine to show that ∼^c is an equivalence relation, by using axioms , , and .If ψ∉s ∈ W^c then there exists a t∈ W^c such that s∼^ct and ψ∈ t. It is routine in normal modal logic to show that if ψ∉s then (s)∪{ψ} is consistent. Now we show that (s)∪{ψ} can be extended to an ^+-maximal consistent set with ∃-property. We follow the general strategy in <cit.> by adding witness formulas one by one. Let θ_0=ψ. We enumerate -formulas as: ϕ_1, ^x_2ϕ_2, … We define θ_k+1 as the formula: θ_k (^x_k+1ϕ_k+1→ϕ_k+1[y x_k+1])where y is the first variable in a fixed enumeration of ^+ such that ϕ_k+1[y x_k+1] is admissible and ^x_k+1ϕ_k+1→ϕ_k+1[y x_k+1] is consistent with {θ_k}∪(s). We now show that such θ_k+1 always exists (i.e., such a y exists), if {θ_k}∪(s) is consistent. Towards a contradiction, suppose that {θ_k}∪(s) is consistent but there is no such a y, i.e., for each y∈^+ such that ϕ_k+1[y x_k+1] is admissible there are χ_1,…,χ_n∈(s) such that⊢χ_1…χ_n→ ((^x_k+1ϕ_k+1→ϕ_k+1[y x_k+1])→θ_k). Byandand the fact that χ_i∈ s for 1≤ i≤ n, it is routine to show that ((^x_k+1ϕ_k+1→ϕ_+1[y x_k+1])→θ_k) is also in s. Byagain, (^x_k+1ϕ_k+1→ϕ_k+1[y x_k+1])→θ_k(⋆)is in s for all y such that ϕ_k+1[y x] is admissible. Note that s has the ∃-property, therefore, ^x_k+1ϕ_k+1→ϕ_k+1[y^* x] is in s for some particular y^* such that ϕ_k+1[y^* x_k+1] is admissible. By Proposition <ref>, (^x_k+1ϕ_k+1→ϕ_k+1[y^* x_k+1]) is in s. By (⋆), we have θ_k is in s, thus θ_k∈(s) which is in contradiction with that {θ_k}∪(s) is consistent. Now since {θ_0}∪(s)={ψ}∪(s) is consistent, we can indeed construct all the θ_k. Let Γ={θ_k | k∈ℕ}∪(s). Γ is consistent and the ∃-property is essentially built-in.We can then extend it into an ^+-maximal consistent set t. It then follows that s∼^ct and ϕ∈ t. Now comes the truth lemma. Let σ^* be the assignment such that σ^*(x)=x for all x∈^+. For any ϕ∈^+, any s∈ W^c: ^c, s, σ^* ϕϕ∈ sWe do induction on the structure of the formula ϕ.Px⃗ By the definition of ρ^c and σ^*, it is obvious. BoolBoolean cases are trivial.ψ Routine, based on Lemma <ref>.ψSupposing ψ∈ s, by ∃-property of s there exists ψ[y x]∈ s for some y such that ψ[y x] is admissible. By , ψ[y x]∈ s. Supposing s∼^c t, we have ψ[y x]∈ t due to the definition of ∼^c. By , ψ[y x]∈ t. By IH, , t, σ^* ψ[y x]. Due to the definition of σ^* and the fact that ψ[y x] is admissible, , t, σ^*[x↦ y] ψ for all t∼ s. Thus ,s, σ^*[x↦ y]ψ, it follows that , s, σ^* ψ. Suppose ψ∉s, by , ψ[y x]∉s for each y such that ψ[y x] is admissible. By IH, ^c, s, σ^*ψ[y x] for each y such that ψ[y x] is admissible. Due to the special assignment σ^* such that σ^*(y)=y, it isclear that ^c, s, σ^*[x↦ y]ψ for each y such that ψ[y x] is admissible. Now consider any y' such that ψ[y' x] is not admissible, then by , we can reletter the modalities of y' in ψ with some fresh variable to obtain ψ' such that ⊢ψψ' and ψ'[y' x] is admissible. Now by , ψ'∉s. We can then repeat the reasoning above to obtain ^c, s, σ^*[x↦ y']ψ' for this y'. Since ⊢ψψ', by soundness, ^c, s, σ^*[x↦ y']ψ. Therefore for each y, no matter whether ψ[y x] is admissible,we have ^c, s, σ^*[x↦ y]ψ. Therefore ^c, s, σ^* ψ.Note that not every ^+-consistent set of formulas can be extended into a world in ^c, e.g., {ϕ(x)}∪{ϕ(x)| x∈^+} cannot be extended consistently to obtain the ∃-property. However, every -consistent set can be extended into a world in W^c by adding the witness one by one using the new variables (cf. <cit.>). Every -consistent set of -formulas can be extended into an ^+-maximal consistent set of ^+-formulas with ∃-property.Now based on Lemma <ref> and Lemma <ref>, every -consistent set is satisfiable by some pointed model and an assignment. The completeness follows:is strongly complete w.r.t.over S5 models.The completeness ofis quite routine based on the completeness proof of . We only sketch the idea here following <cit.>.is strongly complete w.r.t.over S5 models.Unfortunately,over (constant-domain) S5 models is indeed too powerful, we can code first-order formulas by replacing each quantifier in a first-order formula in the prenex form byorrespectively. We can show that this translation preserves the satisfiability.For example, we can translate ∃ x ∀ y ϕ (ϕ is quantifier-free) into anformula ϕ, which is equivalent to the first-order modal formula ∃ x ∀ y ϕ over S5 models, and it implies ∃ x∀ y ϕ by reflexivity. If a first-order formula is satisfiable then we can build a single-world S5 model such that the translated -formula is also satisfiable. On the other hand, if the translated -formula is satisfiable in some pointedS5 model, then we can just pick the designated world in that model as a first-order structure to satisfy the original first-order formula. This leads to theundecidability ofdue to the undecidability of first-order logic: is undecidable over S5 models. § FUTURE WORKDue to limited space, we omit several proofs and some additional results, and leave them to the extended version of this conference paper. We believe that this is only the beginning of an interesting story. On the technical side, we may study potential properties ofsuch as interpolation, frame definability, characterization over finite models, axiomatization and (un)decidability on various frame classes (with or without extra constant symbols). Although the fullover S5 models is undecidable, the concrete “propositional” know-wh logics mentioned in the introduction are usually decidable. One explanation is that the existing know-wh logics are often similar to one-variable fragments of the first-order modal language, which may lead to decidability over S5 and other models <cit.>, e.g., the conditionalknowing value formulas(ϕ,c) discussed in <cit.> are essentially ∃ x (ϕ→ c=x). A detailed comparison with known guarded-like fragments with extra frame constraints will be very useful to understand the new framework more deeply. Moreover, we believe our techniques and results can be generalized to poly-modal (multi-agent) settings. It then makes sense to discuss the mention-some version of common knowledge operator C^x:=∃ x C where C is the propositional common knowledge operator.[This is a rather strong notion of common knowledge, e.g., we commonly know how to prove the theorem in this sense means a fixed proof is also commonly known.] There is also a clear similarity with modal logic over neighbourhood models, which is worth exploring. eptcs | http://arxiv.org/abs/1707.08764v1 | {
"authors": [
"Yanjing Wang"
],
"categories": [
"cs.AI",
"cs.LO"
],
"primary_category": "cs.AI",
"published": "20170727075450",
"title": "A New Modal Framework for Epistemic Logic"
} |
1Department of Astronomy, University of California at Berkeley, Campbell Hall, Berkeley, CA 94720-3411 2Department of Earth and Planetary Science, University of California at Berkeley, McCone Hall, Berkeley, CA 94720-3411 3NASA Sagan Fellow 4The author list is in alphabetical order; all authors contributed equally to the intellectual content and to the work load.email: [email protected], [email protected] We explain the fast-moving, ripple-like features in the edge-on debris disk orbiting the young M dwarf AU Mic. The bright features are clouds of sub-micron dustrepelled by the host star's wind. The clouds are produced by avalanches:radial outflows of dust that gain exponentially more mass as they shatter background disk particles in collisional chain reactions. The avalanches are triggered froma region a few AU across—the “avalanche zone”—located on AU Mic's primary “birth” ring, at a true distance of ∼35 AU from the star but at a projected distance more than a factor of 10 smaller: the avalanche zone sits directly along the line of sight to the star, on the side of the ring nearest Earth, launching clouds that disk rotation sends wholly to the southeast, as observed. The avalanche zone marks where the primary ring intersects a secondary ring of debris left by the catastrophic disruption of a progenitor up to Varuna in size, less than tens of thousands of years ago.Only where the rings intersect are particle collisions sufficiently violent to spawn the sub-micron dust needed to seed the avalanches. We show that this picture works quantitatively, reproducing the masses, sizes, and velocities of the observed escaping clouds. The Lorentz force exertedby the wind's magnetic field, whose polarity reverses periodically according to the stellar magnetic cycle, promises to explain the observed vertical undulations. The timescale between avalanches, about 10 yr, might be setby time variability of the wind mass-loss rateor, more speculatively, by some self-regulating limit cycle.§ INTRODUCTIONDust in debris disks originates from collisional cascades. The largest bodies, comprising the top of the cascade, have lifetimes against collisional disruption equal to the system age,tens of Myrs or longer. They grind down into particles micron-sized or smallerat the bottom of the cascade. These tiny particulates are blown out of the system, typically by stellar radiation pressure, on orbital timescales of tens to thousands of years. For a general review of debris disks, see <cit.>.Quasi-steady cascades, in which the rate of mass erosion is constant from top to bottom (e.g., ; ; ), offer a ready framework for modeling debris diskson the longest of evolutionary timescales <cit.>.At the same time, there are increasingly many observations demanding that we resolve our theories more finely, both in space and time, and accommodate more stochastic phenomena.Mid-infrared excesses that are unusually strong given the Gyr ages of their host stars are thought to signal recent catastrophic collisions or sudden comet showers<cit.>. Fast time variability in the infrared, on timescales of months to years, has been interpreted as tracing the immediate aftermath of a giant plume-inducing impact (; see also, e.g., ; ; ).Of all the short-timescale phenomena reported for debris disks, perhaps the most surprising and least understood is the discovery by the SPHERE (Spectro-Polarimetric High-contrast Exoplanet REsearch) team of fast-moving features in the AU Mic edge-on debris disk <cit.>. The features appear as intensity variations at projected stellocentric separations of ∼10–50 AU, and are seen only on the southeast ansa of the disk. They travel away from the star at projected speeds comparable to—and for the most distant features exceeding by a factor of ∼2—the system escape velocity. The features also appear undulatory; the ones closest to the star are elevated by an AU or so above the disk midplane (see alsoand ).Taken at face value, the faster-than-escape velocities, and the trend of increasing velocity with increasing stellar separation, suggest that the brightest features are coherent “clouds” of dust accelerated radially away from the star by a force stronger than the star's gravity by a factoron the order of 10 <cit.>. We will adopt this simple interpretation.The fact that the clouds are seen to only one side of the star, together with the recognition that most of the mass in the underlying disk is concentrated in a “birth ring” of radius ∼35 AU (; ), suggests that the clouds are launched from the birth ring—from the side of the ring nearest the observer, so as to appear bright in forward-scattered starlight—at a special azimuthal location lying directly along the observer's line of sight to the star. Then the clouds simply inherit the orbital motion of the birth ring, whose near side must rotate from the northwest to the southeast to send the clouds to the southeast.The host star's windcan generate, for sufficiently small grains, the required radially outward force, of magnitude β_ w relative to stellar gravity:β_ w ≡3 Ṁ_∗ v_ wind/16 π G M_∗ρ_ p s∼ 4 ( Ṁ_∗/10^3 Ṁ_⊙) ( v_ wind/400 km/s) ( 0.1μ m/s) where G is the gravitational constant; M_∗ = 0.6 M_⊙ is the stellar mass <cit.>; grains are assumed spherical with internal bulk density ρ_ p∼ 1 g/cm^3 and radius s; and v_ wind and Ṁ_∗ are the stellar wind's speed and mass-loss rate, with the latter scaledto the solar mass loss rate Ṁ_⊙ = 2 × 10^-14M_⊙/yr <cit.>. Grain porosity in AU Mic <cit.> may boost β_ w by an extra factoron the order of 2. Augereau & Beust (; see also )lay out the many reasons why the mass loss rate from this young, active M dwarf is orders of magnitude larger than the solar mass loss rate. See in particular their Figure 11, which attests that β_ w can be as large as ∼40 when AU Mic flares.[<cit.> argue against high mass loss rates in AU Mic, relying instead on stellar radiation pressure to blow out grains. However, they idealized the grain cross section to radiation pressure as geometric; this is an overestimate given real-life optical constants (e.g., Figure 1 of ).]We propose here an explanation for the escaping clouds in AU Mic. We posit thatthey are the outcome of dust avalanches: exponential rises in dust production caused by small grains moving on unbound trajectories and shattering bound disk material in their path <cit.>. In an avalanche, sub-micron grains accelerated to high radial speeds (in this case by the powerful stellar wind) collide with larger parent bodies in the birth ring to create still more sub-micron grains; these collisional progeny are themselves brought up to high speed, leading to a collisional chain reaction and exponential amplification of the escaping dust column. We propose that each of the bright, fast-moving features observed by <cit.> results from an avalanche, launched from a small region (a few AU in size) in the birth ring lying directly along the line of the sight to the star (see above). Only in this localized region—what we call the “avalanche zone”—are sub-micron grains produced that can seed the avalanche. In our model, the avalanche zone marks where the birth ring is intersected by another structure: a secondary ring, much less massive than the primary, substantially inclined and/or eccentric, and composed of debris from the catastrophic disruption of a planetesimal. Avalanches are triggered at the intersection point of these two rings, where collisions are especially violent.Most of the rest of this paper is devoted to reproducing the sizes, masses, and velocities of individual escaping clouds using avalanches. Regarding what sets the periodicity of the avalanches, we have less to say. To match the observations, the avalanche period must be the time between cloud ejections, i.e.,the projected separation between clouds divided by their projected velocity. An approximate, characteristic value for the cycle period ist_ cycle ∼10 AU/5 km/ s∼ 10 yr .We can imagine two mechanisms that can set this period. The first is time variability in the stellar mass loss rate. This proposal is admittedly somewhat ad hoc. Although AU Mic flares dramatically at ultraviolet and X-ray wavelengths (e.g., ;and references therein), these bursts of high-energy radiation last mere minutes, whereas equation (<ref>) indicates that we are interested in modulating stellar activity on timescales of years. It is not clear whether the similarity between our required value for t_ cycle and the Sun's 11-year period for magnetic field reversalsshould be regarded as encouraging or irrelevant. An argument in favor of the latter is that the solar wind mass-loss rateṀ_⊙ betrays no correlation with solar magnetic cycle <cit.>. On the other hand, the rate of solar coronal mass ejections (CMEs) increases by an order of magnitude from solar minimum (when the CME rate is 0.5/day) to solar maximum (when the rate is 6/day; ). Magnetic activity cyclesare just beginning to be photometrically detected for main-sequence stars en masse (; there are rumors of a weak correlationbetween magnetic cycle period and stellar rotation period). For pre-main-sequence stars like AU Mic, there seem to be no useful data on magnetic cycles or on the time evolution of their mass loss rates,although the situation may be changing with long-term monitoring bythe Las Cumbres Observatory Global Telescope network (LCOGT; ).An alternative idea is that the stellar wind blows strongly (as is demanded by the observations which point to large β_ w) but steadily on yr-to-decade timescales, and that the avalanches undergo some kind of self-regulating limit cycle. It takes time for avalanches to clear and for enough material to re-seed them; this time could be identified with t_ cycle. We will briefly attempt to make this identification at the end of this paper, after quantifying some of the avalanche dynamics in <ref>.We do not, however, provide an actual limit-cycle model; in particular, we do not answer the key question of why avalanches would not simply unfold in a steady fashion if the stellar wind were steady.Since the idea of a self-regulating cycle is nothing more than a speculation at this point, we will assume throughout this paper that the avalanche period t_ cycle is set by theperiod of a variable stellar mass-loss rate. Fortunately, many of the remaining elements of our proposal do not depend on this assumption. We flesh our model outin <ref>. A summary, including some predictionsand a recapitulation of unresolved issues, is given in <ref>. In this first cut at a theory, we aim throughout for order-of-magnitude accuracy only.§ MODELWe present an order-of-magnitude understanding of the escaping clouds in the AU Mic system.We set the stage in <ref> by estimating individual cloud masses and mass ejection rates: these are the observables which any theory must explain. The heart of our paper is in <ref>, where we sketch our picture of dust avalanches, detailing quantitatively the creation of an avalanche zone from an ancient catastrophic collision; the current properties of the zone; and how the zone can give rise to the observed escaping clouds. In <ref> we briefly consider magnetic levitation of grains in an attempt to explain the observed vertical displacements of the clouds. To illustrateand provide a proof of concept of our ideas, we offer a numerical simulation in <ref>. §.§ Cloud Mass and Mass Loss RateWe estimate the mass of a given cloudusing feature“B” <cit.> as a fiducial. The V-band surface brightness of cloud B is comparable to that of the local disk, about B ∼ 16 mag/ arcsec^2∼ 0.5 erg/( cm^2 ssr)(; ). We take the line-of-sight column density of grains in the cloud to be N; the scattering cross section per grain to be Q π s^2; the relative power scattered per grain per steradian to be P (normalized such that its integral over all solid angle equals unity); and a to be the true (not projected) distance between the cloud and the star of luminosity L_∗∼ 0.1 L_⊙. ThenB ∼L_∗/4π a^2 N P Q π s^2. The cloud mass isM_ cloud∼4/3πρ_ p s^3 N Awhere A ∼ 4 AU (length) × 2 AU (height)is the projected area of the cloud. Combining (<ref>) and (<ref>), we haveM_ cloud ∼16 πρ_ p B A s a^2/3QPL_∗∼ 4 × 10^-7 M_⊕( s/0.1μ m) ( 0.3/Q) ×(2/4π/P) ( a/35 AU)^2where we have placed the cloud near the birth ring ata ∼ 35 AUand allowed for some forward scattering of starlight by assigning P to be the isotropic scattering value 1/4π multiplied by a factor of 2. We adopt throughout this paper a nominal cloud grain size of s ∼ 0.1μ m ,small enough for grains to enjoy high acceleration (equation <ref>) but still large enough to scatter starlight with reasonable efficiency.The implied mass loss rate from the debris disk—from clouds only[Equation (<ref>) accounts only for the mass ejected in clouds (overdensities). There is also the mass ejected in inter-cloud regions, which we will account for in <ref>. ]—isṀ_ cloud∼M_ cloud/t_ cycle∼ 4 × 10^-8 M_⊕ /yr .This is a large rate: it would take only ∼0.25 Myr, or ∼1% of the stellar age of t_ age∼ 23 Myr <cit.>,to drain the disk of ∼0.01 M_⊕, the total disk mass inferred from millimeter-wave observations <cit.>. This calculation suggests that the episodic ejections we are currently observing have not persisted for the system age, but reflect instead a transient phase.We will provide additional support for this idea in <ref> and <ref>. §.§ Avalanche Dynamics§.§.§ The Azimuth of the Avalanche Zone, Where Clouds are Launched We consider a β_ w-avalanche composed of 0.1 μm grains accelerated radially across the birth ring. The avalanche occurs in a restricted region on the birth ring—the“avalanche zone”—that is fixed in inertial space. In other words, the azimuth of the zone does not rotate at Keplerian speed (the next section <ref> explains why).Because (i) the birth ring has a radius of ∼35 AU while thecloud closest to the star (“A”) is located at a projected separation of ∼8 AU, and (ii) all moving features are located on the southeast ansa and travel further southeast, we position the azimuth of the avalanche zonepractically directly along the line of sight to the star, at a projected stellar separation ≪ 8 AU, so that clouds launched from there can be delivered by disk rotation to the southeast. We also locate the avalanche zone on the half of the ring nearest the observer so that the dust clouds it produces appear bright in forward-scattered light.§.§.§ The Avalanche Zone Lies Where a Secondary Debris Ring Intersects the Birth Ring; Violent Collisions Here Produce Avalanche SeedsIn our model, the avalanche zone is rooted where the birth ring—hereafter the “primary”—intersectsa much less massive “secondary” ring composed of debris from a catastrophically disrupted body (we will place an upper bound on its mass in <ref>).The node where the rings intersect is stationary in inertial space (aside from an insignificant precession).A similar set-up was considered in generic terms by <cit.>; see also the “static” case of <cit.> (neither of these studies considered avalanches, and the latter focused on matching the velocity profile of AU Mic's escaping clouds, not their sizes and masses).See Figure <ref> for a big-picture schematic.We imagine the secondary ring to have a semimajor axis and therefore an orbital period comparable to that of the primary,t_ orb∼ 300 yr ,and to be substantially inclinedrelative to the primary, reflecting the aftermath of a collision between a projectile in the primary ring and a target (the progenitor of the secondary ring) that once moved on an orbit inclined to the primary by ∼1 rad. Alternatively, instead of a large mutual inclination, we could just as well posit a large eccentricity for the secondary progenitor. The secondary ring could then be nearly co-planar with the primary, and be so eccentric that it intersects the primary at the same special azimuthal location of the avalanche zone. The only real requirement on the relative ring geometry is thatthis “intersection region” be just a few AU large, in order to match the observed sizes of the escaping clouds (we elaborate on these considerations of size in the next section <ref>).[We are positing only one intersection region, but there can beeither one or two if the rings are coplanar and eccentric, or two if the rings are circular and inclined and have identical radii. There is no particular reason to think there are two intersection regions based on the <cit.> observations, which indicate only a single launch site for grains, but we suppose it is possible that clouds launched from a second intersection region, on the side of the primary ring farther from the observer, could go undetected in back-scattered light.] We note that the asteroid and Kuiper belts,which are solar system analogues of debris disks,contain bodies commonly moving on highly inclinedand eccentric orbits. Given either a large inclination or large eccentricity for the secondary ring, the relative velocities between secondary and primary particles in the intersection region are large:v_ sec,pri∼ v_ K/2 ∼ 2 km/s ,i.e., within factors of a few of the local Keplerian velocityv_ K∼ 4 km/s . The intersection regionis where 0.1-μm “seeds” for the avalanche are generated from catastrophic collisions between primary and secondary ring particles. The large relative velocities in the intersection region, which approach if not exceed the elastic wave speed in solid rock, readily lead to shattering ofprimary and secondary ring particlesdown to the small, ∼0.1-μm sizes suitable for strong radial acceleration by the stellar wind (equation <ref>). We quantify these statements further in <ref>.Outside the intersection region, in the rest of the primary ring, particle relative velocities are too low to generate sub-micron seeds. In the bulk of the primary, which is composed of bound and more nearly micron-sized particles (those dominating the primary's optical depth), relative velocities are of orderv_ pri,pri∼ 100 m/sas judged from the observed vertical thickness of the ring <cit.>. Collisions at such velocitieswill chip and erode, but do not lead to catastrophic disruption, as they correspond to specific kinetic energies1/2 v_ pri,pri^2 ∼ 5 × 10^7 erg/gthat fall short of S^∗∼ 2 × 10^8erg/gthe threshold for catastrophic disruption of micron-sized, relatively flaw-free targets (; ). Thus in non-intersection regions, collisional cascades are not expected to proceed past particle sizes for which β_ w∼ 0.5, the minimum threshold for unbinding particle orbits. By contrast, the violence of collisions in the intersection region (equation <ref>)permits the creation of especially small grains attaining β_ w≫ 1.[At the risk of belaboring the point, an analogy would be with high-energy collisions in a particle accelerator; the greater the energy of the collision, the smaller the constituent particles that are unleashed. See also <cit.>.]§.§.§ Sizes of the Intersection Regionand of the Avalanche Zone By definition, the avalanche zone comprises all regionswhere ∼0.1-μm grains (accelerating projectiles) and primary ring particles (field targets that shatter into moreprojectiles) co-exist. The avalanche zone includes the intersection region where ∼0.1-μm seeds are created from collisions between primary and secondary ring particles. Avalanches can also extend beyond the intersection region because seeds can travel azimuthally(at the Keplerian speeds which they inherit at birth from their primary parents), out of the intersection region into the rest of the primary ring. Thus the characteristic dimensions of the avalanche zone and of the intersection region obey (see Figure <ref>):Δ l_ avalanche > Δ l_ intersect . The size of the intersection region scales with the thickness of the secondary ring (really, a torus); that thickness, in turn,scales with the ejectavelocities v_ ej of the catastrophic collision that gave birth to the secondary ring:Δ l_ sec∼v_ ej/v_ Ka ∼ 4 ( v_ ej/400 m/s) AU(the thickness scales with the dispersion of orbital elements of the secondary fragments, and that dispersion scales with the deviation of fragment orbital velocities from the progenitor's Keplerian velocity). Ejecta velocities v_ ej of ∼200–400 m/s are realistic; see, e.g., modeling of the Haumea collisional family in the Kuiper belt (). Our nominal estimate for Δ l_ sec is comparable to the radial (Δ a) and vertical (H) thicknesses of the primary ring, each of which is about 3 AU <cit.>. The intersection region between secondary and primary rings should be about Δ l_ seclarge (lower if the originating collision occurred toward the margin of the primary, and higher if the inclination and/or eccentricity of the secondary ring are smaller; see section <ref>):Δ l_ intersect∼Δ l_ sec . An upper limit on Δ l_ avalanche is given by the distance that seeds travel azimuthally through the primary ring between avalanches (since a given avalanchetriggered during the high phase of the stellar wind flushes the primary of all seeds):Δ l_ avalanche < v_ K t_ cycle< 8 AU .The actual length Δ l_ avalanche will be smaller than this because before seeds have hada chance to cover an azimuthal distance of 8 AU,the stellar wind (during its high phase) will have blown them radially out of the primary.Equations (<ref>)–(<ref>) constrain Δ l_ avalanche to be several AU. This is the right order of magnitude: <cit.> observe that the bright fast-moving features are Δ l_ cloud∼ 4 AU in length. Computing Δ l_ cloud from first principles, starting from the considerations outlined here, requires that we fold in the detailed time history of the stellar wind and of the resultant avalanches—this is what we do in the numerical model of <ref>, where we will find that, because of strong radial outflows and projection effects, the intersection region practically single-handedly determines the observed cloud size (i.e., Δ l_ cloud∼Δ l_ avalanche∼Δ l_ intersect).Our main takeaway point for this subsection is that, putting aside the variousorder-unity details,a secondary ring of debris created from a catastrophic collision has the right thickness, namely a few AU (equation <ref>), to be relevant for the observations by <cit.>.§.§.§ The Secondary Ring: Lifetime and Upper Mass LimitThe ejecta velocities of the originating collision place an upper limit on the surface escape velocity of the progenitor: v_ ej∼ 400 m/s implies a progenitor radius ≲ 400 km or equivalently a progenitor massM_ sec≲ 10^-4 M_⊕where we have assumed a progenitor bulk density of ∼2 g/cm^3 (only for equation <ref>; elsewhere, for less compressed grains, we adopt ρ_ p∼ 1 g/cm^3). A progenitor radius of ∼400 km is comparable to thoseof large asteroids (e.g., Vesta) and Kuiper belt objects (e.g., Varuna).The secondary ring has a finite lifespan because its particlesare destroyed by collisions with the primary ring. Every time a secondary ring particle executes an orbit, it has a probability of colliding with a primary particle equal to τ_ pri, the optical depthtraversed through the primary ring. That optical depth is on the order ofτ_ pri∼ 0.01based on detailed models derived from scattered light images <cit.>. Then the secondary ring disintegrates on a timescalet_ sec ∼ t_ orb/τ_ pri∼ 3 × 10^4 yr( 0.01/τ_ pri) .By “disintegrate” we refer only to those secondary ring particlessmall enough to be shattered by the μm-sized particles comprising the bulk of the primary's optical depth. Suchsecondary ring particles are likely to have sizes smaller than several microns. Despite their restriction in size, such particles may still carry a fair fraction of the mass of the secondary progenitor, for two reasons. The first is that in the immediate aftermath of the progenitor's destruction, ejecta mass is expected to be distributed equitably across logarithmic intervals in fragment size. This expectation arises from the “crushing” law for catastrophic single collisions <cit.>.[The crushing law should not be confused with the better known (and often abused) equilibrium cascade describing the long-term comminution of bodies as derived by Dohnanyi (; see also ). The latter does not distribute mass logarithmically evenly, but concentrates it in the largest fragments; its use is not appropriate for either short-timescale, non-equilibrium dynamics, or for particle sizes close to the blow-out limit <cit.>.] Second, as secondary ring bodies collide with one another and establish an equilibrium cascade, particles near the bottom of the cascade—i.e., μm-sized particles on marginally bound orbits—grow enormously in population because they spend much of their time at the apastra of orbits made highly eccentric by the stellar wind, away from destructive collisions in the secondary ring (see Figure 3 of ). Thus our estimate of the total secondary ring mass M_ sec may not be that much greater than the mass in μm-sized secondary ring particles; we will assume in what follows that they are within an order of magnitude of one another.Qualified to refer only to secondary particles small enough to be susceptible to disruption, the lifetime of the secondary ring t_ sec (which notably does not depend on the secondary ring mass) is some three orders of magnitude shorter than the system age. This aligns with our earlier suspicion (<ref>) that the phenomenon of escaping dust seen today is transient.In fact, the avalanches may not even last as long as t_ sec—see <ref> for the reason why.§.§.§ Seed Mass and Avalanche Mass Within the intersection region, primary and secondary ring particles destroy each other to produce 0.1-μm avalanche seeds at a rateṀ_ seed∼M_ sec/t_ sec .A couple of comments regarding this estimate: first, although (<ref>) appears superficially to account only for the destruction of secondary ring particles, it actually accounts for the destruction of primary ring particles as well, by symmetry (Ṁ_ seed∼ M_ sec / t_ sec∝ M_ secτ_ pri∝ M_ sec M_ pri, where M_ pri∼ 0.01 M_⊕ is the primary ring mass). Second, underlying statement (<ref>) is the assumption that when primary and secondary ring particles shatter each other, they invest an order-unity fraction of their mass into 0.1-μm grains. Here again we appeal to the crushing law for catastrophic single collisions, which tends to distribute mass logarithmically uniformly across particle sizes; we are not appealing to any equilibrium cascade law like Dohnanyi's (see the discussion in the penultimate paragraph of section <ref>).The 0.1-μm avalanche seeds are accelerated radially outward by the stellar wind over some fraction of t_ cycle. If the “high” phase of the stellar wind lastst_ high∼ t_ cycle/4then the seeds attain a radial velocityv_β ∼β_ w,high GM_∗/a^2 t_ high ∼ v_ K(β_ w,high/20) ( t_ high/t_ cycle/4).Other values for t_ high and β_ w,high are possible; only their product matters in (<ref>). (Of course, the product cannot be so high that the larger bodies comprising the primary and secondary rings also become unbound.)The 0.1-μm seeds slam into more typically μm-sized primary parent bodies, creating more 0.1-μm grains in an exponentially amplifying avalanche. By the time the avalanche has propagated across the radial width of the primary ring, it has traversed an optical depth τ_ pri and acquired a massM_ avalanche∼ M_ seedexp(ητ_ pri)(see, e.g., the order-of-magnitude description of avalanches by ). Here η is the number of fragments produced per catastrophic collision:η∼(1/2)m_ projv_β^2/S^∗ m_ frag∼v_β^2/2S^∗where the projectile mass m_ proj and the individual fragment mass m_ frag are assumed comparable—both are imagined to correspond to 0.1-μmgrains.Inserting (<ref>) into (<ref>) yieldsη∼ 400.An upper limit on η can be estimated by noting that the catastrophic disruption of a1-μm primary parent particle can yield no more than η_ max = 1000 fragments each of size 0.1 μm.The seed mass underlying a given cloud is that generated during the high phase of the stellar wind (seeds generated during the low phase give rise to the inter-cloud emission; see section <ref>):M_ seed ∼Ṁ_ seed× t_ high∼ 10^-8 M_⊕( M_ sec/10^-4 M_⊕) ( τ_ pri/0.01) ( t_ high/t_ cycle/4).Putting (<ref>), (<ref>), and (<ref>)together, we derive a single-avalanche mass ofM_ avalanche∼ 5 × 10^-7 M_⊕[ exp (ητ_ pri)/50] ( M_ sec/10^-4 M_⊕)which is a remarkably good match to the observationally inferred cloud mass M_ cloud∼ 4 × 10^-7 M_⊕ (equation <ref>), considering that we have not fine-tuned any of the input parameters.Of course, uncertainties in η and τ_ pri will be exponentially amplified in the avalanche gain factor exp(ητ_ pri). Increasing the gain factor would require that we reduce the secondary ring mass M_ sec to maintain the agreement between M_ avalanche and M_ cloud. Thus we can do no better than re-state our upper bound of M_ sec≲ 10^-4 M_⊕ (equation <ref>), which in turn implies that avalanche gain factors exp(ητ_ pri) ≳ 50.Although equation (<ref>) oversimplifies the avalanche dynamics (among other errors, it neglects the finite acceleration times and differing velocities of grains), our conclusions do not depend on the specific implementation of a simple exponential to describe avalanches. Stripped to its essentials, our reasoning can be recapitulated as follows: the mass of an individual cloud is M_ cloud∼ 4 × 10^-7 M_⊕ byequation (<ref>); generically, the mass unleashed in an avalanche is M_ avalanche∼ M_ seed × Gain, where Gain > 1 need not take the form of a simple exponential; M_ seed≲ 10^-8 M_⊕ by equations (<ref>) and (<ref>); then for M_ avalanche to match M_ cloud, we need Gain ≳ 40. Numerical simulations of avalanches by Q. Kral & P. Thébault (personal communication 2017; see also )indicate that such gain factors are possible, even though they are not described by the simplistic exponential in equation (<ref>).§.§.§ Avalanche Propagation TimeThe timescale for the avalanche to propagate radially across the zone (whose size is comparable to the radial width of the primary ring; see <ref>) ist_ rad,esc ∼Δ l_ avalanche / v_β∼ 5 yr( Δ l_ avalanche/4 AU) ( 20/β_ w,high)which is both shorter than t_ cycle∼ 10 yr, implying that only one avalanche occurs per stellar cycle, and also longer than our assumed acceleration time of t_ cycle/4 ∼ 2.5 yr, as required for consistency.§.§.§ Total Mass Budget: Cloud + Inter-Cloud Regions, and Starving the Avalanche That we can reproduce the observed M_ cloud (equation <ref>) using our theory for M_ avalanche (equation <ref>) is encouraging to us. The theory relies on a variety of estimates, several of which were made a priori, and it is heartening that the numbers hang together as well as they do.Here is another check on our work. Suppose (just for the sake of making this check; we will see at the end of this subsection why this supposition is probably not realistic) that avalanches continue for the entire lifetime of the secondary ring at their current pace and magnitude. Then the total mass lost from the system should equal the mass of the secondary ring multiplied by the avalanche gain factor:max M_ total,1∼ M_ secexp (ητ_ pri) ∼ 0.005 M_⊕ .(See the caveats regarding our use of M_ sec in section <ref>.) We want to check whether this (maximum) mass matches that inferred more directly from observations of mass loss (i.e., ). To make this accounting complete, we must include not only the mass lost in clouds (Ṁ_ cloud from equation <ref>, multiplied by the secondary ring lifetime t_ sec from equation <ref>) but also the mass lost from inter-cloud regions. We appeal to our model to account for the latter. The inter-cloud regions represent avalanches from seeds produced during the “low” phase of the stellar wind.Because the duration of the low phase is much less than the orbital time (a.k.a. the system dynamical time), these seeds are not accelerated much beyond their Keplerian speeds during the low phase, and so they stay roughly within the primary ring during this time. Our numerical simulation in section <ref> confirms this point—the low-phase seeds are distributed along streams that are longer azimuthally than radially. They do not escape radially until the stellar wind attains its high phase, at which time they undergo their own avalanche. Thus while inter-cloud regions are produced from seeds generated during the low phase lasting an assumed 3t_ cycle/4, and cloudsare produced from seeds generated during the high phase lasting t_ cycle/4, both sets of seeds are amplified by about the same avalanche gain factor, because avalanches only occur when the wind is in its high phase. Since the seed production rate is constant (equation <ref>), the mass-loss rate from inter-cloud regions must be 3× the mass-loss rate from clouds; the total must be 4× the latter. Hence for our second estimate of the total mass lost from the system, we havemax M_ total,2∼ 4Ṁ_ cloud t_ sec∼ 0.005 M_⊕ .The match between max M_ total,1 and max M_ total,2 is better agreement than we probably deserve.The maximum total mass lost, max M_ total, is still less than what the primary ring contains, M_ pri∼ 0.01 M_⊕, but only by about a factor of 2. The prospect of losing an order-unity fraction of the total disk mass over a small fraction of the stellar age highlights the destructive power of avalanches. But there is good reason to believe that avalanches will not continue unabated for the full lifespan of the secondary ring. If avalanche targets are strictly those at the bottom of a conventional cascade in the primary ring (∼μm-sized particles in our simple model, i.e., those dominating the primary ring's optical depth τ_ pri), then avalanches could be “starved” if the primary cascade does not supply such small targets at a fast enough rate. The primary cascade rate might only beM_ pri/t_ age∼ 0.01 M_⊕/(20 Myr), much lower than the current avalanche mass loss rate M_ total/t_ sec∼ 0.005 M_⊕/(0.03 Myr). Conceivably avalanches weaken well before t_ sec elapses as the population of μm-sized targets in the primary ring dwindles. Forecasting the long-term evolution of avalanches is left for future work. §.§ Vertical Oscillations Driven by the Magnetized Stellar WindThe stellar wind, moving with a radial velocity v_ windr̂ and carrying a magnetic field B, exerts a Lorentz force on dust grains of charge q and velocity v:F_ L = q/c[ (v - v_ windr̂) ×B]where c is the speed of light.[Working in cgs units, in which electric and magnetic fields have the same units, and in which the electrical capacitance of a spherical grain equals its radius s.] At the large stellocentric distances of interest to us, well outside the wind's Alfvén point, the magnetic field is tightly wrapped and primarily azimuthal:B≃ B_ϕϕ̂ <cit.>. The field strength around AU Mic is unknown, but we expect it to be larger than in the solar system, as AU Mic is a rapidly rotating, strongly convective, active young star—properties that all point to a strong stellar magnetic field. For reference, in the solar system, B_ϕ∼ 40μ G(a/ AU)^-1 <cit.>. Thus we expect that for the AU Mic system at a ∼ 35 AU, B_ϕ > 1μG.Given that the field is dominated by its azimuthal component, and that grain velocities |v| are much smaller than v_ wind∼ 400 km/s, the Lorentz force is dominated by the term proportional to -v_ windr̂× B_ϕϕ̂ = - v_ wind B_ϕẑ.This term is equivalent to a vertical electric field E_z ẑ = - (v_ wind/c) B_ϕẑ (the electric field seen by a grain when a magnetic field moves past it). We explore in this subsection how we might use this vertical electric field to generate the cloud vertical offsets observed by <cit.>. We state at the outset that our Lorentz force model will be found wanting in a few respects when confronted with observations (<ref>).In a simple conception of a stellar magnetic cycle, the magnetic field varies sinusoidally with period t_ mag = 2π/ω_ mag:B_ϕ = B_ϕ,0cos (ω_ mag t).The vertical equation of motion for a grain of mass m reads:z̈ = q E_z,0/mcos (ω_ mag t) = -q v_ wind B_ϕ,0/mccos (ω_ mag t).The solution for the displacement z is oscillatory with a phase that depends on initial conditions, in particular the phase in the magnetic cycle at which the grain is born. A grain born with z=ż=0 at t=0 (when the field B_ϕ is strongest with magnitude B_ϕ,0) is displaced according toz (t≥ 0) = - q v_ wind B_ϕ,0/m ω_ mag^2 c [1 - cos(ω_ mag t)];it oscillates vertically on one side of the disk, never crossingthe midplane to the other side. At the other extreme, a grain bornwith z=ż=0 at t = π/ω_ mag (when B_ϕ is strongest with the opposite polarity -B_ϕ,0) obeysz (t≥π/ω_ mag) = + q v_ wind B_ϕ,0/m ω_ mag^2 c [1 + cos(ω_ mag t)]and oscillates on the other side. Intermediate cases cross the midplane.§.§.§ Magnitude of Vertical Displacements and Parameter ConstraintsThe maximum vertical displacement is given byz_ max = | (q / m) B_ϕ,0 v_ wind t_ mag^2/2π^2 c| =2 AU ( v_ wind/400 km/s) ( q/m/5 × 10^-8 e/m_ p) ×( B_ϕ,0/30μ G) ( t_ mag/ yr)^2where the charge-to-mass ratio q/m is scaled to the proton value e/m_ p, andwe have chosen all input parameters toyield a vertical offset z_ max similar to that observed for feature “A” by <cit.>. We can relate q/m to the grain surface potential Φ:q/m = sΦ/4πρ_ p s^3/3∼ 5 × 10^-8e/m_ p( Φ/2volt) ( 0.1μ m/s)^2.Grains are charged positively by ultraviolet (UV)photoelectric emission; the grain charge equilibrates when the rate at which photoelectrons are ejected (a process that becomes less efficient as the grain increases its positive charge) balances the rate at which ambient stellar wind electrons are absorbed. A surface potential of 2 volts (SI units; equivalent to 2/300 statvolts in cgs) would be comparable to potentials for solar system grains <cit.>, and would suggest that when scaling from the solar system to AU Mic, the increased number of electrons from the stronger stellar wind nearly balances the heightened UV radiation field. A first-principles calculation of q/m is reserved for future work.Grains are “picked up” by the magnetized wind to reach stellar wind velocities if they are allowed to complete a magnetic gyration. Since the features are observed to move with velocities much less than v_ wind, pick-up must not have occurred (or at least not fully developed; cf. <ref>). This constrains1> ω_ gyro t_ mag> qB_ϕ,0/mc t_ magwhich re-written in terms of (<ref>) states that2π^2 z_ max/v_ windt_ mag < 1 0.5 ( z_ max/2 AU) ( 400 km/s/v_ wind) (yr/t_ mag)< 1implying t_ mag > 1 yr.§.§.§ Unresolved Issues with Vertical Deflections While our little Lorentz force model predicts an oscillatory vertical motion that recalls the “wavy” structure seen in images of AU Mic <cit.>, the details of this model do not fit the observations. A key unknown is the period of the magnetic cycle, t_ mag. One hypothesis sets t_ mag = 2 t_ cycle = 20 yr, in the belief that if the stellar mass-loss rate varies with time—which it would do with period t_ cycle by definition—then it would peak twice per magnetic cycle. In other words, it is imagined that dust avalanches are launched at t = n π / ω_ mag for integer n, when the stellar field is strongest irrespective of polarity.The hypothesis that t_ mag∼ 20 yrruns into the immediate difficulty that the images of AU Mic, spaced in time by significant fractions of t_ mag (they were taken in July 2010, August 2011, and August 2014) betray no vertical motion for features A and B; these clouds appear to float at a practically constant height of ∼2 AU above the midplane at three separate epochs.Faced with this phasing problem, we might hypothesize instead that t_ mag∼ 1yr, i.e., the observations just happen to catch the clouds at the same phase. A magnetic cycle period as short as ∼1 yr marginally satisfies (<ref>). Itis not obviously compatible with a stellar-mass cycle period as long as ∼10 yr (but then again, the stellar-mass loss rate might not vary with this period in the first place; see <ref> and <ref>).Other mysteries include the observed lack of moving features below the midplane, and the observation that the vertical offsetsappear smaller for the most distant clouds. The expected 1/a decay in the azimuthal magnetic field strength helps to explain this drop off, but might not be sufficient.These problems notwithstanding, Lorentz forces stillappear the most natural way of explaining the observed vertical undulations. There is no doubt that the stellar wind is magnetized, emanating as it does from a low-mass star, just as there is no doubt that sub-micron grains are charged, bathed as they are in a relatively intense stellar ultraviolet radiation field. Moreover, the sign of the field must periodically reverse, because if it did not, the magnetized wind would eventually pick up grains and accelerate them to speeds and heights far exceeding those observed. What we seem to be missing is an understanding ofthe detailed time history of the field, and how it is phased with the avalanche history. §.§ Numerical SimulationsWe construct numerical simulations to illustrate our ideas, omitting magnetic fields for simplicity. We simulate particles that represent packets of 0.1-μm grains produced in theintersection region and amplified by avalanches. In cylindrical coordinates centered on the star, the particle equations of motion read:r̈ = -GM_∗(1-β_ w)/(r^2+z^2)^3/2 r + l^2/r^3 z̈ = -GM_∗(1-β_ w)/(r^2+z^2)^3/2 z,where M_∗ = 0.6 M_⊙ and l = r^2 ϕ̇ is the specific angular momentum, conservedbecause there are no azimuthal forces (torques).The force ratio β_ w cyclesin a step-function manner between a high value β_ w,high = 20 (40) lasting t_ high = t_ cycle/4 = 2.5 yr, and a low valueβ_ w,low = β_ w,high/10 = 2 (4) lasting t_ low = 3t_ cycle/4 = 7.5 yr. All of the numerical parameters of our simulation are inspired by the various estimates made in <ref>.The simulation particles are initialized as seeds freshly produced in theintersection region. They begin their trajectories at r=r_0=35 AU, moving at circular Keplerian speed. Their spatial densityin the {ϕ, z} plane follows a two-dimensional Gaussian representing the intersection region:I_ϕ,z∝exp[ -r_0^2(ϕ-ϕ_0)^2 + z^2/2σ^2] ,where ϕ_0 points to the observer. The size of the intersection region is characterized by σ; wechoose σ = 1 AU (the corresponding full width at 2σis then 4 AU).We do not explicitly simulate the avalanche process but model it as follows.Each simulation particle is initialized with a fixed baseline mass. When that particle experiences β_ w = β_ w,high for the first time at t = t_1, its mass (read: light-scattering cross section) is subsequently amplified by an avalanche gain factor of 50 at t = t_1 + 5 yr. The delay time of 5 yr represents the finite propagation time of the avalanche across the primary ring.We integrate the equations of motion using a second-order leap-frogscheme with a fixed time step of t_ cycle/500 = 0.02yr. New particles (seeds) are generated in the intersection region at a constant rate of ∼15000 particles per year.When constructing surface brightness maps of the disk viewed edge-on, we employ a Henyey-Greenstein scattering phase function with asymmetry parameter g = 0.25.§.§.§ Simulation ResultsFigure <ref> plots the face-on column density of particles from our simulation using β_ w,high = 20. The particles trace a zig-zag path as they flow out of the primary ring. The azimuthal segments (zigs) correspond to seeds (+ their subsequent avalanche products) born during the stellar wind's low state: these particles exited the intersection region moving primarilyazimuthally. Each initially azimuthal segment was thenblown outward when the stellar wind later entered a high state, retaining their azimuthal orientation for subsequent wind cycles (once a zig, always a zig). Conversely, radially oriented segments (zags) contain the seeds (+ their subsequent avalanche products) born during the stellar wind's high state: these particles exited the intersection region moving primarily radially, and remain nearly radial in their orientation as they are blown outward, aside from a small rotational shear (once a zag, always a zag).Remarkably and encouragingly, Figure <ref> shows that the radial segments appear as clouds when viewed edge on. The radial segments have greater line-of-sight column densities than the azimuthal segments do; thus the radial segments appear as bright clouds, while the azimuthal segments represent the inter-cloud regions. We label our brightest clouds A, B, C and D; they seem to compare well with features A through D identified in Figure 2 of <cit.>. In particular, the projected separations between our simulated clouds agree with observation.Projected velocities of the simulation particles, both for β_ w,high = 20 and 40, are plotted in Figure <ref>, together with observational data from Figure 4 of <cit.>.The agreement is good for observed features A, B, and C, and less good for D and E (note that feature E as identified byis not a bright cloud but an inter-cloud region). Feature D in particular appears to be something of an outlier, not much helped by increasing β_ w,high. To improve the fit, we might look to (i) decreasing the launch radius r_0 (thereby increasing our model velocities); (ii) incorporating Lorentz forces from the stellar wind (perhaps we are seeing the onset of magnetic pick-up; see the last paragraph of <ref>); and (iii) independent re-measurement of feature velocities and re-analysis of the uncertainties (see Figure 2 ofwhich reports a new and relatively large error bar on the velocity of feature D.)§ SUMMARY AND DISCUSSIONWe have interpreted the fast-moving features observed in the AU Mic system as coherent clouds of dust produced by periodic avalanches. The clouds are composed of ∼0.1-μm grains, small enough to experience a radially outward ram pressure force from the stellar wind up to ∼20× stronger than stellar gravity. The clouds are launched from a region a few AU across—the “avalanche zone”—situated on the primary ring encircling the star at ∼35 AU, and lying at an orbital azimuth directly along the line of sight to the star. The avalanche zone marks where a body of radius ≲ 400 km and mass ≲ 10^-4 M_⊕ was catastrophically disrupted less than ∼3 × 10^4 yr ago. The debris from that event, now strewn along a secondary ring that intersects the primary ring at the location of the avalanche zone, continues to collide withprimary ring particles at km/s speeds, generating the sub-micron grains that seed the avalanche.This picture can reproduce the individual feature sizes (a few AUs) and masses (∼4 × 10^-7 M_⊕) inferred from the SPHERE and Hubble Space Telescope observations, using standard collision parameters(e.g., specific energies ∼2 × 10^8 erg/g for catastrophic disruption of competent targets, and ejecta velocitiesof a few hundred m/s) and ring parameters validated byprevious modeling of the AU Mic disk (e.g., primary ring optical depths on the order of ∼0.01). Avalanche amplification factors exceed exp(4) ∼ 50. If avalanches of the kind seen today continue unchecked over the ∼3× 10^4 yr lifetime of the secondary ring, then a total mass of ∼0.005 M_⊕ would be blown out, roughly half the total mass of the primary ring. In reality, the avalanches may weaken substantially well before the secondary ring disintegrates, as the number of available targets for disruption (∼μm-sized parents) in the primary ring drops. The rate at which mass is lost through avalanches may eventually asymptote to the rate at which the primary ring erodes mass through a conventional cascade.Our theory underscores the potential of dust avalanches to transfigure debris disks on timescales much shorter than the age of the star. Dust avalanches leverage the power of chain reactions and exponential amplification to process orders of magnitude more mass than they invest. In our story for AU Mic, avalanches began when a single body less than a few hundred km in size was shattered. That single micro-event opened a “wound” in the parent ring that is now hemorrhaging mass on macro-scales.The stellar wind is expected to be magnetized, with a field polarity that switches sign according to the stellar magnetic cycle. The resultant oscillating vertical Lorentz force (equivalently, the oscillating vertical electric field seen by grains as the magnetized wind blows past) will alternately lift and lower cloud grains, which are small enough to be significantly charged by stellar ultraviolet photoelectric emission. For plausible input parameters—ambient field strengths on the order of 10s of μG; grain surface potentials on the order of a volt; a magnetic cycle period of a few years—we can reproduce the observed magnitude of the vertical displacements. However, the detailed phasing of the vertical oscillations with time, and their observed decay with increasing projected separation, require further investigation.We have not specified with confidence the mechanism regulating the avalanche period t_ cycle (equation <ref>). We have supposed that it could set by time variability of the host stellar wind. Certainly AU Mic's wind is known to be at least some two orders of magnitude stronger than the solar wind, as judged by the star's flaring activity, and from detailed models of the primary ring which point to significant sculpting by the wind <cit.>. But to explain how the dust avalanches turn on and off, the wind would need to sustain ∼yr-long gusts every ∼10 yr, causing the star to lose mass at peak rates several thousand times larger than the solar mass loss rate.Whether such extreme and sustained episodes of stellar mass loss are actually realized, and whether we can reconcile the stellar mass-loss cycle with the stellar magnetic cycle (as traced by the cloud vertical displacements; see above), are outstanding issues.We wonder whether we might dispense with the need fordecadal-timescale variability in the stellar mass-loss rate by positing instead some kind of limit-cycle instability in the avalanche zone. The order-of-magnitude similarity between the required cycle period (∼10 yr) and the time it takes forthe avalanche to propagate radially across the parent ring (∼5 yr from equation <ref>) suggests that perhaps the avalanche zone regulatesitself—that avalanches are triggered when some threshold conditionis periodically satisfied in the intersection region. It must be some condition on the seed optical depth. Conceivably the avalanche evacuates the zone so thoroughly of seeds that the system needs time to re-fill. Avalanches are characterized by exponential amplification,and with exponentials there is extreme sensitivity to environmental conditions. At the moment we are unable to say more than this, but thepossibility of a self-regulating limit cycle (and a stellar windthat is steady but that still needs to blow strongly to achieve the large force ratios β_ w∼ 10implied bythe observed cloud velocities) seems deserving of more thought. Note that while distinct clouds are created from line-of-sight projection effects in our time-variable stellar-windmodel (<ref>), they would be created instead bytime variability in the avalanche dust production rate in the limit-cycle picture.Regardless of what drives the time variability, the avalanche zone from which dust clouds are launched remains fixed in inertial space. By contrast, orbiting sources of escaping dust (e.g., a planet—putting aside the separate problem of how a planet could be a source of dust in the first place) tend to follow the dust that they eject, since the observed velocities of the featuresare comparable to orbital velocities at ∼35 AU. This similarity of velocities leads to immediate difficulties in using a moving source to reproduce the observed spacing between features; the resultant clouds will be too closely spaced if launched from a moving source near the primary ring at ∼35 AU. Our model avoids this problem altogether because the dust launch zone is located at the intersection of two rings; this node does not move, aside from a negligible precession.Although our proposal contains significant room for improvement, it points to a few observational predictions: (i) the escaping cloud grains should have smaller sizes (the better to be accelerated outward, and to be electrically charged) than their counterparts bound to the primary and secondary rings; the size difference could be confirmed by measuring color differences between the fast-moving features and the rest of the disk (a pioneering attempt to spatially resolve color differences has been made using the Hubble Space Telescope by ); (ii) assuming there is only a single avalanche zone (i.e., a single intersection point between the primary and secondary rings) located directly along the line of sight to the star, all escaping clouds will always be seen on the southeast ansa of the disk; (iii) the primary ring should rotate such that the northwest ansa approaches the Earth while the southeast ansa recedes from it. This sense of rotation is not a crucial detail of our model, but is preferred because it situates the avalanche zone on the side of the primary ring nearest the observer, so that the clouds launched from there are seen more easily in forward-scattered than in back-scattered starlight. Spectral line observations in CO gas can test our expectation; (iv) the secondary ring has a mass < 1% that of the primary, and might be distinguished from the primary (this is admittedly ambitious) in ultra-deep exposures if the rings are mutually inclined; and (v) what goes up should come down: thefeatures should be observed to vary their vertical positions according to the stellar magnetic cycle.We thank Pawel Artymowicz, Jean-Charles Augereau, Anthony Boccaletti,Rob de Rosa, Tom Esposito, James Graham, Meredith Hughes,Paul Kalas, Quentin Kral, Eve Lee, Jamie Lomax,Maxwell Millar-Blanchaer, Ruth Murray-Clay, Élie Sezestre,Kate Su, Philippe Thébault, Jason Wang, and Mark Wyattfor discussions. Erika Nesvold provided a lucid, helpful, and encouraging referee's report. This work was performed under contract with the Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. | http://arxiv.org/abs/1707.08970v3 | {
"authors": [
"Eugene Chiang",
"Jeffrey Fung"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170727180006",
"title": "Stellar Winds and Dust Avalanches in the AU Mic Debris Disk"
} |
Laboratoire de Neurosciences Cognitives,Inserm UMR No. 960, Ecole Normale Supérieure,PSL Research University, Paris, FranceDepartment of Statistics and Department of Neurobiology,University of Chicago, Chicago, IL 60637, USA Department of Neurobiology and Department of Physics,Duke University, Durham, NC 27710, USALaboratoire de Neurosciences Cognitives,Inserm UMR No. 960, Ecole Normale Supérieure,PSL Research University, Paris, FranceNetworks of randomly connected neurons are among the most popular models intheoretical neuroscience. The connectivity between neuronsin the cortex is however not fully random, the simplest andmost prominent deviation from randomness found in experimental databeing the overrepresentation of bidirectional connections amongpyramidal cells. Using numerical and analytical methods, weinvestigated the effects of partially symmetric connectivity onthe dynamics in networks of rate units. We consider the two dynamicalregimes exhibited by random neural networks: the weak-couplingregime, where the firing activity decays to a single fixed pointunless the network is stimulated, and the strong-coupling or chaoticregime, characterized by internally generated fluctuating firingrates. In the weak-coupling regime, we compute analytically, for anarbitrary degree of symmetry, the autocorrelation of networkactivity in the presence of external noise. In the chaotic regime, weperform simulations to determine the timescale of the intrinsicfluctuations. In both cases, symmetry increases the characteristicasymptotic decay time of the autocorrelation function and thereforeslows down the dynamics in the network. Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks Srdjan Ostojic December 30, 2023 ========================================================================================================== § INTRODUCTIONThe dynamics and function of a network of neurons is to a largeextent determined by its pattern of synaptic connections. In themammalian brain, cortical networks exhibit a complex connectivitythat to a first approximation can be regarded as random. Thisconnectivity structure has motivated the study of networks ofneurons connected through a random synaptic weight matrix withindependent and identically distributed (i.i.d.) entries, which have become acentral paradigm in theoreticalneuroscience <cit.>.Randomly connected networks of firing-rate units exhibit a chaotic phase <cit.>, which can be exploited as asusbstrate for complexcomputations <cit.>.Networks of randomly connected spiking neurons also exhibit rich dynamics that can account forthe highly irregular spontaneous activity observed inthe cortex invivo <cit.>.Importantly, these models are to a large extent amenable to a mathematicalanalysis, which allows for a thorough understanding of themechanisms underlying their dynamics. Detailed analyses of experimental data on cortical connectivity have howeveridentified patterns of connectivity that strongly deviate from thei.i.d. assumption <cit.>. Themost prominent of such deviations is the overrepresentation ofreciprocalconnections <cit.>,and the fact that synapses of bidirectionally connected pairs ofneurons are on average stronger than synapses of unidirectionallyconnected pairs. These observations are consistent with a partiallysymmetric connectivity structure, intermediate between full symmetryand full asymmetry. How partial symmetry in the connectivity impactsnetwork dynamics is not yet understood, in part because suchpartial symmetry renders the mathematical analyses morechallenging <cit.>. Here we study theimpact of partial symmetry in the connectivity structure on the dynamicsof a simple network model consisting of interacting rateunits. Depending on the overall strength of coupling, such a networkcan display either a stable or a chaotic regime of activity, as inthe random asymmetric case <cit.>. We examined how thedegree of symmetry in the network influences the temporal dynamicsin both regimes. For the stable regime, we exploited recent resultsfrom random matrix theory <cit.> toderive analytical expressions for the autocorrelationfunctions. These expressions demonstrate that increasing thesymmetry in the network leads to a slowing down of thedynamics. Numerical simulations in the chaotic regime show a similareffect, with time scales increasing far more substantially withsymmetry than in the fixed point regime. Altogether, our results indicate that symmetry in the connectivity can act as an additionnal source of slow dynamics, an important ingredient for implementing computations in networks of neurons <cit.>. § DESCRIPTION OF THE MODELWe consider a network of N fully connected neurons, each described by an activation variable (synaptic current) x_i, i=1,…,N, obeying x̣_i/ṭ = -x_i + g ∑_j=1^NJ_ijϕ(x_j), where g is a gain parameter that modulates the strength ofrecurrent connections, and where ϕ(·) is the input-outputtransfer function that transforms activations x_i into firingrates. This transformation is non-linear and we model it asϕ(x)=tanh(x) for mathematical convenience (see <cit.> for studies of network models withdifferent choices of ϕ). The elements J_ij of theconnectivity matrix are drawn from a Gaussian distribution with zeromean, variance 1 / N, and correlation [J_ij J_ji]_J = η / N, with the square brackets [·]_J denoting an average overrealizations of the random connections. The parameter η is thecorrelation coefficient between the two weights connecting pairs ofneurons, and quantifies the degree of symmetry of theconnections. For η=0 the elements J_ij and J_ji areindependent and the connectivity matrix is fully asymmetric; forη=1 the connectivity matrix is fully symmetric; For η=-1 itis fully antisymmetric.In Secs. I–IIIA we study the full rangeη∈[-1,1], while in Secs IIIB–IV we focus on η∈[0,1]. § DYNAMICAL REGIMES OF THE NETWORKFor fully asymmetric matrices, previous work has shown that thenetwork activity described by (<ref>) undergoes a phasetransition at g=1 in the limit of large N <cit.>. For g < 1 theactivity for all units decays to 0, which is the unique stable fixed point of thedynamics <cit.>, while for g > 1 the activity is chaotic. Such atransition can be partially understood by assessing the stability ofthe fixed point at x_i=0 for i=1, …, N. If we linearizeEq. (<ref>) around this fixed point we obtain thestability matrix, with components M_ij = -δ_ij + gJ_ij. The eigenvalues of M_ij are therefore those of the matrix J_ij, scaledby the gain g and shifted along the real axis by -1. In the limitN→∞, for a connectivity matrix J_ij whose entries arei.i.d. Gaussian random variables of zero mean and variance 1/N, eigenvaluesare uniformly distributed in the unit disk of the complexplane <cit.>.This implies that the eigenvalues of the stability matrix have a negative realpart as long as g < 1, and therefore that the fixed point at 0 is stable inthat range. An analogous transition occurs when connections are partiallysymmetric. The presence of correlations among weights deforms thespectrum of eigenvalues into an ellipse, elongating its major radiusby a factor of 1 + η and shortening the minor radius by afactor 1 - η <cit.>[Fig. <ref>a]. This property is usually referred to as theelliptic law. For the network described by (<ref>) such adeformation causes the fixed point at x_i = 0 for i=1,…,Nto lose its stability at g = 1 / (1 + η)[Figs <ref>b and <ref>c]. In other words, symmetry lowers thecritical coupling. Our goal is to characterize how the degree of symmetry in theconnections affects the network activity on each side of theinstability: the relaxation response of the network at low gains andthe chaotic self-generated activity observed at stronggains. Our description of the network activity will be based on theaverage autocorrelation function, C(τ) = 1/N∑_i=1^N[x_i(t) x_i(t + τ)]_J, where the average is over both the population and the realizations of theconnectivity matrix <cit.>, and wherewe are assuming for now that the system is stationary.§ DYNAMICS IN THE FIXED-POINT REGIME§.§ Derivation of the autocorrelation functionIn the fixed-point regime, the activity decays to zero unless the network is stimulated by external inputs. To characterize the dynamics of the network in this regime, we induce network activity by feeding each neuron with independent Gaussian white noise <cit.>. The amplitude of this noise is assumed to be small enough so that the synaptic activation of all neurons lies within the linear range of their input-to-rate transfer function [see Fig. <ref>f for the range of validity of that approximation]. Under these conditions, ϕ(x) can be approximated by its first order Taylor expansion ϕ(0) + ϕ'(x)|_x=0 x = x, and the dynamical equations become x̣̣⃗(̣ṭ)̣/ṭ = (-1 + g𝐉)x⃗(t) + σξ(t), where x⃗(t)=(x_1(t), …, x_N(t))^T, 1 is theidentity matrix, J⃗ is the connectivity matrix, andξ(t) = (ξ_1(t), …, ξ_N(t))^T is a vectorof independent white noise sources of zero mean and unit variance:ξ_i(t) = 0, ξ_i(t) ξ_j(t') = δ_ijδ(t - t'), with angular brackets representing averages overnoise realizations. The parameter σ is the standard deviationof the white noise injected into neurons. The time scales displayed by a linear system like (<ref>)are strongly affected by the real part of the eigenvalues of thesystem's stability matrix and, in particular, they get longer aseigenvalues get closer to the imaginary axis. To disentangle thistype of slowing down from the effects due to symmetry alone, we varythe parameter η while keeping the spectral gap fixed. Byspectral gap we mean the distance between the spectrum ofeigenvalues of the stability matrix M_ij,Eq. (<ref>), and the imaginary axis [Fig. <ref>a]. From the elliptic law, the eigenvalue of thestability matrix with the largest real part is z = -1 + g(1 + η),and we can keep the spectral gap at δ by setting the gain tog = (1 - δ) / (1 + η).The system described by (<ref>) is linear and can be solved by diagonalizing the connectivity matrix. The matrix J⃗ admits a set of right eigenvectors {R⃗_1, …, R⃗_N} that obey J⃗R⃗_i = λ_i R⃗_i for i=1,…,N. These eigenvectors are in general complex-valued and, except for the symmetric case η=1, not orthogonal to one another, which implies that J⃗ cannot be diagonalized through a unitary transformation. Matrices of this kind are called non-normal and do not commute with their transpose conjugate: J⃗J⃗^†≠J⃗^†J⃗ <cit.>. Even if non-normal matrices cannot be diagonalized by an orthogonal set of eigenvectors, it is always possible to form a biorthogonal basis by extending the set of right eigenvectors with the set of left eigenvectors, which obey L⃗_i^†J⃗ = λ_i L⃗^†_i. This extended basis is biorthogonal in the sense that L⃗^†_i R⃗_j = δ_ij. We can summarize all these properties in a compact way by defining the square matrices R⃗ and L⃗ that result from adjoining in columns the set of, respectively, right and left eigenvectors, and by introducing the diagonal matrix Λ that contains the eigenvalues λ_i of J⃗ in its diagonal entries. In this notation the biorthogonality condition is L⃗^†R⃗ = 1⃗ and the eigenvalue equations for the right and left eigenvectors read J⃗R⃗ = R⃗Λ and L⃗^†J⃗ = ΛL⃗^†.We can now write the formal solution of (<ref>):x⃗(t) = σ∫_-∞^t^(-1⃗ + gJ⃗)(t - s)ξ(s) ṣ = σR⃗∫_-∞^t^(-1⃗ + g Λ)(t - s)R⃗^-1ξ(s) ṣ ,where in the last equality we used the basis of right eigenvectors to write J⃗ = R⃗ΛR⃗^-1 and we implicitly expanded the exponential in its power series to obtain the final result. From this expression we can derive the population-average autocorrelation for a particular realization of the connectivity:C_J(τ) = 1/Nx⃗^†(t)x⃗(t + τ) =1/Nx⃗(t + τ)x⃗^†(t) = σ^2/N∫_0^∞^-2 u - τ{R⃗^†R⃗ ^gΛ(u + τ)L⃗^†L⃗ ^gΛ^†u} ụ.In the second equality we used the cyclicity of the trace, and in the last line we changed the integration variable to u=t - s and we used the biorthogonality condition to write R⃗^-1 = L⃗^†. The average over noise amounts to applying the identity ⟨ξ(t)ξ^†(t') ⟩ = σ^2 1δ(t - t'). Note that the σ^2 appears as an overall factor, so we can set σ=1 without loss of generality.We can simplify (<ref>) by introducing the so-called overlap matrix, with componentsO_ij = (L⃗^†L⃗)_ij (R⃗^†R⃗)_ji,and which characterizes the correlations between left and right eigenvectors <cit.>. Equation (<ref>) then becomesC_J(τ) = 1/N∫_0^∞^-2 u - τ∑_i=1^N∑_j=1^N^g λ_i(u + τ)O_ij^g λ̅_ju ụ.If the connectivity matrix were normal, the overlap would be the identity matrix and the autocorrelation (<ref>) would just be a sum of independent contributions—one per eigenvalue. These contributions are coupled for non-normal matrices.We can make further analytical progress by studying the autocorrelation (<ref>) in the limit N →∞, in which the differences of C_J(τ) across realizations of the connectivity matrix disappear, and the autocorrelations of all units become close to the population average [Fig. <ref>g]. In that limit sums over indices are replaced with integrals over eigenvalues, while the overlap matrix is replaced with the local average of the overlap, defined asD(z_1, z_2) = lim_N→∞4/N[ ∑_i=1^N∑_j=1^N O_ijδ^2(z_1 - λ_i) δ^2(z_2 - λ_j)]_J.Here z=x +y are complex numbers and we defined the complex Dirac delta as δ^2(z) ≡ (1/2) δ(x) δ(y) so that it satisfies the normalization condition ∫δ^2(z) z≡∫δ^2(z) ẓẓ̣̅ = 1. After taking the limit N→∞, Eq. (<ref>) becomesC(τ) = ∫_0^∞^-2 u - τ A(u, τ) ụ.where we definedA(u, τ) = 1/4∬^g(z_1 + z̅_2) u + g z_1 τ D(z_1,z_2) z_1z_2.Each of the integrals in Eq. (<ref>) is over complex values, and involves the expression of D(z_1, z_2) for the ensemble of Gaussian random matrices with partial symmetry, which was derived using diagrammatic techniques in <cit.> and whose functional form can be found in Appendix <ref>. We used the result of <cit.> to evaluate the double complex integral A(u, τ) in Eq. (<ref>). The details of the evaluation are given in Appendix <ref>, and the result isA(u, τ) = A_1(u, τ) + A_2(u, τ),withA_1(u, τ) = (1 + η^2) I_0(g ψ(u, τ; η) ) - 2 η(1 + 2(1 -η)^2τ^2/ψ(u, τ; η)^2)I_2(gψ(u, τ; η)),A_2(u, τ) = -1/g^2 u (u + τ)∑_k=1^∞η^k k^2 I_k(2g√(η)u)× I_k(2g√(η)(u + τ)),where I_k(·) is the modified Bessel function of order k, and where in Eq. (<ref>) we definedψ(u, τ;η) = 2 √((1 + η)^2 u (u + τ) + ητ^2).The autocorrelation is finally computed from Eq. (<ref>), integrating numerically over u.Expressions. (<ref>), (<ref>), (<ref>) are valid for full range -1 ≤η≤ 1. For negative η we replace √(η) with √(|η|) and apply the identity I_ν( z) = ^ν J_ν(z), which is valid for integer ν.The analytical prediction given by Eqs. (<ref>) and Eqs. (<ref>)–(<ref>) matches with the autocorrelation estimated from numerical simulations [Fig. <ref>b], although for long time lags the numerical estimate becomes noisy due to finite-size effects. To check the validity of our prediction also at long time lags, we compared our analytical prediction with three alternative derivations [Fig. <ref>c]. One such derivation consists of estimating the autocorrelation for large but finite N, by computing numerically the eigenvalues and eigenvectors of randomly generated matrices, evaluating the time integral of Eq. (<ref>), which givesC_J(τ) = -1/N∑_i=1^N∑_j=1^NO_ij^-(1 - gλ_i) τ/2 + g(λ_i + λ̅_j),and then by averaging C_J(τ) over multiple realizations of the connectivity matrix. Another derivation is based on dynamical mean-field theory <cit.>, which gives rise to a set of integro-differential equations involving C(τ) that can be solved numerically (Appendix <ref>). Finally, we numerically computed the inverse Fourier transform of the power spectrum derived in <cit.> for this same system. <cit.> used a perturbative method to derive the system of integro-differential equations (<ref>) and (<ref>), which they solved for the correlation and reponse functions by using a Laplace transform. All derivations yield the same result, except for the deviations we observe when applying Eq. (<ref>) at long τ, which are caused by finite-size effects.Our results show that an increase in symmetry tends to spread autocorrelations toward longer time lags, and that this effect gets larger the closer the system gets to the onset of chaos [Fig. <ref>e]. An intuitive explanation for this slowing down is that the deformation of the eigenspectrum caused by symmetry increases the density of eigenvalues with small imaginary parts, thereby enlarging the contribution of low-frequency modes.§.§ Behavior at long time lagsWhile equations (<ref>)–(<ref>) are exact, they provide little analytical insight into how the autocorrelation depends on parameters. A more explicit dependence can be obtained by evaluating C(τ) in the limit of long τ. We relegate the details of the calculation to Appendix <ref> and summarize the main results here. The analysis shows that, in the fixed point regime, there exist two subregimes of activity that differ in how the asymptotic decay rate of the autocorrelation depends on the symmetry parameter η and the spectral gap δ. For small values of η and δ, the autocorrelation decays as a pure exponential at long τ (regime I),C(τ) = F_I(η, δ) ^-τ G_I(η, δ),withF_I(η, δ) = δ^-1/2(1 - η)^2/2√(2) (1 - δ), G_I(η, δ) = 1 - η/1 + η√(2δ - δ^2).Conversely, for sufficiently large values of η and δ autocorrelation for long τ can be approximated by a power multiplied by an exponential decay (regime II):C(τ) = τ^-3/2 F_II(η, δ) ^-τ G_II(η, δ),withF_II(η, δ) = 1/4√(π)( 1 + η/1 - δ)^3/2[2 η^-1/4(1 + η^2)/δ(1 + η) - [1 - √(η)]^2aa- η^5/6 (1 + η)/(1 - √(η))^2 + 2 √(η)δ]G_II(η, δ) = (1 - √(η))^2 + 2δ√(η)/1 + η.A comparison between the asymptotic expression in Eq. (<ref>) and the full expression for the autocorrelation function reveals, however, that the power law is not observed in practice because the range below the cutoff falls below the values of τ where the asymptotic approximation starts matching the exact expression.Figure <ref>a shows the exact parameter region of each asymptotic regime, after transforming the spectral gaps into gains. In both regimes the autocorrelation's asymptotic decay rate matches the exact result for time lags longer than a few time units [see Fig. <ref>a, lateral panels]. It seems therefore reasonable to associate the time scale of the autocorrelation with the inverse of G_I,II(η,δ) [see Eqs. (<ref>) and (<ref>)], where the subindex I, II is chosen according to the subregime found at the parameter values (η, δ). The asymptotic time scale of the autocorrelation increases monotonically with symmetry regardless of the subregime the network operates in [Fig. <ref>b], although this dependence is convex in the exponential subregime and concave in the power-law-with-cutoff regime [in Fig. <ref>b see the curves split by the red dots, which mark the boundary between subregimes]. Note also that as the spectral gap δ shrinks to 0 the system enters the exponential regime and timescales diverge as δ^-1/2, according to Eq. (<ref>). §.§ Effect of overlaps As shown in Eq. (<ref>), the autocorrelation function in general depends on two factors, the full eigenspectrum of the connectivity matrix and the overlaps between eigenvectors, both of which are modified when η is changed. To disentangle the effects of the changes in the eigenspectrum and the changes in eigenvector overlaps, in this section we compare the autocorrelation we derived in Sec. <ref> with the autocorrelation we would obtain if we assumed that the eigenvectors of J⃗ were orthogonal. If that were the case, the autocorrelation is computed as a sum of decoupled contributions associated with the different eigenvalues and, in particular, the distribution of eigenvectors would play no role in the result (see Appendix <ref> for details). Figure <ref>a shows the predicted autocorrelations, both including and excluding the contribution from the overlap (<ref>). As expected, both predictions coincide for η=1 and they increasingly depart from each other for decreasing values of η. To better characterize this difference we show the variance C(0) as a function of the symmetry parameter for several values of the spectral gap [Fig. <ref>b]. Remarkably, the variance decreases with symmetry, but the opposite occurs when we remove the contribution from eigenvector overlaps [see 'with' and 'without' overlap curves in Fig. <ref>b]. In both cases the variance increases as spectral gaps get smaller, which is consistent with the fact that the restoring drive towards the fixed point gets weaker as the spectral gap gets smaller. This effect is, however, much subtler when overlaps are not taken into account <cit.>.The overlap also contributes to the overall time scale of the autocorrelation, which we define by the quantity τ̂ = ∫_0^∞ t C(t) ṭ / ∫_0^∞ C(t) ṭ. This definition guarantees that for an exponential autocorrelation C(τ) ∝exp(-|τ|/τ_0) the overall time scale is exactly τ_0, and provides a rough estimate of a natural time scale for autocorrelations with more complex dependences. The numerical evaluation of T shows that the overall time scale is systematically smaller if the contribution of the eigenvalues is removed [Fig. <ref>c]. Unsurprisingly, either with or without the overlap contribution the timescale gets longer as the spectral gap gets smaller. Note also that the overall time scale T varies non-monotonically with the symmetry parameter [Fig. <ref>c], unlike the asymptotic dependence shown in Fig. <ref>b.§ DYNAMICS IN THE CHAOTIC REGIME In the chaotic regime, the network generates its own fluctuating activity without the need for external noise. Recall that chaotic activity emerges as soon as the largest of the real parts of the eigenspectrum, given by -δ and usually called spectral abscissa, becomes positive. We follow the strategy of the preceding section and we keep the spectral abscissa fixed while we vary the symmetry parameter η.The evolution of firing activities shown in Fig. <ref>a suggests that in the chaotic regime the self-generated fluctuations get slower as η increases. This slowing is accompanied by an increasing tendency of firing rates to linger around the extreme values of their dynamical range, as reflected by an increasingly bimodal distribution of currents x and rates ϕ(x) when η increases [Fig. <ref>b]. We quantified the slowing down of the fluctuations with the population-average autocorrelation. For η=0 the autocorrelation can be derived self-consistently in the limit of infinitely large networks, using the dynamical mean-field approach (, see also Appendix <ref> for a general derivation). Unfortunately, this method does not lead to a closed-form solution for the autocorrelation as soon as η >0 (see Appendix <ref>) and we have to resort to numerical estimates, summarized in Fig. <ref>c for several values of η. For completeness we also include the autocorrelation functions for fixed gain, rather than fixed spectral abscissa [Fig. <ref>d].The numerical estimates show that the time scale associated with the autocorrelation increases strongly as a function of η and is considerably longer than in the fixed-point regime [Fig. <ref>e]. Such a slowing is rather insensitive to whether we fix the spectral abscissa or the gain, despite the fact that the variance C(0) varies far more strongly when gain is fixed [Fig. <ref>d]. Quite strikingly, for η=1 fluctuations become slower as time goes by, and our initial assumption that the activity is stationary does not hold. The population-averaged autocorrelation C(t, t+τ)=[x(t) x(t + τ)]_J,N at different points in time shows that the characteristic timescale of the autocorrelation grows with t [Fig. <ref>d], a signature of aging dynamics <cit.>. For lower values of η, the dependence on the autocorrelation on the two timescales is less clear. Due to strong finite-size effects, it is difficult to determine from simulations alone whether aging appears also when the connectivity is not fully symmetric.§ DISCUSSION In this work we examined the effect of partially symmetric connectivity on the dynamics of randomly connected networks composed of rate units. We have derived an analytical expression for the autocorrelation function in the regime of linear fluctuations around the fixed point, and shown that increasing the symmetry of the connectivity leads to a systematic slowing-down of the dynamics. Numerical simulations confirm that a similar phenomenon takes place in the chaotic regime of the network.The impact of the degree of symmetry of the connectivity matrix on the dynamics of neural networks has been a long-standing question in theoretical neuroscience. Theorists initially focused on fully symmetric networks of binary spin-like neurons <cit.> for which tools from equilibrium statistical mechanics could be readily applied <cit.>. After these initial studies, the realization that brain networks are not symmetric led physicists to investigate the dynamics of networks whose connectivity matrix has a random antisymmetric component. It was found that departures from full symmetry destroys spin-glass states, while retrieval states in associative memory models were found to be robust to the presence of weak asymmetry <cit.>.Theorists also studied fully asymmetric networks, using rate models <cit.>, networks of binary neurons <cit.> and networks of spiking neurons <cit.>. In all these models, chaotic states were shown to be present for sufficiently strong coupling. In networks of spiking neurons, chaotic states are characterized by strongly irregular activity of the constituent neurons, with self-generated fluctuations that evolve on fast time scales. Motivated by experimental findings, recent studies have considered synaptic connectivity matrices where bidirectionally connected pairs are overrepresented with respect to a random network. In contrast with our model, in which no structure exists beyond the level of pairs of neurons, these studies have considered structured connectivity matrices in which partial symmetry is a consequence of a larger-scale structure. <cit.> considered a connectivity clustered into groups of highly connected neurons and demonstrated that clustered connectivity could lead to slow firing-rate dynamics generated by successive transitions between up and down states within individual clusters. An overrepresentation bidirectional connections can also arise in networks with broad in- and out-degree distributions, which affect the dynamics and the stability of asychronous states in such networks <cit.>. Other works have considered connectivities with non-trivial second-order connectivity statistics, and studied the resulting network dynamics. <cit.> analyzed how the presence of connectivity patterns involving two connections (not only bidirectionally connected pairs) affected the tendency for a neuronal network to synchronize, while <cit.> focused on the oscillatory activity generated by partially antisymmetric, delayed interactions. Taking a completely different approach, <cit.> showed that maximizing the number of patterns stored in a network entails an overrepresentation of bidirectionnally connected pairs of neurons, which suggests that partially symmetric connectivity may be a signature of optimal information storage.An important ingredient in our analysis is the fact that partially symmetric interaction matrices are non-normal, i.e., they are not diagonalizable by a set of mutually orthogonal eigenvectors. The influence of non-normal connectivity on network dynamics has recently received a considerable attention in the neuroscience community <cit.>. Particularly relevant to our study is the work by <cit.>, who quantified the effects of non-normality on the amplitude of the autocorrelation function in random networks. Here we extend their results by studying the full temporal shape of the autocorrelation function and by characterizing how this shape is affected by the partial symmetry of connections.The present work is also related to models of disordered systems and spin glasses <cit.>. Most studies in that field were inspired by physical phenomena and considered fully symmetric interaction matrices. In that context, a major result has been the discovery of aging, the phenomenon by which dynamics become slower the longer the system evolves <cit.>. This phenomenon has been observed in a broad class of complex systems characterized by configuration spaces with extremely rugged energy landscapes, composed of many local minima surrounded by high barriers. In these systems a random initial condition is very likely to set the system far from a stationary state and initiate a very slow relaxation towards a fixed point. The relaxation takes infinitely long for N →∞ because, loosely speaking, the longer the system evolves, the deeper it wanders in the valleys of the energy landscape, and the harder it becomes for it to find configurations of lower energy <cit.>.Whether fully symmetric interactions are necessary to observe aging does not seem to be entirely understood, as to the best of our knowledge only a few works seem to have considered partially symmetric coupling <cit.>. Fully asymmetric networks have received more attention, but they do not exhibit any aging phenomena. Here we interpolated between fully asymmetric and fully symmetric networks, and have been able to obtain mathematical results only in linear networks, in the non-chaotic regime. Interestingly, we found that the partially symmetric case is mathematically more complex than the symmetric or asymmetric limits. This can be seen in the form of autocorrelation function (<ref>), which simplifies considerably when η=0 or η=1, but also in the Dynamical Mean Field Theory (Appendix <ref>), where a coupling between the autocorrelation and the response function appears for η>0. This additional complexity results from the fact that the influence of a single neuron's activity on all the other neurons is fed back through couplings that are correlated with the neuron's activity, due to the partial symmetry of the connections. More specifically, the inputs received by neuron i are given by terms ∑_j J_ijϕ(x_j), which are themselves influenced by the activity of neuron i. As a result, neuron i influences its own activity by an amount proportional to the sum ∑_j J_ijJ_jiϕ(x_i), a random number of mean ηϕ(x_i). The effect of this feedback loop is that the individual input terms exhibit correlated fluctuations. When η=0, the inputs received by neurons are uncorrelated and their sum can be approximated by a Gaussian random variable whose mean and variance can be determined self-consistently <cit.>. At the other extreme, when η=1, the inputs received by neurons are correlated, but the dynamics of the network can be described as a relaxation of an energy function and the standard machinery of statistical mechanics can be used. For other values of η, none of these analytical strategies can be applied and the analysis becomes more complex. Demonstrating analytically whether aging dynamics are present in partially symmetric, non-linear networks seems an outstanding open problem.Our results on the autocorrelation function in the linear network are closely related to recent results published by <cit.>, who used a different set of methods to compute the power spectrum of the network activity, i.e., the Fourier transform of the autocorrelation function of the same model we investigated. Unlike <cit.>, we obtained the autocorrelation directly in real time, although our results are fully consistent with theirs in that we obtain the same two regimes with the same asymptotic timescales, depending on the symmetry and the gain (or leak, in their case. cfr. Fig. 1 of <cit.> with Fig. <ref>a).Our work provides a potential bridge between two seemingly unrelated observations in neuroscience. The first is the observation of strong correlations between the synaptic strengths in pairs of cortical pyramidal cells, the main excitatory neuronal type in cerebral cortex, by multiple groups using in vitro electrophysiological recordings <cit.>. These correlations are a consequence of two features of the connectivity: First, there exists an overrepresentation of bidirectionally connected pairs, compared to a Erdős-Rényi network with the same connection probability.For instance, <cit.> found a connection probability of c=0.116 in pairs of neurons whose somas are less than 100 μm apart, while the probability that a pair of such neurons are connected bidirectionally is approximately 4c^2. This degree of overrepresentation has been found in multiple cortical areas, except in barrel cortex where no such overrepresentation exists <cit.>. Second, synaptic connections in bidirectionally connected pairs are on average stronger than those in unidirectionally connected pairs, and are significantly correlated <cit.>. These observations lead to estimates of η∼ 0.5, a value that, according to our model, would lead to a significant increase in autocorrelation time scales compared to a random asymmetric connectivity.The second is the observation of long time scales in the autocorrelations of neuronal activity from in vivo electrophysiological recordings (see e.g. <cit.>). Interestingly, the time scales of these autocorrelations increase from sensory to higher level areas such as the prefrontal cortex. Several mechanisms have been proposed to account for this phenomenon: differences in the level of expression of slow NMDA receptors <cit.>, or an increase in the strength of recurrent connectivity <cit.> which could in particular lead to the presence of multiple fixed points that can slow down the dynamics <cit.>. Our results suggest that this increase in time scale could also be due to an increase in the degree of symmetry of cortical connectivity. This would be consistent with the study of <cit.>, who showed that the overrepresentation of bidirectionally connected pairs of neurons is significantly stronger in prefrontal cortex than in visual cortex.From a neuroscience point of view, the model considered here is an extremely simplified model of cortical networks because it lacks the fundamental constraint that neurons are either excitatory or inhibitory, and because it does not constrain firing rates to be positive. These simplifications were made for the sake of mathematical tractability. A few recent studies have investigated how these two constraints influence the dynamics of such networks <cit.>. Extending those works to connectivity with segregated excitation and inhibition and partial symmetry is an important direction for future work that might be facilitated by recent developments in random matrix theory <cit.>.§.§.§ Acknowledgments We thank Johnatan Aljadeff for his comments on a previous version of the manuscript. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007–2013/ under REA grant agreement 301671. This has also been funded by the Programme Emergences of City of Paris, and the program “Investissements d'Avenir” launched by the French Government and implemented by the ANR, with the references ANR-10-LABX-0087 IEC and ANR-11-IDEX-0001-02 PSL* Research University.The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. § DERIVATION OF THE DOUBLE COMPLEX INTEGRAL We summarize here the derivation of the double complex integral of Eq. (<ref>). Before doing so, we sketch the derivation of the local density of the overlap done in <cit.>, as this will let us introduce some notation and pave the way for our calculation.For a complex variable z=x +y, with x and y real and with conjugate z̅=x -y, we define the Wirtinger derivatives ∂/∂ z = (∂ / ∂ x - ∂/ ∂ y) / 2 and ∂/∂z̅ = (∂ / ∂ x + ∂/ ∂ y) / 2, which obey ∂ z / ∂ z = ∂z̅ / ∂z̅ = 1 as well as ∂z̅/ ∂z = ∂ z / ∂z̅ = 0. The complex differential is defined to be z≡ẓ ẓ̣̅ = 2 x̣ ỵ, where the factor 2 comes from the Jacobian. We also define the complex Dirac delta so that it obeys the relation ∫δ^2(z) z≡ 1, which implies δ^2(z) = (1/2) δ(x)δ(y) given our convention for the complex differential. Two useful identities for the δ function in the complex plane areδ^2(z) = 1/2π∂/∂z̅1/z = 1/2π∂/∂ z1/z̅,which can be checked by integrating over z and applying the complex version of Green's theorem:∫(∂ v/∂ z + ∂v̅/∂z̅) z = ∮ (v dz̅ - v̅ dz),where v and v̅ are to be considered independent functions.The resolvent, defined for any matrix 𝐉 as (z1 - 𝐉)^-1, is a key quantity in the analysis of random matrices because it can be ensemble-averaged using standard methods and can be related to quantities of interest. So, for example, the empirical density of eigenvalues of a given 𝐉,ρ_J(z) = 1/N∑_i=1^Nδ(x - λ_i) δ(y - λ_i) = 2/N∑_i=1^Nδ^2(z - λ_i),can be expressed thanks to the identities (<ref>) asρ_J(z) = 1/π∂/∂z̅1/N∑_i=1^N1/z - λ_i = 1/π∂/∂z̅1/N (z1 - 𝐉)^-1.This quantity is hard to compute for any particular realization at finite N, but it becomes easier to handle in the limit of large N, where all empirical densities converge to the average densityρ(z) = [ ρ_J(z) ]_J = 1/π∂/∂z̅[ 1/N (z1 - 𝐉)^-1]_J.Deriving the average density, therefore, amounts to computing the function G(z) = [ (z1 - 𝐉)^-1 / N]_J in the large N limit.The local density of the overlap can be derived in a similar manner using the spectral decomposition (z1 - 𝐉)^-1 = ∑_i=1^N𝐑_i (z - λ_i)^-1𝐋^†_i, where R⃗_i and L⃗_i are the right and left eigenvectors of J⃗, respectivelyt. If we substitute the definition of the overlap matrix into Eq. (<ref>) and we use the identities (<ref>) we obtain D(z_1, z_2) = 4/N[ ∑_i,j=1^N(L⃗^†L⃗)_ij (R⃗^†R⃗)_jiδ^2(z_1 - λ_i) δ^2(z_2 - λ_j) ]_J = 1/Nπ^2∂/∂z̅_1∂/∂ z_2[ ∑_i,j=1^N R⃗_i1/z_1 - λ_iL⃗^†_iL⃗_j1/z̅_2 - λ̅_jR⃗^†_j]_J = 1/π^2∂/∂z̅_1∂/∂ z_2[ 1/N 1/z_11 - 𝐉1/z̅_21 - 𝐉^†]_J,and the problem reduces to computing the quantityG(z_1, z_2) = [ 1/N 1/z_11 - 𝐉1/z̅_21 - 𝐉^†]_J.The expression of G(z_1, z_2) for the ensemble of Gaussian random matrices with partial symmetry was derived by <cit.>. The basic idea behind their calculation is to expand resolvents in power series, average over the disorder term by term, and organize the sums so that a recursive relation can be established and ultimately solved (for a thorough description of the method, see also <cit.>). The result is a complex function that takes the valueG(z_1, z_2) = 1/1 - η^2 ×( (1 - η^2)^2 + η (z_1^2 + z̅_2^2) - (1 + η^2) z_1 z̅_2/|z_1 - z_2|^2 - 1)when both z_1 and z_2 lie inside the ellipse centered at the origin and which has major and minor radii 1 + η and 1 - η, respectively. We will call this ellipse E_η for later convenience. When z_1 and z_2 lie outside E_η we have insteadG(z_1, z_2) = h_1 h̅_21 - h_1 h̅_2,whereh_1 = z_1 - √(z_1^2 - 4η)/2η,h̅_2=z̅_2 - √(z̅_2^2 - 4η)/2η.Right on the ellipse E_η, |h_i| = 1. When both z_1 and z̅_2 lie outside the ellipse, the function G(z_1, z_2) is analytic on z_1 and z̅_2. This analyticity implies, from (<ref>), that the local density of the overlap vanishes outside the ellipse.We now proceed to compute A(u, τ) [Eq. (<ref>)]. Inserting the identity (<ref>) into (<ref>) leads toA(u, τ) = 1/4π^2∬^g(z_1 + z̅_2) u + g z_1 τ∂/∂z̅_1∂/∂ z_2 G(z_1, z_2) z_1z_2.Because the exponential prefactor is analytic in z_1 and z̅_2, it commutes with the two partial derivatives. We can therefore apply Green's theorem twice to obtainA(u, τ) = 1/4π^2∮_E∮_E^g (z_1 + z̅_2) u + g z_1 τ G(z_1, z_2) ẓ_̣1̣ẓ̣̅_̣2̣,where both contour integrals are around the ellipse E, at whose boundary G(z_1, z_2) stops being analytic. To compute A(u, τ) we follow the approach of <cit.> and use the linear transformation w=(z - ηz̅) / (1 - η^2) (or, equivalently, z = w + ηw̅) to reshape the contour of integration from the ellipse E_η into the unit circle. Applying this transformation to both z_1 and z_2, the surface integrals in (<ref>) become countour integrals on the unit circle |w|^2=w w̅ = 1. On this contour we can replace every w̅ in the integrand by w^-1 and we can use the standard tools of complex analysis to carry out the integrals. We describe in more detail our derivation in the following.We start by performing the integral over z̅_2, expressing Eq. (<ref>) in terms of w_1, w̅_1, and w_2, and replacing all w̅_2 by w_2^-1. After some simplifications, we obtainG(z_1(w_1), z_2(w_2)) = -1 + 1/η(α_+ - α_-) ×( 1 - η w_1 α_+/w_2 - α_+ - 1 - η w_1 α_-/w_2 - α_-),where we defined the polesα_±(w_1) = w̅_1 + η w_1 ±√((w̅_1 + η w_1)^2 - 4η)/2 η.These poles depend on w_1 and and can be shown to map the unit disk onto an annulus of inner radius 1 and outer radius 1/|η| [Fig. <ref>]. This information will be relevant when we use residue calculus. The double integral (<ref>) has to be regularized because the integrand diverges at z_1 = z_2. Our regularization consists of first integrating w_2 on the unit circle while constraining w_1 to be on a concentric circle of smaller radius |w_1| = 1 - ϵ, with ϵ>0 small. Once the integral over w_2 is done, we take the limit ϵ→ 0 and perform the second integral over w_1.Under this regularization, we decompose the double integral (<ref>) asA(u, τ) = lim_ϵ→ 01/4π^2∮_|w_1|=1 - ϵ^g (w_1 + ηw̅_1)(u + τ)𝒜(w_1, u) × (ẉ_1 + ηẉ̣̅_1)where we used z_1 = w_1 + ηw̅_1 and we defined𝒜(w_1, u) = ∮_|w_2|=1^g (w^-1_2 + η w_2)uG(z_1(w_1), z_2(w_2)) × (-1/w_2^2 + η) ẉ_̣2̣.Note that here we used w̅_2 = w_2^-1 to express the integrand and the differential ẓ̣̅_2 = ẉ̣̅_2 + ηẉ_2 in terms of w_2 only. The integrand of Eq. (<ref>) contains one singularity inside the contour of integration, at w_2=0. This singularity is associated with the essential singularity from the exponential, ^1/w_2, as well as with the pole of second order 1 / w_2^2. Because we are assuming that |w_1| < 1, the poles of G(z_1(w_1), z_2(w_2)) at w_2=α_± lie outside the contour and therefore do not contribute to the integral. We are thus left with the task of computing the residue at the origin. We do that by expanding the integrand in Laurent series around w_2 = 0, using the relations ^(z^-1 + η z)t = ∑_k=-∞^∞( z √(η))^k I_k(2√(η)t),forz ≠ 0, (z - z_0)^-1 = -1/z_0∑_k=0^∞(z/z_0)^k,for|z| < |z_0|, with I_k(z) being the modified Bessel function of order k. We use the last power series to expand the terms (w_2 - α_±)^-1 in G(z_1(w_1), z_2(w_2)) [Eq. (<ref>)]. This power series converges because |w_2| < |α_±| when |w_1|<1, as we assume in our regularization scheme. After expanding, applying Cauchy's residue theorem, and taking the limit ϵ→ 0, we obtain 𝒜(w_1, u) = -2π/gu∑_k=0^∞ w_1^-kη^-k/2kI_k (2g√(η)u),The final step is to compute the integral in (<ref>) with the same strategy we used for 𝒜(w_1, u, τ). In this case we express the integrand in terms of w_1 only and we expand the exponential factor in (<ref>) with the identity^(z+ η z^-1)t = ∑_k=-∞^∞( z/√(η))^k I_k(2√(η)t).We then pick the residue from the expansion and apply Cauchy's theorem. The result isA(u, τ) = A_1(u, τ) + A_2(u, τ),withA_1(u, τ) = ∑_k=-∞^∞η^k/2 I_k(2g√(η)τ) [ (1 + η^2) I_k(2g (1 + η) u)- η(I_k-2(2g (1 + η) u) + I_k+2(2 g(1 + η) u)) ]and A_2(u, τ) given by Eq. (<ref>).The expression (<ref>) for A_1(u, τ) can be further simplified with the identity <cit.>∑_k=-∞^∞^ kα J_k(w) J_k+ν (z) = ( z - w^-α/z - w^α)^ν/2 × J_ν(√(w^2 + z^2 - 2wzcosα)),which we can transform into a more convenient expression for our problem, using J_ν( z) = ^ν I_ν(z) and taking α = π - ( / 2) lnη so that ^α = -√(η). The identity (<ref>) then becomes∑_k=-∞^∞η^k/2 I_k(w) I_k+ν (z) = ( z + wη^-1/2/z + w η^1/2)^ν/2 × I_ν(√(w^2 + z^2 + wz(η^1/2 + η^-1/2))),which allows us to rewrite Eq. (<ref>) as the final expression (<ref>).The series A_2(u, τ) does not seem to have a closed expression for general η. For η=1, however, we can exploit the identity∑_k=-∞^∞ I_k(w) I_n - k(z) = I_n(w + z)to conclude thatA_2(u, τ) = -I_0(2g(2u + τ)) + I_2(2g(2u + τ)) (for η=1). § EVALUATION OF THE TIME INTEGRAL FOR LONG Τ The exact average autocorrelation is given in (<ref>) as a time integral that can be decomposed asC(τ) = ∫_0^∞^-2u - τ[A_1(u, τ) + A_2(u, τ)] ụ,where A_1(u, τ) and A_2(u, τ) are defined in Eqs. (<ref>) and (<ref>). To gain more analytical insight we will evaluate C(τ) when τ is sufficiently large, a limit that allows us to invoke Laplace's method and approximate the integral with a closed-form expression <cit.>. Before applying the limit it is convenient to express this integral in terms of a new variable ξ≡ u / τ, which is well defined for τ > 0. With this definitionC(τ)= ∫_0^∞^-τ (2ξ + 1)[A_1(u(ξ), τ) + A_2(u(ξ), τ)] τξ̣,≡ C_1(τ) + C_2(τ).which we split into the two terms composing the integrand. We start with the asymptotic dependence of C_1(τ), ignoring for the moment C_2(τ). The integrand of C_1(τ) contains I_0(·) and I_2(·), whose argument is large in the long-τ limit. We can therefore use the asymptotic expansion of the modified Bessel functions of order ν,I_ν(x) = ^x/√(2π x)[1 - 4ν^2 - 1/8x + O(x^-2)]for x ≫ 1.At this order, definingψ(ξ;η) ≡ψ(u(ξ), τ; η) / τ = 2 √((1 + η)^2 ξ (ξ + 1) + η),we obtainC_1(τ) = √(τ/2π)∫_0^∞exp{ -τ[ (2ξ + 1) - g ψ(ξ;η) ]}/√(g ψ(ξ; η)) [ (1 + η^2)(1 + 1/8 τ g ψ(ξ; η))-2 η(1 + 2(1 -η)^2/ψ(ξ; η)^2) (1 - 15/8 τ g ψ(ξ; η)) ]ξ̣.with g = (1 - δ) / (1 + η). This integral is of the formA(τ) = ∫_0^∞ f(ξ) ^τ b(ξ) ξ̣.In the limit of large τ, only the infinitessimal interval around the maximum of b(ξ) contributes to the integral because the contribution of the remaining intervals is exponentially suppressed. In our particular case the maximum of b(ξ) is atξ^* = 1/2( -1 + 1 - η/(1 + η)√(2δ - δ^2)).We distinguish two cases. For large enough values of η and δ, ξ^* is negative [Fig. <ref>], which means that in the integration range [0, ∞) of Eq. (<ref>) the maximum value of b(ξ) is at ξ=0. Conversely, for low values of η and δ, the maximum of b(ξ) occurs within (0, ∞). These two cases will lead to different time dependences and will be studied separately in the following. For values of η and δ such that ξ^* < 0, the limit τ→∞ of Eq. (<ref>) can be approximated by <cit.>A(τ)∼lim_ϵ→ 0∫_0^ϵ f(0)^τ[b(0) + b'(0) s] ṣ∼∫_0^∞ f(0) ^τ[b(0) + b'(0) s] ṣ∼ -f(0)^τ b(0)/τ b'(0),where b'(0) denotes the derivative of b(ξ) evaluated at ξ^*=0. Applying this approximation to (<ref>), we obtainC_1(τ) = τ^-3/2η^-1/4/2√(π)1 + η^2/δ(1 + η) - [1 - √(η)]^2(1 + η/1 - δ)^3/2 ×exp{-τ(1 - √(η))^2 + 2δ√(η)/1 + η}Conversely, if η and δ are such that ξ^* > 0, the maximum ξ^* falls within the integration region and we can approximate the large-τ limit of the integral (<ref>) by <cit.>A(τ) ∼lim_ϵ→ 0∫_ξ^* - ϵ^ξ^* + ϵ f(ξ^*)^τ(b(ξ^*) + (s - ξ^*)^2 b”(ξ^*) / 2) ṣ ∼∫_-∞^∞ f(ξ^*)^τ(b(ξ^*) + (s - ξ^*)^2 b”(ξ^*) / 2) ṣ ∼√(2π) f(ξ^*)^τ b(ξ^*)/√(-τ b”(ξ^*)).with b”(ξ^*) denoting the second derivative of b(ξ) evaluated at ξ^*. In this case Eq. (<ref>) is approximately given byC_1(τ) = δ^-1/2(1 - η)^2/2√(2) (1 - δ)exp{-τ1 - η/1 + η√(2δ - δ^2)}. We now turn to the asymptotic dependence of C_2(τ). The original form for C_2(τ) isC_2(τ) = ∫_0^∞{^-τ (2ξ + 1)∑_k=0^∞η^k+2 (k+1)^2 ×I_k + 1(2 g τ√(η)ξ) I_k+1(2 g τ√(η)(ξ + 1))/ητ g^2 ξ (ξ + 1)}ξ̣.The integrand contains a product of Bessel functions that grows exponentially with τ (see Eq. (<ref>)), but this growth is kept in check by the exponential prefactor. To see this, we introducethe scaled modified Bessel function,I_ν^*(x) ≡^-x I_ν(x),in terms of which (<ref>) becomes C_2(τ) = ∫_0^∞exp[-τ(1 - √(η))^2 + 2δ√(η)/1 + η]∑_k=0^∞η^k+1 (k+1)^2×I^*_k + 1(2 gτ√(η)ξ)I^*_k+1(2 g τ√(η)(ξ + 1))/τ g^2 ξ(ξ+1) ξ̣. The integral converges because the exponential decays to zero for large τ and trumps the power-law decay of I^*_k(·). Equation (<ref>)also has the same form as (<ref>) and contains exactly the same exponent as in Eq. (<ref>); because the exponential attains its maximum at ξ=0, we can use the approximation (<ref>). To evaluate at ξ=0 we use the power expansion I^*_k + 1(2 g τ√(η)ξ) ξ∼ 0= (g τ√(η)ξ)^k+1, and identify f(ξ) in (<ref>) withf(ξ) ≡∑_k=0^∞η^3(k + 1)/2/ξ + 1 (k+1)^2 g^k-1ξ^kτ^kI^*_k+1(2g τ√(η) (ξ + 1)).All the terms of f(ξ) with k>0 vanish at ξ=0, and Eq. (<ref>) reads in this caseC_2(τ)∼τ^-3/2η^5/6/4√(π)(1 + η)^5/2/(1 - δ)^3/21/(1 - √(η))^2 + 2 √(η)δ ×exp{ -τ(1 - √(η))^2 + 2δ√(η)/1 + η}Summing this contribution to that in Eq. (<ref>) leads to the result reported in Eqs. (<ref>)–(<ref>).§ AUTOCORRELATION WITHOUT OVERLAPS Here we compute the autocorrelation ignoring the effect of the overlaps between eigenvectors. In this case, we need to compute the individual contribution of a single eigenvalue to the autocorrelation, and then sum over the contributions of all eigenvalues. We start with the one-dimensional version of Eq. (<ref>)x̣/ṭ = α x + σξ(t),where the parameter α would be the (single) eigenvalue of the system, assumed to have negative real part to prevent x(t) to grow unbounded, and where ξ(t) is a source of standard Gaussian white noise. The solution of (<ref>) isx(t) = σ∫_-∞^t^α (t - s)ξ(s) ṣ,from which we can derive the autocorrelation:x(t) x(t + τ)= σ^2 ∫_-∞^t∫_-∞^t + τ^α (2t + τ - s - u)ξ(s)ξ(u) ṣ ụ = - σ^2^ατ/2α≡σ^2 C_α(τ).The eigenvalue α determines both the time scale and the amplitude of the autocorrelation. We set the overall factor σ^2 to 1, without loss of generality.The average autocorrelation for the high-dimensional system in the absence of overlaps is the sum of (<ref>) over all the eigenvalues. In the large-N limit we would haveC(τ) = 1/N∑_i=1^N C_α_i(τ) ∫ C_α(τ)ρ(α) α̣α̣̣̅,where ρ(α) is the probability density of eigenvalues and the integral is on the complex plane. For the system (<ref>) and for the connectivity matrices we consider, the density of the eigenvalues α is uniform and has support on an ellipse centered at z=-1 with major radius g(1 + η) and minor radius g(1 - η). The integral (<ref>) can be computed in that case and readsC(τ) = -1/π g^2 (1 - η^2)∫_E^ατ/2α α̣α̣̣̅,where we used Eq. (<ref>) and the prefactor is the constant value that ρ(α) takes on the elliptic support E. To evaluate the integral we use the parametrizationα = -1 + r (1 + η) cosθ +r (1 - η) sinθand integrate over r ∈ [0, g] and θ∈ [0, 2π]. Noting that∫_Eα̣α̣̣̅ = (1 - η^2) ∫_0^2π∫_0^g r ṛ θ̣,Eq. (<ref>) becomesC(τ) = 1/π g^2∫_0^2π∫_0^gexp{-τ[1 - r ψ (θ) ]}/2 [1 - r ψ(θ)] r ṛ θ̣.where for convenience we definedψ(θ) ≡ (1 + η)cosθ +(1 - η) sinθ.The integral (<ref>) is hard to compute, but we can make progress by taking the derivative of C(τ) with respect to τ, C'(τ) = -^-τ/2π g^2∫_0^2π∫_0^g^τψ(θ)r r ṛ θ̣, which is easier to evaluate. Equation (<ref>) can be integrated over r by parts, yielding an integral over θ only that, excluding prefactors, reads B(g, τ) ≡∫_0^2π{g ^τψ(θ) g/τψ(θ) - ^τψ(θ) g - 1/τ^2 ψ^2(θ)} θ̣.Again, this integral is hard to compute but we can use the same trick we used before, noting that the partial derivative of A(g, τ) with respect to g simplifies considerably,∂ B(g, τ)/∂ g = g ∫_0^2π^τψ(θ) g θ̣ = 2 π g I_0(2gτ√(η)),where in the last equation we used <cit.>. We recover the expression for A(g,τ) by integrating along g, with the initial condition A(0,τ)=0:B(g,τ) = 2π∫_0^g x I_0(2x τ√(η)) x̣ = 2 π/4 τ^2 η∫_0^2 g τ√(η) y I_0(y) ỵ.The last integral can be computed with the help of the recurrence relation zI_0(z) = z I'_1(z) + I_1(z). An integration by parts of the term zI_1'(z) leads to the final identity ∫ x I_0(x) x̣ = x I_1(x) and therefore toB(g,τ) = π g/τ√(η) I_1(2 τ g √(η)).Equation (<ref>) then readsC'(τ) = -^-τ/2τ g √(η) I_1(2 τ g √(η)),which we have to integrate to recover C(τ). Such an integration is subject to the initial condition C(0):C(0)= 1/2π g^2∫_0^2π∫_0^gr/1 - r ψ(θ) ṛ θ̣.= -1/2π g^2∫_0^2π{g/ψ(θ) + 1/ψ^2(θ)ln[1 - g ψ(θ)]} θ̣,which can be evaluated numerically.§ SUMMARY OF THE DYNAMIC MEAN FIELD DERIVATION The starting point of the calculation is the moment generating functional for the state variables x_i(t) obeying Eq. (<ref>). We consider the more general case where the activation variable is driven by recurrent inputs as well as independent external white noise:ẋ_i(t) = -x_i(t) + g ∑_j=1^N J_ij r_j(t) + σξ_i(t), i=1,…,Nwhere we defined r_j(t) ≡ϕ(x_j(t)) to simplify the notation. The white noise sources ξ_i(t) have zero mean and unit variance. The moment generating functional for such a system can be shown to be Z[l, l̃; 𝐉] = ∫𝒟x(t) 𝒟x̃(t) exp(-S[x, x̃; 𝐉]+ ∑_i=1^N∫l̃_i(t) x_i(t) ṭ + ∑_i=1^N∫ l_i(t) x̃_i(t) ṭ, )where 𝒟x(t) 𝒟x̃(t) = ∏_i=1^N𝒟x_i(t) 𝒟x̃_i(t) is the functional measure for all possible paths for all variables and we introduced the actionS[x, x̃; 𝐉] = ∑_i=1^N∫x̃_i(t) ×{ẋ_i(t) + x_i(t) - g ∑_j=1^N J_ij r_j(t) - σ^2/2x̃_i(t) } ṭIn this definition we assume that the auxiliary fields x̃(t) are purely imaginary, so we do not have to write explicit imaginary units all along. By construction the generating functional satisfies the normalization condition Z[0,0;𝐉] = 1. The fact that Z[0,0; 𝐉] does not depend on 𝐉 allows us to compute the quenched average directly on Z <cit.>,Z[l, l̃] ≡∫Z[l, l̃; 𝐉]/Z[0, 0; 𝐉] P̣(̣𝐉̣)̣ = ∫ Z[l, l̃; 𝐉] P̣(̣𝐉̣)̣,which simplifies considerably the average, now reduced to computing [exp(-S[x,x̃, 𝐉])]_J. To do so, we use the decomposition of partially symmetric connectivity matricesJ_ij = J_ij^s + k J_ij^a,where J_ij^s = J_ji^s, J_ij^a = -J_ji^a, and where both J_ij^s and J_ij^a are Gaussian random variates with zero mean and variance[ (J_ij^s)^2 ]_J = [ (J_ij^a)^2 ]_J = 1/N1/1 + k^2so that [J_ij^2]_J = J^2 / N. With these matrix decompositions, the correlation between bidirectional weights is <cit.>[J_ij J_ji]_J = 1/N1 - k^2/1 + k^2,which must equal η / N by our definition of η. This leads to the relation k^2 = (1 - η)/(1 + η). To integrate over the disorder we use the Gaussian measures:P̣(𝐉^s) = ∏_i ≤ jP̣(J_ij^s) ∝exp{-N/1+η∑_i ≤ j (J^s_ij)^2}𝐉̣^̣ṣ,P̣(𝐉^a) = ∏_i < jP̣(J_ij^a) ∝exp{-N/1+η∑_i < j (J^a_ij)^2}𝐉̣^̣ṣ,with 𝐉̣^̣ṣ = ∏_i ≤ jJ̣_̣ịj̣^̣ṣ and 𝐉̣^̣ṣ = ∏_i < jJ̣_̣ịj̣^̣ạ. We will ignore the contribution of diagonal elements of the synaptic matrix because it is negligible in the limit of large N. We can now integrate out the terms linear in J_ij that appear in Eq. (<ref>), by separating symmetric and antisymmetric components. Excluding prefactors and time integrals, these terms are of the formL(𝐉, t) ≡∑_i, j i≠ jx̃_i(t) J_ij r_j(t) =∑_i, j i≠ jx̃_i(t) [J^s_ij + k J^a_ij] r_j(t) = ∑_i < j{J^s_ij[x̃_i(t) r_j(t) + x̃_j(t) r_i(t)]+ k J^a_ij[x̃_i(t) r_j(t) - x̃_j(t) r_i(t)]}so that∫exp{g ∫ L(𝐉, t) ṭ} P̣(𝐉^s) P̣(𝐉^a) = exp{g^2/2N∑_i, j i ≠ j∬{[x̃_i(t) r_j(t) x̃_i(t') r_j(t')] + η[x̃_i(t) r_j(t) x̃_j(t') r_i(t')]} ṭ ṭ'̣},where we used the property that, for a Gaussian variable z of zero mean and variance σ^2, the expected value of exp(λ z) is exp(λ z)_z = exp(λ^2σ^2/2), which can be checked by completing the square in the exponential.Putting back together all the pieces, the average generating functional (<ref>) is thereforeZ[l, l̃] =∫𝒟x(t) 𝒟x̃(t) exp(-S_0[x(t), x̃(t)]+ σ^2/2x̃·x̃ + l̃· x_i + l ·x̃+ g^2/2N∑_i, ji ≠ j∬{[x̃_i(t) r_j(t) x̃_i(t') r_j(t')] + η[x̃_i(t) r_j(t) x̃_j(t') r_i(t')]} ṭ ṭ'̣)where we defined the free actionS_0[x, x̃] ≡∑_i=1^N∫x̃_i(t)[ẋ_i(t) + x_i(t)] ṭ.and we introduced the notationf · g ≡∑_i=1^N∫ f_i(t) g_i(t) ṭ. As a result of averaging out the disorder, we obtained a coupling involving four fields with different indices and at different times. To proceed it is convenient to introduce auxiliary fields that involve terms local in space (i.e., with the same index) but not in time:q_1(t, t')= g^2/N∑_i=1^Nx̃_i(t) x̃_i(t'),q_2(t, t')= g^2/N∑_i=1^N r_i(t) r_i(t'), q_3(t, t')= g^2/N∑_i=1^Nx̃_i(t) r_i(t'),q_4(t, t')= g^2/N∑_i=1^N r_i(t) x̃_i(t'),so Eq. (<ref>) now readsZ[l, l̃] = ∫𝒟x(t) 𝒟x̃(t) (∏_α=1^4N/g^2 𝒟q_α) ×δ(N/g^2 q_1 - ∑_i=1^Nx̃_i(t) x̃_i(t')) δ(N/g^2 q_2 - ∑_i=1^N r_i(t) r_i(t'))×δ(N/g^2 q_3 - ∑_i=1^Nx̃_i(t) r_i(t')) δ(N/g^2 q_4 - ∑_i=1^N r_i(t) x̃_i(t')) ×exp(-S_0[x, x̃] + σ^2/2x̃·x̃ + l̃· x_i + l ·x̃ + N/2 g^2∬{ q_1(t, t') q_2(t, t') + ηq_3(t, t') q_4(t, t') } ṭ ṭ'̣)We now express the Dirac functionals in their integral representation. The first Dirac functional appearing in Eq. (<ref>) can be written asδ(N/g^2 q_1 - ∑_i=1^Nx̃_i(t) x̃_i(t')) = 1/2π∫𝒟q̂_1(t, t')×exp{∬q̂_1(t, t') [N/g^2 q_1(t, t') - ∑_i=1^Nx̃_i(t) x̃_i(t') ] ṭ ṭ' },where the integral over q̂ is understood to be along the imaginary axis. The other Dirac functionals in Eq. (<ref>) are rewritten analogously. Equation (<ref>) then becomesZ[l, l̃] = ∫𝒟X 𝒟Q exp(-S_0[x, x̃] + σ^2/2 x̃·x̃ + l̃· x + l ·x̃+ N/g^2∬{∑_α = 1^4q̂_α(t, t') q_α(t, t')+ 1/2[ q_1(t, t') q_2(t, t') + ηq_3(t, t') q_4(t, t')] - g^2/N∑_i=1^N[ q̂_1(t, t') x̃_i(t) x̃_i(t') + q̂_2(t, t') r_i(t) r_i(t')+ q̂_3(t, t') x̃_i(t) r_i(t') + q̂_4(t, t') r_i(t) x̃_i(t')]} ṭ ṭ'̣)where we introduced the shorthand notation𝒟 Q≡∏_α=1^41/2πN/g^2𝒟 q_α𝒟q̂_α, 𝒟 X≡𝒟x(t) 𝒟x̃(t) = ∏_i=1^N𝒟x_i(t) 𝒟x̃_i(t). Equation (<ref>) can now be expressed as <cit.>Z[l, l̃] = ∫𝒟Q ^N f(q, q̂, x, x̃),wheref(q, q̂, x, x̃)≡ G(q, q̂) + 1/Nlog∫𝒟X exp[ℒ(q, q̂, x, x̃)], G(q, q̂)≡1/g^2∬{∑_α=1^4 q_αq̂_α + 1/2[ q_1 q_2 + ηq_3 q_4 ] } ṭ ṭ'̣, ℒ(q, q̂, x, x̃)≡ -S_0[x, x̃] + σ^2/2 x̃·x̃ + l̃· x + l ·x̃ - ∑_i=1^N∬[ q̂_1(t, t') x̃_i(t) x̃_i(t') + q̂_2(t, t') r_i(t) r_i(t')+ q̂_3(t, t') x̃_i(t) r_i(t') + q̂_4(t, t') r_i(t) x̃_i(t')] ṭ ṭ'̣.In the limit of large N we can apply the saddle-point method to Eq. (<ref>), which amounts to making the following approximation Z[l, l̃] = ∫𝒟Q ^N f(q, q̂, x, x̃)≈^N f(q^0, q̂^0, x, x̃) where q^0 and q̂^0 are the values that extremize f. Requiring δ f/δq̂_α = 0 leads to q^0_1(t, t')= g^2/N∑_i=1^N⟨x̃_i(t) x̃_i(t')⟩_ℒ , q^0_2(t, t')= g^2/N∑_i=1^N⟨ r_i(t) r_i(t')⟩_ℒ , q^0_3(t, t')= g^2/N∑_i=1^N⟨x̃_i(t) r_i(t')⟩_ℒ , q^0_4(t, t')= g^2/N∑_i=1^N⟨ r_i(t) x̃_i(t')⟩_ℒ , with the average ⟨·⟩_ℒ defined as𝒪_ℒ≡∫𝒪(X) exp[ℒ(X)] X̣∫exp[ℒ(X)] X̣.Similarly, from the saddle-point conditions for q_α we obtainq̂^0_1(t, t')= -1/2 q^0_2(t, t'),q̂^0_2(t, t')= -1/2q^0_1(t, t'), q̂^0_3(t, t')= - η/2 q^0_4(t, t'),q̂^0_4(t, t')= - η/2 q^0_3(t, t'). Now the right hand side of Eq. (<ref>) readsZ[l, l̃] = ∫𝒟X exp(-S_0[x, x̃] - S_int[x, x̃] + l̃· x + l ·x̃)withS_int[x, x̃] ≡1/2∬{N/g^2[q^0_1(t, t') q^0_2(t, t')ηq^0_3(t, t') q^0_4(t, t') ] - ∑_i=1^N{[q^0_2(t, t') + σ^2 δ(t - t')]x̃_i(t) x̃_i(t')+ q^0_1(t, t') r_i(t) r_i(t') ηq^0_4(t, t') x̃_i(t) r_i(t') ηq^0_3(t, t') r_i(t) x̃_i(t')}} ṭ ṭ'̣. The auxiliary fields defined in Eqs. (<ref>)–(<ref>) are related to physically observable quantities.First, q_2^0(t, t') is related to the population-averaged autocorrelation functionC(t, t') ≡1/N∑_i=1^N⟨ r_i(t) r_i(t') ⟩ ,by q_2^0 (t, t') = g^2 C(t, t').Second, the auxiliary fields q_3^0(t, t') and q_4^0(t, t') are related to the so-called response function, which characterizes the response of the system when it is perturbed by a weak field. More specifically, in our context the response function at site i would beG(t, t') ≡. δ⟨ r_i(t) ⟩/δ h_i(t')|_h=0,where h_i(t') is a time-dependent external field, and angular brackets denote the average over the effective action S[x,x̃] = S_0[x,x̃] + S_int[x,x̃] that appears in Eq. (<ref>). Note that from the definition of response function G(t, t') has to be 0 whenever t < t', due to causality. To see the link between G(t, t') and q_3(t,t') and q_4(t,t'), we add an external field h_i(t) for each neuron in Eq. (<ref>), and evaluate (<ref>). With the new field the action becomes S_h[x,x̃] = S[x,x̃] -∑x̃_i(t) h_i(t) and. δ⟨ r_i(t) ⟩/δ h_i(t')|_h=0= δ/δ h(t')∫𝒟Xr_i(t) exp(-S_h[x, x̃]) |_h=0 = -⟨ r(t) . δ S_h/δ h_i(t')|_h=0⟩ = ⟨ r_i(t) x̃_i(t') ⟩.Defining the population-averaged response function asG(t, t') ≡1/N∑_i=1^N⟨ r_i(t) x̃_i(t') ⟩ ,we obtain q_4^0(t, t') = q_3^0(t', t) = g^2 G(t, t').As for q_1^0(t,t'), it can be shown that the presence of vertices like r_i(t) r_i(t') in the action necessarily leads to violation of causality <cit.>. We thus need to impose q_1^0(t,t')=0 to obtain a physical solution.We can finally write the interacting action in Eq. (<ref>) in terms of the physical quantitiesS_int[x,x̃] = -∑_i=1^N∬{1/2Γ(t, t') x̃_i(t) x̃_i(t')+ η g^2G(t, t') x̃_i(t) r_i(t') } ṭ ṭ'̣,where we ignored the term containing G(t, t')G(t, t'), which vanishes due to causality, and where we definedΓ(t, t') ≡ g^2 C(t, t') + σ^2 δ(t - t') .Note that the final action involves only interactions that are local in space, which implies that all units are equivalent. This equivalence comes as no surprise, because all units are equivalent once we average over all realizations of the connectivity matrix. We can thus drop the irrelevant indices i and focus on the single relevant dynamical variable x(t). §.§ Equation of motion for the average activityThe local action S[x, x̃] = S_0[x, x̃] +S_int[x, x̃], with S_0 and S_int given by Eqs. (<ref>) and (<ref>), has the formS[x, x̃] = ∬{x̃(t) G^-1_F(t - t') x(t') - η g^2 x̃(t) G(t, t') r(t')- 1/2Γ(t, t') x̃(t)x̃(t')} ṭ ṭ'̣where for later convenience we have introduced the inverse of the free propagator, G^-1_F(t - t'). The free propagator G_F(t - t') is just the Green's function associated with the operator /ṭ + 1,(/ṭ + 1) G_F(t - t') = δ(t - t'),and is related to its inverse through ∫ G_F^-1(t - s)G_F(s - t') ṣ = δ( t - t'), which is automatically satisfied ifG_F^-1 (t - t') = δ(t - t') (/ṭ'̣ + 1). From the original stochastic system (<ref>) and its associated Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) action (<ref>), we infer that the equation of motion associated with the action (<ref>) isẋ(t) = - x(t) + η g^2 ∫_-∞^t G(t, s) r(s) ṣ + φ(t),where φ(t) is a source of noise with autocorrelation⟨φ(t) φ(t') ⟩ = Γ(t, t') = g^2 C(t, t') + σ^2 δ(t - t').This relation has to be consistent with the dynamics generated by Eq. (<ref>), that is, the noise φ(t) has to be such that the firing activity r(t) has autocorrelation C(t, t').We can go further and write a self-consistent relation involving the two-point functions C(t,t') and G(t, t'). A starting point to derive them are the identitiesδ x(t)/δ x(t') = δx̃(t)/δx̃(t') = δ(t - t'),δx̃(t)/δ x(t') = δ x(t)/δx̃(t') = 0from which we can obtain relations such as⟨δ x(t)/δ x(t')⟩ ≡∫𝒟X δ x(t)/δ x(t')exp{-S[x(t), x̃(t)]} = ⟨ x(t) δ S/δ x(t')⟩ = δ(t - t').Other relations follow analogously:⟨ x(t) δ S/δ x(t')⟩= δ(t - t'),⟨ x(t) δ S/δx̃(t')⟩= 0, ⟨x̃(t) δ S/δx̃(t')⟩= δ(t - t'), ⟨x̃(t) δ S/δ x(t')⟩= 0, We now apply the identities (<ref>) and (<ref>) for the action (<ref>). In particular, we use the identities involvingδ S/δx̃(t') = ẋ(t) + x(t) - η g^2 ∫_-∞^t G(t,s) r(s)ṣ - ∫Γ(t, s) x̃(s) ṣand we define the autocorrelation and response function of the activation field x(t)Δ(t, t') ≡x(t) x(t'), R(t, t') ≡x(t) x̃(t').The last equation in (<ref>) and the first equation in (<ref>) then become, respectively, ∂/∂ tΔ(t, t') = -Δ(t, t') + σ^2 R(t', t)η g^2 ∫^t_0 G(t, s) r(s)x(t') ṣ + g^2 ∫^t'_0 R(t',s) C(t,s) ṣ, ∂/∂ t R(t, t') = -R(t, t') + δ(t - t') + η g^2 ∫^t_t' G(t, s) G(s,t') ṣ,where in (<ref>) we have used x̃(t)x̃(t') = 0. It can be shown that the remaining identities in Eqs. (<ref>) and (<ref>), which involve δ S / δ x, do not provide additional information <cit.>. Note that Δ(t,t') has a cusp at t=t' due to the term σ^2 R(t', t), which from (<ref>) we know it must be of the form R(t,t') ∝Θ(t - t'), with Θ(t) being the step function. More specifically,[∂/∂ tΔ(t, t') ]_t'= t^+^t'= t^- = σ^2 [R(t',t)]_t'= t^+^t'= t^- = -σ^2.Moreover, the symmetry of Δ(t, t') around t=t' implies lim_t'→ t^-∂_t Δ(t, t') = - lim_t'→ t^+∂_t Δ(t, t'), which leads to the relation lim_t'→ t^-∂_t Δ(t, t') = -σ^2 / 2. The amplitude of external noise thus determines the slope of the autocorrelation of x(t) at zero time lag. This is the only dependence on σ^2 of the solutions of (<ref>) and (<ref>).Equations (<ref>) and (<ref>) cannot be solved in a closed-form except for η=0 <cit.>, but perturbative solutions can be found by expanding the nonlinearity r(t)=ϕ(x) in power series of x(t) and then solving the resulting hierarchy of equations, which involve correlations and response functions of increasingly larger order. The problem becomes unwieldly except for the linear case where r(t)=x(t). In that case, C(t,t')=Δ(t,t'), G(t,t')=R(t, t'), and Eqs. (<ref>) and (<ref>) form a closed system of integro-differential equations:∂/∂ tΔ(t, t') = -Δ(t, t') + σ^2 R(t', t) + η J^2 ∫^t_0 R(t, s) Δ(s, t') ṣ + g^2 ∫^t'_0 R(t',s) Δ(t,s) ṣ, ∂/∂ t R(t, t') = -R(t, t') + δ(t - t') + η g^2 ∫^t_t' R(t, s) R(s, t') ṣ.apsrev4-1 | http://arxiv.org/abs/1707.08337v3 | {
"authors": [
"Daniel Martí",
"Nicolas Brunel",
"Srdjan Ostojic"
],
"categories": [
"q-bio.NC"
],
"primary_category": "q-bio.NC",
"published": "20170726094125",
"title": "Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks"
} |
9.3in by -1.0in 6.7in by -0.8in propertiesl[2][1] | http://arxiv.org/abs/1707.08781v1 | {
"authors": [
"Lin Gao",
"Giorgio Battistelli",
"Luigi Chisci",
"Ping Wei"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170727085431",
"title": "Consensus-based joint target tracking and sensor localization"
} |
1Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China2Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China*[email protected] quantum communications, vortex photons can encode higher-dimensional quantum states and build high-dimensional communication networks (HDCNs). The interfaces that connect different wavelengths are significant in HDCNs. We construct a coherent orbital angular momentum (OAM) frequency bridge via difference frequency conversion in a nonlinear bulk crystal for HDCNs. Using a single resonant cavity, maximum quantum conversion efficiencies from visible to infrared are 36%, 15%, and 7.8% for topological charges of 0,1, and 2, respectively. The average fidelity obtained using quantum state tomography for the down-converted infrared OAM-state of topological charge 1 is 96.51%. 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Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, Orbital angular momentum-entanglement frequency transducer, Phys. Rev. Lett. 117(10), 103601 (2016).mancinelli2017mid M. Mancinelli, A. Trenti, S. Piccione, G. Fontana, J. Dam, P. Tidemand-Lichtenberg, C. Pedersen, and L. Pavesi, Mid-infrared coincidence measurements on twin photons at room temperature, Nat. Commun. 8,15184(2017).§ INTRODUCTION Orbital angular momentum (OAM),a remarkable photonic freedom that inherently has multiple dimensions, can enhance the information capacity in communications. The pioneering work in 1992<cit.> of Allen et al. demonstrated that light beams with azimuthal phase dependence of exp(iLϕ) can carry OAM of Lħ. These vortex beams have interesting characteristics that have stimulated major research interest in several fields, including optical manipulation and trapping<cit.>, high-capacity communications<cit.>, spiral imaging, and high-precision optical measurements<cit.>. In quantum information fields, photons can be encoded in the OAM space because of the basis of OAM with its infinite dimensions in Hilbert space, which can then generate high-dimensional entanglement states<cit.> and can be used to build OAM quantum memory for high-dimensional communication networks (HDCN)<cit.>. To date, most atom-based quantum repeaters operate in the visible spectrum. However, fiber-based quantum communication networks generally work in the telecommunications band, which lies within the low-loss communication windows <cit.>. Therefore, a coherent frequency bridge is required to connect the two spectra. In 1990, Kumar et al. first proposed the concept of a quantum frequency converter using second-order nonlinearity to change the frequency of the quantum state while maintaining other quantum properties<cit.>. Subsequently, many quantum frequency converters have been constructed to connect different systems<cit.>.Two reversible processes are usually required to realize the interface between visible and telecommunications-band photons: quantum frequency up-conversion (QFUC) and quantum frequency down-conversion (QFDC).QFUC, which is based on sum frequency generation (SFG), can convert infrared photons into visible photons. Many previous researchers have realized QFUC experimentally on the single photon level in different material systems, including periodically poled lithium niobate (PPLN) and potassium titanyl phosphate (PPKTP) nonlinear crystals and PPLN waveguides<cit.>.In contrast to up-conversion, the reverse process of QFDC, which is based on difference frequency generation, has been rapidly developed in recent years <cit.>. There are two possible configurations for frequency conversion of OAM modes: one is the single pass configuration; the other is the cavity-enhanced single resonance scheme. In the first configuration, the two input fields make single passes through the crystal. Shao et al. presented a theoretical model and a simulation of OAM frequency conversion using a quasi-phase-matching (QPM) crystal<cit.>. Later, Li et al. provided an analytical expression for frequency conversion in the SFG process<cit.>.Steinlechner et al. proposed an SFG-based conversion scheme to convert structured light from the near infrared (803 nm) to the visible range (527 nm) <cit.>, although the power conversion efficiency achieved was low (0.09% for L=1). The second cavity-enhanced configuration can increase the conversion efficiency obviously when compared with the first scheme, where the basic Gaussian pump beam is resonant with the cavity and the input photons make single passes through the crystal. Using the second setup, Zhou et al. realized frequency up-conversion of OAM modes from the infrared to the visible band based on SFG<cit.>, in which the maximum conversion efficiency of the OAM photon for topological charge 1 was 8.3%. However, down-conversion of OAM modes has not been reported to date. Additionally, because superconducting detectors have near-unity quantum efficiency at 1550 nm, the down-conversion process has the potential to become much more useful.In this work, we present the first demonstration of down-conversion for the OAM mode from a visible laser operating at 525 nm to an infrared laser beam at 1550 nm using a strong pump beam at 794 nm. Based on the nonlinear coupling equation, we propose an analytical expression to describe the conversion efficiency of the OAM down-conversion frequency bridge. The single-pass efficiencies realized for conversion from visible to infrared are 1.6%, 0.52%, and 0.4% for L=0, 1, and 2, respectively, and the corresponding maximum quantum conversion efficiencies are 36%, 15%, and 7.8%, respectively. In order to test the coherence of the converted OAM photons, we analyze the density matrix of the infrared OAM photons using quantum tomography<cit.>. The average output OAM photon fidelity for topological charge 1 is 96.18%. The high fidelity and high conversion efficiency indicate that the OAM down-conversion frequency bridge is both reliable and useful, which will pave the way for use of this bridge at the interface between two high-dimensional quantum systems with different spectra. § PRINCIPLEDifference frequency generation (DFG), which is based on a second-order nonlinearity, involves mixing of three waves such that they interact in a quasi-phase-matching periodically poled nonlinear crystal. During this process, the energy (ω_v-ω_p=ω_i), momentum (k_v-k_p-k_i-2π/Λ=0), and OAM (L_v-L_p-L_i=0) must be conserved, regardless of the forms of the interacting fields. Here, k_v, k_p, and k_i represent the properties of the input fundamental visible laser, the strong pump laser and the output infrared laser, respectively. In the DFG-based down-conversion experiment, the basic strong pump laser P_p has a Gaussian beam, and the output infrared laser P_i is dependent on the profile of the input visible beam P_v. By ignoring the time components of the strong Gaussian field, the spatial field can be written as<cit.>:E_p(r,z)=√(P_p/πϵ_0n_pc)1/w_0p(1+iτ_p)exp(-r^2/w_0p^2(1+iτ_p))Here, w_0p is the beam waist; τ_p (=2z/b_p) is a variable related to the propagation distance; b_p (=k_pw_0p^2) is the confocal parameter of the Gaussian beam; and n_p is the refractive index of the pump laser.The input visible OAM beam has a similar expression with the exception of the topological charge L<cit.>:E_v(r,z,L)=√(P_v/πϵ_0n_vcL!)(√(2)r)^2/w_0v(1+iτ_v)exp(-r^2/w_0v^2(1+iτ_v)+iLϕ) Based on the nonlinear coupled equation, the evolution of the output fields can be written as<cit.>:dE_i/dz=K_iE_p^*E_ve^-iΔ kzwhere K_i (=2id_effω_i/n_ic) is the coupling coefficient of the infrared laser. d_eff is the effective nonlinear efficiency of the crystal. The two beams are assumed to be focused at the crystal's center at z=0. On the output surface of the crystal, we obtain the output power as follows:P_i(L/2)=2ϵ_0cn_i∫ E_iE_i^*ds=16π^2d_eff^2l2^L/ϵ_0cn_vn_iλ_i^2λ_p h(α,β,ξ,σ)· P_pP_vwhere n_v and n_i are the refractive indices of the visible (λ_v) and infrared laser (λ_i) in crystal, respectively; l is the length of the crystal; and h(α,β,ξ,σ) is the focusing function that is determined based on the waist ratio of the two beams:h=1/ξ∫_-ξ^ξ∫_-ξ^ξ(1-ix)^-1(1+iy)^-1e^-iσ(x-y)/(αβ^2(β+ix)(β-iy)(1/1-ix+1/1+iy) +β^3(2β-iy+ix))^L+1dxdyHere, α (=w_0v^2/w_0p^2) and β (=b_v/b_p) are the ratio of the squares of the beam waist and confocal parameters, respectively; and ξ (=l/b_p) and σ (=Δ kb_p/2) are the pump focusing and spatial phase-mismatching parameters, respectively. When the passive transmission loss and the crystal's absorption δ(=T_in^vexp(-α _Ll)T_out^i)<cit.> are considered, Eq. (4) can be simplified to read:P_i=K_Lhδ· P_pP_vHere, K_L represents a constant with the topological charge and the parameters of the crystal that were used in Eq. (4). Based on Eq. (6), the single pass conversion efficiency (SPCE), the quantum conversion efficiency (QCE) and the maximum power of the strong pump power (P_p_Max) for unit one conversioncan be calculated. The QCE from the visible spectrum to the infrared is defined as<cit.>:η=N_i/N_v=sin(π/2√(P_p/P_p_Max))where P_p_Max (=λ_v/λ_iK_Lh(α,β,ξ,σ)δ) is the maximum pump power. Because of the high divergence angle for the higher-order OAM mode<cit.>, we increase the pump power for a higher conversion efficiency. In this down-conversion experiment, a resonant cavity is used, as shown in Fig. 1(a). When the cavity length is locked to the master laser, the circulating power can then be written approximately as P_circ=F/2π· P_p, where F is the finesse of the cavity. In Fig. 1(b), we plot the theoretical SPCE and P_p_Max for the different input topological charges L. Here, the x-axis represents L, ranging from 0 to 10, and the left and right y-axes are the SPCE and P_p_Max with logarithmic scales, respectively. In our calculations, the waist sizes of the two beams are w_p=60 μ m and w_v (=50×√(L+1) μ m). The crystal used is a bulk PPLN crystal, for which the passive loss δ=0.95^3. From Fig. 1(b), we find that the relationships of t he SPCE and P_p_Max with L are both approximately quadratic. For L=0 (Gaussian beam), L=1, and L=2, P_p_Max has values of 20 W, 50 W, and 166 W, respectively. For higher-order OAM modes, the values of P_p_Max are numerous.§ EXPERIMENTThe design and testing of the coherent OAM down-frequency bridge are introduced in this section, including the detailed experimental setup and the conversion results.§.§ Experimental setup A detailed schematic of the difference frequency generation (DFG) process is shown in Fig. 2. There are four modules: the photon source, state preparation, state conversion and state tomography modules. The first setup shown in Fig. 2 is that of the photon source. The 794 nm laser beam comes from a Ti: sapphire laser, which is resonant with the cavity to produce strong pump power. The seeded 1550 nm infrared laser, which comes from a diode laser, is amplified via the EDFA and a single pass through the PPKTP crystal. When the quasi-phase-matching (QPM) condition is satisfied for the two fundamental laser beams in PPKTP, a high-quality 525 nm Gaussian beam is generated. The next part shows the state preparation structure. A modified Sagnac interferometer is used to produce an arbitrary OAM state, and includes a vortex phase plate, wave plates (H1, Q), and a polarizing beam splitter. After the interferometer, the two beams acquire the opposite and equal spatial vortex phases exp(iLϕ), and the superposition state generated can be written as<cit.>: |ϕ⟩=1/√(2)·(|h,L⟩+e^iθ|v,-L⟩)where h,v represent the horizontal and vertical polarizations of the input state, respectively. In particular, when the optical axes of H2 are rotated by 22.5^∘ along the horizontal direction, the preparation state can be expressed as: |ϕ⟩_pre=1/2·((|L⟩+e^iθ|-L⟩)|h⟩+(|L⟩+e^i(θ+π)|-L⟩)|v⟩)The next section is the state down-conversion design, which is crucial to realization of down-conversion of the OAM mode from a visible laser (VL) beam to an infrared laser (IL) beam. This section consists of a single resonant cavity and a nonlinear crystal. The cavity parameters are optimized based on the theory of Boyd and Kleinman <cit.>. We select the focusing parameters ξ=0.93,μ=0.66, and the corresponding beam waist of the 794 nm pump beam is 56 μ m at the center of the PPLN. The PPLN that was used in this experiment with a type-0 (e+e->e) quasi-phase-matching (QPM) condition for DFG has a cross-section of 50×7.9×0.5 mm^3 (L× W× T), and was placed within a homemade oven. The selected period of the PPLN is 7.30 μ m. The ring cavity has four mirrors, where M1 is the input mirror with 97% reflection at 794 nm, and the plane mirror M2 and the two curved mirrors M3 and M4, with their 80 mm radius of curvature , have high-reflection(>99.8%) coating for 794 nm. The crystal surface and the two curved mirrors have anti-reflection (<1%) coating for the two laser beams. By measuring the leaked power after M2, we can then estimate the power that is circulating in the cavity. In order to test the coherence of the output infrared OAM state, we use two strategies. On the one hand, the shapes and the power of the infrared fields are observed using an infrared charge-coupled device (CCD) and a power meter. On the other hand, we calculate the density matrix and its fidelity using quantum state tomography<cit.>. The infrared light is filtered using a long pass filter, projected onto the spatial light modulator (SLM), and then collected into the fiber using collimators. The four projections selected on the basis of the SLM are |R⟩, |L⟩, |H⟩,and |A⟩, where |R⟩ and |L⟩ represent the eigenstates in self-representation for the OAM, and |H⟩=1/√(2)·(|R⟩+|L⟩) and |A⟩=1/√(2)·(|R⟩-i|L⟩)are the supposition state. §.§ DFG-based OAM quantum-frequency down-conversion First, we measured the single pass conversion efficiency (SPCE) from the visible range to the infrared in frequency down-conversion of the OAM mode, where the two laser beams make a single pass through the PPLN. The pump laser waist is approximately 60 μ m, and the input laser is different green OAM state. The optimal temperature is 29.50^∘ C. The main results are shown in Fig. 3(a), where the x-axis represents the input fields, including the Gaussian (|0⟩), the pure OAM state (|1⟩, |2⟩, as shown in Eq. (8)), and the supposition state (|1⟩+|-1⟩, |2⟩+|-2⟩, as shown in Eq. (9)). The y-axis is the measured SPCE. The green bars are the measured average values of the SPCE. In this process, the visible power is fixed at 40 mW, and the pump power is increased linearly from 50 mW to 500 mW. Because of system jitter, error bars must be inserted on top of the data, and these error bars represent the standard deviation in a group of measurements. Because of the type-0(e+e->) quasi-phase-matching condition, the conversion efficiency of the supposition state (|1⟩+|-1⟩;|2⟩+|-2⟩) is half of that of the corresponding pure state (|1⟩;|2⟩), based on Eq. (9). The conversion efficiency is lower than the theoretically predicted figure shown in Fig. 1(b), which is based on the uncertainty of the beam waist and the imperfect mode overlap. From Fig. 3(a),we find that the conversion efficiency for higher-order OAM modes is very weak in the single-pass configuration. To obtain high conversion efficiency, we place the crystal within a single resonant ring cavity. Second, we present the main results of the cavity-enhanced configuration in Fig. 3(b), which shows that the intra-cavity power is a function of the circulated pump power. Here, the x-axis represents the measured intra-cavity power, and the y-axis is the intra-cavity quantum conversion efficiency (QCE). The data represented by blue asterisks, red circles, and green diamonds are the QCEs for L=0, L=1, and L=2, respectively, and the three corresponding lines are the fitting results based on the least squares method. To estimate the intra-cavity QCE, linear attenuation factors of approximately 92% are taken into account. Here, the finesse of the cavity for the pump laser is 85, and thus the enhanced factor is approximately 14 when the cavity is locked onto the master laser. Under lower pump powers, the relationship between the QCE and circulated power show good linearity. For the Gaussian beam, unit conversion efficiency can be reached when the intra-cavity power increases by three times with respect to the current maximum power, which is the same as the theoretical prediction given in Fig. 1(b). Finally, we test the fidelity of the output infrared OAM-mode. The 3D graph shown in Fig. 4(a) represents the density matrix of the infrared output state for topological charge 1 that are determined by quantum state tomography, where the components on the left are the intensities and phase distributions of the input infrared OAM fields, and the components on the right are the corresponding density matrices. In this process, the input states are |R⟩, |L⟩, |V⟩, and|D⟩. For each input state, the four corresponding projection bases that are loaded on the measured SLM are (|R⟩, |L⟩, |A⟩, and |H⟩). The average fidelities ⟨ϕ|ρ|ϕ⟩ of |R⟩, |L⟩, |V⟩, and |D⟩ are 98.01%, 98.82%, 97.01% and 92.23%, respectively. The fidelity of |D⟩ is lower than that of the other input fields, which represents the imperfect preparation of the input state and the deflected positions of the projection bases in the SLM. Nevertheless, the high fidelity and the high conversion efficiency for down-conversion of the OAM mode indicate the reliable performance of the method. Without generality, we show the intensity profiles of the two fields in Fig. 4(b), as measured using visible and infrared CCDs. The colorized graphs shown on the left of Fig. 4(b) are the input pure or superposition states |1⟩, |1⟩+|-1⟩, |2⟩, and |2⟩+|-2⟩, and the corresponding gray graphs on the right are the corresponding infrared OAM modes. It can been seen that the intensity profiles of the input and output fields are highly similar. Based on observation of the interference of the superposition state, we can also determine that the topological charge of the output state is equal to that of the input state, i.e., the OAM is conserved in this process. § CONCLUSIONIn summary, we have demonstrated coherent OAM frequency down-conversion from the visible range to the infrared based on difference frequency generation in a single resonant cavity. We first propose a theoretical model to describe the OAM frequency down-conversion process. Then, we demonstrate OAM frequency down-conversion experimentally for different OAM modes. The maximum quantum conversion efficiencies that were obtained for OAM modes with topological charges of 0, 1, and 2 were 36%, 15%, and 7.8%, respectively. Using quantum state tomography, the average fidelity achieved was determined to be 96.5%. We also showed that the OAM is conserved and that the coherent property is preserved during the DFG process. The high fidelity and high QCE values obtained show that our OAM frequency-down converter is reliable and has potential applications in construction of high-dimensional quantum networks. When compared with recently reported results in OAM frequency up-conversion, our OAM frequency down-conversion process has shown higher conversion efficiencies<cit.>. The remaining problem that must be solved is further enhancement of the quantum conversion efficiency for the higher-order modes. Use of strong pump laser is one possible way, but the best method would involve realization of mode-independent conversion efficiency for the higher-order modes with a fixed pump power. This issue will be investigated in our future research. When this mode-independent conversion efficiency is realized, frequency conversion of higher-dimensional OAM states would then be feasible. We will study frequency down-conversion of the single photon OAM states, two-dimensional OAM entangled states and high-dimensional OAM-entanglement states in the near future. Because of the near optimal quantum detection efficiency of superconducting detectors operating at 1550 nm, this frequency conversion process will be very useful in the frequency conversion detection of visible, mid-infrared and far infrared light beams <cit.>.§ ACKNOWLEDGMENTSWe would like to thank Yu Zhang of Hangzhou Normal University for help with the experiments. National Natural Science Foundation of China (11174271, 11604322, 61275115, 61435011, 61525504, 61605194); China Postdoctoral Science Foundation (2016M590570); Fundamental Research Funds for the Central Universities. We thank David MacDonald, MSc, from Liwen Bianji, Edanz Group China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript. | http://arxiv.org/abs/1707.08787v2 | {
"authors": [
"Shi-long Liu",
"Shi-kai Liu",
"Yin-hai Li",
"Shuai shi",
"Zhi-yuan Zhou",
"Bao-sen Shi"
],
"categories": [
"physics.optics",
"quant-ph"
],
"primary_category": "physics.optics",
"published": "20170727091432",
"title": "Coherent frequency bridge between visible and telecommunications band of vortex light"
} |
firstpage–lastpage On the “Poisson Trick” and its Extensions for Fitting Multinomial Regression Models [ December 30, 2023 ===================================================================================is a γ-ray emitting X-ray binary with periodic radio outbursts with time scales of one month. Previous observations have revealed microflares superimposed on these large outbursts with periods ranging from a few minutes to hours. This makes , along with Cyg X-1, the only TeV emitting X-ray binary exhibiting radio microflares. To further investigate these microflaring activity in we observed the source with the 100-m Effelsberg radio telescope at 4.85, 8.35, and 10.45 GHz and performed timing analysis on the obtained data. Radio oscillations of 15 hours time scales are detected at all three frequencies. We also compare the spectral index evolution of radio data to that of the photon index of GeV data observed by . We conclude that the observed QPO could result from multiple shocks in a jet. Radio continuum: stars – X-rays: binaries – X-rays: individual ()§ INTRODUCTION It is a known fact that a subclass of X-ray binaries and a subclass of Active Galactic Nuclei (AGN) are sources of radio emission.There is evidence that radio outbursts in these systems are superimposed by microflaring activity of lower amplitude and short time scales.These short timescale variations, characterised as Quasi Periodic Oscillations (QPO) <cit.>, may change period between different epochs or are present in a relatively short interval of time, with few oscillations.In X-ray binaries with radio jets, i.e., microquasars <cit.>, QPO were first observed in the black hole systemwhere a range of 22–120 min sinusoidal radio variations were observedduring the decay of a radio outburst <cit.>, and more recently <cit.> found correlation between radio and X-ray variability on minute time scales. Short-term radio variability on time scales of ∼1 hour was observed in Cyg X-1 <cit.>. QPO inhave been extensively studied, e.g., by <cit.>, who revealed QPO with periods in the range of 20–40 min; observations by <cit.> showed oscillations of 30 min. Also, <cit.> performed an analysisat dual radio frequencies and again revealed QPO with periods of ∼30 min following the decay of a major outburst. Moreover, the radio spectral index α oscillates from negative to positive values (see Fig. 2 in ).<cit.> demonstrated that infrared oscillations precede the radio ones, and both oscillations have similar period and shape. In simultaneous radio and X-ray observations of , <cit.> showed that X-ray dips were associated with radio peaks.Oscillations with time scales of days are known to occur in <cit.>.These longer period QPO are present in both total flux density andradio spectral index (Fig. <ref>).QPO seem to be a general phenomenon associated to the ejections in accreting systems and have in fact been observed also in AGN. X-ray QPO of 55 minutes have been indicated in the flat spectrum radio quasar 3C 273 <cit.> and QPO of ∼60 minutes have been observed in the narrow-line Seyfert 1 Galaxy RE J1034+396 <cit.>. <cit.> found evidence for optical QPO of 15 minutes in the blazar S5 0716+714 in R-band.The physical process behind QPO is still a matter of debate, three explanations are discussed. The first hypothesis explains QPO as the result of a single shock propagatingdown a helical jet and producing increased flux each time the shock meets another twist of the helix at the angle that provides the maximum Doppler boosting for the observer <cit.>. In the second scenario, QPO could be related to shocks passing down the jet and accelerating particles in situ <cit.>.As a third explanation, QPO have been related to discrete ejections of plasma. Multi-wavelength QPO observations of(see references in ) have been interpreted as periodic discrete ejections of plasma, with a mass of about ∼ 10^19 g and at relativistic speeds, with subsequent replenishment of the inner accretion disk. Testing possible models for the origin of QPO is still complicated because of insufficient statistics. Remaining open questions include: How stable are these “quasi” periodic oscillations? Why are QPO present in the radio spectral index? Why does the radio spectrum oscillate between optically thin and thick emission inand , and is that a general property of QPO? Do the “flat” radio spectra in microquasars and AGN indeed arise from the combination of emission from optically thick and thin regions as suggested in <cit.>? In order to answer these questions, it is requisite to increase the sample of X-ray binaries exhibiting radio QPO. For this purpose we investigate on which is one of the few radio emitting X-ray binaries which also emits in γ-rays (GeV, , and TeV, , ).Moreover,is periodic at all wavelengths on the orbital time scale (26.5 d)and has revealed microflaring activity (Sect. 2).Aimed at investigating QPO inwe performed new observations with the Effelsberg 100-m radio telescope. Our new radio observations and data analysis are described in Sect. 3 along with the description ofdata reduction. In Sect. 4 we present our results and in Sect. 5 our conclusions.§ THE BINARY SYSTEMThe X-ray binaryis composed of a Be star <cit.> and a black hole <cit.>, exhibiting radio outbursts <cit.> ocurring periodic, related to the orbital period of ∼ 26.5 days.With an SED peaking above 1 GeV,can further be classified as a γ-ray binary (see Table 2 in ). Among the black hole binaries discussed above, onlyand Cyg X-1 have the peculiarity of emitting at TeV,this, however, being a transient phenomenon for Cyg X-1 <cit.>, and alsohas episodes of TeV non-detection <cit.>.Two magnetar-like signals were detected from a large region of sky crowded by other potential sources beside<cit.>. On that basis <cit.> put forward their working hypothesis thatcould be the first magnetar detected in a binary system, and studied the implications. In the magnetar scenario, as well as for the pulsar scenario <cit.>, variability of the Be star would trigger and explain observed long-term variability in the emission ofat all wavelengths <cit.>. However, following the well-studied (i.e., over 100 years) case of the binary system ζ Tau, i.e., also a Be star in a binary system (Sect. 4 inand references therein), Be star variations last 2-3 cycles only and are of different lengths <cit.>. Timing analysis of 37 years of radio data performed by <cit.> reveals thatdoes not show merely 2-3 cycles which are of different lengths but a repetition of 8 full cycles of an identical length of 1628 days, well in agreement with the scenario of a microquasar with a precessing jet <cit.>. In addition, as discussed in <cit.>, the orbital shift in the equivalent width of the Hα emission line <cit.>, points tovariations caused byaprecessingjet.Optical polarization observations determined a rotational axis for the Be star of 25 degrees <cit.>. For parallel orbital and Be spin axes and the mass function determined through orbital motions measurements <cit.> a 25 degree inclination implies a black hole of 4 solar masses, as argued by <cit.>, who further probe from X-ray observations that the photon index vs. luminosity trend ofis very different from that of the non-accreting pulsar binary PSR B1259-63,whereas its trend agrees with that of moderate-luminosity regime black holes in general and with the two black holes in the same X-ray luminosity range: Swift J1357.2-0933 and V404 Cygni, in particular.The source is well-known for its strong radio outbursts with orbital periodicitymonitored for ∼ 40 years <cit.>. Along with this strong radio outburst there is also evidence of occurrance of microflaring activity with time scales of minutes to hours.During the decaying phase of one of the radio outbursts of , a step-like pattern was observed for the first time with the Westerbork Synthesis Radio Telescope (WSRT) with a characteristic timescale of ∼ 10^3 s <cit.>. <cit.> performed the first systematic study of this short-term radio variability and found a period of 1.4 hours for these microflares with an amplitude of ∼4 mJy in VLA observations related to the decay of one radio outburst.Recently, <cit.> observed during the decay of one outburst ofsub-flares with a characteristic time-scale of two days (see their Fig. 1). Furthermore, the radio spectral index oscillatedin a quasi-regular fashion and the local peaks in spectral index roughly coincide with the peaks of the sub-flares seen in the total intensity light curves. Finally, at higher energy, <cit.> discovered a periodicity of ∼40 minutes in an X-ray ASCA observation associated to the onset of a radio outburst.§ OBSERVATION AND DATA ANALYSIS §.§ Effelsberg radio telescope Our multi-frequency flux density measurementswere performed about every 45 minutes for almost 100 hours on April 17-21, 2014 (MJD 56764.726 until MJD 56768.763). The orbital phase Φ is defined asΦ = t-t_0P_orb - int(t-t_0P_orb),where, t_0 = MJD 43366.275, orbital period P_orb=26.4960±0.0028 d <cit.>, giving Φ=0.68-0.83 for our observations. The secondary focus receivers of the Effelsberg 100-m telescope were used at three frequencies, namely4.85, 8.35, and 10.45 GHz (6.0, 3.6, 2.8 cm wavelengths respectively). Flux density measurements were performed using the “cross-scan” technique, i.e., progressively slewing over the source position in azimuthal and elevation direction with the number of sub-scans matching the source brightness at a given frequency (typically 4 to 12).At 4.85 and 10.45GHz, the “beam switch”, realised through multiple-feed systems, removed most of the tropospheric variations, allowing for more accurate measurements. The cross-scan technique on the other hand allows instantaneous correction of small, remaining pointing offsets. The data reduction, from raw telescope data to calibrated flux densities/spectra, was done in the standard manner as described in <cit.>. Problems with the 8.35 GHz receiver caused the large flagging of data at this frequency.The best data set for sampling rate and SNR is that at 4.85 GHz. The light curves obtained for all three frequencies are shown in Figure <ref> (a, c, e) along with their spectral index computed as α = log(S_1/S_2)/log(ν_1/ν_2) for Fig <ref> b, and as a linear fit to the fluxes vs. frequency in double logarithmic scale for every time bin of 45 minutes, shown in Fig. <ref> d.In order to analyse short-term periodicities, we removed the long-term trend from the light curves by subtracting a quadratic function f_1(t)=a_1 (t-t_0)^2 + b_1,with best-fit parameters listed in Table <ref>. The rectified data were then analyzed using wavelet analysis <cit.>, auto-correlation function and Lomb-Scargle periodogram <cit.>. We test the significance of found periodic signals by employing the Fisher randomisation test <cit.> where the flux is permuted a thousand times and thousand new randomised time series are created and their periodograms calculacted. The proportion of randomised time series that contain a higher peak in the periodogram than the original periodogram at any frequency then gives the false alarm probability p of the peak. If p < 0.01, the period is significant, and if 0.01 < p < 0.1 the period is marginally significant.The data were then folded on the resulting significant period, and the folded data were fitted with a sine-function f_2(ϕ)=Asin2π(ϕ-ϕ_0)+B,with its best-fit parameters in Table <ref>.§.§data reduction We compare our radio data with GeV γ-ray data. For this purpose we use the GeV data from thein the energy range 0.1–3.0 GeV from MJD 56764.125 until MJD 56768.875. For the analysis ofdata we used version v10r0p5 of the Fermi ScienceTools[available from <http://fermi.gsfc.nasa.gov/ssc/data/analysis/software/>]. We used the instrument response function P8R2_SOURCE_V6 and the corresponding model gll_iem_v06.fits for the Galactic diffuse emission and the template iso_P8R2_SOURCE_V6_v06.txt. Model files were created automatically with the script [available from <http://fermi.gsfc.nasa.gov/ssc/data/analysis/user/>] from the thirdpoint source catalog <cit.>. The spectral shape ofin the GeV regime is a power law with an exponential cut-off at 4–6 GeV <cit.>. Here we restrict our analysis to the power law part of the GeV emission by fitting the source withdn/ dE = n_0(E/E_0)^-Γ [ countscm^2sec dE] with all parameters left free for the fit, and including data in the energy range E = [0.1 - 3]GeV. All other sources within a radius of 10^∘ and the Galactic diffuse emission were left free for the fit. All sources between 10^∘-15^∘ were fixed to their catalog values. The light curves were computed performing this fit for every time binof width half a day for data from 2014 April 17 (MJD 56764.125) till 2014 April 21 (MJD 56768.875). On average, the test statistic forwas 40, which corresponds to a detection of the source at the 6.3σ level on average in each time bin. § RESULTS The light curves at all three frequencies (Fig. <ref> a, c, e) show small-scale oscillations. Amplitude and width seem to vary from one peak to the other. The higher sensitivity of 4.85 GHz data give rise to significant timing analysis results: the wavelet analysis in the top panel of Fig. <ref> shows that the oscillations at 4.85 GHz have a periodicity of about 16 hours.The auto-correlation coefficient in the middle panel of Fig. <ref> shows peaks at 15 hours, 30 hours and at about 55 hours. Lomb-Scargle analysis in the bottom panel of Figure <ref> givesa dominant and significant feature at 15.4 ± 0.6 h. Indeed, data at all three frequencies fold with the 15.4 hours period (see Fig. <ref> and fitting parameters in Table <ref>) with significance of the oscillations clearly above 8σ. Oscillations are also present in the radio spectral index α (Fig. <ref> d) corroborating the observed spectral index oscillations by <cit.>. The oscillations create a sort of flattening of the spectral index. and a zoom-in centered at MJD 56766.3 clearly shows α = 0 (Fig. <ref> f).§ CONCLUSIONS AND DISCUSSION We observed one radio outburst ofwith the Effelsberg 100-m telescope in April 2014 for approximately 100 h at 4.85, 8.35, and 10.45 GHz. We analysed these data along with simultaneousGeV data. Our results reveal the following: *QPO previously observed inshowed time scales of minutes, hours and days <cit.>. Our study determines periodicities of ∼ 15 h. There are three hypotheses to explain the physical mechanism behind the occurrence of these periodicities. The first is associated to the geometry of the jet <cit.>, the second one to multiple shocks <cit.>, and the third one implies discrete plasma ejections <cit.>. In the first scenario the reason for oscillations is the helical topology of the magnetic field in the jet associated to Doppler boosting effects.Since in a conical jet of a microquasar the 10 GHz emission originates from a jet-segment nearer to the central engine of the system than the 5 GHz jet-segment <cit.>, this scenarioimplies alonger period for 5 GHz oscillations. This is not compatible with our results in Table <ref> and Fig. <ref>,giving the same period for the oscillations at all three observed frequencies.Following the <cit.> model of adiabatically expanding spheroidal ejecta of plasma, the low-frequency emission peak is delayed and weaker than that at higher frequencies.Our result of a larger amplitude at lower frequency is in contradiction with this model.Finally, the shock-in-jet model proposed by <cit.> and generalised by <cit.> predicts flare amplitudes to increase towards lower frequency, as observed.*As previously observedin X-ray binariesand , the radio spectral index ofoscillates in phase with total flux variations, the radio spectral index modulation being superimposed on a longer modulation whose peak corresponds to the flattening of the spectral index. This finding is consistent with the hypothesis that the “flat” radio spectra in microquasars arise from the combination of emission from optically thick and thin regions <cit.>.The radio spectrum after the oscillations becomes optically thin with p ∼ 2.0, the γ-ray emission follows a similar trend with a photon index Γ∼ 2.0. § ACKNOWLEDGEMENTS We thank Eduardo Ros for carefully reading the manuscript. This work has made use of public Fermi data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA Goddard Space Flight Center.mnras | http://arxiv.org/abs/1707.08537v1 | {
"authors": [
"Frederic Jaron",
"Richa Sharma",
"Maria Massi",
"Lars Fuhrmann",
"Emmanouil Angelakis",
"Ioannis Myserlis",
"Guang-Xing Li",
"Xun Shi"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726165413",
"title": "Radio QPO in the $γ$-ray-loud X-ray binary LS I +61${^\\circ}$303"
} |
Stellar Astrophysics Centre (SAC). Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus, Denmark [email protected] School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK California Institute of Technology, 1200 E. California Blvd, MC 249-17, Pasadena, CA 91125, USA Leiden Observatory, Leiden University, 2333CA Leiden, The Netherlands Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Previously [Y/Mg] has been proven to be an age indicator for solar twins. Here, we investigate if this relation also holds for helium-core-burning stars of solar metallicity.High resolution and high signal-to-noise ratio (S/N) spectroscopic data of stars in the helium-core-burning phase have been obtained with the FIES spectrograph on the NOT 2.56m telescope and the HIRES spectrograph on the Keck I 10m telescope. They have been analyzed to determine the chemical abundances of four open clusters with close to solar metallicity; NGC 6811, NGC 6819, M67 and NGC 188. The abundances are derived from equivalent widths of spectral lines using ATLAS9 model atmospheres with parameters determined from the excitation and ionization balance of Fe lines. Results from asteroseismology and binary studies were used as priors on the atmospheric parameters, where especially the log g is determined to much higher precision than what is possible with spectroscopy.It is confirmed that the four open clusters are close to solar metallicity and they follow the [Y/Mg] vs. age trend previously found for solar twins.The [Y/Mg] vs. age clock also works for giant stars in the helium-core burning phase, which vastly increases the possibilities to estimate the age of stars not only in the solar neighborhood, but in large parts of the Galaxy, due to the brighter nature of evolved stars compared to dwarfs. The [Y/Mg] clock works for evolved solar metallicity stars Based on spectroscopic observations made with two telescopes: the Nordic Optical Telescope operated by NOTSA at the Observatorio del Roque de los Muchachos (La Palma, Spain) of the Instituto de Astrofísica de Canarias and the Keck I Telescope at the W.M. Keck Observatory (Mauna Kea, Hawaii, USA) operated by the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.D. Slumstrup 1 F. Grundahl 1 K. Brogaard 1,2 A. O. Thygesen 3 P. E. Nissen 1 J. Jessen-Hansen 1 V. Van Eylen 4 M. G. Pedersen 5 Received 3 July 2017 / Accepted 20 July2017============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION Stars in an open cluster can be assumed to originate from the same molecular cloud and are therefore expected to have the same chemical composition and age. Age is an especially difficult parameter to determine for single field stars and different methods yield ages that are not necessarily in agreement <cit.>, whereas stars in clusters have ages determined to much higher precision (e.g., <cit.>).<cit.>, <cit.> and <cit.> investigated trends of different chemical abundances with stellar age and found that [Y/Mg] is a sensitive age indicator for solar twins. <cit.> did an independent study extending the sample to a much larger range in [Fe/H]. They found that the relation depends on [Fe/H]. For solar metallicity stars they clearly see the relation, but for stars with [Fe/H] ∼ -0.5 dex the relation is insignificant and cannot be used to determine age. Our targets have solar metallicity but are evolved (in the helium-core-burning phase) and therefore do not have the same atmospheric properties as the solar twins. They are also located at larger distances than previously studied. Ages are well determined from cluster studies, which allows us to determine if the relation found for solar twins persists for helium-core-burning stars.Previously, yttrium abundances of open clusters have been studied to find possible trends with age, for example,<cit.> found no clear trend of [Y/Fe] with age most likely due to large uncertainties, whereas <cit.> find a declining trend, but the scatter is large, which could be affected by the large range in [Fe/H].We have carried out a detailed fundamental parameter and abundance analysis of six targets in four open clusters (NGC 6811, NGC 6819, M67 and NGC 188) based on high-resolution and high signal-to-noise ratio (S/N) spectroscopic data from the Nordic Optical Telescope (NOT) and the Keck I Telescope. For NGC 6811, NGC 6819 and M67 we have asteroseismic data available from the Kepler <cit.> and K2 missions <cit.>, which put a strong constraint on the log g value and can thereby constrain the analysis. In Sect. <ref> we present the data taken for this project. In Sect. <ref>, we describe the analysis. In Sect. <ref> we present the results. In Sect. <ref>, we test the [Y/Mg] vs. age relation by <cit.> for our sample of solar metallicity helium-core-burning stars. Finally, we conclude on the results in Sect. <ref>. § TARGETS, OBSERVATIONS AND DATA REDUCTIONThe metallicity of NGC 6819 is still debated <cit.> and we aim to establish it securely through the analysis in this project. M67 is a very well studied nearby solar-like cluster, which has also been observed with the K2 mission. NGC 188 is also a solar metallicity cluster that is older and fairly well studied, however not as well as M67. For these three clusters, we have only chosen one target in each cluster, but they are all confirmed members, <cit.>. The data used in our analysis for both NGC 188 and NGC 6819 is of higher resolution and higher S/N than previously used. NGC 6811 is also a solar-metallicity cluster and it is the youngest in the sample. For this we have three targets, all confirmed members <cit.>. All targets, except the NGC 188 target, is confirmed by asteroseismology to be in the helium-core-burning phase <cit.>. We have adopted literature values for reddening, distance modulus and age for each cluster (see Table <ref>). The observations of NGC 6819, M67 and NGC 188 were collected with the Nordic Optical Telescope (NOT) on La Palma, Spain. Spectra for the three stars were obtained in the summer of 2013 and 2015 with the high resolution FIbre-fed Echelle Spectrograph (FIES) covering the wavelength region from 3700-7300Å, see <cit.> for a detailed description of the spectrograph. All observations were carried out in the high resolution mode, R=67,000.The individual spectra were reduced with FIEStool[<http://www.not.iac.es/instruments/fies/fiestool/>], an automated data reduction software for FIES. Lastly, the spectra for each target were shifted to a common wavelength scale and merged.The observations of the three targets in NGC 6811 were carried out on the night of Aug. 23, 2016 using the HIRES spectrograph <cit.> on Keck I. All targets were observed with the red cross-disperser, covering the wavelength region from 4000-7900Å, with a few inter-order gaps for the reddest orders. We used the ”C5” decker, providing a resolution of R=37 000. The targets were observed with the exposure meter in operation <cit.>to ensure a uniform S/N in all spectra.The reductions were performed using the MAKEE pipeline[<http://www.astro.caltech.edu/ tb/makee/>]. All observations are presented in Table <ref>.The co-added spectrum for each star was normalized order by order using RAINBOW[<http://sites.google.com/site/vikingpowersoftware>], which uses appropriate synthetic spectra to identify continuum points in the observed spectrum, which are then fitted with a spline function. The S/N values given in Table <ref> were estimated from the rms variation of the flux in a region around 6150 Å.§ DATA ANALYSISParameters derived from the photometry presented in Fig. <ref> were used as starting estimates, listed in Table <ref>. The effective temperatures for NGC 6819, M67 and NGC 188 were calculated from the color-temperature calibration presented by <cit.> for different filter combinations. For each target, an average of the different filter combinations was used. For NGC 6811 we used the effective temperatures from <cit.> as first estimates. The log g for the stars in NGC 6811, NGC 6819 and M67 in Table <ref> is from asteroseismology, ( respectively) calculated with the ν_max scaling relations (Brown1991, Kjeldsen1995): log g = log( ( ν_max/3100) ·( T_eff/5777) ^1/2) + 4.44.The asteroseismic log g values are determined to very high precision and have been shown to be in very close agreement with the physical log g (e.g., ). This provides a strong constraint on the analysis. For NGC 188, the log g is calculated with the equation from <cit.> using a mass of 1.1M_⊙ from <cit.> assuming no significant mass loss on the RGB <cit.>.We have tested different line lists and different programs to calculate the equivalent widths, which will be discussed in more detail in a forthcoming paper. Based on external constraints from especially asteroseismology, the final choice of line list is from <cit.> with astrophysical log gf values based on solar abundances from <cit.>. We have omitted lines stronger than 100mÅ for Fe and 120mÅ for other elements. A few additional magnesium and yttrium lines were added to do a more robust determination of [Y/Mg] for the [Y/Mg] vs. age relation discussed in Sect. <ref>. The final line list will be given in a forthcoming paper with the measured equivalent widths for all lines. The equivalent widths were measured with DAOSPEC <cit.> and the auxiliary program Abundance with SPECTRUM <cit.> was used to calculate the atmospheric parameters and abundances based on solar abundances from <cit.> and ATLAS9 stellar atmosphere models <cit.>. Local thermodynamic equilibrium (LTE) is assumed. There may be non-LTE effects on the derived abundances, but since the stars have similar parameters, differential abundances between the stars are reliable.The atmospheric parameters were determined by requiring that [Fe/H] has no systematic dependence on the excitation potential or the strength of the FeI lines and that the mean [Fe/H] values derived from FeI and FeII lines are consistent. The slope of [Fe/H] as a function of excitation potential is sensitive to the effective temperature and the slope of [Fe/H] as a function of the reduced equivalent widths of the lines (log (EW)/λ) depends on the microturbulence. The surface gravity is determined via its effect on the electron pressure in the stellar atmosphere with the FeI and FeII equilibrium, as the FeII lines are more sensitive to pressure changes than the FeI lines. This is however also affected by the temperature and heavier element abundances and it was necessary to make a number of iterations. For NGC 6811, NGC 6819 and M67, we also calculated a new asteroseismic log g with the newly found effective temperature, but the variation in effective temperature is low enough, that the asteroseismic log g was not significantly affected.§ ATMOSPHERIC PARAMETERS AND ABUNDANCESThe final result for the atmospheric parameters are presented in Table <ref>. The uncertainties are only internal and calculated by varying a parameter until at least a 3σ uncertainty is produced on either of the two slopes, [Fe/H] vs. excitation potential and [Fe/H] vs. reduced equivalent width, or on the difference between FeI and FeII. The change in the parameter is then divided by the highest produced uncertainty to give one standard deviation, provided in Table <ref>. The errorbars on [Fe/H], [α/Fe] and [Y/Mg] is the standard error of the mean. The spectroscopic log g values are, within the errorbars, in agreement with the results from asteroseismology. The metallicities for all targets are close to solar, with NGC 188 having a slightly higher-than-solar abundance and NGC 6811 having a slightly lower-than-solar abundance, which along with the temperature differences can be seen in the line depths in Fig. <ref>.The metallicity of NGC 6819 is lower than that found by <cit.> ([Fe/H]=+0.09 ± 0.03 dex found by analyzing three giants with high resolution spectroscopy), but fits better with the value from <cit.> who performed an analysis of multiple targets but with low resolution spectroscopy. Their value of [Fe/H]= -0.02 ± 0.02 dex is found using main sequence and turnoff stars.The abundances for [Fe/H], [α/Fe], and [Y/Mg] are given in Table <ref> while several additional elements will be presented in a forthcoming paper. The magnesium lines used are at wavelengths 5711.09Å, 6318.70Å, 6319.23Å and 6319.48Å. The yttrium lines are at 4883.70Å, 4900.13Å and 5728.89Å. Not all of the lines were usable for each star. The magnesium line at 5711.09Å (see Fig. <ref>) is, for some of the targets, stronger than the limit of 120mÅ for non-iron lines, but it is well isolated for all stars, and we therefore chose to include it to do a more robust determination of Magnesium. The final value for [Y/Mg] used for NGC 6811 is a the mean of the three targets.§ [Y/MG] VS. AGEMagnesium is an alpha-element mostly originating from type II supernovae explosions, which gives an increase of [Mg/Fe] with increasing stellar age because iron is also produced in the later type Ia supernovae explosions. Yttrium is an s-process element and [Y/Fe] is observed to decrease with increasing stellar age. This is likely a consequence of intermediate mass asymptotic giant branch stars not yet being important for the production of Y at early times. The slope of [Y/Fe] with age is steep and opposite to that of [Mg/Fe]. For solar twin stars, <cit.> found the relation:[Y/Mg] = 0.170(±0.009) - 0.0371(± 0.0013) ·Age [Gyr].This relation is plotted in Fig. <ref>. <cit.> confirmed the relation for dwarfs of solar metallicity but found that it disappears for stars with [Fe/H]≈ -0.5 and below. We have extended the parameter range to helium-core-burning giants at close to solar metallicity, and they also follow the relation from <cit.> as seen in Fig. <ref>.This result is of particular interest for galactic archaeology studies as giants are much brighter than dwarfs, which allows us to study farther regions of the galaxy and not only the solar neighborhood.<cit.> carried out an abundance analysis of 14 stars in M67 at different evolutionary stages with high resolution spectra. They find an average metallicity of [Fe/H]=+0.06 with an average [Y/Mg]=-0.04 ± 0.05, which is a little lower than our result, but still close to being in agreement with the relation from <cit.>, marked in Fig. <ref>§ CONCLUSIONAtmospheric parameters and abundances have been determined for the four open clusters, NGC 6811, NGC 6819, M67 and NGC 188 with an equivalent width analysis of individual spectral lines from high-resolution, high S/N observations from the NOT and the Keck I Telescope. The parameters obtained fit very well with the literature, and especially, the log g values fits with predictions from asteroseismology. The metallicities of all four clusters are nearly solar, with NGC 6811 being slightly sub-solar.The empirical relation between [Y/Mg] and age as presented by <cit.> was found to hold also for helium-core-burning giants of close to solar metallicity. This is of great importance to galactic chemical evolution studies, as the brighter nature of giants allows us to probe the Galaxy to greater distances and not only the solar neighborhood. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.We are most fortunate to have the opportunity to conduct observations from this mountain. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant DNRF106). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. MGP is funded from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation program (grant agreement N^o 670519: MAMSIE). aa | http://arxiv.org/abs/1707.08585v1 | {
"authors": [
"D. Slumstrup",
"F. Grundahl",
"K. Brogaard",
"A. O. Thygesen",
"P. E. Nissen",
"J. Jessen-Hansen",
"V. Van Eylen",
"M. G. Pedersen"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170726180117",
"title": "The [Y/Mg] clock works for evolved solar metallicity stars"
} |
Sectoring in Multi-cell Massive MIMO Systems Shahram Shahsavari, Parisa Hassanzadeh, Alexei Ashikhmin, and Elza Erkip S. Shahsavari, P. HassanzadehandE. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {shahram.shahsavari,ph990, elza}@nyu.eduA. Ashikhmin is with Bell Labs, Nokia, Murray Hill, NJ, USA. Email:[email protected] December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================== One important feature of the mammalian immune system is the highly specific binding of antigens to antibodies. Antibodies generated in response to one infection may also provide some level of cross immunity to other infections. One model to describe this cross immunity is the notion of antigenic space, which assigns each antibody and each virus a point in ℝ^n. Past studies have suggested the dimensionality of antigenic space, n, may be small. In this study we show that data from hemagglutination assays suggest a high dimensional random walk (or self avoiding random random walk). The discrepancy between our result and prior studies is due to the fact that random walks can appear low dimensional according to a variety of analyses. including principal component analysis (PCA) and multidimensional scaling (MDS). 1cm § INTRODUCTION §.§ Antigenic SpaceDuring a viral infection, antibodies bind to viral antigens by recognizing specific epitopes on their surface. The same antibody may provide protection against other strains of the same virus if the antigens are not too dissimilar. In 1979, Alan Perelson and George Oster defined the idea of antigenic space <cit.>. They supposed that each antibody and antigen might be described by a vector in ℝ^n. This idea was used as a basis to study the dynamics of the immune response, with n assumed to be a small number <cit.>. Subsequent work to estimate n based on the frequency of cross reactivity resulted in the conclusion that n was around five to eight <cit.>.The notion of antigenic space has proven particularly popular for understanding the evolution of influenza H3N2 <cit.>. This strain has been circulating in the human population since 1968 and gradually mutating. These mutations can in principle be represented as the movement of the virus through antigenic space. As the antigen moves it can evade the antibodies elicited by older strains and thus reinfect individuals. The distance between a viral strain and an antibody can be measured via the hemagglutination inhibition (HAI) assay, in which a viral strain and a serum of antibodies are both added to a culture of red blood cells. If the antibodies are ineffective against the viral strain then the virions stick to the red blood cells causing them to cluster together (hemagglutinate). However if the antibodies are effective, they will neutralize the virions and inhibit their hemagglutination of the red blood cells. In the former case, the strain and the serum are distant antigenically, whereas in the latter case they are close. By performing serial dilutions of the antibody serum, one can quantify just how close a serum and antibody are.Points in antigenic space can be inferred from a distance matrix via multidimensional scaling (MDS).Low dimensional reconstructions of antigenic space can reproduce the HAI data with high fidelity, and adding new dimensions beyond n=5 does not improve the quality of the fit <cit.>. Thus it may be tempting to conclude that influenza is evolving in an antigenic space of no more than five dimensions. §.§ Outline of resultsIn this work we will argue that influenza H3N2 is evolving in a very high dimensional space, and that it may appear to be low dimensional due to the nature of random walks. Our argument consists of three parts. * High dimensional random walks contain most of their variance in a small number of dimensions. Specifically, one would expect at least 60% of the variance to occur in a single dimension. We show this via principal component analysis. However, we note that the true dimensionality of random walks can be revealed by out of sample PCA.* This apparent low dimensionality also occurs with multidimensional scaling — the method used to analyze HAI titer data. We simulate HAI data generated using a high dimensional random walk and find we can accurately reproduce the data with points taken from a low dimensional space; increasing the dimensionality of our representation does not improve our fit. We also show that non-metric MDS can be used to reconstruct an infinite dimensional random walk in a single dimension.* Finally, we show that H3N2 data has characteristics of a high dimensional random walk, indicating that it was unlikely to result from a random walk of dimensionality less than n=10. This is even the case when we consider that the random walk of H3N2 is likely self avoiding.§.§ Why a random walk?Throughout this paper we argue for a high dimensional random walk as a model for influenza evolution. A random walk may seem a priori to be a poor model for viral evolution, as immunological memory should prevent a virus from revisiting areas of antigenic space. Therefore we should expect the path of viral evolution to be self avoiding. In high dimensions an unbiased random walk and self avoiding random walk will behave very similarly, because a high dimensional random walk is already extremely unlikely to cross itself. In an n dimensional random walk the distance between points i and j is a random variable D_ij. Its distribution isD_ij^2∼α |i-j|χ^2_nwhere α is a constant of proportionality. This means that for large n the distances increase in a very predictable manner as the χ_n^2 distribution narrows. The probability of the random walk approaching a previous point is essentially zero, so we need not include any further tendency for self avoidance. However, in the latter part of the paper we will address the question as to whether low dimensional self avoiding random walk could also be consistent with the data. §.§ True dimensionality vs effective dimensionalityLet x_i ∈ℝ^n represent distinct viral strains and/or antisera. The antigenic dissimilarity of the two different strains x_i and x_j is the euclidean distance D_ij=x_i-x_j_2. D_ij represents the true distance between these two strains and n is the true dimensionality of antigen space.If we can represent each point x_i with a corresponding point y_i ∈ℝ^k such that the distances from y_i to y_j is approximately D_ij, then we say that the effective dimensionality of {x_i} is k.There are several possible ways to obtain the points y_i and to evaluate how closely the reconstructed distances match the true or measured distances. In this paper we shall make use of three: classical MDS, metric MDS and nonmetric MDS. Classical MDS is also commonly known as principal component analysis (PCA). The points y_i are found by orthogonally projecting the points x_i onto the dimensions that contain the most variance. Metric MDS simply minimizes the residual between the true distances and the reconstructed distances. Non-metric MDS is similar to metric MDS but the relationship between the true distances and the reconstructed distances is only assumed to be monotonically increasing as opposed to directly proportional.§ RESULTS §.§ Truly high dimensional random walks have low effective dimensionality according to PCAWe will now show that random walks with high true dimension can have a very low effective dimension. Our analysis will focus on infinite dimensional random walks, but we will show via simulations that finite random walks have similar behavior. Let x_i∈ℝ^n be the ith step in our random walk in n dimensions. Let X be an m by n matrix whose ith row is x_i. The rows of x_i are determined by a seres of random steps b_i ∈ℝ^n, each entry of which is an independent sample from 𝒩(0,1/√(n))x_1 =b_1x_i-x_i-1 =b_ii>1We can state this relationship compactly as LX=B, where the rows of B are b_i and L is an m by m matrix with one on the diagonal and negative one on the subdiagonal.To perform PCA, we first center each column of X by premultiplying be the projection matrix P=I-^T. Note that centering each column is just a translation of the data, so all pairwise distances are preserved. We then compute the eigenvalues of A=PXX^TP, λ_1>λ_2>⋯λ_m. These eigenvalues indicate the variance in each principal component of X. For this section we shall define effective dimension asED_a=min{d:∑_i=1^k λ_k≥ a tr(A)}where 0<a<1 is the fraction of variance that we require within the first k dimensions.The random matrix A=PL^-1BB^TL^-TP is full rank and has a wishart distribution if the number of steps is no greater than the number of dimensions, i.e. m ≤ n. If number of steps is greater than the number of dimension (n>m), A is singular and has a pseudo-wishart distribution.§.§.§ Infinite dimensional random walksThe matrix W=BB^T is an uncorrelated Wishart matrix. The joint probability distribution of the eigenvalues of Wishart matrices is known, but the formula is cumbersome <cit.>. Therefore we will focus our analysis on the limiting behavior as n→∞, i.e. infinite dimensional random walks.As n increases, W approaches the m by m identity matrix. (To see this, either note that the i,jth entry of W is <b_i,b_j>). Using this simplification, the matrix A approachesA=PL^-1L^-TPNoting that L^-1 is an upper triangular matrix with all ones, we can calculate tr(A)=(m-1)(m+1)/6.We can first observe that this matrix has one eigenvalue v= with corresponding eigenvalue λ_1=0. To find the remaining eigenvalues we use the pseudo-inverse of A, A which has the simple formA= [1 -10⋯0; -12 -1⋱⋮;0⋱⋱⋱0;⋮⋱ -12 -1;0⋯0 -11 ]The eigenvalues of this matrix are μ_s=2-2cos(π s/m). Therefore the eigenvalues of A areλ_k=1/(2-2cos(π s/m)) The fraction of the variance in the first k dimensions is∑_s=1^k λ_s/tr(A) =∑_s=1^k (1-cos(π s/m))^-1/(m-1)(m+1) Finite m=6/π^2∑_s=1^k 1/s^2Infinite mFor infinite dimensional random walks, the first principal component contains at least6/π^2 or roughly 60% of the total variance. The first two components contain 80% of the variance, but the subsequent convergence is slow as 12 components are needed to account for 95% of the variance (Fig. <ref>A,B). These numbers are the limiting behavior as the length of the random walk grows. For shorter walks, more of the variance is in the first few components (Fig. <ref>C, E) and thus the effective dimensionality may be even lower.§.§.§ Finite dimensional random walksIn a finite number of dimensions n, the matrix A is a random variable with a correlated Wishart or Pseudo-Wishart distribution. We therefore use simulation to investigate the effective dimensionality. As the true dimensionality increases, the effective dimensionality asymptotically approaches the value for the infinite case, in which 12 dimensions contain 95% of the variance. However, the effective dimensionality for finite random walks is stochastic and in a few cases we find that it may exceed 12, although the mean is lower (Fig <ref>A).Like the effective dimensionality, the percentage of variance in the first principal component is a random variable, Ψ_1. Regardless of the length of the walk or the true dimensionality, the mean of Ψ_1 is very similar to the infinite dimensional case, and the variance of Ψ_1 decreases with the true dimensionality but is independent on the length of the walk (Fig <ref>C,D). This gives the somewhat counter intuitive result that lower dimensional random walks may have less variance in their first principal component than higher dimensional walks. When we consider the percentage of variance in the first 5 components Ψ_5, this pattern disappears (Fig. <ref>E,F). The values of Ψ_5 tend to decrease monotonically as both the true dimensionality and the number of steps increase, asymptotically approaching the prediction of roughly 89% from the infinite dimensional case.§.§.§ True dimensionality of random walks can be revealed by subsamplingWe have demonstrated that when we perform PCA on a random walk, we find most of the variance in only one dimensoin. Therefore, it is be difficult to tell whether the data originate from a high or low dimensional random walk using PCA. In this section we demonstrate a way around this problem using out of sample PCA.We consider two scenarios: a uniform random walk in which movement in every direction is equally likely, and a nonuniform random walk in which movement is ten times greater — and thus the variance is 100 times greater — in 14 dimensions compared to the other 86. PCA finds the majority of variance is in the first dimension for both of these scenarios (Fig <ref>). However if we take the projection matrix derived from performing PCA on the first half of the random walk and apply it to the second half, we see that the bias towards the first dimension largely disappears. The approximation improves further if we divide the second half into subsequences, apply the projection to each subsequence and then average the results (see methods for further details). This latter method shows clearly that in the uniform random walk, each principal component has roughly the same variance. It also captures the fact that 14 dimensions contain significantly more variance in the nonuniform random walk. Therefore out of sample PCA can in principle allow one to recover the true dimensionality of a random walk. §.§ Truly high dimensional random walks look low dimensional according to MDS§.§.§ H3N2 evolutionDerek Smith, Alan Lapedes and colleagues have used dimensional reduction techniques to reconstruct two-dimensional antigenic maps of the evolution of Influenza H3N2 <cit.>. In these maps, some points represent viruses whereas others represent antisera. Distances can only be computed between a virus and an antisera as opposed to between different viruses directly. Thus classic MDS, which requires measured distances between all pairs of points, cannot be used for this problem. Instead, they use metric MDS with a slight modification to handle sub-threshold values. Using this method, they have reconstructed a two-dimensional map of antigenic space. To validate this map, they challenged it to predict the antigenic distance between strains and antisera that were not used as inputs to build the map. They report a good correlation (roughly 80%) between the predicted and actual values, and furthermore that the accuracy of their predictions is similar for a two dimensional map and for higher dimensions. This is consistent with prior work showing that no more than five dimensions is required to satisfactorily reconstruct HAI data <cit.>. Although these authors make no claims about the true dimensionality of the underlying space, it may be tempting to conclude that antigenic space has a relatively low dimensionality. Our work, however, suggests that the antigenic space of H3N2 may appear to have low dimensionality even though the true dimensionality is very high. We therefore created artificial data to to mimic the evolution of H3N2 via a high dimensional random walk.We then followed the procedure for evaluating the quality of reconstructions outlined in <cit.>.* We partitioned the real and artificial distance matrices into a training set (90% of the data) and a test set (10%)* Using only the distances in the training set we reconstructed points in antigenic space corresponding to each virus and serum. We varied the dimensionality of antigenic space (k) between 2 and 20. Note our artificial data was created using points in 100 dimensional space.* We then uses the reconstructed points to predict the antigenic distances in the test set. We evaluated the quality of these fits both by the root-mean-square error and the correlation between the actual and predicted distance. These were the same criteria used in <cit.>. We find that increasing k beyond 2, i.e. adding more dimensionality to our reconstructed antigen space, did not improve the quality of the distance estimates (Fig <ref>A,B). This suggests that even data originating from a high dimensional space can be approximated in two dimensions.§.§.§ An extreme case: High dimensional random walks can look one dimensional in non-metric MDSThe goal of metric MDS is to find points x ∈ℝ^k whose pairwise euclidean distances approximately match a set of given distances. In particular, the points x must minimize the stressMetric Stress=√((x_i-x_j-d_ij)^2/∑ d^2_ij)where d_ij is the distance between point i and j.In practice, the d_ij used as an input to MDS may not represent physical distances but rather a generic measure of dissimilarity that does not behave like a metric. For example, consider three points with dissimilarities d_12=1, d_23=2 and d_13=100. The dissimilarity cannot be a metric as it violates the triangle inequality. However, there may be an underlying metric space that generated the points, and the underlying dissimilarity represents a transformation, f, of that metric. The goal of non-metric MDS is to find both f and values for x that minimizeNon-metric Stress=√((f(x_i-x_j)-d_ij)^2/∑ d^2_ij) The distance-squared between steps in an n dimensional random walk follows a χ^2_n distribution.D_ij^2∼α |i-j|χ^2_nwhere α is a constant of proportionality. In this case we let the dissimilarity d_ij be the euclidean distance D_ij. As n →∞, the dissimilarity becomes exactly proportional to √(|i-j|). Thus we can construct a one dimensional map of our random walk viaDissimilarity between point i and point j d_ij^2=α |i-j| Location of points x_i=α i Monotonic Transform f(·)=√(·)Plugging (<ref>) into (<ref>) we find that the stress is identically equal to zero. Therefore an infinite dimensional random walk can be exactly represented in one dimension using non-metric MDS.§.§ Distinguishing between low dimensional and high dimensional random walks using distance measuresSo far we have demonstrated that high dimensional random walks can appear low dimensional using methods such as MDS or PCA. We therefore ask whether there is any good way to distinguish a low dimensional from a high dimensional random walk. One characteristic of high dimensional random walks is the curve that forms when the first two principal components are plotted against each other <cit.>. This pattern emerges as the number of dimensions approaches infinity. Fig. <ref>A shows the first two principal components by year, whose path is reminiscent of the quadratic shape predicted in <cit.>. This pattern is suggestive of a high (FIg <ref>B), rather than low dimensional (Fig <ref>C) random walk. We next sought an objective measure to distinguish a low dimensional from high dimensional walk. Once again we take advantage of the fact that the square of the distance between the ith and jth step in an n dimensional random walk is proportional to a χ^2 distribution with n degrees of freedom.D_ij^2∼α |i-j|χ^2_nLet D̅^2_s denote the average distance between two points separated by s steps, i.e.D̅^2_s=1/m-s∑_i=1^i-m+sD^2_i, i+swhere m is the number of steps. As n→∞, the coefficient of variation of χ^2_n → 0, so we should expect D̅^2_s ≈α s for some α. In other words, for high dimensional random walks plotting D̅^2_s vs s should give a linear relationship. We therefore grouped all antigenic map points in the H3N2 data by year, representing both viral strains and anti-sera, and computed the mean of each group. Computing the distance-squared between the annual means and plotting versus the number of years elapsed does indeed produce a linear pattern. In this regard, the H3N2 data once again qualitatively resembles a high dimensional random walk rather than a low dimensional one.(Fig <ref>D-F).To quantify this resemblance, let τ be the coefficient of variation of x_s=1/sD̅^2_s, i.eτ=√(∑_s=1^m-1 x^2_s/(∑_s=1^m-1 x_s)^2-1) For high dimensional random walks, we expect τ to be close to zero. We simulated τ for large numbers of random walks and compared to the value computed for the H3N2 data (Fig <ref>). We find that the H3N2 data is unlikely to have been produced by a random walk of less than ten dimensions (n<10). Note also that the value of τ for the data may be inflated due to any number of sources of variability not accounted for in the model, such as measurement error or a heavy tailed distribution in annual step size. Therefore d=10 is the lower bound for the dimensionality antigenic space and dimensionality of 30 or more is quite likely. These findings extend to self avoiding random walks, as decribed in the methods section, which behave similarly in this regard.§ DISCUSSIONHigh dimensional random walks can appear to be low dimensional. We have shown that most of their variance will appear in a single principle component, and that their pointwise distances can be well approximated by embedding in a two dimensional space. Therefore when using principal component analysis (PCA) or multidimensional scaling (MDS) we may erroneously find that only a few dimensions are important. We demonstrate that by splitting the data set into two, we can use out of sample PCA to potentially recover the true dimensionality of the random walk. If only MDS is possible, perhaps due to missing data, we demonstrate that high dimensional and low dimensional random walks can be distinguished by testing the relationship between distance and step number.We apply these ideas to the evolution of influenza H3N2. Using data from the HAI assay different strains from 1968 to 2003 can be readily represented in a two dimensional antigenic map. The distances on this map represent the level of cross immunity that an antibody response to one strain may provide against another. Given this it may be tempting to conclude that influenza is constrained to move in a two dimensional space or at least is biased to move in two dimensions. However, we caution that antigenic space may in fact be very high dimensional or even infinite dimensional and we would still expect to be able to reconstruct it in two dimensions. In fact, we find that a low dimensional random walk or self avoiding random walk is very unlikely to have produced the H3N2 HAI titer data.One interesting aspect of influenza H3N2 evolution is that despite the annual variation in strain, the strains have not been diverging. Different strains circulating in a given year tend to be close together antigenically. There have been many models proposed to explain this phenomenon <cit.>. Here we have only shown that the data is consistent with a high dimensional random walk or self avoiding random walk. We assume that the direction of the next step is equally likely to be in any direction, provided that this does not bring the walk back to an already visited region. Therefore our model is consistent with antigenic drift hypotheses as described in <cit.> but not the antigenic thrift hypothesis described in <cit.>.§ METHODS §.§ Out of sample PCA* Start with a length 2m random walk in n dimensions.* Split the random walk into two halves.* Apply PCA to the first half. This will yield an orthogonal projection matrix P∈ℝ^n× n which projects any point onto the principal component axis.* Split the second half of the walk into subsequences with a short length, μ. These subsequences consist of steps m+1 to m+μ, m+μ+1 to m+2μ, etc. Let S_i ∈ℝ^n×μ denote the ith such subsequence.* Apply P to each subsequence S_i, i.e. compute B_i=PS_i.* Let σ^2_i be the variance of each row of B_i.* Let σ^2=1/ν∑_i=1^νσ^2_i be the sum of all i, where ν is the number of subsequences (i.e. νμ=m).§.§ Generating simulated dataWe want to generate N_v viral strains and N_s antisera. We will represent each strain and antisera as a point in ℝ^n, where d is our chosen dimensionality of antigenic space. The generation of the artificial data makes repeated use of random vectors in ℝ^n. We draw these random vectors from N(0,I_d), i.e. their entries are i.i.d unit normal random variables. We first generate the viral strains via a random walk. Let V_i ∈ℝ^n represent the ith viral strain in the walk, thenV_1 =b_1V_i-V_i-1 =b_ii>1where b_i N(0,I_n)Next we use these strains to randomly generate antisera. Let S_j ∈ℝ^n be the jth antiserum, thenS_j =V_p_j+bwhere p_j is a random integer between 1 and N_v and c_j N(0,I_n).§.§ Multi Dimensional ProcedureIn this paper we use the form of multidimensional scaling described in <cit.>. We perform this scaling in two distinct cases: starting from a matrix of measured HI and starting from artificial data generated as described in the previous section.Not every viral strain is measured against each serum. Let N_Meas be the number of measurements and let 0<l≤ N_Meas index those measurements. We can then definei_lStrain corresponding to the lth measurementj_lSerum corresponding to the lth measurementd_lDistance between strain i_l and serum j_l In the case of real HI titer data, we convert the titer to a distance using the method described in <cit.>. When using artificial data, we simply compute d_l=V_i_l-S_j_l.Next we seek a set of points V̂_i∈ℝ^k and Ŝ_j∈ℝ^k that minimizes the objective functionF({V̂_i},{Ŝ_j}|{d_l}, {i_l}, {j_l})=∑_l=0^N_Meas(d_l-V̂_i_l-Ŝ_j_l)^2We use values of k between 2 and 20.We minimize F numerically using the optim function in R. We use the BFGS method and provide the analytically derived gradient of F for efficient computation. This is an iterative algorithm that requires an initial guess. We generate these guesses using classic MDS, which is equivalent to principal component analysis. The classic MDS algorithm requires a complete distance matrix. In the case of the artificial data, we can calculate this distance matrix directly. In the case of the real data, we construct a plausible distance matrix by finding the shortest path between any two points whose distance was not directly measured. Note that although this plausible distance matrix is likely incorrect, it is only used to find a good starting guess for our algorithm. The solution found using this method compared favorably to the solution found in <cit.> §.§ Self avoiding random walkThe self avoiding random walk behaves similarly to an unbiased random walk accept that the new steps are excluded from getting to close to previous ones. Each new step in the walk is determined according to the following algorithm. * Given a current step x_i, propose a new step x̂=x_i+b∈ℝ^n, where the entries of b are i.i.d with distribution 𝒩(0,1/√(n).* Calculate the probability of acceptance viaP =∏_j=0^i g(x̂|x_i)g(x̂|x_i) =1-e^-x̂-x_i^2 * With probability P accept the new step and let x_i+1=x̂. With probability 1-P return to step one and propose a new step.alpha | http://arxiv.org/abs/1707.09361v1 | {
"authors": [
"James Moore",
"Hasan Ahmed"
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"categories": [
"q-bio.PE"
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"published": "20170727212058",
"title": "High Dimensional Random Walks Can Appear Low Dimensional: Application to Influenza H3N2 Evolution"
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"authors": [
"Ho-Ung Yee"
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"hep-th",
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"primary_category": "hep-ph",
"published": "20170726174751",
"title": "Dynamic Universality Class of Model H with Frustrated Diffusion: $ε$-Expansion"
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Destruction of Refractory Carbon in Protoplanetary Disks Dana E. Anderson1, Edwin A. Bergin2, Geoffrey A. Blake1, Fred J. Ciesla 3, Ruud Visser 4, Jeong-Eun Lee 5December 30, 2023 ==============================================================================================================§ INTRODUCTION There are few ways to analytically studythe low temperature and density behavior of QCD-like quantum field theories. [ By “QCD-like” we mean 4D asymptotically free SU(N) gauge theories, possibly containing fermions but without light fundamental scalar fields. We assume that the fermion content is such that the theory, when defined on ℝ^4, has a confining phase characterized by some strong scale Λ. ] Near the chiral limit (in theories containing light fermions), chiral perturbation theory may be used to systematically characterize the low energy consequences of spontaneously broken chiral symmetry using a small number of low energy parameters. (See, e.g., Ref. <cit.> for a review.) But the demonstration of chiral symmetry breaking and determination of these low energy constants requires other methods, such as large scale lattice gauge theory simulations or input of experimental data. Gauge-gravity duality <cit.> has provided insight into some 4D confining gauge theories <cit.>, but is usefully applicable primarily in theories which are strongly coupled at all scales, not asymptotically free, and have a large number N of colors. For 4D confining, asymptotically free gauge theories, analytic methods based on controlled approximations are generally unavailable.In this paper, we study properties of 4D confiningQCD-like theories, at finite N, in a regime which allows controlled analytic calculations. Specifically, we consider theories on ℝ^3 × S^1, with one dimension compactified on a circle of circumference L which is small compared to the inverse strong scale of the theory, L ≪Λ^-1 (and henceforth denoted S^1_L). This is a very old idea (see, e.g., Ref. <cit.> for a review) but interest has been renewed in recent years with the realization that a wide range of QCD-like theories may be engineered to possess a phase diagram in which the small-L regime is continuously connected to the large-L or decompactified regime. Achieving such “adiabatic compactification” requires non-thermal boundary conditions and suitable matter content (or the addition of double trace deformations) <cit.>.Compactifying one direction on a small circle does, obviously, change properties of a theory. Lorentz invariance is reduced from SO(1,3) to SO(1,2) and physical quantities will depend on the newly introduced scale L. But if one can engineer compactifications where the L dependence is smooth (“adiabatic”), then studies of the small-L regime may teach one qualitative lessons which remain valid in the large-L limit. Previous work <cit.> has examined symmetry realizations at small L and studied the properties of the very lightest excitations. One finds that it is possible to prevent the spontaneous breaking of the ℤ_N center symmetry of pure Yang-Mills (YM) theory, which would signal a deconfinement transition. With massless quarks present, one finds that chiral symmetry is spontaneously broken. The mechanism of confinement, the generation of a non-perturbative mass gap (without massless quarks), and the spontaneous breaking of chiral symmetry (with massless quarks) all may be nicely understood in the small-L regime using semiclassical methods. All evidence supports the view that these center-stabilized compactifications are, indeed, adiabatic. [ Consistency of symmetry realizations between small and large L is, of course, necessary but not sufficient for physics to be smooth in L. Phase transitions not involving any change in symmetry realization could always be present at some intermediate value of L. For center-stabilized QCD, with light quarks, the careful lattice studies which would be needed to rule out this possibility are not yet available. In the absence of any evidence to the contrary, we proceed assuming that for the compactifications we study below, physical properties are smooth in L. ]Given the weight of evidence that adiabatic compactifications exist, it is interesting to use these calculable settings to explore properties of QCD-like theories in more detail. In this paper we initiate efforts in this direction by investigating qualitatively, and where possible quantitatively, the spectrum and properties of glueballs, mesons, and baryons in the small-L regime of adiabatically compactified theories. Some of the hadronic states we find are stable, but naturally most are resonances. In the weakly coupled small-L regime, hadronic resonances are narrow with parametrically small decay widths. Portions of the spectrum have interesting parallels with what one obtains from naive quark models, but in a context where the dynamics of the quantum field theory are under systematic theoretical control.We mention here two especially curious aspects of our results. First, we find that the lightest glueballs (or dual photons in the small-L description) form bound states whose binding energies are given by iterated exponentials of the Yang-Mills coupling, Δ E ∼exp (-A g^kexp(B/g^2)). Second, we find that the density of states of both glueballs and mesons exhibits Hagedorn (or exponential) growth with energy, but this growth has an unusual origin. Hagedorn scaling of the density of mesonic states is typically attributed to the fluctuations of a long, highly excited confining string, and can only be established systematically in the large N limit where mesons cannot decay. The origin of Hagedorn scaling in our context is quite different. The extra scale L introduced by the adiabatic compactification modifies the potential experienced by heavy test quarks separated by a distance r, and introduces a parametrically large regime where the potential is logarithmic, as illustrated in Fig. <ref>. The compactified theory has many narrow resonances which can be described using non-relativistic quantum mechanics with this logarithmic potential, leading to a Hagedorn spectrum. The fact that stringy dynamics are not the only way to obtain a Hagedorn spectrum, and in particular that such a spectrum arises in ordinary quantum mechanics with logarithmic potentials does not seem to be widely appreciated. [ However, the notion of a limiting temperature for systems with exponential densities of states was first introduced by Rumer in 1960 <cit.>, precisely in quantum mechanics with a logarithmic potential, several years before Hagedorn's suggestion <cit.> that such a density of states may arise in hadronic physics. ]To make our presentation reasonably self-contained, we begin in Section <ref> with a summary of center-stabilized adiabatic compactifications. Section <ref> discusses the light sector of the compactified theory, with a focus on the spectrum of bound states. In Section <ref>, we formulate the 3D non-relativistic effective field theory (EFT) which efficiently describes the dynamics of heavy quark and gluon degrees of freedom. Section <ref> describes how the various symmetries of the underlying 4D gauge theory act within our 3D effective field theory. In Section <ref> we examine the resulting spectrum of heavy bound states, while Section <ref> discusses decay processes. We summarize our findings in Section <ref> and discuss some of their consequences, including large N scaling relations and implications for the thermodynamics of QCD-like theories. Several appendices contain technical details. § ADIABATIC COMPACTIFICATION Consider SU(N) Yang-Mills theory compactified on ℝ^3 × S^1, with the spatial circle having circumference L,S_YM=1/4g^2∫_ℝ^3× S^1d^4x (F_μν^a)^2.If all matter fields added to the theory transform in the adjoint representation of the gauge group, then the theory has a ℤ_N center symmetry. (We discuss below the addition of fundamental representation fermions.) Order parameters for center symmetry are built from the holonomy of the gauge field in the compact direction (or “Polyakov loop”),Ω≡𝒫 e^i ∫_0^L dx_3A_3 .Center symmetry transformations multiply the (fundamental representation) trace of the holonomy by a phase factor equal to an N'th root of unity. The defining transformation isΩ→ω Ω ,ω≡ e^2π i/N .At large L center symmetry is unbroken, implying that ⟨Ω^n ⟩ = 0 for all integer n ≠ 0N. This is a hallmark of a confining phase. At small L the realization of center symmetry is analytically calculable <cit.>. We require that the theory is engineered to prevent spontaneous breaking of the ℤ_N center symmetry in the L → 0 limit, so that the theory is not in a deconfined plasma phase at small L. This can be achieved by adding suitable double trace deformations of the form |Ω|^2 (plus higher windings) to the action of pure Yang-Mills theory <cit.>. Alternatively, the center symmetry at small L can be stabilized by the addition of massless or sufficiently light adjoint representation fermions <cit.>. [ If center symmetry is stabilized with adjoint fermions, we assume that 2 ≤ n_ adj≤ 5 species of adjoint Majorana fermions are added, so the theory is asymptotically free but non-supersymmetric (in the massless limit). We also take the adjoint fermion mass m_ adj to be large compared to the mass gap scale m_γ discussed below. ] If the adjoint fermions are massive, center stabilization for small L requires that their mass m_ adj satisfy the constraint m_ adj≲ 2π/NL <cit.>.With center symmetry stabilized, the one-loop effective potential V_ eff(Ω) for the holonomy, obtained by integrating out field modes with non-zero Kaluza-Klein (KK)momentum in the compact direction, has a unique (up to gauge equivalence) ℤ_N symmetric minimum,Ω = ω^-(N-1)/2diag (1, ω, ω^2, ⋯, ω^N-1).For sufficiently small L, the gauge coupling at the compactification scale is weak and quantum fluctuations are suppressed. Hence, one may regard the holonomy Ω as a nearly constant SU(N) matrix with eigenvalues which are all N'th roots of unity for N odd, and all N'th roots of -1 for N even. The holonomy acts like an adjoint representation Higgs field, “breaking” the non-Abelian gauge symmetry (using typical sloppy perturbative language) down to the U(1)^N-1 Cartan subgroup. We will refer to the N-1 diagonal Cartan components of the gauge field as “photons.” The off-diagonal components of gauge field (charged under the Cartan subgroup) will be termed “W-bosons” and receive masses given by positive integer multiples of≡ 2π/(NL). Fluctuations in the eigenvalues of the holonomy will have an effective mass m_Ω whose value depends on the details of the center symmetry stabilization. One may regard m_Ω∼√(λ) as a characteristic fiducial value, with λ≡ g^2 N the usual 't Hooft coupling. This is the typical size resulting from modifications to the one-loop effective potential for the holonomy, unless one fine-tunes the stabilization mechanism, for instance by considering a nearly supersymmetric limit of the theory. The dynamical Higgs mechanism and resulting Abelianization induced by the center-symmetric holonomy is the key feature responsible for the analytic tractability of the theory at small L. All charged degrees of freedom have masses of orderor more, so the 4D 't Hooft coupling λ does not continue to run below the scale . If ≫Λ, or equivalentlyη≡ N L Λ≪ 1,then the long-distance value of the 't Hooft coupling will be small, λ() ≪ 1. We focus on this regime in what follows and, unless stated otherwise, the value of g^2 is taken at the scale .Previous work on adiabatically compactified QCD-like theories has focused exclusively on the lightest subsector in the small L limit, with characteristic energies and momenta much less thanand m_Ω. On these scales, the physics can be described by an effective field theory of N-1 Abelian photons living in three dimensions. Non-perturbative monopole-instanton effects generate small but relevant interactions between the photons. The Euclidean action for the diagonal components of the gauge field has the schematic form [ Perturbative corrections generate photon mixing terms (as well as higher derivative terms which are irrelevant at long distances). The photon mixing matrix has been calculated in 𝒩=1 SYM theory to first order in λ <cit.>. This photon mixing is diagonalized by the same ℤ_N Fourier transform mentioned below, and does not affect the following discussion.]S_ light = L ∫ d^3x[14g^2 (F_μν^a)^2 + ℒ^ monopole_ int] .A three-dimensional Abelian duality transformation leads to the Coulomb gas representation, [ A redundant field component has been introduced in this representation, as if the original gauge group were U(N) instead of SU(N). The unphysical components, ∑_a=1^N F_μν^a and ∑_a=1^N σ^a, exactly decouple and can be ignored. See, e.g., Ref. <cit.> for more detailed discussion. Appendix <ref> contains details of our conventions, normalizations, and duality transformation. ]S_ light = ∫ d^3x[ λ/16π^3(∇)^2 - ζ∑_i=1^N cos(_i · + θ/N) ] .The field = {σ^i } is an N-component compact scalar field; in our basis it is independently periodic in every component with period 2π. The fundamental domain ofis the unit cell of the weight lattice, generated by the shifts → + 2π_i where {_i } are the fundamental weight vectors of SU(N) and {_i } are the corresponding root vectors. The “fugacity”ζ = A^3λ^-2e^-8π^2 /λ ,where A is an Ø(1) coefficient which depends on the choice of regularization scheme. Although not immediately apparent, the action (<ref>) is invariant, as it must be, under shifts in the QCD θ angle by multiples of 2π.To obtain an expression for the masses of the dual photons, note that the potential V = - ζ∑_i=1^N cos(_i · + θ/N) has N extrema in the unit cell of the weight lattice located at ⟨⟩_k = 2 π k/N for k=0,⋯,N-1, where = ∑_i=1^N-1_i is the Weyl vector. [ To see this, use _i · = 1 for i=1,⋯, N-1, together with _N · = 1-N. ] For θ =0 the minimum lies atk=0. For general θ, the vacuum energy density is given byV_0 = - N ζ max_k(cos2 π k +θ/ N ) .Expanding thepotential around each of the N extrema and diagonalizing the curvature (via aFourier transform) yields the θ-dependent mass spectrum in each of the N extrema (not all of which are minima). At the lowest-energy minimum, which determines the physical mass spectrum, one findsm_p^2 = m_γ^2sin^2 (π p/N) max_k(cos2 π k + θ/ N ) ,for p = 1, 2, ⋯, N-1, withm_γ≡ C λ^-3/2e^-4π^2/λ .The Ø(1) coefficient C is determined in terms of the coefficient A in the fugacity (<ref>). The label p can be viewed as the charge under ℤ_N center symmetry transformations; this is discussed in Sec. <ref>. One may also show that expectation values of large fundamental representation Wilson loops (not wrapping the compactified direction) have area law behavior, with a string tension <cit.>T = C'λ m_γ ,with C' another Ø(1) coefficient.The dual photon mass m_γ can be expressed in terms of the strong scale Λ by using the renormalization group to relate λ at the scale ofto Λ. The specific form of this relation depends on the value of the beta function, and hence on whether center symmetry is stabilized by double trace deformations, or by the addition of adjoint fermions. If center symmetry is stabilized by a double trace deformation,then parametrically <cit.>m_γ∼Λ (NLΛ)^5/6 = Ø(Λ η^5/6) ,and m_γ/ = Ø(η^11/6). [ If center symmetry is stabilized by the addition of n_ adj light adjoint Majorana fermions with mass comparable to , then m_γ/ = Ø(η^(11 - 2 n_ adj)/6). ]§.§ Addition of fundamental quarksWe will consider center-stabilized adiabatically compactified QCD in addition to pure Yang-Mills theory. This entails addingflavors of quarks — fundamental representation Dirac fermions. We restrict our discussion to ≤ N and, for simplicity, focus on the massless quark limit,m_q = 0,where the uncompactified theory has an SU()_L × SU()_R × U(1)_V continuous chiral symmetry. [ For > N, it is not currently known how to ensure that chiral symmetry realizations coincide at large and small L. ] When compactifying the theory on ℝ^3 × S^1, one must specify the boundary conditions on the quark fields. Instead of simply choosing periodic, or antiperiodic, boundary conditions for all quark flavors, we will consider flavor-twisted boundary conditions, or equivalently introduce a non-dynamical flavor holonomy Ω_F ∈ U()_V. If one regards the quark fields q as an N × matrix of spinors, then in A_3 = 0 gauge (where the gauge holonomy becomes encoded in boundary conditions), the boundary conditions on quarks areq(t,,L) = Ωq(t, , 0)Ω_F^† .We specifically choose the flavor holonomy Ω_F to have a set of eigenvalues which are invariant under ℤ_ cyclic permutations.The symmetry structure of QCD with such boundary conditions was discussed in Ref. <cit.> (see also Refs. <cit.>). To preserve reflection (in the compactified direction) and charge conjugation symmetries, we also require that complex conjugation leave this set of eigenvalues unchanged. These two conditions imply that the eigenvalues of Ω_F are either given by all 'th roots of +1, or by all 'th roots of -1. Finally, to simplify our discussion and leave unchanged the relevant degrees of freedom in the non-perturbative analysis of the light sector, we want all flavors of quarks to receive non-zero effective masses from the compactification. This requires that no eigenvalue of the gauge holonomy coincide with an eigenvalue of the flavor holonomy.Solutions to these just-stated constraints depend on the values of N and , in particular whether N is even or odd and (when N is even) whether N andhave common divisors. For simplicity of exposition we will henceforth assume that N is odd, unless stated otherwise, so that the eigenvalues (<ref>) of the gauge holonomy Ω are N'th roots of unity. To avoid coinciding gauge and flavor eigenvalues, this implies that the flavor holonomy eigenvalues must equal 'th roots of -1. Consequently, we chooseΩ_F = diag (ξ^1/2, ξ^3/2, ⋯, ξ^-1/2), ξ≡ e^2π i/ .When the gauge holonomy is encoded in a non-zero value of A_3 (so that the gauge field satisfies simple periodic boundary conditions), the resulting quark boundary conditions areq^A(t,,L) = ξ^1/2 -Aq^A(t,,0),where A = 1,⋯, is a flavor index. The effect of these boundary conditions is to shift the moding (i.e., the allowed values of the momentum in the compact direction), in a flavor-dependent fashion which is detailed below. The boundary conditions (<ref>) reduce the non-Abelian flavor symmetry to the Abelian subgroup [ More precisely, the unbroken subgroup is U(1)^-1_L × U(1)^-1_R × U(1)_V / ℤ_. Henceforth, we will not be explicit with the discrete identification needed to avoid double counting ℤ_ phase rotations. ]U(1)^-1_L × U(1)^-1_R × U(1)_V.Note that this residual flavor symmetry of our compactified theory contains the axial subgroup U(1)_A^-1 which differentially rotates the phases of left and right handed quarks in a flavor-dependent fashion.In the center-stabilized regime of YM theory, the addition of massless quarks with the boundary conditions (<ref>) produces fermion zero modes localized on the monopole-instantons. The presence of these zero modes modifies the non-perturbative long distance dynamics. After a 3D duality transformation, one may show that -1 of the dual scalar fields remain exactly massless <cit.>, while the remaining N - dual scalar fields develop non-perturbative masses just as in center-stabilized YM theory without fundamental quarks. The mechanism causing -1 dual scalars to become massless in the presence of fermion zero modes involves their acquisition of non-trivial transformation properties under the anomaly-free U(1)^-1_A axial symmetry, as explained in Ref. <cit.>. Consequently, these exactly-massless fields are precisely the expected Nambu-Goldstone bosons (or `neutral pions') produced by spontaneous breaking of the chiral symmetry (<ref>) down to the diagonal vector-like U(1)_V^ subgroup <cit.>.If a small quark mass m_q is added to the theory, then some of the dual photons, or neutral pions, become massive. For example, when N one finds <cit.> (at θ = 0) thatm_p = C √( m_q)e^-4π^2/λsinπ p/N .(Here p is the charge of the pion under cyclic flavor permutations.) One may again relate m_p to the strong scale Λ by taking into account the contribution of the fundamental fermions to the running of the coupling at the scale . With the pure-YM center symmetry stabilized via double trace deformations and N, one findsm_p = Ø( η√(m_q Λ)),where, once again, η≡ NL Λ.§ LIGHT SECTOR BOUND STATESAs noted in the introduction, when the color holonomy has the center symmetric form (<ref>), a rich spectrum of hadronic states is present in the small-L regime of the compactified theory. These states fall into two categories based on the scale of their rest masses.One set of states have rest masses of order of the light scale m_γ, while the other set has rest masses of order of the heavy scale .As will be shown below, in both sectors the binding momenta are small compared to the rest masses of constituents, so the most efficient way to describe each sector of the theory involves constructing an appropriate non-relativistic effective field theory. In this section we describe the effective field theory for the light `dual photon' sector and discuss the resulting light bound state spectrum.§.§ N=2 bound states To illustrate the relevant physics in the simplest setting, consider adiabatically compactified pure YM theory with N = 2 and θ =0. The relativistic 3D effective theory describing interactions of the single (physical) dual photon field σ≡σ_1-σ_2, to leading non-trivial order in the semiclassical expansion, isS_3D, rel = ∫ d^3x [ λ/32π^3 (∂_μσ)^2 - 2 ζcos(σ) ] .Introducing a canonically normalized field ≡σ (λ/16 π^3 )^1/2, and expanding the potential, one findsS_3D, rel = ∫ d^3x [ 12 (∂_μ)^2 + 12 m _γ^2^2 - 23ϵ m_γ ^4 + 1645ϵ^2 ^6 - 32315ϵ ^3m_γ^-1 ^8 + ⋯] ,whereϵ≡π^3 m_γ/λ m_W = Ø( λ^-5/2e^-4π^2/λ).At first glance it is tempting to assume that the interaction terms in (<ref>) have negligible consequences. To our knowledge, effects of these weak interactions have not previously been considered, either in the literature on adiabatically compactified 4D theories starting with Ref. <cit.>, or in the original literature on the Polyakov model in three dimensions <cit.>. As we now discuss, this presumption overlooks interesting physics. The ^8 and higher terms in the action (<ref>) are irrelevant and can be ignored when focusing on the long distance behavior of the theory.The ^4 coupling is relevant, but its coefficient is exponentially small in units of the σ mass. The ^6 coupling is marginal and infrared-free <cit.>. It is also exponentially small and stops running below the mass gap scale m_γ. These considerations might naively be interpreted to imply that all interaction effects in the low energy theory (<ref>) are tiny.But consider interactions ofmodes with low momenta p ≪ m_γ. Such interactions can be described by a non-relativistic effective field theory. Writing σ̃= (2m_γ)^-1/2 e^- i m_γ t Σ + (h.c.), where Σ is the non-relativistic field, and integrating out rapidly oscillating terms leads to the non-relativistic description, [ Here and in Eq. (<ref>) below, we flip the overall sign so that the nonerelativistic action S_ 3D,NR has the conventional T-V form.]S_3D, NR = ∫ dt d^2x [ Σ^†(i∂_t + ∇^2/2m_γ) Σ + ϵ/m_γ(Σ^†)^2Σ^2 - 8 ϵ^2/ 9 m_γ^3(Σ^†)^3Σ^3 + ⋯] .The scaling dimension assignments appropriate to non-relativistic theories in spacetime dimension d are [t] = -2, [x]=-1, [Σ] = d-12, and [m_γ] = 0. This implies that the coefficients of the (Σ^†Σ)^2 and (Σ^†Σ)^3 interactions have dimensions d-3 and 2(d-2), respectively. In d=3, this shows that the two particle (Σ^†Σ)^2 interaction becomes marginal in non-relativistic dynamics, while the three particle (Σ^†Σ)^3 interaction becomes irrelevant. In fact, the (Σ^†Σ)^2 coupling ϵ runs logarithmically with scale <cit.>, as may be seen (for example) by calculating the two particle scattering amplitude. Consequently, the definition (<ref>) should be interpreted as the value of the running interaction strength ϵ at the UV momentum cutoff μ_ UV∼ m_γ. In the non-relativistic limit the only diagrams which contribute to the renormalization group (RG) evolution of ϵ beyond tree level are iterated bubble diagrams. Summing them yields the exact beta function for ϵ. Using dimensional regularization, one simply finds <cit.>μd ϵ(μ)/d μ = -1/π ϵ(μ)^2.When the initial coupling ϵ(μ_ UV) is positive, corresponding to an attractive interaction, ϵ(μ) diverges at the momentum scale = μ_ UVexp[-π/(ϵ(μ_ UV))]. As a function of momentum, the two particle scattering amplitude 𝒜(k) becomes singular at k^2 = -^2. A pole develops at this position, indicating thatcan be interpreted as the binding momentum for a two-body bound state of dual photons. [ One may also directly solve the quantum mechanical problem a particle of reduced mass m_γ moving in the attractive potential -2 ϵ/m_γ δ^(2) (). The bound state wave function equals K_0(r/r_B), with the bound state size r_B = |m_γ Δ E_2|^-1/2 and Δ E_2 equaling the binding energy (<ref>). ] The two particle binding energy is thusΔ E_2 = -k^2/m_γ = - μ_ UV^2 / m_γe^-2 π/ϵ(μ_ UV) = -14 c^2m_γe^-2 λ / π^2 m_γ .In the final form we used the bare value (<ref>) of ϵ and set the ultraviolet cutoff to the reduced mass m_γ times an Ø(1) coefficient c, whose determinationrequires a more careful matching calculation and is left for future work. The two dual photon bound state has a rest massm_2 = 2 m_γ + Δ E_2 = m_γ ( 2 - 14 c^2 e^-2 λ m_W /π^2 m_γ).Expressed in terms of the original gauge coupling, the fractional binding energy involves a non-perturbative double exponential,Δ E_2/2m_γ = - 14c^2exp( -2π^2 C λ^5/2e^4π^2/λ) ,whose appearance is quite peculiar in the context of the 4D gauge theories. [ However, the existence of double-exponential non-perturbative scales in gauge theory has been previously advocated <cit.>, based on quite different considerations from those discussed here. ]In addition to a two particle bound state, an attractive two-body interaction in two space dimensions also binds higher multi-body bound states. (See, for example, Refs. <cit.>.) The magnitude of the k-body binding energy Δ E_k increases exponentially with k, with Δ E_k+1/Δ E_k∼ 8.6 for large k <cit.>. In our context, we thus deduce the presence of a very large number of bound states of dual photons, one slightly below each k-particle threshold at E = k m_γ for k = 2, 3, ⋯, with fractional binding energies proportional to the non-perturbative double exponential (<ref>). [ This weak coupling non-relativistic description breaks down when k (ln 8.6) becomes exponentially large and comparable to 2 λ / π^2 m_γ∼λ^5/2 e^+4π^2/λ. ]§.§ N>2 bound statesWe now briefly consider the generalization to arbitrary N, still with θ = 0. Using a ℤ_N Fourier transform to diagonalize the mass terms, σ_i ≡(λ/8π^3)^-1/2∑_p=1^N-1ω^ i p σ̃_p/√(N) (with σ̃_p^* = σ̃_N-p), the generalization of the action (<ref>) isS_ 3D = ∫ d^3x ∑_p=1^N-1( |∂_μσ̃_p|^2 + m_p^2 |σ̃_p|^2 )- 4ϵm_γ/ 3 N∑_p_1⋯ p_4 = 1^N-1δ_p_1+p_2+p_3+p_4,0 e^iπ(p_1+p_2+p_3+p_4)/N ×[ ∏_i=1^4 sin(π p_i/N) ]_p_1_p_2_p_3_p_4 + Ø(σ̃^6),where all center charges {p_k} are understood to be defined modulo N. The masses { m_p } and coupling ϵ are given by Eqs. (<ref>) and (<ref>), respectively. [Recall that the field σ̃_0 ∝∑_i σ_i decouples, and is omitted. Expression (<ref>) reduces to the earlier form (<ref>) for N 2, as it should.]The sign of the quartic interaction depends on the values of the center charges of the particles under consideration. For elastic scattering of dual photons with arbitrary charges p_1 and p_2, the relevant piece of the quartic interaction has an overall minus sign, which corresponds to attraction. The effective theory which follows from a non-relativistic reduction of the action (<ref>), and generalizes the earlier form (<ref>) to arbitrary N, isS_3D, NR = ∫ dt d^2x[ ∑_p=1^N-1^†_p (i∂_t + ∇^2/2 m_p) _p + 2 ϵ/Nm_p ^2 / m_γ^3(^†_p)^2_p^2+ ∑_p_1 < p_28 ϵ/Nm_p_1 m_p_2/ m_γ^3 ^†_p_1^†_p_2_p_2_p_1 + ⋯] ,where we have included only those terms contributing to elastic 2 ↔ 2 scattering. [ The interaction (<ref>) also includes charge exchange processes which lead to mixing among bound states with differing constituents but the same total center charge.For generic values of N and choices of p_1 and p_2the effects of such interaction terms on binding energies are suppressed in the non-relativistic limit, because the masses of the dual photons depend on their center charge. Charge exchange processes can only become relevant if states with differing constituents and the same total charge also have the same total constituent mass. Such mixing will deepen the binding of the lowest energy bound states of a given total charge. We defer a complete multi-channel treatment to future work. ] Note the factor of 4 difference in the coefficients of the quartic terms responsible for scattering of identical vs. non-identical particles.Applying the earlier analysis (either solving the two-particle Schrödinger equation with a delta function potential, or resumming bubble diagrams and locating the resulting pole in the scattering amplitude) to states containing particles of center charge p_1 and p_2, one finds the binding energyΔ E_2^p_1p_2 = - 2c^2 mexp( - π N / 4 ϵm_γ^3 / m_p_1 m_p_2 m) ,if p_1p_2. Here m ≡ (m_p_1^-1 + m_p_2^-1)^-1 is the reduced mass of the two constituents. If the two constituents are identical, then the result isΔ E_2^p_1 = p_2 = - c^2 m_p_1 exp( - π N /ϵm_γ^3 / m_p_1^3 ). Bound states composed of equal mass constituents can have either equal or opposite charge constituents. For the first case, with charges p_1 = p_2 = p, the identical particle binding energy (<ref>) gives a total massm_2^p,p = m_p[ 2 -c^2 e^- π N /ϵ (m_γ/m_p)^3] .For opposite charges, p and N-p, the non-identical binding energy (<ref>) with m_p_1 = m_p_2 = 2m = m_p gives total massm_2^p,N-p = m_p [ 2 -c^2 e^- π N / 2 ϵ (m_γ/m_p)^3](except for the special case of pN/2 with N even, where the first result (<ref>) applies). In other words, the fractional binding energy for non-identical particles is Ø( e^- π N / 2 ϵ (m_γ/m_p)^3) = Ø( e^- π N / 2 ϵ |sinπ p/N|^-3), while bound states of identical constituents have twice the exponential suppression in their binding energy.§ HEAVY SECTOR EFFECTIVE FIELD THEORYWe now consider states with rest masses of orderand above, and characteristic binding momenta p in the rangem_γ≪ p ≪ .This section describes the construction of a non-relativistic effective theory suitable for the description of such states. We begin with the effective theory characterizing pure gauge, or glueball, dynamics, and then discuss the addition of fundamental representation quarks. §.§ Gauge field contributions The center-symmetric holonomy (<ref>) may equivalently be regarded as a non-vanishing constant diagonal gauge field in the compact direction, A_3, together with conventional periodic boundary conditions. The [A_3,A]^2 term in the classical Yang-Mills action generates tree-level masses of orderfor the charged W-bosons. The efficient description of the interactions of these massive charged degrees of freedom with the Cartan photons is provided by a non-relativistic effective field theory with action:S_heavy = ∑_a,b=1^N_n=-∞^∞∫ dt d^2x [ (ϕ⃗_n^ ab)^† i∂_tϕ⃗_n^ ab - M_n^ab|ϕ⃗_n^ ab|^2 - |ϕ⃗_n^ ab|^2/2 m_n^ab]+ λ/4π∑_a=1^N∫ dtd^2x d^2y ρ^a(t,) G (-)ρ^a(t,),whereG(-) ≡ 12π ln (μ |-|)is the two dimensional Laplacian Green's function. The derivation of this effective theory is detailed in appendix <ref>. Higher order (in λ) corrections, such as magnetic moment interactions, are omitted for simplicity.The two-dimensional vector fields ϕ⃗^ ab_n are the non-relativistic reduction of the n'th Fourier component (in the compact direction) of the (ab) component of the SU(N) gauge field, viewed as an N × N Hermitian matrix. The color (or `Cartan') indices a,b run from 1 to N, and the Kaluza-Klein index n is an arbitrary integer. In the action (<ref>), the prime on the sum over n is an indication to omit the n=0 term when a = b, but not otherwise. The vector field ϕ⃗_n^ ab annihilates W-bosons with charges (+1,-1) with respect to the a'th and b'th unbroken U(1) gauge groups. The spatial gradientis a two-dimensional U(1)^N covariant derivative defined by()_i (ϕ^ab_n)_j ≡[∇_i - i g_3 (A_i^a - A_i^b) ] (ϕ^ab_n)_j.Here i,j = 1,2 label the two non-compact spatial directions and {A⃗^a} are N independent spatial gauge fields. We have introduced N Abelian gauge fields, instead of N-1, as if the original gauge group were U(N) instead of SU(N). This simplifies notation, and makes no difference as the unphysical extra photon, A̅_i ≡∑_a A^a_i, will exactly decouple from all physical states. We have also reverted to a perturbative normalization for the gauge fields, with a dimensionless gauge coupling g_3 appearing inside the covariant derivative, and a corresponding 3D Maxwell action given by L ∫ d^3x 1/4 (F^a_ij)^2. The 3D gauge coupling is, to lowest order, just the 4D gauge coupling evaluated at the scale ,g_3^2 ≡ g^2().Due to the non-trivial holonomy Ω, momentum in the compact direction carried by individual field components is quantized in units of , not N= 2π/L. The Kaluza-Klein reduction of the (ab) component of the gauge field yields a sum of modes with momentump_3= k,where k= a-b + n N,n ∈ℤ. For any given value of a = 1,⋯,N specifying a row of the SU(N) gauge field, there is a one-to-one mapping between the momentum index k and the corresponding values of the column b and KK index n,b-1 = (k-a+1)N, n = (k-a+b)/N.In the following, we will sometimes write expressions involving the relabeled fieldϕ⃗^ a_k ≡ϕ⃗^ ab_n,with the implicit understanding that momentum index k is related to the (antifundamental) column and KK indices {b,n} via relations (<ref>). The momentum index k may take on any integer value other than zero. For charged W-bosons, kN0.The “diagonal” operators ϕ⃗_n^ aa with n 0 annihilate the neutral (uncharged under U(1)^N) gauge bosons carrying non-zero KK momentum. These gauge bosons form the Kaluza-Klein tower whose n=0 modes (excluded from S_ heavy) are the U(1)^N light Abelian photons.The rest and kinetic mass parameters appearing in the effective theory (<ref>) only depend on the Cartan and KK indices via the combination k, and equal the magnitude ofthe compact momentum p_3, up to higher order radiative corrections. In other words, M_n^ab = M_k≡( |k| + Ø(λ) ) =|a - b + nN| + Ø(λ), m_n^ab = m_k≡( |k| + Ø(λ) ) =|a - b + nN| + Ø(λ). Although they coincide at lowest order, the kinetic and rest masses appearing as parameters in our 3D non-relativistic effective field theory (<ref>), or any other non-relativistic EFT, may differ when subleading corrections are included, even when the underlying theory retains full 2+1 dimensional Lorentz invariance.In the effective action (<ref>), the time components of the U(1)^N Abelian gauge fields have been integrated out, producing non-local Coulomb interactions. The operatorsρ^a ≡∑_b = 1^N_n=-∞^∞[ (ϕ⃗^ ab_n)^†·ϕ⃗_n^ab - (ϕ⃗^ ba_n)^†·ϕ⃗_n^ba],are the U(1)^N charge densities. (Note that ρ̅≡∑_a ρ^a vanishes identically.) The conserved charges defined by spatial integrals of these charge densities must vanish,Q^a ≡∫ d^2x ρ^a() = 0,when acting on any physical, gauge invariant state. Because of this, the dependence of the 2D Laplacian Green's function (<ref>) on the arbitrary scale μ inside the logarithm cancels in any physical state, since the variation of the Lagrangian with respect to μ is proportional to (Q^a)^2.The non-relativistic effective theory (<ref>) describes the dynamics of all modes of the non-Abelian gauge field which are charged under the U(1)^N Cartan subgroup, namely W-bosons, plus the uncharged gauge field modes which carry non-zero KK momentum, which we will term “heavy photons.” However, we have not included any fields describing fluctuations of the eigenvalues of the holonomy in the effective field theory. These could easily be included as N-1 additional neutral scalar fields (not 2D vectors like ϕ⃗_n^ ab) with Ø(√(λ)) masses whose precise values depend on the matter content or double trace deformations used to stabilize the center symmetry. These scalar fields only interact with ϕ⃗_n^ ab via higher dimension local operators, suppressed by powers of λ. For the physics we choose to focus on, holonomy fluctuation fields will not play any significant role and may be neglected. If adjoint fermions are used to stabilize the center symmetry, then these fermions are also missing from our non-relativistic effective theory. They could be easily included but, for simplicity, we will limit our attention to states where adjoint fermions (and eigenvalue fluctuations) play no significant role.Reading off the quantum Hamiltonian from the effective action (<ref>) is trivial, except for one UV subtlety. The Hamiltonian of the second quantized non-relativistic theory (with rest energies included) isĤ = ∑_a,b=1^N_n=-∞^∞∫ d^2x ϕ_n^ab()_i^†[ -^2/2m_k + M_k(μ) ] ϕ_n^ab()_i - ∑_a,b,c=1^N_m,n=-∞^∞∫ d^2x d^2yλ/8π^2ln (μ|-|)××[ ϕ_n^ ab ()_i^†( ϕ_m^ ac ()_j^† ϕ_m^ ac()_j - ϕ_m^ ca ()_j^† ϕ_m^ ca()_j ) ϕ_n^ ab()_i . - . ϕ_n^ ba ()_i^†( ϕ_m^ ac ()_j^† ϕ_m^ ac()_j - ϕ_m^ ca ()_j^† ϕ_m^ ca()_j ) ϕ_n^ ba()_i ] .where the field operators satisfy canonical commutation relations,[ ϕ_n^ ab()_i^ , ϕ_n'^ cd()_j ] = 0,[ ϕ_n^ ab()_i , ϕ_n'^ cd()_j^†] = δ^ac δ^bd δ_nn' δ_ij δ^2 (-).In the Hamiltonian (<ref>) we have written out the charge densities ρ^a explicitly and normal ordered the results. In the quartic terms, normal ordering removes the UV sensitive self-energy of each charged W-boson. The price of that removal is that the μ dependence of the Coulomb interaction terms no longer vanishes identically. Instead, this unphysical dependence on the scale μ is canceled by explicit dependence on μ which has been introduced into the bare rest masses (of charged W's only),μd/dμM_k(μ) = - λ/4π^2(1-δ^0_kN). The effective action (<ref>), and corresponding Hamiltonian (<ref>), depend on the 3D gauge coupling g_3, or equivalently the 't Hooft coupling λ, both in the coefficient of the Coulomb interactions and inside the spatial covariant derivatives. But when considering phenomena for which the coupling to the transverse Cartan gauge fields {A⃗^a } may be neglected, the remaining dependence on λ takes a very simple form. To see this, rescale all spatial coordinates, →'/s, →'/s, and then redefine ϕ⃗_k^ a('/s) = sφ⃗_k^ a('). This is a unitary transformation; the rescaled operators {φ⃗_k^ a()} satisfy the same canonical commutation relations as the original operators {ϕ⃗_k^ a()}. In the Hamiltonian, the effect of this rescaling is to change the relative coefficients of the kinetic and Coulomb energy terms. LetN̂_n^ab≡∫ d^2x ϕ⃗_n^ ab()^†·ϕ⃗_n^ ab()denote the number operator which counts the number of constituents of the indicated type, and defineĤ_ NR (λ;μ) ≡. Ĥ|_A⃗^a = 0 - ∑_a,b=1^N_n=-∞^∞ M_k(μ)N̂_n^ abas the non-relativistic Hamiltonian with rest energy contributions removed, the spatial Abelian gauge fields set to zero, and dependence on λ and the scale μ made explicit. If one chooses s = √(λ), then a short exercise shows thatĤ_ NR(λ;μ)≅λ Ĥ_ NR(1,μ/√(λ)) = λ Ĥ_ NR(1,μ) - λlnλ/8π^2 N̂_ W ,where ≅ denotes unitary equivalence andN̂_ W≡∑_a,b=1 a b^N_n=-∞^∞N̂_n^abis the total number of charged W-bosons. The scaling relation (<ref>) shows that the spectrum of the 2D Coulomb Hamiltonian Ĥ_ NR(λ;μ) is simply proportional to the 't Hooft coupling λ, up to an overall additive shift proportional to λlnλ times the number of charged constituents. This relation may equivalently be expressed as1/λ Ĥ_ NR(λ;μ) ≅1/λ' Ĥ_ NR(λ';μ) - /8π^2 ln (λ/λ')N̂_W.§.§ Quark contributionsThe quark fields modify the light and heavy sectors of the theory in several ways. In addition to their effects on the non-perturbative large distance dynamics, already mentioned in the previous section, the compactified quark fields contain massive degrees of freedom which play a role in physics on the scale ofand above. Specifically, every flavor and color component of a fundamental representation Dirac fermion leads, in a non-relativistic description, to a pair of two-component spinor fields which we will denote as ψ_n^aA and χ_n^aA. The field ψ_n^aA annihilates quarks of flavor A which have charge +1 under the a'th U(1) gauge group (and are neutral with respect to all other U(1) gauge group factors). The field χ_n^aA annihilates antiquarks of flavor A and charge -1 under the a'th U(1) gauge group (and are neutral with respect to the other U(1) gauge group factors). It will be convenient to define quark KK indices as half-integers, n ∈ℤ+. These fields satisfy canonical anticommutation relations,{ψ_n^aA()_s, ψ_n'^bB()^†_s'} = {χ_n^aA()_s, χ_n'^bB()^†_s'} = δ^ab δ^AB δ_nn' δ_ss' δ^2(-),where s,s' = ± are spin-1/2 spinor indices. All other anticommutators vanish. To describe the dynamics of the quarks, one must add another set of terms to the effective theory (<ref>) describing W-bosons, namelyS_ quark = ∑_a=1^N∑_A=1^∑_n∈ℤ + 1/2∫ dt d^2x[ (ψ_n^aA)^†i∂_tψ_n^aA - M_n^aA |ψ_n^aA|^2 -|ψ_n^aA|^2/2m_n^aA. +. (χ_n^aA)^†i∂_tχ_n^aA - M_n^aA |χ_n^aA|^2 -|χ_n^aA|^2/2m_n^aA] ,where the covariant spatial gradients acting on fermions are defined by()_iψ^aA_n ≡[∇_i - i g_3 A_i^a ] ψ^aA_n, ()_iχ^aA_n ≡[∇_i + i g_3 A_i^a ] χ^aA_n.The compact momentum p_3 carried by a quark created by (ψ_n^aA)^† isp_3 = [ (a-) - (A-) N/ + n N ],while the antiquark created by (χ_n^aA)^† carries the opposite momentum -p_3. The rest and kinetic quark masses equal |p_3|, the magnitude of the compact momentum, up to higher order radiative corrections,M_n^aA = |p_3| (1 + Ø(λ)), m_n^aA = |p_3| (1 + Ø(λ)).Note that these fermion masses in the effective theory have nothing to do with chiral symmetry breaking quark masses in the underlying 4D theory, which we have assumed vanish. Our EFT fully respects the chiral symmetry (<ref>) of the compactified theory. Nevertheless, the non-relativistic quark masses (<ref>) are non-vanishing for all values of n ∈ℤ+, a = 1,⋯,N, and A=1,⋯,. (Recall that we have assumed that N is odd.) Our explicit calculations in Sec. <ref> will focus on the special case of = N, for which the allowed values of the compact momentum of a quark become half-integers (times ),p_3 = k, k ≡ a-A + n N.For a given Cartan index a, relation (<ref>) gives a one-to-one mapping between the flavor and KK indices {A,n} and the quantized momentum index k. When discussing the = N theory, it will often be convenient to use the momentum index k ∈ℤ + in place of the (equivalent) values of the the flavor and KK indices and relabel the quark fields asψ^a_k ≡ψ^aA_n,χ^a_k≡χ^aA_n,with the implicit understanding that the flavor, KK and momentum indices are connected via relation (<ref>). In other words, ψ^a_k annihilates a quark with compact momentum p_3 =k and charge +1 under the a'th U(1) gauge group, while χ^a_k annihilates an antiquark with compact momentum p_3 = - k and charge -1 under the a'th U(1) group.In addition to the above quark kinetic terms, the Abelian charge densities ρ^a appearing in the Coulomb interactions of the effective theory (<ref>) must be augmented to include the quark contributions,ρ^a ≡∑_b = 1^N_n∈ℤ[ (ϕ⃗^ ab_n)^†·ϕ⃗_n^ab - (ϕ⃗^ ba_n)^†·ϕ⃗_n^ba] + ∑_A=1^∑_n∈ℤ+1/2[ (ψ^aA_n)^†ψ_n^aA - (χ^aA_n)^†χ_n^aA] ,and the form of the Coulomb interactions appearing in the action (<ref>) must now have the contribution from the unwanted extra U(1) gauge group removed,S_ Coulomb = λ/4π∫ dt d^2x d^2yG(-) [ ∑_a=1^N ρ^a(t,)ρ^a(t,) - 1/N∑_a,b=1^N ρ^a(t,)ρ^b(t,) ] .(Without the subtraction of the second term in this expression, the Coulomb energy would be that of a U(N) gauge theory instead of SU(N).) With quarks added to the theory, all the conserved Abelian charges Q^a, when acting on physical states, equal the baryon number,Q^a = N_B ≡1/N∑_a,A,n∫ d^2x [ (ψ^aA_n)^†ψ_n^aA - (χ^aA_n)^†χ_n^aA] . Conversion of the effective action for quarks (<ref>) to the corresponding quark contribution of the non-relativistic Hamiltonian proceeds as described earlier. As with the W-bosons, normal ordering the Coulomb interactions induces logarithmic dependence on the scale μ in the quark rest masses,μd/dμM_n^aA(μ) = -λ/8π^2(1- 1N).In the presence of quarks the rescaling relation (<ref>) becomes1/λ Ĥ_ NR(λ;μ)≅1/λ' Ĥ_ NR(λ',μ) - /16π^2ln(λ/λ') [ 2 N̂_ W + (1- 1N)N̂_ q+q̅] ,whereĤ_ NR (λ;μ) ≡. Ĥ|_A⃗^a = 0 - ∑_a, b=1^N_n∈ℤ M_n^ab(μ)N̂_n^ ab - ∑_a=1^N∑_A=1^∑_n∈ℤ+1/2 M_n^aA(μ)N̂_n^ aAis the non-relativistic Hamiltonian with all rest energies removed,N̂_n^aA≡∫ d^2x [ ψ_n^ ab()^†ψ_n^ ab() + χ_n^ ab()^†χ_n^ ab() ]counts the number of quarks plus antiquarks of the specified type, and the operator N̂_ q +q̅≡∑_A=1^∑_a=1^N ∑_n∈ℤ+1/2N̂_n^aA is the total number of quarks plus antiquarks. § SYMMETRIES As already noted, physical states in an SU(N) gauge theory must be gauge invariant. In the compactified theory, this is trivially enforced dynamically: gauge invariant states are those which do not have divergent Coulomb energies. This is equivalent to the just-stated condition (<ref>) that all U(1) changes equal the baryon number, Q^a = N_B. To see this connection more explicitly, it may be helpful to note that our effective W-boson fields, ϕ⃗_n^ ab, which weredescribed earlier in a basis-dependent fashion as coming from a specified row and column of the 4D gauge field — when the holonomy has the specific form (<ref>) — could have been introduced in a manifestly basis-independent fashion by first defining the operators_a ≡1/N∑_n=0^N-1ω^-(a-1/2 (N+1))nΩ^n,The operators (<ref>) are mutually orthogonal Hermitian projection operators, _a_b = δ_ab _a, when Ω lies at the center-symmetric minimum (<ref>) and the eigenvalues of Ω are all N'th roots of -1 or +1. Our effective 3D fields correspond to pieces of the original 4D fields extracted by these projection operators, [ These are leading order relations. As with any effective field theory, field redefinitions and matching corrections complicate higher order relations between fields in the effective and original theories. ]F^a_μν∝ (_a F_μν),ϕ⃗^ab_n ∝_a D⃗ _b,ψ^aA_n ∝_a q^A,χ^aA_n ∝q̅^A _a,(neglecting details of the KK decomposition, spinor structure, etc.).This highlights the point that the Cartan gauge fields are associated with manifestly gauge invariant 4D operators, while the W-boson and quark fields are gauge covariant, as one would expect.With the aid of such expressions, it is easy to see that composite operators in the 3D theory which map onto manifestly gauge invariant 4D operators are precisely those satisfying the condition Q^a = N_B. As examples, the operators G^ab ≡ϕ⃗_0^ ab·ϕ⃗_0^ ba∼ (D_i_b D_i_a) , M^a_AB ≡χ_1/2^aA ψ_1/2^aB∼q̅^B _aq^A, B_A≡ψ_1/2^1,A ψ_1/2^2,A⋯ψ_1/2^N,A∼ (_1q^A) (_2q^A) ⋯ (_Nq^A), (with no implied sums on Cartan indices, and extraneous structure suppressed) are prototypical glueball, meson, and baryon operators, respectively.The global symmetries which are respected by our compactification and under which eigenstates of the Hamiltonian may be classified include the spacetime symmetries of 2+1 dimensional Minkowski space, leading to conserved total 2D spatial momentum (P⃗) and angular momentum (J_z). States with vanishing J_z may be further classified by their behavior under 2D spatial reflections. [ Reflections are only a symmetry of the theory when θ = 0 (or π), but the violation of reflection symmetry induced by a non-zero θ only affects the long distance non-perturbative physics. For a more thorough discussion of the action of various symmetry transformations in the 3D effective theory, refer to Appendix <ref>. ] Translation invariance in the compactified direction implies conservation of the total compact momentum,P_3≡∫ d^2x {∑_a,b=1^N ∑_n∈ℤ (a-b+nN)(ϕ⃗_n^ ab)^†ϕ⃗_n^ ab+ ∑_a=1^N ∑_A=1^∑_n∈ℤ+1/2( (a-)- N(A-) + nN) [ (ψ_n^ aA)^†ψ_n^ aA - (χ_n^ aA)^†χ_n^ aA] } .As discussed earlier, our individual fields carry compact momentum quantized in units of(for ϕ⃗^ ab_n) or linear combinations ofand (N/) (for ψ^aA_n and χ^aA_n). Physical glueball and flavor singlet mesons statesmust have total compact momentum equal to an integer multiple of 2π/L = N, as these states remain invariant when translated once around the compact dimension. Due to our flavor-twisted boundary conditions for quarks, flavor non-singlet mesons can have P_3 equal to integer multiples of 2π/( L). The allowed values of P_3 for flavor singlet (non-singlet) baryons are integer or half-integer multiples of 2π/L (or 2π/( L)) depending on whether N is even or odd.When quarks are present, the unbroken U(1)^_V flavor symmetry transformations are generated by the conserved flavor chargesN^A ≡∫ d^2x ∑_a=1^N ∑_n∈ℤ+1/2[ (ψ^aA_n)^†ψ_n^aA - (χ^aA_n)^†χ_n^aA] .The sum of these flavor charges equals the total number of quarks minus antiquarks, or N times the baryon number N_B.Axial U(1)^_A flavor symmetry transformations act as spin rotations on the EFT fermions and are generated by the axial chargesN^A_5 ≡∫ d^2x ∑_a=1^N ∑_n∈ℤ+1/2[ (ψ^aA_n)^†σ_3ψ_n^aA + (χ^aA_n)^†σ_3χ_n^aA] .The perturbative dynamics conserves these charges but the long range non-perturbative dynamics violates conservation of N_5 ≡∑_A N^A_5 (and the non-perturbative vacuum is not annihilated by the other axial charges).In the absence of quarks, the compactified theory is invariant under the ℤ_N center symmetry which, by construction, remains unbroken. The defining center symmetry transformation (<ref>) multiplies the holonomy by an N'th root of unity, Ω→ω Ω. This permutes the projection operators (<ref>), _a →_a-1 (with _0 ≡_N), and also acts as a cyclic permutation on our 3D fields,σ^a →σ^a-1 ,ϕ⃗_k^ a→ϕ⃗_k^ a-1 .Here, Cartan indices are to be understood to be defined modulo N (so a-1≡ N when a 1). Glueball operators such as G^a_k ≡ϕ⃗^ a_k ·ϕ⃗^ a-q_-k (with kN ≡ q) are likewise cyclically permuted by center symmetry transformations. To diagonalize center symmetry, one must perform a discrete ℤ_N Fourier transform and define, for example,σ̃^p ≡1/√(N)∑_a=1^Nω^a p σ^a,G^p_k ≡1/√(N)∑_a=1^Nω^a pG^a_k.These operators now have definite center symmetry charge p = 0,⋯,N-1, meaning that under the center symmetry transformation (<ref>) they transform into themselves multiplied by the eigenvalue ω^p = e^2π i p/N.Adding fundamental representation quarks to the theory generally breaks the ℤ_N center symmetry. However, in the special case of N, the theory retains an intertwined ℤ_N color-flavor center symmetry (see, e.g., Refs. <cit.>). [ More generally, if d ≡gcd(,N) > 1, then a ℤ_d color-flavor center symmetry remains <cit.>. For simplicity, we will focus on the case of N. ] This symmetry combines the usual center transformation (<ref>) with a cyclic permutation of quark flavors. In terms of our 3D fields, this flavor-intertwined center symmetry acts asσ^a →σ^a-1 ,ϕ⃗_k^ a→ϕ⃗_k^ a-1 ,ψ_k^a→ψ_k^a-1 ,χ_k^a→χ_k^a-1 ,and again may be diagonalized by a discrete ℤ_N Fourier transform.Because the sets of eigenvalues (<ref>) and (<ref>) of the gauge holonomy Ω and our chosen flavor holonomy Ω_F are invariant under complex conjugation, both charge conjugation and reflection of the compactified dimension (x_3 → -x_3) remain symmetries of theory provided they are combined with global gauge and flavor transformations which suitably permute the Cartan and flavor indices. The ordering (<ref>) of the eigenvalues of the gauge holonomy was chosen so that the required global gauge transformation V is just a permutation which flips Cartan indices, a → N+1 - a, reflecting the fact thatΩ^* = VΩV^† ,with V ≡δ_a+b,N+1 an anti-diagonal transposition. Similarly, given the order (<ref>) of the flavor holonomy eigenvalues, the required flavor transformation V_F also corresponds to a simple flip of flavor indices, A →+1 - A, sinceΩ_F^* = V_FΩ_F V_F^† ,with V_F ≡δ_A+B,+1. This redefined charge conjugation symmetry acts on the fields of our our dimensionally reduced EFT as σ^a→ -σ^a̅ ,ψ^aA_n→χ^a̅A̅_-n ,ϕ⃗^ ab_n→ -ϕ⃗^ b̅a̅_n ,χ^aA_n→ψ^a̅A̅_-n , where a̅≡ N+1-a, A̅≡+1-A. [ The form of this transformation relies on our simplifying assumption that N is odd, so that eigenvaluesof Ω are roots of +1 and Ω_F eigenvalues are roots of -1. If N is even then both ± 1 can be eigenvalues of the flavor holonomy Ω_F for some values of ≤ N. When two eigenvalues of Ω_F are real, the required flavor transformation V_F which must be combined with the naive action of charge conjugation no longer corresponds to the simple flip A →A̅ of flavor indices. ] Note that center symmetry does not commute with charge conjugation. In choosing a basis for degenerate levels of the Hamiltonian, one must choose between specifying center symmetry charge, or the sign under the (appropriately redefined) charge conjugation symmetry; we will generally opt for the former.Finally, reflection in the compact direction, x_3to -x_3, when combined with the same global gauge and flavor transformations V and V_F, remains a symmetry. This redefined reflection symmetry acts on our 3D EFT fields as σ^a→σ^a̅ ,ψ^aA_n→-iσ_2ψ^a̅A̅_-n ,ϕ⃗^ ab_n→ϕ⃗^ a̅b̅_-n ,χ^aA_n→ iσ_2χ^a̅A̅_-n . The combined symmetry of charge conjugation times x_3 reflection does not involve any global gauge or flavor transformations and acts as σ^a→ -σ^a ,ψ^aA_n→iσ_2χ^aA_n ,ϕ⃗^ ab_n→ -ϕ⃗^ ba_-n ,χ^aA_n→ -iσ_2ψ^aA_n . This is the same as a CP transformation times a 180^∘ rotation in the uncompactified directions. § HEAVY SECTOR SPECTRUM §.§ Overview Three basic types of bound states can be formed from the constituents of our non-relativistic effective theory: glueballs, mesons, and baryons. Here, “bound state” means either a genuine single particle eigenstate of the full theory, or a narrow resonance whose fractional decay width vanishes in the L → 0 (and correspondingly λ→ 0) limit. In this section, we neglect the coupling to the Abelian gauge fields contained in the spatial covariant derivatives, as well as higher dimension operators not shown explicitly in our effective theories (<ref>) and (<ref>). Effects of these terms are discussed in Sec. <ref> which discusses decay processes.By glueballs we mean bound states of two or more charged W-bosons, and no quarks or antiquarks. Mesons are, of course, bound states of a quark and antiquark, possibly containing additional W-bosons, while baryons are bound states of N quarks (perhaps with additional charged W-bosons). In our weakly coupled small-L regime, mixing between glueballs and flavor singlet mesons is suppressed, so they are clearly distinguishable. Manifestly gauge invariant interpolating operators for simple examples of such states were shown in Eq. (<ref>). Further possibilities, which we will not focus on in this paper, include multi-meson or multi-glueball “molecules” and multi-baryon bound states.As discussed above, all physical (gauge invariant) states must satisfy Q^a = N_B. Hence, glueballs and mesons must be composed of combinations of constituents for which all U(1)^N charges sum to zero. The simplest glueballs are two-body bound states of a W-boson and its oppositely charged antiparticle, created by operators such as(ϕ⃗_0^ ab)^†· (ϕ⃗_0^ ba)^† ,with ab. Two different U(1) gauge group factors contribute to the logarithmic interaction between these constituents, giving an attractive interaction of relative strength 2. The explicit two-body Hamiltonian, and its spectrum, is examined in Sec. <ref> below. Bound states of more than two W-bosons can also form. States of this type which cannot be decomposed into two or more separately gauge invariant glueballs consist of W-bosons whose charge assignments lead to a ring-like color structure with nearest-neighbor logarithmic interactions. Examples of operators creating such states are(ϕ_0^ ab)^†_i (ϕ_0^ bc)^†_j (ϕ_0^ ca)^†_k,(ϕ_0^ ab)^†_i (ϕ_0^ bc)^†_j (ϕ_0^ cd)^†_k (ϕ_0^ da)^†_l,etc., with up to N constituents and Cartan indices a,b,c,⋯ all distinct. We will refer to these as “closed string” glueballs. These are all single trace operators when expressed in terms of the original 4D fields (as in Eq. (<ref>)). In these multi-body states, a single U(1) factor generates an attractive logarithmic interaction (of relative strength 1) between each pair of neighboring constituents in the cyclic list. This is illustrated schematically in Fig. <ref>.We note that there is an amusing similarity between these states and the picture advocated long ago in Ref. <cit.>.The situation with mesons is similar. The simplest mesons are two-body bound states, created by operators such as(χ_1/2^aA)^† (ψ_1/2^aB)^† .The attractive logarithmic interaction between the quark and antiquark has relative strength of (1-1/N), with the reduction from 1 coming from the subtraction of the unwanted “extra” U(1) contribution in the Coulomb interaction (<ref>). There are also mesons in which one or more additional W-bosons are present. States of this type which cannot be decomposed into meson-glueball products have charge assignments implying an “open string” color structure. Examples of operators creating such states include(χ_1/2^aA)^† (ϕ_0^ab)^†_i (ψ_1/2^bB)^† ,(χ_1/2^aA)^† (ϕ_0^ab)^†_i (ϕ_0^bc)^†_j (ψ_1/2^cB)^† ,etc, with up to N-1 W-bosons inserted between the quark and antiquark and Cartan indices a,b,c,⋯ all distinct. There are attractive logarithmic interactions of relative strength 1 between each pair of neighboring constituents, along with a repulsive logarithmic interaction of strength 1/N between the quark and antiquark (with differing Cartan charges). This is illustrated schematically in Fig. <ref>.Finally, baryons containing N quarks, potentially with additional W-bosons as well, are present as finite energy bound states because our gauge group is SU(N), not U(N). The simplest non-exotic baryons are created by operators like(ψ_1/2^1,A)^† (ψ_1/2^2,B)^† (ψ_1/2^3,C)^†⋯ (ψ_1/2^N,Z)^† .In such states, every pair of quarks has an attractive logarithmic interaction of relative strength 1/N. Two such baryon states, as well as a baryon state containing an additional W-boson, are illustrated schematically in Fig. <ref>.The stability of these various hadronic states will depend on their relative energy differences and the resulting radiative transition and short distance annihilation rates. These are discussed below in Sec. <ref>. §.§ Two-body states Neglecting couplings to the spatial Abelian gauge fields (which are relevant for radiative decays but not the leading order spectrum), the dynamics of all two-body sectors of our effective theory (<ref>), namely glueballs composed of oppositely charged W-bosons, quark-antiquark mesons, and diquark baryons in the special case of N2, are described by a common first-quantized two-dimensional non-relativistic Hamiltonian,Ĥ = _1^2/2m_1 + _2^2/2m_2 + κ ln(μ|_1-_2|),with a logarithmic potential and positive interaction strength, κ>0. Before discussing our specific application to glueball, meson, and N2 baryons in compactified QCD, we first summarize properties of the spectrum of this quantum theory.§.§.§ 2D logarithmic QM Starting with the two particle Hamiltonian (<ref>), separating the center of mass motion and working in the center-of-mass frame leads to a one-body Hamiltonian for the relative motion,Ĥ_ relative = ^2/2m + κ ln(μ||),where m ≡ m_1 m_2 / (m_1+m_2) is the reduced mass. Non-relativistic dimensional analysis (with ħ≡ 1) shows that κ m/μ^2 is the only dimensionless combination of parameters appearing in the Hamiltonian (<ref>), so its eigenvalues must have the form E = κf(κ m/μ^2) for some univariate function f. The manifestly trivial μ dependence, ∂ E/∂μ = κ/μ, then implies that the energy eigenvalues of Ĥ_ relative are given byE = κ[ ϵ - lnκ m/μ^2] ,where ϵ is an eigenvalue of the theory with κ = m = μ≡ 1. Introducing a dimensionless radial variable r = √(κ m)|𝐱|, eigenstates with orbital angular momentum L_z ≡ℓ = 0, ±1, ±2, ⋯ satisfy the one-dimensional radial Schrödinger equation,[ -d^2/dr^2 + V_ℓ(r) ] χ(r) = ϵ χ(r),with effective radial potentialV_ℓ(r) ≡ℓ^2- 14/2r^2 + ln r. Solutions to the Schrödinger equation (<ref>) are not expressible in terms of familiar special functions. The equation was analyzed numerically over 40 years ago <cit.> (see also Refs. <cit.>), but we will present our own more accurate and extensive results. Calculations of low-lying energy levels are fairly straightforward using variational methods and a suitable basis set, or alternatively using pseudo-spectral methods <cit.> with a Gauss-Laguerre grid for the semi-infinite radial domain. [ A simple choice of basis for a variational calculation consists of 2D harmonic oscillator eigenstates with definite angular momentum ℓ. Given a suitable adjustment of the scale of the harmonic oscillator basis functions, a truncated basis of 40 harmonic oscillator states is sufficient to find the lowest energy level of the logarithmic Hamiltonian (<ref>) to an accuracy of a few parts in 10^4.However, pseudo-spectral discretization using a Laguerre grid turns out to provide significantly better accuracy for a given basis size. (This is because harmonic oscillator wavefunctions with their Gaussian envelope decrease too rapidly at large r; as discussed below eigenstate wavefunctions in a logarithmic potential decrease much more slowly.) To obtain the eigenvalues shown in Table <ref> and compute transition matrix elements for radiative decays, discussed in Sec. <ref>, we used Gauss-Laguerre grids with 100–200 points. To avoid excessive precision loss in the evaluation of the spectral differentiation matrices and the resulting eigenvalue computation, we used extended precision arithmetic with slightly over twice as many digits as the number of grid points.] The first ten levels, for each |ℓ| = 0, ⋯, 9, are listed in Table <ref>. The spectrum is shown graphically in Fig. <ref>. Notice that levels at neighboring values of ℓ are interleaved, ϵ_n,|ℓ| < ϵ_n,|ℓ|+1 < ϵ_n+1,|ℓ|. As |ℓ| increases, the minimum of the potential moves to larger values, with r_ min∼ |ℓ| + Ø(ℓ^-2). When |ℓ| ≫ 1, a quadratic approximation to the potential is sufficient to find low-lying states. For fixed level number n (starting from 0),ϵ_n,ℓ = ln (|ℓ|) +12 + 2n+1/√(2)|ℓ| + Ø (ℓ^-2). Standard WKB methods may be used to study more highly excited states. When the energy ϵ is large compared to max(1,ln |ℓ|), the classically allowed region of the Schrödinger equation (<ref>) extends out to a turning point at r_* ≡exp(ϵ). For r > r_*, the WKB solution which decays as r →∞ isf_ I(r) = [ln(r)/ϵ-1]^-1/4exp[ -√(2ϵ)|Q_0(r)| + Ø(ϵ^-1/2) ] ,whereQ_0(r) ≡∫_r^r_* dr' √(1-ln(r')/ϵ) .The usual Airy function matching across the turning point (or analytic continuation around the turning point) shows that this solution matches onto the allowed region WKB solutionf_ II(r) = [1-ln(r)/ϵ]^-1/4cos[ √(2ϵ)Q_0(r) - π4 + Ø(ϵ^-1/2) ] .This WKB approximation is valid down to r = Ø(1), wheref_ II(r) ∼cos[ √(2ϵ)r - I(ϵ) + π4 + Ø(ϵ^-1/2) ] × (1 + Ø(ϵ^-1)),withI(ϵ) ≡√(2ϵ)Q_0(0) = √(π 2) exp(ϵ).For parametrically small values of r, the centrifugal term in the potential cannot be neglected but the logarithmic term is subdominant. In this region, the appropriate solution satisfying regularity at the origin isf_ III(r) = (ϵ)^1/4√(π r) J_|ℓ|(√(2ϵ)r).When r ≫ϵ^-1/2, f_ III(r) ∼cos(√(2ϵ)r - π/2|ℓ| - π/4) + Ø((√(ϵ)r)^-1). For Ø(1) values of r, this matches onto the the classically allowed WKB solution (<ref>) providedI(ϵ) =(2n + |ℓ| + 1)π + Ø(ϵ^-1/2) ,for some integer n. Inserting the result (<ref>), one finds that eigenvalues ϵ_n,ℓ of the radial Schrödinger equation (<ref>) are given byϵ_n,ℓ = ln ( 2n + |ℓ| + 1 ) + lnπ 2,up to corrections vanishing faster than Ø(1/n). One may verify that n equals the number of nodes in this solution, so n is level number when counting from 0.Numerically, the accuracy of the WKB approximation(<ref>) to energy levels is surprisingly good for modest values of the level number n. For ℓ 0 and n10, the difference between our numerical and WKB results is less than 2 parts in 10^4. The relative deviation grows with increasing ℓ at fixed n, reaching 2% for ℓ n10.The WKB result (<ref>) shows that the level spacing (at fixed ℓ) decreases with increasing level number, dϵ/dn = 2/(2n+|ℓ|+1). Inverting this relation, one finds that the asymptotic density of states with fixed orbital angular momentum ℓ rises exponentially with energy,∂ n_ℓ/∂ϵ∼e^ϵ/√(2π) .(This neglects any spin degeneracy of the constituents.) The integral of this density of states gives the total number of quantum states, with fixed ℓ, below a given energy, and asymptotically equals the area of the classically allowed region in phase space (in units of 2πħ),n_ℓ(ϵ)= ∫dp/2πdr Θ(ϵ- p^2 - V_ℓ(r)) = √(2)/π∫_r_ min^r_ max dr √(ϵ- V_ℓ(r)) = √(2)/π[ ∫_0^e^ϵ dr √(ϵ-ln r)] + Ø(|ℓ|-) = e^ϵ/√(2π) + Ø(|ℓ|-). The total number of states below energy ϵ (with vanishing total momentum, but no projection onto definite ℓ), N(ϵ) = ∑_ℓ n_ℓ(ϵ), coincides asymptotically with the classically allowed phase space volume of the 2D relative dynamics. This grows exponentially at twice the rate of the fixed-ℓ result,N(ϵ) = ∫d^2p/(2π)^2d^2r Θ(ϵ- p^2 -ln r) = ∫_0^e^ϵ r dr(ϵ-ln r) = 14 e^2ϵ .This exponential growth is a direct consequence of the slow increase of the confining logarithmic potential with distance. Bound states spread over rapidly growing spatial regions as their energy increases. The exponential behavior (<ref>) of the fixed-ℓ number of states is nothing but linear dependence on the turning point radius r_*, while the total number of states (<ref>) is, up to a factor of 1/(4π), just the spatial area of the allowed region, π r_*^2. §.§.§ Glueballs For every pair of oppositely charged W-bosons there is a manifold of bound states described by the two-body logarithmic interaction Hamiltonian (<ref>) with interaction strengthκ = λ/2π^2 .This is analogous to the ro-vibrational states associated with each electronic level in molecular spectroscopy. For a pair of W-bosons with compact momentum indices k and k' (defined by the relation (<ref>) and satisfying the constraint k+k'= 0N so that the W-bosons have opposite Cartan charges), the resulting bound state energies are given byE_WW = M_k() + M_k'() + λ/2π^2( ϵ_n,ℓ -lnλm_kk'/2π^2) ,where the reduced mass m_kk'≡ m_k m_k' /(m_k + m_k'), and we have chosen to set the arbitrary scale μ equal to .The lightest glueballs are composed of W-bosons with one unit of compact momentum, |k| = |k'| = 1, and tree-level constituent mass , leading to glueball energiesE = 2 M_1() + λ/2π^2( ϵ_n,ℓ -lnλ/4π^2) . Neglecting higher order relativistic corrections, as well as non-perturbative physics on the scale of m_γ, two-body glueball states have a degeneracy of 4N if they are ℓ = 0 and CP self-conjugate. (Center symmetry gives a factor of N, and there is a spin degeneracy of 4 since each massive W-boson has two spin states.) There is an additional factor of 2 degeneracy for states with non-zero orbital angular momentum (corresponding to positive and negative values of ℓ, which are exchanged by 2D spatial reflections), and a separate additional factor of 2 degeneracy for states which are not CP self-conjugate. The lightest glueball level (<ref>) contains CP self-conjugate ℓ=0 states, and hence has the minimal degeneracy of 4N.Relativistic corrections to the above results contribute Ø(λ^2 ) energy shifts, or relative Ø(λ) corrections to binding energies. Spin-orbit corrections give an energy shift proportional to ℓS_z (where S_z ≡ s^(1)_z + s^(2)_z), with a positive coefficient. In our dimensionally reduced effective theory, spin-spin (or hyperfine) interactions are local and proportional to s_z^(1) s_z^(2) δ^2 (𝐱), also with a positive coefficient. [ In two spatial dimensions, spin-spin interactions do not have a long range dipolar form since the magnetic field produced by a current loop is localized inside the loop. ] This spin-spin correction only has a non-zero expectation value in ℓ = 0 states. Hence, first order relativistic corrections produce an energy shift of the formΔ E_ finestructure = λ^2 [ AℓS_z + Bδ_ℓ^0 (S_z^2 - 2) ],where A and B are positive Ø(1) coefficients (depending on n and |ℓ|). For a given n and ℓ 0, the spin-orbit correction splits the four possible spin states, {|↑↑⟩, |↑↓±↓↑⟩, |↓↓⟩}, into three sublevels with the S_z = -2 ℓ/|ℓ| state moving lower in energy, the S_z = +2 ℓ/|ℓ| state moving higher, and the two S_z = 0 states unchanged. For ℓ = 0 levels, the spin-spin interaction produces two sublevels, with the energy of the S_z = ± 2 states shifted upward, and the S_z = 0 states downward. The degeneracy between the spin symmetric and antisymmetric S_z = 0 states, |↑↓±↓↑⟩, is not lifted by these leading relativistic corrections, but should be removed at higher orders.Short distance effects will also induce higher order corrections to the rest and kinetic masses, leading to further spin-independent Ø(λ^2) energy shifts. Operators producing Ø(λ^2) corrections are listed in Appendix <ref>, which discusses the relevant power counting rules. The structure of higher dimensional operators that appear in our non-relativistic EFT follow the same pattern known, for example, from studies of hydrogenic spectra or heavy quark physics in QCD <cit.>, but quantitative evaluation of these higher order effects is left to future work.The factor of N degeneracy associated with center symmetry would be lifted by the non-perturbative long distance physics on the scale of m_γ but, more importantly, this degeneracy is first lifted by one loop perturbative corrections which generate photon mixing terms (mentioned earlier in footnote <ref>). Such mixing arises from vacuum polarization corrections which are sensitive to the differing masses M_n^ab of the charged virtual W-bosons. This mixing (when rediagonalized) induces Ø(λ) variations in the coupling strengths of different light photons. Eigenstates of bound W-bosons will have definite center charge and are constructed by a ℤ_N Fourier transform, as in Eq. (<ref>). The energies of states with differing values of center charge will be split by Ø(λ^2 ), or in other words additional Ø(λ) relative corrections to binding energies.§.§.§ Mesons Differences between the two-body meson and glueball spectra arise from the differing constituent masses and the strength of the logarithmic interaction. For an oppositely charged quark-antiquark pair, the interaction strength is given byκ = (1- 1N) λ/4π^2 .The allowed values of compact momentum (<ref>) depend on both N and . As mentioned earlier, a particularly simple case which we will focus on is = N. For this number of flavors the tree-level constituent quark masses (<ref>) become half-integers times ,M_n^aA = M_k ≡ (|k| + Ø(λ)) ,m_n^aA = m_k ≡ (|k| + Ø(λ)) ,with k = a - A + nN and n ∈ℤ+. The resulting bound state energies are given byE_q̅ q = M_k() + M_k'() + (1- 1N) λ/4π^2( ϵ_n,ℓ -ln(1-1/N)λm_kk'/4π^2) ,where, once again, m_kk' is the reduced mass. The lightest mesons have |k|=|k'| =, leading toE_q̅ q = 2M_1/2() + (1- 1N) λ/4π^2( ϵ_n,ℓ -ln(1-1/N)λ/16π^2) .Neglecting higher order relativistic corrections, the lightest two-body meson levels (<ref>) have a degeneracy of 16N if they have ℓ 0, with an additional factor of 2 if ℓ 0. (Four factors of 2 coming from the choice of spin for quark and antiquark, plus the choice of sign of each momentum index, and a factor of N from one choice of flavor, or equivalently from the choice of which U(1) photon provides the binding.) Higher order spin-orbit, spin-spin and other radiative effects partially lift this degeneracy in the same manner discussed above for glueballs. §.§.§ N = 2 baryons Finally, in the special case of two-color QCD, the simplest baryons are bound states of two quarks (with no additional W-bosons). The interaction strength κ equals 1/Nλ/(4π^2) which, for N = 2, coincides with the quark-antiquark interaction strength. Consequently, the resulting diquark baryon spectrum is identical to the meson spectrum (<ref>) and (<ref>) given above, when specialized to N = 2. The degeneracy of the lightest baryon levels (neglecting relativistic corrections) is 16 for ℓ = 0 states, with an additional factor of two for ℓ 0. §.§ Multi-body states §.§.§ Glueballs As noted in the overview, in addition to two-body W-boson bound states, multi-body bound states containing three or more W-bosons with a ring-like color structure can also form, such as those illustrated in Fig. <ref>. The spectrum of such “closed string” states is quite rich.The rest mass of W-bosons is given by Eq. (<ref>), reproduced here for convenience,M_n^ab = M_k ≡ |k| =|a-b+nN|,up to Ø(λ) corrections. To form a physical (gauge invariant) bound state, the U(1)^N Cartan charges of all W-bosons in the bound state must sum to zero. For closed-string glueball states which are not decomposable into multiple separate glueballs, this means that each neighboring pair of W's in the ring is bound together by a distinct Abelian gauge interaction.Bound states containing 3 ≤ P ≤ N constituents having compact momentum indices { k_1,k_2, ⋯, k_P } exist, consistent with this constraint, provided that∑_i=1^P k_i = 0N.For this state to be non-decomposable, no partial sum of the momentum indices should vanish modulo N. In addition to specifying the momentum index of each constituent, one may specify one Cartan index of a single constituent; together this information completely determines the Cartan and KK indices of all constituents around the cycle. The tree-level mass of such a closed string state is justM_ tot = ∑_i=1^P |k_i|. “Near extremal” states:An interesting subset of states are those with non-zero compact momentum P_3 and whose tree-level mass equals the minimal value consistent with this compact momentum,M = |P_3|.This implies that the momentum indices of all constituents have the same sign. One simple case, satisfying the constraint (<ref>) (plus non-decomposability), are “pearl necklace” bound states containing N W-bosons, all with momentum indices equal to unity, k_i = 1, or all equal to minus one, k_i = -1. For these states P_3 = ∑_i k_i = ± N = ± 2π/L and the (tree level) rest mass M = |P_3| = N. The middle example in Fig. <ref> illustrates this type of pearl necklace state (with P_3 = -2π/L) in the case of N4. Such a stateis created by the N-body operatorA^i_1 i_2 ⋯ i_N(ϕ_-1^1)^†_i_1(ϕ_-1^2)^†_i_2⋯ (ϕ_-1^N-1)^†_i_N-1(ϕ_-1^ N)^†_i_N ,where the coefficients { A^i_1 ⋯ i_N} (defining a rank-N 2D spatial tensor) determine the spin wavefunction.There are also near-extremal states with fewer constituents. One can imagine fusing together any neighboring pair of constituents in the operator (<ref>) and replacing them with a single W-boson having the same Cartan charges and compact momentum as the pair. Or doing the same fusing process with a neighboring triplets of constituents, etc. The resulting states are also near-extremal, and are created by N-1 or N-2 body operators such as A^i_1 i_2 ⋯ i_N-1(ϕ_-1^1)^†_i_1(ϕ_-1^2)^†_i_2⋯ (ϕ_-2^N-1)^†_i_N-1 ,or A^i_1 i_2 ⋯ i_N-2(ϕ_-1^1)^†_i_1(ϕ_-1^2)^†_i_2⋯ (ϕ_-3^ N-2)^†_i_N-2 . Continuation of this fusing process leads to near-extremal states with any number of constituents from N down to 1. Three and two body examples are A^i_1 i_2 i_3(ϕ_-1^1)^†_i_1(ϕ_-1^2)^†_i_2(ϕ_-(N-2)^ 3)^†_i_3 ,and A^i_1 i_2(ϕ_-1^1)^†_i_1(ϕ_-(N-1)^2)^†_i_2 , while the endpoint of this process is a neutral “heavy photon” state created by a one-body operator such asA^i(ϕ_-N^ 1)^†_i .More generally, ignoring spin and center degeneracies there are NP-δ^P_1 distinct categories of near-extremal states containing P constituents associated with different contiguous fusing of the fields in the N-body operator (<ref>), or altogether 2^N-N types of non-decomposable near-extremal states having the same value of P_3 = ± N.“Non-extremal” states: Bound states containing constituents with oppositely signed momentum indices are “non-extremal.” Such states have rest masses which exceed their compact momentum, M > |P_3|, by an Ø() amount or more. This includes all bound states of W-bosons having vanishing total compact momentum, P_3 = 0, such as the lightest glueballs (<ref>).Binding energies: Calculating the Ø(λ) binding energies of multi-body glueball states requires one to find eigenvalues of the first-quantized Hamiltonian which describes the sector of the theory (<ref>) with the chosen number of constituents. For “closed string” bound states composed of P ≤ N W-bosons, this isĤ = ∑_i=1^P [ 𝐩_i^2/2m_i + λ/4π^2ln (μ |𝐱_i-𝐱_i-1|) ] ,with the understanding that 𝐱_0 ≡𝐱_P. The scaling relation (<ref>) allows one to remove the dependence on λ, but eigenvalues will be non-trivial functions of constituent mass ratios,E_ binding = Pλ/8π^2[ f({m_i/m_j}) - ln (λm /μ^2) ] ,where f is a dimensionless Ø(1) function (depending on the chosen energy level as well as mass ratios), and m is the harmonic mean of the constituent masses.For modest values of P (three or four), an accurate variational calculation should be feasible despite the fact that computational effort will rise as a rather high power of the number of single particle states included in the truncated basis. We leave such calculations to future work.An interesting limiting case partially amenable to analytic analysis concerns low-lying states with large orbital angular momentum, ℓ≫ 1, and constituents all having the same mass m. Such states include rotating “pearl necklace” configurations in which each constituent contributes equally to the total orbital angular momentum. A semiclassical analysis of such states is straightforward. The classical Hamiltonian (for fixed ℓ) has a local minimum in which the constituents lie at the vertices of a regular P-sided polygon whose circumscribed circle has radius r = 2 πℓ/(P √(λ m )), rotating at angular velocity Ω = ℓ/(P m r^2) = P λ / (4 π^2 ℓ). Semiclassical quantization of vibrations about this configuration leads to energy levels whose binding energies (ignoring center of mass motion) are given byE_ binding = P λ/4π^2[ + ln(4 πℓμsin(π/P)/P √(λ m )) ] + ∑_i=-(P-2)^P-2 (n_i +)ω_|i| + Ø(ℓ^-2),where the P-1 vibrational frequencies {ω_i} are Ø(λ/ℓ). [ One mode, here labeled i=0, is a uniform “breathing” mode with ω_0 = √(2) Ω. All other modes (present only for P > 2) are higher frequency doubly-degenerate asymmetric stretching modes. For P2, the form (<ref>) agrees as it must with the prior results (<ref>) and (<ref>). ]The result (<ref>) grows logarithmically with increasing angular momentum ℓ, with a coefficient of P λ/4π^2 proportional to the number of constituents. This linear increase with P implies that these semiclassical “pearl necklace” states are not the minimal energy states with a given large orbital angular momentum. “Core-halo” states will exist in which P-1 constituents are clumped together in a region of size √(P/λ m ) while a single constituent circles at a distance of order Ø(ℓ/√(λ m)) and contributes (nearly) all the orbital angular momentum. The binding energy of such states will increase with ℓ just like the two-body case, namely E_ binding∼ (λ/2π^2)lnℓ as ℓ→∞. Computing the sub-dominant ℓ-independent contribution coming from the core wavefunction requires a full quantum calculation. §.§.§ Mesons Largely identical considerations apply to multi-body mesons. Focusing, once again, on the case of N, bound states containing a quark and antiquark having half-integer compact momentum indices k_q and k_q̅, plus P W-bosons with momentum indices {k_1,⋯,k_P}, will have total compact momentumP_3 = (k_q - k_q̅ + ∑_i=1^P k_i ).For the state not to be decomposable into a glueball-meson molecule, no partial sum of the W-boson momentum indices should vanish modulo N. With tree-level mass M_ tot =(|k_q| + |k_q̅| + ∑_i |k_i|), it is immediate that M_ tot≥ |P_3|. Any of the multi-body “closed string” glueball states discussed above may be converted into an “open string” meson state by replacing any one of the W-boson constituents by a q q̅ pair collectively having the same Cartan charges and compact momentum. As an example, one analogue of the near-extremal N-body glueball operator (<ref>) is the near-extremal meson operatorB^s_q̅ s_q i_1 i_2 ⋯ i_N-1(χ_+1/2^ 1)^†_s_q̅ (ϕ_-1^1)^†_i_1(ϕ_-1^2)^†_i_2⋯ (ϕ_-1^N-1)^†_i_N-1(ψ_-1/2^ N)^†_s_q ,(with s_q and s_q̅ denoting two-component spinor indices of the quark and antiquark, respectively), in which N-1 W-bosons are inserted between the quark and antiquark.The Ø(λ) binding energies of (non-decomposable) multi-body meson states containing P W-bosons are given by eigenvalues of the first-quantized HamiltonianĤ = ∑_i=0^P+1𝐩_i^2/2m_i + λ/4π^2[ -1N ln (μ |𝐱_0-𝐱_P+1|) + ∑_i=1^P+1ln (μ |𝐱_i-𝐱_i-1|) ] ,where 𝐱_0 ≡𝐱_q̅ and 𝐱_P+1≡𝐱_q refer to the antiquark and quark, respectively, and likewise for the momenta 𝐩_0 and 𝐩_P+1 and masses m_0 ≡ m_q̅ and m_P+1≡ m_q. The resulting energy levels have the formE_ binding = (P+(1- 1N))λ/8π^2[ f({m_i/m_j}) - ln (λm /μ^2) ] ,with f some Ø(1) function, differing from the glueball case (<ref>) just in the prefactor.Just as with closed-string glueballs, it is interesting to consider open-string mesons with large orbital angular momentum, ℓ≫ 1. Among such states are semiclassical “rotating wire” states. The classical Hamiltonian (for fixed orbital angular momentum ℓ) has local minima in which all constituents are arrayed along a straight line which rotates uniformly with some angular velocity ω, with the positions of constituents along this line adjusted so that the sum of forces (falling with inverse separation) acting on each constituent provides the required centripetal acceleration, and the common angular velocity ω is suitably adjusted to yield the chosen angular momentum ℓ. Solving for this minimum analytically, for arbitrary P, is not easy, but a numerical determination for chosen values of P is straightforward. Semiclassical quantization of such a stationary configuration will lead to energy levels which, as in the glueball case (<ref>), grow logarithmically with increasing ℓ, with a coefficient which increases with the number of constituents. Hence, for the same reasons discussed above, lower energy “core-halo” mesonic states will exist in which all but one constituent are clumped together and collectively carry little or no angular momentum while a single constituent (which may be either a quark or a W-boson) circles the core at a large Ø(ℓ/√(λ m )) distance and carries (nearly) all the orbital angular momentum. §.§.§ Baryons Baryonic bound states containing quarks with no additional W-bosons (“non-exotic baryons”) may be formed from a collection of N quarks, each having a distinct color (Cartan) index. Focusing, once again, on the case of = N, the momentum indices { k_1,⋯, k_N} of the quarks are arbitrary half-integers (with k_i the momentum index of the quark with Cartan index i). The total compact momentum P_3=∑_ik_i and the tree-level mass M_tot=∑_i=1^N|k_i|.Note that, for large values of N, baryons which are composed of the lightest quark constituents with Ø(1) momentum indices will have a total mass M_ tot which scales linearly with N. Such baryons contain quarks of (nearly) all N different flavors. Baryons which are solely composed of quarks of a single flavor will have a total mass which is at least Ø(N^2), because the momentum indices of quarks must, in this case, all be distinct and hence will, at a minimum, have magnitudes ranging fromup to ⌊ N/2 ⌋.The strength of the attractive logarithmic interaction between two quarks of differing colors is 1/N, so the first-quantized non-relativistic Hamiltonian for non-exotic baryons isĤ = ∑_i=1^N_i^2/2m_i + 1/N∑_i<j=1^Nλ m_W/4π^2 ln(μ|_i-_j|) ,with m_i =|k_i| the i'th constituent quark mass.For the lightest class of baryons, each quark has momentum index ± and the minimal constituent mass m_i = m_q ≡. Such states are created by operators of the formC^s_1s_2⋯ s_N(ψ_± 1/2^1)_s_1^†(ψ_± 1/2^2)_s_1^†⋯(ψ_± 1/2^N)_s_N^† ,with s_i denoting the two-component spinor index of the ith quark. (Fig. <ref> illustrates one such state for N4.) For simplicity of presentation, we will focus our discussion on this lightest class of baryons.For baryons with equal mass constituents, the Hamiltonian (<ref>) is completely symmetric under permutations of constituents. The rescaling relation (<ref>) implies thatĤ≅λ m_W/4π^2( ∑_i=1^N_i^2 + 1/2N∑_i j=1^Nln|_i-_j| ) - (N-1)λ m_W/16π^2 ln(λ m_q/4π^2 μ^2).The spectrum of this Hamiltonian was already discussed in Sec. <ref> in the special case of N2. We now examine the opposite extreme, N ≫ 1.As discussed by Witten <cit.>, a Hartree approximation to the many-body wavefunction is asymptotically accurate as N →∞. The appropriate N-body Hartree wavefunction for the ground state is just a product of identical one-body wavefunctions,Ψ(_1,⋯,_N) = ∏_i=1^Nψ(_i),with the one-body wavefunction ψ() determined by minimizing the expectation value of the Hamiltonian (subject to the normalization constraint ∫ d^2x|ψ()|^2 = 1). [ A better approximation would project this state onto vanishing center-of-mass momentum. However, such projection only affects Ø(1) contributions to the total energy of the state, which we neglect. ] The resulting ground state baryonic mass grows linearly with N and is given byE_ baryon/N = M_1/2() + λ/4π^2( ϵ̅-14 lnλ m_q/4π^2) + Ø(1/N),whereϵ̅≡min_ψϵ[ψ],ϵ[ψ] = 𝒯[ψ] + 𝒱[ψ].Here, 𝒯[ψ]≡∫ d^2 |∇ψ()|^2 / 𝒩[ψ],𝒱[ψ]≡∫ d^2d^2' ln|-'| |ψ()|^2 |ψ(')|^2 / 𝒩[ψ]^2, with 𝒩[ψ] ≡∫ d^2 |ψ()|^2. The ground state wavefunction which minimizes ϵ[ψ] satisfies the Hartree equation,[ -∇^2 + U() ] ψ() = λ ψ(),with the self-consistent potentialU() ≡∫ d^2' ln|-'| |ψ(')|^2 / 𝒩[ψ].This wavefunction is guaranteed to be nodeless, and hence is spherically symmetric, ψ() = ψ(||). After angular averaging of the logarithm, the potential (<ref>) becomes a convolution with the radial Green's function,U(||) ≡∫_0^∞ r' dr' ln(max(||,r')) ψ(r')^2 / ∫_0^∞ r' dr' ψ(r')^2 . We minimize the functional ϵ[ψ] numerically, using pseudospectral methods <cit.>. We write ψ(r) = e^-μ r/2 f(r) and then represent the function f as an order M-1 polynomial determined by its values { f_k } on the Gauss-Laguerre grid points { r_k } which are the roots of the Laguerre polynomial L_M(μ r). This is equivalent to, but much more computationally convenient than using the coefficients { c_k } in the orthogonal polynomialexpansion f(r) = ∑_k=0^M-1 c_k L_k(μ r). The radial integrals in expressions (<ref>)–(<ref>) are evaluated using M-point Gauss-Laguerre quadrature. Radial derivatives become dense M× M matrices acting on the M-component vector f⃗≡ ( f_k ), and the Hartree equation (<ref>) becomes an M-dimensional linear eigenvalue equation. Starting with a simple pure exponential initial guess for ψ(r), we compute the Hartree potential (<ref>), solve for the lowest eigenvalue of the Hartree equation (<ref>), and iterate these two steps until convergence. [ Demanding stationarity of ϵ[ψ] under a rescaling ψ() →ξψ(ξ) at ξ = 1 shows that 𝒯[ψ] =14 at extrema of ϵ. This is the analogue of the usual virial theorem for our logarithmic potential. Choosing the scale μ = √(2) in our spectral representation gives our initial guess this correct value of 𝒯. ]Due to the non-analyticity in the Green's function (<ref>), the truncation error only falls with increasing basis size as Ø(1/M). Six points suffice for 5% accuracy, thirty points yield better than 1%, and several hundred are needed to achieve 0.1% accuracy. For large M, the spectral matrices become quite ill-conditioned and extended precision arithmetic with roughly 2M digits is needed to avoid precision loss. A very stable extrapolation in 1/M yields the result,ϵ̅= 0.449558. The degeneracy of this lightest baryon level, before taking into account splittings due to higher order radiative corrections, is 4^N, growing exponentially as N increases. (For each quark, there is one factor of two for the choice of spin and another factor of two from the compact momentum k = ±.)To compare our N=2 and N ≫ 1 results for ground state baryons in a coupling independent fashion, consider the binding energy scaled by N-1, with the exactly known λlnλ contribution removed,δ E_ binding(N) ≡1/N-1[ E_ baryon - N M_1/2() ] + λ/16π^2lnλ/8π^2 .Our results,δ E_ binding(∞)/δ E_ binding(2) = ϵ̅/ (ϵ_00 + ln 2) = 1.0298,show stunningly little dependence on N. It would be interesting to see if this near-constancy is a coincidence, or remains true for other values of N.At large N, the probability density to find a quark at positionrelative to the baryon center of mass equals the square of the Hartree single particle wavefunction, p() = |ψ()|^2. To compare this with the corresponding distribution in N = 2 ground state baryons, recall that the Hamiltonian for relative motion (<ref>) was expressed in terms of the separation between constituents, so the corresponding distribution relative to the center of mass is p() = 4|ψ_ rel(2)|^2. One finds that the single particle distribution is more highly concentrated at N = 2 than at N = ∞. The mean square deviations differ by just about a factor of two,⟨^2 ⟩ = 8π^2/λ^2 × 1.0907, N = 2; 2.0294, N = ∞.Fig. <ref> compares the N = ∞ single particle radial probability density || p() with the corresponding N = 2 distribution when distance is rescaled by a factor of √(2), that is || p(/√(2)). As one sees from the figure, with this rescaling the two radial distributions are very similar. Above the baryon ground state level there is a manifold of vibrationally excited baryon levels. For N ≫ 1, energy levels in which a small number of quarks are excited may be computed using a product wavefunction with a few of the factors in the ground state wavefunction (<ref>) replaced by excited single particle wavefunctions. Low lying levels with a single excited quark may be labeled by the number of radial nodes n and orbital angular momentum ℓ of the excited quark, and have excitation energiesΔ E_n,ℓ = λ/4π^2( λ_n,ℓ - λ_0,0) ,where λ_n,ℓ is an eigenvalue of the Hartree equation (<ref>) containing the mean field generated by all the unexcited quarks. The subtraction of λ_0,0 accounts for the decrease in the number of quarks in the lowest single particle level. Table <ref> lists the eigenvalues λ_n,ℓ for the lowest few levels. Excitation energies to baryon levels with multiple excited quarks are, up to 1/N corrections, just the sum of the individual excitation energies (provided the number of excited quarks is a negligible fraction of N). Lastly, in the same manner discussed above for mesons, it is also possible to form exotic baryons containing N quarks plus one or more W-bosons. For the bound state to be non-decomposable into baryon-glueball molecules, no partial sum of the W-boson momentum indices should vanish. Such states can be progressively built from non-exotic baryons by replacing a quark with a quark plus one or more W-boson(s) which collectively have the same Cartan charge and compact momentum as the removed quark. One example of such a state is shown in Fig. <ref>. By suitably repeating this process one may, for example, build baryons in which all N quarks have the same color while N-1 W-bosons mediate attractive interactions between these quarks.§ DECAY PROCESSES Higher order perturbative interactions turn most of the hadronic states discussed in the previous sections into narrow resonances. Examining the systematics of the various decay processes is our next topic. First, however, we detail those states which cannot decay.§.§ Stable statesIn the light sector of the quarkless theory, individual dual photons are exactly stable. Each dual photon has a non-zero center charge p = 1,⋯,N-1, and is the lightest state with that value of center charge. [ Recall that a p=0 dual photon was artificially added to the light sector effective theory (<ref>) to simplify the presentation, but this extra degree of freedom exactly decouples from all physical degrees of freedom. The physical particles of the SU(N) gauge theory do not include a p = 0 dual photon. ] To see this, note that the mass formula (<ref>) is a subadditive function of the center charge, m_p_1+p_2 < m_p_1 + m_p_2. This implies that any splitting of a dual photon into two or more photons with the same total center charge is kinematically forbidden. The formation of k-body light sector bound states discussed in Sec. <ref> does not affect this conclusion, as the k-body binding energies are exponentially small compared to the relevant differences in photon masses.The two-body bound state of dual photons with center charges 1 and N-1, whose binding energy is given by Eq. (<ref>), is the lightest center charge zero excitation and is likewise exactly stable.If θ = 0 then the theory is CP invariant. [ This paragraph assumes that N ≥ 3. Because SU(2) is pseudo-real, charge conjugation is a distinct symmetry in SU(N) pure YM theory only for N > 2. ] Individual dual photons are CP odd. The lightest CP even states with non-zero center charge p are bound states of two dual photons with charges q and p-q and minimal total mass M_p = min_q (m_q + m_p-q). Specifically, these are the (q,p-q) bound states withq =1, p = 2,⋯,⌊N/2⌋, ; N-1, p = ⌊N+1/2⌋ ,⋯,N-2, .Similarly, the lightest CP odd state with vanishing center charge is a bound state of three dual photons with charges ( 1,1,N-2 ) (or their conjugates). These bound states are necessarily stable at θ = 0. Moreover, the charged two particle bound states (<ref>) remain absolutely stable at θ 0 for purely kinematic reasons. These bound states are heavier than a single dual photon of the same total center charge, but are lighter than all other multiparticle bound states of the given charge, and hence have no allowed decay channels which can conserve both energy and momentum.Turning now to the theory with quarks, as discussed in Sec. <ref> with ≤ N massless quark flavors, -1 of the dual photons become exactly massless and are the Goldstone bosons of spontaneously broken U(1)^-1_A symmetry. When =N, this means all N-1 dual photons are massless. These massless Goldstone bosons are stable.In the heavy sector, exactly stable states are those protected by conservation of the U(1)^_Vflavor charges (<ref>) and/or compact momentum (<ref>). With =N, mesons composed of a quark and antiquark having the minimal mass, m_q = m_q̅ =, and opposite compact momentum indices, k_q = -k_q̅ = ±, have flavor charges (+1,-1) under two different U(1) flavor subgroups and non-vanishing total compact momentum P_3 = ±. Such mesons (with vanishing vibrational and rotational excitations) are the lightest states with these flavor quantum numbers, and hence are stable. [ More precisely, such mesons with opposite spins and total S_z = 0 are stable. As noted in Sec. <ref>, hyperfine interactions shift the S_z = ± 1 mesons up in energy relative to the S_z = 0 states. A light S_z = ± 1 meson can decay to its corresponding S_z = 0 partner via emission of a dual photon — the QCD analog of 21 cm radiation from hydrogen. ] These mesons are the small-L avatars of charged pions and kaons (in the chiral limit).Baryons (or antibaryons) composed of N quarks (or antiquarks) all with mass m_q = are the lightest states with non-vanishing baryon number, and a subset of these states (those with minimal energy after including hyperfine interactions) are stable. Whether there are additional bound, and hence stable, di-baryons or higher multi-baryon states is an interesting open question.Whether the heavy photons created by our EFT operators ϕ⃗_± N^ aa are stable is also an interesting open question. These states have P_3 = ± N and tree-level mass M = N. This is the same value of P_3 and the same tree-level mass as a flavor singlet meson containing a quark and antiquark with k_q = -k_q̅ = ± N/2, or of a collection of N lightest mesons each with identical values of P_3 = ± 1 and flavor charges summing to zero, or a variety of other “near-extremal” flavor singlet multi-constituent states. Whether heavy photons decay into flavor singlet mesons, or collections of flavored mesons, or vice-versa, depends on which of these near-extremal states have the lowest energy. To determine this one must, at a minimum, take into account the leading Ø(λ) perturbative energy shifts. These include the binding energies computed in Sec. <ref> for two-body mesons. But Ø(λ) energy shifts also include corrections to the tree-level constituent rest masses. Evaluation of such corrections requires an improved one-loop matching of the EFT parameters to the underlying 4D gauge theory, and this matching calculation has not yet been completed. Consequently, we are not yet able to determine which transitions among near-extremal states are kinematically allowed. §.§ Light sector resonancesLight sector bound states other than those discussed above (which are stable due to the absence of any symmetry and kinematically allowed decay channels) will decay via emission of one of more dual photons. Such decays are induced by the cubic and higher order terms in the expansion of the effective Lagrangian (<ref>) about its minimum. The relative decay widths of all of these states are doubly exponentially small. Not only are the non-linear couplings within the dual photon sector (<ref>) exponentially small, Ø(e^-4π^2/λ), more importantly the binding momentum (<ref>) is so tiny that the probability for two constituents of a bound state to be within a Compton wavelength of each other is comparable to the relative binding energy (<ref>). Consequently, the logarithm of the relative decay width is exponentially large and negative,-ln (Γ/m_γ) = Ø(e^4π^2/λ)(neglecting powers of λ). We have not attempted to compute any such decays quantitatively.§.§ Heavy sector resonances The primary decay processes for heavy sector resonances are direct analogues of familiar processes in QED and atomic physics: radiative decays and particle-antiparticle annihilations. The key differences are the reduced dimensionality, additional conserved quantities (compact momentum and center charge), and multiple U(1) gauge groups.There are also more unusual decay processes involving splitting or joining of W-boson constituents within hadrons. These include, in particular, transitions among “near-extremal” states whose tree-level masses are identical. As noted above, understanding such processes requires a higher order determination of rest masses in the non-relativistic EFT. We leave explorations of such transitions to future work, and focus here on radiative and annihilation processes, specifically in two-body states.§.§.§ Radiative decays The relevant photon momenta for radiative decays will be in the range m_γ≪ p ≪, so the non-perturbative physics of the light sector may be wholly ignored and photons treated as massless. Excitation energies of low-lying heavy sector states are Ø(λ). Photons of such energies have wavelengths parametrically large compared to the characteristic Ø(λ^-1/2 ^-1) size of these states. Consequently, the usual multipole expansion of the photon field applies. The fastest radiative decays will be electric dipole transitions. Adapting the standard logic for hydrogenic decays to our 2D multi-photon situation, one finds that the total dipole transition rate from some initial state |I⟩ to lower energy final states {|F⟩} is given byΓ_ tot = π2 κ∑_F Δ E_IF^2 | ⟨ F || I ⟩|^2 ,where Δ E_IF≡ E_I - E_F and κ equals to the strength of the logarithmic potential binding the constituents, so κ=λ/(2π^2) for glueballs and (1-1/N) λ/(4π^2) for mesons.Parametrically, dipole decay rates for low-lying states are Ø(λ^2 ). To obtain quantitative results, including state dependence, one must evaluate the precise dipole matrix elements. We evaluated these matrix elements, for level numbers n up to 100, using radial wavefunctions computed using pseudo-spectral methods (as briefly described in footnote <ref> and Sec. <ref>), with up to several hundred grid points. Figure <ref> shows the resulting total dipole decay rates, in units of κ^2/m (with m the reduced mass of the two-body bound state), for orbital angular momentum ℓ = 0, 1, 2 and 4. As seen in the figure, decay rates at fixed ℓ grow with increasing level number n and appear to asymptote to a finite limit. At fixed level number n, decay rates also grow with increasing ℓ, and quickly appear to reach a limiting value. Our numerical results are consistent with a limiting value of π/4 κ^2/m in either case, with subleading Ø(1/ℓ) corrections if ℓ increases at fixed n, and Ø(n^-1/2) corrections if n increases at fixed ℓ, although this inverse power of n is not well-constrained by our data on the first 100 levels.Consider states with positive orbital angular momentum, ℓ > 0. The interleaving of energy levels, ϵ_n,|ℓ| < ϵ_n,|ℓ|+1 < ϵ_n+1,|ℓ|, implies that the |0,ℓ⟩ minimal energy states (for a given angular momentum) decay down to the |0,0⟩ ground level by sequential |0,ℓ⟩→ |0,ℓ-1⟩ transitions, with each emitted photon carrying off one unit of angular momentum. States with non-zero angular momentum and non-minimal energy, n > 0 and ℓ > 0, have multiple possible dipole allowed final states, including both Δℓ+1 and Δℓ-1 transitions. Examining transition rates to specific final states, one finds that the total decay rates for states with n,ℓ > 0 are highly dominated by decays to the nearest lower levels, either |n,ℓ⟩→ |n,ℓ-1⟩ or |n,ℓ⟩→ |n-1,ℓ+1⟩. Of these two decay channels, the decay decreasing ℓ is significantly more likely than the decay increasing ℓ. All other decays channels are smaller by one or more orders of magnitude. (The predominance of transitions decreasing |ℓ| over those increasing |ℓ| is visible in Fig. <ref> as the smaller values of the ℓ 0 points compared to ℓ 1.) Consequently, an excited state |n,ℓ⟩ with n ≫ 1 will cascade stepwise down to n0, with ℓ undergoing a random walk biased toward ℓ = 0.For high angular momentum, ℓ≫ 1, one may regard the n0 eigenstate as a quasiclassical circular orbit. In two dimensions, the power radiated by an electric dipole of magnitude eR rotating at frequency ω isP =18 e^2 R^2ω^3.For our high-ℓ bound states with e^2 = 2πκ, R = ℓ (κ m)^-1/2, and orbital frequency ω = κ /ℓ, this gives P = π/4 κ^3 / (m ℓ). The power radiated must equal the photon frequency times the decay rate, so this classical result implies an ℓ-independent asymptotic decay rate,Γ = π4 κ^2 / m ×( 1 + Ø(ℓ^-1) ).Decay rates from states with fixed n nicely converge to this value as ℓ increases. §.§.§ Annihilation decay In addition to radiative decays, two-body bound states having ℓ = 0 and composed of particle-antiparticle pairs can annihilate into two or more light sector photons. This is a short-distance process, represented by higher dimension operators in our non-relativistic EFT. Annihilation rates are parametrically smaller than dipole-allowed radiative transition rates, and hence only significant for the lowest ℓ =0 energy levels. Constituents with masses of orderhave Compton wavelengths which are comparable (for small N), or larger (for large N), than the compactification size L. Consequently, annihilation rates are most easily calculated using a dimensionally reduced relativistic EFT, having the form (<ref>) for W-boson bound states or 2+1 dimensional QED for mesons. The annihilation rate may be expressed asΓ_ annih = (lim_v→ 0 σ v) |ψ(0)|^2,where σ v is the flux-weighted cross-section in two spatial dimensions (a quantity with dimensions of length) and ψ() is the wavefunction for relative motion, so |ψ(0)|^2 is the 2D probability density for coincident constituents. Parametrically, σ v ∼λ^2/ for CP even states which can annihilate to two photons having momenta of order , while |ψ(0)|^2 ∼κ m since this is the inverse mean square size of the lowest ℓ = 0 two-body bound states. HenceΓ_ annih = Ø(λ^3 ),which is one power of λ smaller than radiative decay rates.Evaluating the cross section in the relativistic 2+1D relativistic EFT, we find [ We consider decays from bound states with vanishing total compact momentum and equal mass constituents. Higher KK modes (i.e., heavy photons) may be neglected. For WW annihilation, each W-boson couples to two different U(1) photons and consequently there are three different processes which contribute (γ_A γ_A, γ_Bγ_B, and γ_Aγ_B). Evaluating the leading order seagull, t, and u-channel diagrams and taking the non-relativistic limit yields the result shown. ]σ_WW→2γ = 11 π/64 v κ^2/m^3[ 1 + Ø(𝐩^2/m^2 ) ]for annihilation of W-bosons with mass m and interaction strength κ = λ / (2π^2), andσ_qq̅→2γ = 5π/128vκ^2/m^3[ 1 + Ø(𝐩^2/m^2) ]for qq̅ annihilation with mass m and interaction strength κ = (1-1/N)λ/(4π^2).For the lowest n = ℓ = 0 level of our two-body logarithmic quantum mechanics, the probability at the origin is|ψ(0)|^2 = 2.68915κ mwith m the reduced mass of the two constituents. Consequently, for the lightest CP-even glueballs and mesons (with constituent masses equal toand , respectively) we findΓ_ annih = 5.80815 (λ/4π^2)^3 , 0.660017 (λ/4π^2)^3 ( 1- 1N )^3 ,§ DISCUSSION §.§ Adiabatic continuation Recent studies have shown that it is possible to engineer circle compactifications of 4D SU(N) YM theory and QCD in such a way that symmetry realizations for large and small circle sizes coincide <cit.>. Available evidence is consistent with the natural conjecture that the weakly coupled small-L regime is smoothly connected — that is, without intervening phase transitions — to the strongly coupled large-L regime. The small circle regime offers a rare luxury: controlled analytic calculations in a phase of the theory with confinement and chiral symmetry breaking. Taking advantage of this tractability, we have studied the behavior of glueballs, mesons, and baryons, with a focus on the spectrum of resonances and their decays.Our results are broadly consistent with the conjecture of continuity between small and large L.Much physics in adiabatically compactified theories depends on the circle size L through the parameter η = N LΛ. To place our small η results into perspective, first recall that whenη≫ 1, the dynamics of QCD-like theories are insensitive to the scale L. (Finite volume effects vanish at least as fast as L^-2.) With fundamental representation fermions (≲ N) with a common mass m_q ≪Λ, at large L there are multiple characteristic scales for the masses of particles: the pseudo-Nambu-Goldstone (pNGB) mass scale m_pNGB∼√(m_q Λ), the glueball and meson mass scale m_M∼Λ, and the baryon mass scale m_B∼ N Λ. [ If ≪ N and m_q/Λ≪ 1/N≪ 1, then there is an additional scale Λ N^-1/2 associated with the mass of the η' meson <cit.>. ]In the weakly coupled regime η≪ 1, we find a similar picture, but with particle masses depending on L through the combination η = NLΛ. In adiabatically compactified QCD with, e.g., = N earlier work <cit.> found that the pNGB masses lie in the range m_pNGB∼[ Ø(1/N) Ø(1) ] ×η√(m_q Λ) at small η (if double trace deformations stabilize the color-flavor center symmetry). Our results in this paper show explicitly that m_M∼Λη^-1 andm_B∼ N Λη^-1. This is clearly similar to the large L pattern, apart from the natural appearance of dependence on the parameter η when L is small. [ The dependence of pNGB masses (<ref>) at small L on the charge of the particle under cyclic flavor permutations may, at first sight, seem surprising. But such dependence is also present when L is large but finite in adiabatically compactified theories with flavor twisted boundary conditions, as seen explicitly in the results of Ref. <cit.>. ] In Fig. <ref> we sketch a possible simple interpolation of the spectra of light and heavy states as L is varied.The situation at 0 is depicted in Fig. <ref>. At small L, instead of light pNGB mesons there are now light glueball states involving dual photons and their bound states, with masses m_ light∼[ Ø(1/N) Ø(1) ] ×Λ η^5/6 (if double trace deformations stabilize center symmetry). The N-1 dual photons are charged under the center symmetry, indicating that they are topologically non-trivial excitations containing flux wrapping the compactified direction. These states cannot be created by topologically trivial local operators (acting on the vacuum) and will have masses which do not asymptote to finite limits at large L but rather grow linearly, m ∼σ L, with σ the decompactified YM string tension. The bound state of two dual photons with vanishing total center charge is the lightest topologically trivial glueball at small L, and can smoothly connect to the lightest glueball at large L. In the heavy sector at small L, W-boson bound states form nearly degenerate multiplets containing all values of center charge. Within each such multiplet, the vanishing center charge state can evolve into an ordinary topologically trivial glueball at large L, while the remaining states with non-zero center charge will have linearly diverging masses at large L.Finally, when 1 ≤ < N, the overall picture is the same as the sketch shown in Fig. <ref>, except that the light sector at small L now contains -1 pseudo-Nambu-Goldstone bosons, with masses vanishing at m_q = 0, as well as non-pNGB states, namely the remaining N- dual photons and their bound states. These non-pNGB states have masses on the order of m_ light∼Λ η^b with an exponent b > 0 depending on /N. Whether these states should be described as glueballs or mesons, or some admixture, is not clear. There is no symmetry which clearly delineates a distinction. It should be possible to clarify the situation by computing the amplitudes with which these states are created by local fermion bilinears or Polyakov loop operators, but such an analysis has not yet been performed. In any case, these states can smoothly evolve into ordinary glueballs and mesons as L →∞. The same is true of the glueballs and mesons in the heavy sector at small L. Due to string breaking by dynamical quarks, none of these states will have masses which diverge as L →∞. §.§ Large N behavior Our analysis has been carried out with N arbitrary but fixed. The usual large N limit involves sending N to infinity while holding fixed the 't Hooft coupling λ (or equivalently the strong scale Λ). If the compactification size L is also held fixed, then the large N limit takes the compactified theory out of the regime η = NLΛ≪ 1 where a weak coupling analysis is possible and into the strongly coupled domain, η≫ 1, where large N volume independence applies <cit.>. Our small η analysis adds nothing to the understanding of this limit.However, it is interesting to consider an alternate N →∞ limit in which η = NLΛ is held fixed. This is the key parameter which controls the physics of adiabatically compactified QCD-like theories. Viewing Λ as a fixed physical scale, fixing η requires reducing the compactification size as N increases, L ∝ 1/N, or equivalently holding fixed ≡ 2π/(NL). If η is fixed at a small value, then a weak coupling analysis remains valid for all N. §.§.§ Heavy sectorStarting with the heavy non-relativistic sector, our results show that the glueball and meson spectra remain stable as N →∞ (regardless of whether = N, or ≪ N). For example, the value of N simply never enters the result (<ref>) for two-body glueball binding energies, while the only N dependence in meson binding energies (<ref>) comes from the quark-antiquark interaction strength proportional to 1-1/N. So masses of glueballs and mesons become N-independent at large N. The lightest baryon masses, as one would expect, grow linearly with N, but (based only on results at N 2 and N ≫ 1) the ground state baryon binding energy per quark (<ref>) and the shape of the single particle distribution (Fig. <ref>) are quite insensitive to N.Similarly, the only N dependence in the glueball and meson radiative decay (<ref>) and annihilation rates (<ref>) arises from the same 1-1/N interaction strength factor for mesons. Given this, one might guess that glueball and meson scattering amplitudes would also have finite, non-zero large N limits — but this is not entirely correct. If one ignores higher order radiative corrections then, for example, two-body mesons (at small η) may be labeled by the Cartan charge of their constituents. The amplitude for the elastic scattering process M^a + M^b → M^a + M^b arising from the exchange of one or more Cartan photons will include a trivial factor of δ_ab expressing the fact that both mesons must contain constituents charged under the same U(1) factor if they are to scatter via photon exchange. When radiative corrections are included, the actual mass eigenstates are linear combinations of the fixed Cartan charge states which (for = N) have definite center charge (or more precisely, definite color-flavor center charge, as discussed in Ref. <cit.>), M^p = N^-1/2∑_a ω^-a p M^a. The resulting scattering amplitude for M^p + M^q →M^p' + M^q', is Ø(1/N) for all center charges satisfying p+q=p'+q', instead of Ø(1) for coinciding Cartan charges and zero otherwise.The same argument applies to glueballs. Consider, for simplicity, glueballs which are bound states of two W-bosons, with either 0 or N (so the compactified theory has either an ordinary, or intertwined color-flavor center symmetry). As discussed in Sec. <ref>, glueballs in our small η regime, before diagonalizing center symmetry, may be labeled by a single Cartan index plus the ordered compact momenta of their W-boson constituents. (Subsequent Cartan indices are determined by the mass formula (<ref>), which in turn is a consequence of the adjoint Higgs mechanism operative at small η.) The transformation to a mass eigenstate basis with definite center charge involves exactly the same discrete Fourier transform as for mesons, G^p = N^-1/2∑_a ω^-a pG^a. The resulting 2↔ 2 scattering amplitude for G^p + G^q →G^p' + G^q' is suppressed by 1/N for all center charges satisfying p+q=p'+q'.More generally, scattering amplitudes at small η involving K external particles (incoming plus outgoing) scale as Ø(N^1-1/2 K). This holds for processes involving any combination of light dual photons and heavy sector bound states (either mesons or glueballs) with Ø(1) constituents, provided at least one of the particles in the scattering process is a heavy sector bound state. (Scattering involving only dual photons is discussed below.) This relation shows that decay amplitudes into two particle final states are Ø(N^-1/2), so decay rates to exclusive two particle final states are suppressed by 1/N. That may appear inconsistent with the Ø(1) total radiative and annihilation rates computed in Sec. <ref>, but inclusive decay rates sum over all accessible final states. Because the splittings between states with differing center charge are parametrically smaller than heavy sector binding or rest energies (by powers of λ for heavy states, or m_γ/ for light dual photons), inclusive 1 → 2 decay rates pick up a factor of N from summing over all possible center charges of the final state particles consistent with the initial state center charge. [ This assumes the decay channel is not parametrically close to threshold, so that the decay kinematics is insensitive to the splittings between final state particles with differing center charges. ] The same logic shows that while fully exclusive 2 ↔ 2 scattering rates are Ø(N^-2), inclusive 2 ↔ 2 scattering rates for mesons and glueballs are Ø(N^-1) as N →∞.Meson-baryon scattering amplitudes scale as Ø(N^0), since a quark (or antiquark) with any given Cartan index can interact with the quark having the same Cartan index in the baryon. The same scaling holds for glueball-baryon scattering (for both heavy sector bound state glueballs, and light dual photons). Baryon-baryon scattering amplitudes are Ø(N), since every quark in one baryon can interact via an unbroken U(1) gauge group with one of the quarks in the other baryon.These large N scaling relations at small η may be compared with conventional large N behavior when Λ and L are held fixed, and hence η→∞. It will be interesting to compare with conventional behavior in both the 't Hooft ( fixed as N→∞) and Veneziano (/N fixed as N →∞) limits.In all cases, meson and glueball spectra are stable as N →∞, while the lightest baryon masses grow linearly with N. One unusual consequence of our flavor-twisted boundary conditions, at small η, is that baryons composed of only a single flavor of quark (or more generally Ø(1) different flavors) have masses which grow quadratically with N.In the standard 't Hooft large N limit, glueball scattering amplitudes scale as Ø(N^2-K_g), with K_g the number of external glueballs (incoming plus outgoing) <cit.>. For processes involving mesons, possibly with additional glueballs, the scaling of scattering amplitudes becomes Ø(N^1-K_g -1/2 K_m), where K_m is the number of external mesons. Hence, meson decay widths are Ø(N^-1) and glueball decay widths to either two glueball, or two meson final states are Ø(N^-2). Rates for two glueballs to scatter into two glueballs, or into two mesons, are Ø(N^-4), while 2 ↔ 2 meson scattering rates are Ø(N^-2). Baryon-baryon scattering amplitudes are Ø(N) while baryon-meson scattering amplitudes are Ø(1) <cit.>.In the Veneziano large N limit, the additional factors of ∝ N in sums over final states (assuming a common quark mass for all flavors) make both meson and glueball decay rates Ø(1). Hence, except for the lightest states in each symmetry channel, mesons and glueballs remain resonances, with finite lifetimes, as N →∞. The inclusive rate for two mesons to scatter into two mesons is Ø(N^-1), while two glueballs can scatter into two mesons with an Ø(N^-2) inclusive rate, parametrically faster than the Ø(N^-4) rate for pure glueball scattering.Comparing these conventional large η scaling relations with our small η results, one sees that for mesons our Ø(N^0) total decay rates, Ø(N^-1) inclusive two particle scattering rates, and Ø(N^-2) exclusive two particle rates all coincide with the behavior of mesons in the Veneziano limit. The scaling of our baryon-baryon and baryon-meson scatteringamplitudes is the same as in conventional large N limits. But the fact that, at small η, glueball processes have the same large N scaling as mesons is quite peculiar.Two significant features contribute to this change in behavior of glueball dynamics between large and small η. First is the adjoint Higgs mechanism induced by the center-symmetric holonomy at small η. This suppresses fluctuations in off-diagonal components of the SU(N) gauge field, so that only the N-1 gluonic degrees of freedom play a singificant role in resonance formation, scattering, and decay. In contrast, at large η there are huge fluctuations in the holonomy and all N^2 gluonic degrees of freedom contribute to every glueball operator. This leads to the familiar 1/N^2 suppression factors in exclusive decay rates and 2 → 2 scattering amplitudes of glueballs. A second essential difference at large and small η is the contribution of states with non-zero center charge. At = 0, such states have linearly diverging energy as η→∞ (as shown in Fig. <ref>), and play no role in scattering processes involving Ø(N^0) energies. But at small η these topologically non-trivial states become nearly degenerate in energy with vanishing center charge states, and dominate inclusive scattering and decay rates at large N. §.§.§ Light sectorTurning now to the light sector, when ≪ N, the smallest non-zero dual photon mass is Ø(m_γ/N).Holding η fixed as N →∞ implies that the light scale m_γ is also held fixed. Consequently, the lightest (non-Goldstone boson) mass vanishes as N →∞.The interpretation and consequences of the vanishing of the mass of the lightest non-Goldstone boson excitations in the small-η large N limit were the focus of Ref. <cit.>. At very low energies, small compared to m_γ, the theory does not flow to a trivial fixed point. Rather, to all orders in the semi-classical expansion the low energy theory becomes gapless as N →∞. The low energy dynamics at N ∞ is most naturally written as a four-dimensional theory, despite the fact that the “parent” UV theory was compactified on a tiny circle. The fourth dimension in the low energy, large N dynamics is emergent, appearing only on length scales large compared to m_γ^-1.The results in this work are consistent with this picture, but do not shed much additional light on the origin or interpretation of this unexpected phenomena. The quartic interactions of dual photons, shown in Eq. (<ref>), may be interpreted in the large N emergent dimension description as momentum-dependent interactions with vertex factors proportional to 1/N times the product of photon momenta in the emergent dimension. Consequently, for Ø(N^0) momenta (in the original spatial dimensions), dual photon scattering amplitudes scale as Ø(N^-1) at large N, the same as for heavy sector glueballs.As shown in Eqs. (<ref>) and (<ref>), the dual photon binding energies (and momenta) discussed in Sec. <ref> vanish exponentially as N →∞. So these bound states play no significant role at large N, and the emergence of the extra dimension in the light sector of the theory happens just as described in Ref. <cit.>. To understand how, e.g., the glueballs arising from W-boson bound states fit into the large N emergent dimension picture, recall that the emergent dimension appears as an N-site discretized circle with lattice spacing m_γ^-1 <cit.>. A continuum 4D description is only relevant for physics with momenta small compared to m_γ. But at small η, the Ø() W-boson masses, their Ø( λ) binding energies, and the Ø(λ^2 ) radiative corrections to binding energies are all large compared to m_γ. So the large N bound state dynamics does not involve the low energy emergent dimension, and must be treated using a 3D effective field theory, as done in the present paper. §.§ Outlook The analysis and results of this paper raise a number of questions which would be interesting to study in future work. First, as noted near the end of Sec. <ref>, we have not performed the matching calculation necessary to determine the Ø(λ) corrections to the rest mass parameters of the 3D non-relativistic EFT. Differences in the short distance corrections to the EFT rest masses are needed to determine the relative stability of meson, glueball, and heavy photon resonances whose leading order masses are identical. For example, the lightest glueball resonances with mass near 4 might be composed of two W-bosons each with (tree level) mass 2, or from four of the lightest W-bosons each with mass . Such glueball states are nearly degenerate with heavy photons having a tree-level mass of 4. The results of a one-loop matching calculation of EFT rest energies would enable one to determine the relative ordering of these states. In particular, this would allow one to answer the interesting question of which near-extremal states are absolutely stable by virtue of minimizing the ratio of mass to compact momentum, M/|P_3|.Second, as emphasized in Sec. <ref>, the bound state spectra for glueballs, mesons, and baryons have an exponentially rising (Hagedorn) density of states. It is interesting that this Hagedorn scaling emerges as a consequence of a logarithmic potential within the domain of validity of non-relativistic quantum mechanics, in contrast to the common lore that Hagedorn scaling is characteristic of relativistic string dynamics. In any case, the implications of Hagedorn scaling in the density of states for the thermodynamics of adiabatically compactified QCD deserve further study. Previous work <cit.> considered the SU(2) deformed Yang-Mills theory (see also Refs. <cit.>), and argued that a thermal confinement-deconfinement transition occurs near the temperatureβ^-1_c ≃λ/ 4 π^2 .The picture behind this conclusion is that in the regime [ This temperature range is similar to, but slightly more restrictive than the condition for the validity of our non-relativistic EFT analysis, and is needed to justify the treatment ofthe monopole-instanton gas as two-dimensional. ] ζ^1/3≪β^-1≪, the dilute monopole-instanton gas representation of the 3D Euclidean vacuum effectively reduces to a dilute two-dimensional gas of magnetically charged particles subject to binary logarithmic interactions. At the same time, there is also a thermal gas of electrically charged particles, namely W-bosons. The thermal phase transition is believed to be driven by a competition between the effects of these electrically and magnetically interacting gases. However, in Refs. <cit.> the electrically-charged component of the gas was treated classically, and the existence of Hagedorn behavior in the density of states was not taken into account. It would be interesting to revisit these calculations in light of our results here, and clarify whether the temperature (<ref>) is indeed a correct estimate of the phase transition temperature.Next, it would be very interesting if lattice gauge theory simulations could be performed in both pure Yang-Mills and QCD exploring the cross-over regions in Figs. <ref> and <ref>, along the lines of Refs. <cit.>. This would require simulations in a variety of lattice volumes with one dimension having double trace center stabilizing terms and flavor-twisted boundary conditions on quarks.Last, and perhaps most interesting from a phenomenological perspective, is the possibility of studying multi-baryon states at small L. To motivate this, recall that in the real world there is a wide separation between “nuclear” excitation scales relevant in multi-baryon systems and the energy scales characteristic of single baryons. For example, the saturation binding energy per nucleon of nuclear matter, roughly 14 MeV, is tiny compared to the ≈ 300 MeV energy required to excite a single nucleon beyond its ground state. Or, one may compare nuclear binding scales to nucleon masses of nearly a GeV. Both comparisons indicate a wide separation between nuclear and single-baryon energy scales. Moreover, lattice simulations indicate that the nuclear/hadronic scale separation persists even as quark masses are varied <cit.>, and that it also persists when N2 instead of 3 <cit.>, suggesting that this scale separation is robust feature of QCD. This scale separation is vital for essentially all phenomenological understanding nuclear physics, including the modeling of nuclei as a collection of individual nucleons.The puzzle is that there is no fundamental explanation for this important experimentally-observed scale separation from QCD. For example, this scale separation is not an automatic consequence of either the large N or chiral limits. The adiabatic small-L regime allows one to use straightforward numerical and analytic methods to study multi-baryon systems for any quark mass and any number of colors. Further exploration of QCD phenomenology on a small circle may thus yield useful insights into the long-standing and important puzzle of the separation between nuclear and hadronic energy scales in QCD. We are grateful to M. Ünsal for helpful discussions. This work was supported, in part, by the U. S. Department of Energy via grants DE-FG02-00ER-41132 (A.C.) andDE-SC0011637 (K.A. & L.Y.) and by a Discovery Grant of the National Science and Engineering Research Council of Canada (E.P.). L. Yaffe thanks the University of Regensburg and the Alexander von Humboldt foundation for their generous support and hospitality during completion of this work.§ NON-RELATIVISTIC EFT DERIVATION We denote SU(N) indices by a,b,c,d, etc., each running from 1 to N, and define the set of N× N basis matrices {E^ab} by (E^ab)_cd≡δ_ac δ_bd. We use an N-dimensional basis for the root vectors β_ab (a b). The positive roots are β_ab = (0,...,0,1,0,...,0,-1,0,...,0), a < b, with 1 and -1 in the a-th and b-th position, respectively; the negative roots are β_ba = - β_ab, a < b. The indices μ = 0,1,2 denote the noncompact spacetime directions and x^3 ≡ x^3 + L is the coordinate of the compact direction. The circumference L ≡ 2 π R. The Cartan generators are denoted by H^a≡ E^aa. The overall U(1) photon coupling to ∑_a H^a decouples from the SU(N) dynamics and is introduced solely for the convenience of working with an N-dimensional weight basis. Since all weight vectors are orthogonal to the vector (1,1,1,1,...,1), the static interactions discussed below in Appendix <ref> only involve SU(N) charges which are neutral with respect to this overall U(1).Until otherwise specified [just before Eq. (<ref>)], we write Euclidean space expressions in this appendix. The Yang-Mills Lagrangian L = N/4 λtrF_αβ^2, with F_αβ Hermitian. The 't Hooft coupling λ≡ N g^2(), where the scale ≡ 1/(NR) denotes the lightest W-boson mass.Wedecompose the gauge field into components along the compact and noncompact directions, A_3=∑_1≤ a ≤ N A_3^a(x^μ) H^a, A_μ =∑_1≤ a ≤ N A_μ^a (x^μ, x^3) H^a+∑_1 ≤ a < b ≤ NW_μ^ab(x^μ, x^3) E^ab + W_μ^ab *(x^μ, x^3)E^ba . The expansion (<ref>) is written in the unitary gauge, where the only nonzero gauge field components along the S^1 direction are the Cartan components and they have no x^3-dependence. The N real fields A_μ^a describe 3D photons in the Cartan subalgebra, while the (N^2-N) complex fields W^ab_μ (a < b) in the off-diagonal elements describe charged W-bosons.Next, we expand A_3^a around thecenter symmetric expectation value (<ref>) of theholonomy, A_3^a ≡ρ^a/(N R) + ϕ^a, so that ϕ^a represents thefluctuations of the holonomy. [ Here, ρ^a =(N+1)-a are the components of the Weyl vector in our basis. The expectation value ⟨ A_3^a ⟩ = ρ^a/(NR) corresponds to ℤ_N symmetric eigenvalues of the holonomy and produces vanishing traces in the fundamental representation, ⟨ tr_F Ω^k ⟩= 0 for k=1,...,N-1. ] Plugging the expansion (<ref>) into the Yang-Mills Lagrangian one obtains, up to quadratic order in the W-boson fields,L_2W =N / 4 λ{∑_1 ≤ a ≤ NF_μν^a F^μνa +2∂_μϕ^a∂^μϕ^a +2∂_3 A_μ^a∂^3 A^μa + ∑_1 ≤ a < b ≤ N 2 |∂_μW_ν^ab + i ( A_μ^a- A_μ^b ) W_ν^ab - (μ↔ν) |^2 + 4 |( - i ∂_3 + a-bR N + ϕ_b -ϕ_a ) W_μ^ab|^2 + 2i ( F_μν^a-F_μν^b ) W_[μ^abW_ν]^ab *} . The second line shows explicitly that the W-boson field W^ab_μ has charge +1 and -1 under the a-th and b-th Cartan U(1) gauge groups, respectively.Hereafter, we neglect the fluctuations ϕ^a of the holonomy; as explained in Sec. <ref>, they play no role in the dynamics to the order that westudy. (These neutral fluctuations are gapped by the perturbative center-stabilization mechanism.)Next, we derive the leading-order EFT valid for momenta p ≪ (but large compared to the non-perturbatively induced mass gap (<ref>), p ≫ m_γ). This EFT describes the interactions of charged massive W-bosons with the (perturbatively) massless Cartan photons and with the “heavy photons," modes in the Kaluza-Klein (KK) tower containing the Cartan photon fields.As a final step before considering the p ≪ non-relativistic limit, we rewrite the Lagrangian (<ref>) in a mass (or KK) eigenstate basis. The KK expansions are defined as usual, e.g.,A_μ^a (x^ν, x^3) =∑_n = - ∞^∞e^ix^3 n /R A_μ^a, n(x^ν),with A_μ^a, -n(x^ν) = ( A_μ^a, n(x^ν))^*, and similarly for the W^ab_μ fields (without a corresponding reality condition). Inserting these expansions into the 4D Lagrangian (<ref>), integrating over x_3 (and neglecting holonomy fluctuations), leads to an effective three-dimensional LagrangianL^3D =L_2 +L_3+L_3^' +⋯ ,in which we separate, for convenience, quadratic, cubic, and higher order terms. The quadratic part is given byL_2=N L / 4 λ∑_n=-∞^∞( ∑_1 ≤ a ≤ N|F_μν^a, n|^2 + 2 |m_n^aa A_μ^a,n|^2 + ∑_1 ≤ a < b ≤ N 2 |∂_[μ W_ν]^ab,n|^2 + 4 |m_n^abW_μ^ab,n|^2 ),with the KK massesm^ab_n ≡ |a-b + n N| , ≡ (NR)^-1 .The cubic terms contain thecoupling of the Cartan photons to the charge currents of the W-bosons,L_3 =N L/ 4 λ∑_m,n=-∞^∞∑_1 ≤ a < b ≤ N 2 i ∂_[μ W_ν]^ab,n*(A_[μ^a,n-m - A_[μ^b, n-m) W_ν]^ab,m +,as well as theirmagnetic-moment coupling to the spin of the W-bosons,L_3^' =N L / 4 λ∑_m,n=-∞^∞∑_1 ≤ a < b ≤ N 2i ( F_μν^a,n-m -F_μν^b,n-m) W_[μ^ab,mW_ν]^ab,n* .Quartic terms in the Lagrangian, if needed, can be worked out similarly.We shall eventually return to our Lagrangian of interest, L^3D, but first we discuss the construction of a non-relativistic effective field theory (NR EFT) in the simpler case of a single massive charged vector boson. To this end, let W_μ denote a 3D complex vector field with U(1) gauge symmetry, W_μ→ e^i α W_μ, and Lagrangian [ At this point, we revert to Minkowski space expressions using a (-,+,+) metric signature.]L =-1 / 4 e^2 (∂_[μ A_ν])^2 -12|∂_[μW_ν]|^2 -M^2 W_μ W^μ* + i A^μ( W^ν * ∂_[μ W_ν] - W^ν ∂_[μ W_ν]^* ) - 12 |A_[μ W_ν]|^2 - i2 ∂^[μA^ν] W_[μW_ν]^*.This charged vector boson Lagrangian contains precisely the kinds of terms appearing in the Lagrangian (<ref>)–(<ref>) of our full theory. We usee^2 to denote the coupling constant of the massless photon. The leading-order correspondence with our full theory ise^2 =λ/ N L = λ/ 2 π .Note that the vector field W_μ has a conventional normalization, but we have chosen to scale the charge e out of covariant derivatives and define the photon field A_μ as having dimension 1.A 3D massive vector field has two polarization states. Define polarization vectors e_μ^i()̨, i=1,2, obeying e^i_μ()̨e^j μ()̨ = δ^ij and k^μe^i_ μ()̨ = 0, for on-shell momenta k_μ≡ (||̨,)̨. Explicitly,e^1_μ()̨ ≡(0,̨/ ||̨),e^2_μ()̨≡(||̨/ M, / ||̨ω_k / M),where ω_k ≡√(^̨2 + M^2) and ( ̨)_i ≡ϵ_ij ()̨_j (we use i,j=1,2 to denote spatial indices and take ϵ_12 = - ϵ_21=1). The free mode expansion of the second quantized field isW_μ(t, )= ∫d^2 k / (2π)^2√(2ω_k)∑_i=1^2 [ e^i(ω_k t - ·̨)e_μ^i()̨a^i()̨^† + e^- i(ω_k t - ·̨)e_μ^i()̨b^i()̨] , ≡ W_μ^+(t, ) + W_μ^-(t, ),where [a^i ()̨, a^j()^† ] = [b^i ()̨, b^j()^† ] = (2 π)^2δ^ij δ^2 (-)̨, and all other commutators vanish. It is convenient to denote by W^±_μ the positive frequency (∝ e^i ω_k t) and negative frequency (∝ e^- i ω_k t) parts, respectively. The U(1) charge operator Q ≡∫ d^2 x2 Im( W^ν*∂_0 W_ν), after normal ordering, becomes Q = ∫d^2 k / (2 π)^2∑_i[ a^i()̨^† a^i()̨ - b^i()̨^† b^i ()̨], from which it is evident that the operators a^i()̨^† (a^i()̨) are creation (annihilation) operators of positively charged vector bosons while the operators b^i()̨^† (b^i()̨) create (annihilate) negatively charged antiparticles. Polarization index i=1 (i=2) refers to particles with transverse (longitudinal) polarization, respectively. The free Hamiltonian P_0= ∫d^2 k / (2 π)^2ω_k ∑_i[ a^i()̨^† a^i ()̨ + b^i()̨^† b^i ()̨] and has eigenvalue ω_k for all four single-particle states of a given spatial momentum $̨.Apart from explaining the physical content of the massive vector boson theory, the mode expansion (<ref>) provides an easy way to see that an effective theory describing the dynamics of non-relativistic vector bosons can be expressed solely in terms of the spatial componentsW_iof the vector fieldW_μ. For small momenta,||̨ ≪M, the longitudinal polarization vector e^2_ μ()̨ = (0, / ||̨) + O(||̨ / M) , withonly spatial components to leading order. Since thetransverse polarization vectore_μ^1()̨is purely spatial, in the non-relativistic limit the time componentW_0can be eliminated, leading to an effective theory for a spatial vector field.One may construct the Lagrangian of this effective non-relativistic theory by writing all terms consistent with the symmetries and matching the coefficients to terms in the relativistic theory to the desired order in the small coupling and small momentum expansion (treating||/ M ∼||̨/ M ≪1, whereis a spatial gradient.) To carry out this procedure, we introduce two different two-component complex fields,ϕ⃗_+(t, )andϕ⃗_-(t, ). In a second-quantized non-relativistic theory, these fields (and their Hermitian conjugates) annihilate (or create) particles of charges+1and-1, respectively. The two-component vector represents the direction in the two-dimensional polarization space. To leading order in the derivative and small-coupling expansion, the fieldsϕ⃗_±can be considered as scalars underSO(2)spatial rotations, with an emergentSO(2)“flavor" symmetry acting as rotations in the polarization space. Magnetic moment interactions explicitly break thisSO(2) ×SO(2)symmetry down to the diagonalSO(2). (This is completely analogous to the approximate spin rotation symmetry in light atoms and molecules when spin-orbit interactions can be neglected.)Temporarily ignoring the gauge fieldA_μ, to lowest non-trivial order in powers of/ M, the Lagrangian of the NR EFT isL_NR = ϕ⃗_+^ † i∂_tϕ⃗_+ + ϕ⃗_-^ † i∂_tϕ⃗_- - M|ϕ⃗_+|^2- M|ϕ⃗_-|^2 - |ϕ⃗_+|^2/2 m - |ϕ⃗_-|^2/2 m .and the corresponding Hamiltonian isH = ∫ d^2 x ϕ⃗_+ ()^†·( M - ^2 2m) ϕ⃗_+() + ϕ⃗_- ()^†·( M - ^2 2m) ϕ⃗_-().The conserved charge Q = ∫d^2x(ϕ⃗_+)^† ·ϕ⃗_+ + (ϕ⃗_-)^† ·ϕ⃗_- . Mode expansions of the non-relativistic fields readϕ_+^i(t, )^† = ∫d^2 k / (2 π)^2e^i ε_k t - i ·̨a^i()̨^† , ϕ_-^i(t, )^† = ∫d^2 k / (2 π)^2e^i ε_k t- i ·̨b^i()̨^† ,where ε_k ≡M + ^̨2/(2m) , anda^i()̨^†andb^i()̨^†are the same creation operators appearing in the relativistic expansion (<ref>) (and its Hermitian conjugate). The fields (<ref>) satisfy non-relativistic canonical commutation relations, [ϕ^i_+(t, ), ϕ_+^j(t, )^†] = [ϕ^i_-(t, ), ϕ_-^j(t, )^†] = δ^ij δ^2(-) , with other commutators vanishing.To fix parameters in the NR EFT one demands that physical quantities, computed in the EFT and in the IR limit of the full theory, agree with each other order by order in the low energy and weak coupling expansions. At low orders, the matching is rather straightforward. In the free theory (<ref>), single-particle states have energiesε_=̨ M + ^̨2 /2 mand charges±1. This agrees with the energy and charge of low momentum states in the relativistic theory (<ref>) provided both the rest mass parameterM, and the kinetic massm, appearing in the non-relativistic theory (<ref>) equal, at lowest order, the physical massMof the original theory.Note that if one ignores the explicit polarization vector dependence in the relativistic expression (<ref>), then the operatorϕ_+^i(t,)^†corresponds, in the non-relativistic limit and after a trivial rescaling by√(2 M), to the positive-frequency partW^+_iofW_μ, whileϕ_-^i(t,)^†corresponds to the positive frequency part(W^-_i)^†of the conjugate fieldW_μ^†.We now proceed to write down the NR EFT Lagrangian describing the theory (<ref>) to leading order in the small-λand derivative expansion. We choose to work in Coulomb gauge for the photon fieldA_μ.The time componentA_0is not an independent field but is determined by the charge distribution of theW-bosons via Gauss' law. We denote the vector boson charge density byn(t, ) = i W^ν * ∂_[0 W_ν] - i W^ν ∂_[0 W_ν]^*,(neglecting higher order “seagull” contributions). Varying the action, the Lagrangian (<ref>) gives A_0 (t, ) = e^2 ∫d^2 y G(- ) n(t, ) , where the two-dimensional Laplacian Green's functionGwas defined in Eq. (<ref>). Using this result to eliminateA_0from the action, one obtains the Coulomb energy, V_C ≡-e^2/2 ∫d^2 x d^2 y n(t,) G(-) n(t, ) , as a contribution to (minus) the Lagrangian. Ignoring, for the moment, interactions mediated by spatial components of the photons as these are higher order in the non-relativistic limit, our effective theory (<ref>) changes, to leading order, only by the inclusion of the Coulomb energy in the action,S_NR = ∫ dt d^2 x ( ϕ⃗_+^ † i∂_tϕ⃗_+ + ϕ⃗_-^ † i∂_tϕ⃗_- - M|ϕ⃗_+|^2 - M|ϕ⃗_-|^2 - |ϕ⃗_+|^2/2 M - |ϕ⃗_-|^2/2 M)+ e^2/2∫ dt d^2 x d^2 y n(t, ) G(-)n(t, ),where n(t, ) = ϕ⃗_+ (t, )^†·ϕ⃗_+ (t, ) -ϕ⃗_-(t, )^†·ϕ⃗_- (t, ) is the non-relativistic limit of the vector boson charge density (<ref>). The corresponding Hamiltonian is justH= ∫ d^2 x ϕ⃗_+ ()^†·( M - ^2 2M) ϕ⃗_+() + ϕ⃗_- ()^†·( M - ^2 2M) ϕ⃗_-()- e^2/2∫ d^2 x d^2 y n() G(-) n().The action (<ref>) or Hamiltonian (<ref>) include all leading-order terms in the non-relativistic (v/c ≪1) limit using the systematic power counting rules that we discuss next. (In what follows,c≡1.)Higher order terms which can appear in the NR EFT may be classified and ordered using a suitable power counting scheme for the operators and their matrix elements, evaluated in characteristic bound states. [ These are determined by solving the two-particle Schrödinger equation which results from projecting the Hamiltonian (<ref>) into the two-particle Hilbert space. ] This approach is now well-established for 3+1 dimensional Coulombic systems <cit.>. Compared to such systems, several important differences arise in our 2+1D theory. The first is that a particle of massMmoving in a non-relativistic orbit due to a central forceF ∼e^2 /rmoves at speedv ∼e/√(M), for any orbit radius, rather thanv ∼e/√(Mr)as in a three dimensionalF ∼e^2/r^2central force field. The second is the appearance ofe^2 ln(e^2/M)terms, non-analytic in the coupling, in the ground state energy [as seen in Eq. (<ref>)], owing to the scaling properties of the logarithmic potential. Ignoring such logarithmic factors, the appropriate power counting is similar to that detailed in Ref. <cit.>: the size of bound states is of ordera_0 ∼(e^2 M)^-1/2and their characteristic binding energyΔE ∼e^2. For estimating the parametric dependence of matrix elements of arbitrary operators that may arise in the NR EFT Hamiltonian, evaluated in low-lying bound states of the lowest-order theory (<ref>), we take [ These power counting rules for ϕ⃗_± follow, e.g., by demanding that ∫ d^2x ϕ^†ϕ∼ 1 in a bound state of size a_0. For the remaining assignments, the arguments are the same as given in Ref. <cit.>; see also Ref. <cit.>. ] the fieldsϕ⃗_±to scale as√(e^2 M), time derivatives∂_t ∼e^2, spatial derivatives∇∼√(e^2 M), and Coulomb-gauge scalar and vector potentialse A_0 ∼e^2ande 𝐀 ∼e^4 / √(e^2 M). Thus, the field strengths scale as e 𝐄 ∼e^2 √(e^2 M)ande B ∼e^4. (Here, and below, we have rescaled the Maxwell action for the photon bye^2, to give the gauge field a conventional perturbative normalization.) Using these parametric estimates, it follows that all terms in the lowest-order NR Hamiltonian (<ref>), excepting the rest-energy terms, are of ordere^2, as required.To assess the relative importance of higher order terms, we begin withthe magnetic moment coupling of the vector bosons, the last term in the NR Lagrangian (<ref>). Writing the leading terms consistent with the symmetries of the theory which couple the field strength tensorF_ijto the NR vector fieldsϕ^iandϕ^i †, one finds, to leading order in1/M, that there is a unique such term,L_ NR^ mag = - i / 2 Me F_ij ( ϕ^i†_+ ϕ^j_+ - ϕ^i†_- ϕ^j_- ) ,whose coefficient follows by matching to the relativistic form (<ref>) using relations (<ref>), (<ref>), and (<ref>). The above power counting rules show that magnetic moment interactions will shift bound state energy levels by an amount of ordere^4/M, or a relativeØ(e^2/M)correction to binding energies.Given our original choice (<ref>) of polarization vectors, the NR fieldsϕ_±^i,i=1,2annihilate vector bosons which are linearly polarized, either transverse or parallel to their momenta, respectively. However, using operators that create particles in eigenstates ofS_z, the spin of the vector boson fieldS_z = ∫d^2 x (ϵ_ij Ẇ_i W^*_j + h.c.), is typically more convenient when discussing bound states in a central potential. Such operators can be obtained by redefining our NR field operators as follows:ϕ_±^1 = 1 √(2)( ϕ_±^↑ + ϕ_±^↓), ϕ_±^1 † = 1 √(2)( ϕ_±^↑ † + ϕ_±^↓ †),ϕ_±^2= 1 √(2) i( ϕ_±^↑ - ϕ_±^↓), ϕ_±^2† = i √(2)( ϕ_±^↑ † - ϕ_±^↓ †).The new operatorsϕ_±^↑†andϕ_±^↑obey canonical commutation relations and create or destroy vector bosons withS_z = 1, similarly,ϕ_±^↓†andϕ_±^↓create or destroyS_z = -1states. Using these redefined fields, the magnetic moment coupling (<ref>) becomesL_ NR^ mag= -eB / 2 M ( ϕ_+^↑ †ϕ_+^↑ - ϕ_+^↓ †ϕ_+^↓- ϕ_-^↑ †ϕ_-^↑ + ϕ_-^↓ †ϕ_-^↓),showing that the magnetic moment couplingssplit, at ordere^4/M, the level degeneracy of↑↑and↓↓bound states. In particular, the magnetic moment interaction term (<ref>) leads to the spin-spin hyperfine interaction potential (local in 2D), discussed in Sec. <ref>.Coefficients of further operators in the EFT can be found by matching scattering amplitudes between the full and effective theories, as done in continuum NRQED in Ref. <cit.>. (See Ref. <cit.> for matching in NRQCD using lattice gauge theory.) The resulting terms are dimensionally reduced versions of ones listed in the above references and include, for example, e/M^2 [ C_1 ∇·𝐄 ϕ^j †_+ ϕ^j_+ + C_2 ( ∂_i E_j - 1/2 δ_ij∇·𝐄) ϕ^i †_+ ϕ^j_+ + ⋯] , whose coefficients can be found by matching scattering amplitudes in external static electric fields. The contribution of these operators to the bound state energies also scale ase^4/M. Additionally, there are a number of possible contact terms involving four non-relativistic fields, schematically of the forme^2 (ϕ^†ϕ)^2, that also contributeØ(e^4/M)energy shifts. There are, of course, also corrections arising from higher orders in the expansion of the relativistic dispersion relation of the formϕ_+^i † ∇^4 / M^3 ϕ^i_+ + ⋯. According to the power counting rules, these also contribute to bound state energies at ordere^4/M. We have not systematically enumerated all possible higher order terms in the NR EFT and leave their detailed study and matching for future work.To conclude this Appendix, we invite the reader to consider the transition from the non-relativistic effective theory (<ref>) for our toy single vector boson model (<ref>), to the effective theory (<ref>) describing our full theory (<ref>)–(<ref>). The transition from the toy NR EFT (<ref>) to our full EFT (<ref>) is largely one of bookkeeping due to the proliferation of fields in the full theory. In brief, in the NR EFT (<ref>), the fieldsϕ⃗^abwitha>bcorrespond to the fieldϕ⃗_+of the toy model, while the fieldsϕ⃗^abwitha<bcorrespond to theϕ⃗_-field of the single complex vector model. The charge densities (<ref>) are the multi-field generalizations of toy model charge density (<ref>). The Hamiltonian (<ref>) is easily seen to give rise to the complete form (<ref>) (with the same normal ordering issues discussed in Sec. <ref>).§ LIGHT SECTOR DETAILSWe start with the quadratic part (<ref>) of the 4D action to remind the reader about the 3D photon-scalar duality and the normalization of the dual photon field used in dual description (<ref>). The Cartan generators in the fundamental representation have eigenvalues given by theNweight vectors_A,A=1,...,N. In our basis and choice of normalization, the highest weight is_1 =(1-1/N, -1/N, ...,-1/N)and coincides with the first fundamental weight vector_1ofsu(N). Consider a static quark, or fundamental representation probe charge, placed at the origin ofℝ^2and having some weight vectorcharacterizing its color charge. This adds a source to the 3D (Minkowskian, c.f. footnote <ref>) Lagrangian for the static (KK indexn=0) Cartan components of the gauge field, - NL/4 λ F_μν^a F^μνa + A_0^a () ν^a δ^2() (where a sum onais implied, andν^ais thea-th component of the quark's weight).The resultingA_0^aequation ofmotion, NL/λ ∇^2 A_0^a() = ν^a δ^2() , implies Gauss' law, ∮_C dl n̂^i ( NL/ λ F_i0^a ) = ν^a , where the curveCencircles the origin (counterclockwise) andn̂is its outward normal. AnN-component dual photon fieldmay be introduced via the relation NL/ λ F_i0^a = 1/2π ϵ_ij ∂_jσ^a (withϵ_12 ≡1). The choice of coefficient ensures that ∮_C dl n̂^i ϵ_ij ∂_j σ^a = 2πν^a , i.e., the monodromy of the dual photon field is2πtimes the charge (the weight vector). To be consistent with probes in all fundamental representations, the dual photon field is defined to be periodic with a periodicity of2πtimes thesu(N)weight lattice, generated by the fundamental weights{ 2 π_A }. The2+1D Lorentz invariant form of the above duality relation is F_μν^a = λ/ 2πNL ϵ_μνλ ∂^λσ^a = λ/ 4 π^2 ϵ_μνλ ∂^λσ^a (withϵ_0ij≡-ϵ_ij). To implement the duality, we replace the Maxwell part of the quadratic action (<ref>) by - NL / 4 λ F_μν^a F^μνa + 1 / 4 π ϵ_μνλ F^μνa ∂^λσ^a . Treatingσ^aandF_μν^aas independent integration variables and integrating out the field strengthF_μν^a, the resulting kinetic term for the dual photon is λ/ 8 π^2 NL(∂_λσ^a)^2 = λ/ 16 π^3 (∂_λσ^a)^2 , as shown in the light sector action (<ref>).The Coulomb energyV_Cof two static charges with weights_1and_2, separated by a distancer, can also be obtained from the above expressions. One finds V_C = -λ/ 4 π^2(_1 ·_2) lnr . The weights forW-bosons are root vectors, and since roots have length two, the interaction energy of oppositely charged staticW-bosons isλ/ 2 π^2logr, as shown in Eqs. (<ref>) and (<ref>). For a fundamental quark and an antiquark of opposite weights, we have - _1 ·_2 = ·= 1 - 1/ N , hence they experience attraction of that strength, as shown in (<ref>). On the other hand, a quark with weight_1 =and antiquark with weight_2 = - ', with', experience repulsion since - _1 ·_2 = ·' =-1/ N , as shown in Fig. <ref>. Likewise, it follows that quarks (or antiquarks) of different weights attract with strength1/ N, as shown in Fig. <ref>.Finally, a magnetic monopole-instanton of magnetic chargeα(one of the affine roots), is represented in the dual description by insertions ofe^i ·(x)(x ∈ℝ^3). Hence the interaction action between two monopole-instantonsof charges_1and_2can be obtained as ⟨e^i _1 ·(x_1) e^i _2 ·(x_2) ⟩= exp[ - 2π^2/λ _1 ·_2 /|x_1 - x_2| ] , where the expectation value was calculated with the free field portion of the-field Lagrangian (<ref>). A remark relevant for the thermal case is that, when reduced to two dimensions, the corresponding correlator becomes e^ 4 π^2 T / λ _1 ·_2 ln(|x_1 - x_2| T) for|x_1 - x_2| ≫1/T. § SYMMETRY TRANSFORMATIONS Let us choose to work inA_3 = 0gauge, where the holonomyΩis an independent degree of freedom. RegardingA_μ(x)as anti-Hermitian, and viewing the quark fieldqas anN ×matrix of spinors, we will defineΩ_F = diag(ξ^1/2,ξ^3/2,⋯,ξ^N-1/2). Our boundary conditions (in both index-free and component forms) areA_μ(x_3+L)= ΩA_μ(x_3)Ω^† , A_μ(x_3+L)^ab ≃ω^a-bA_μ(x_3)^ab , q(x_3+L)= Ωq(x_3)Ω_F^† , q(x_3+L)^aA ≃ω^-N/2 +(a-1/2) ξ^-(A-1/2)q(x_3)^aA ,where≃means whenΩhas the form (<ref>). §.§ Mode expansions Suppose thatΩhas the form (<ref>) with negligible fluctuations, let≡(y_1,y_2)denote the non-compact spatial coordinates, and ignore interactions. Then:A_μ(t,,x_3)^ab = 1/L∑_n∈𝐙∫d^2p/(2π)^2 √(2 ω)[ e^-i (ω t - · - k_n^ab x_3) e^i_μ(p⃗)ϕ_i()_n^ab - e^i (ω t + · + k_n^ab x_3)e^i_μ(-p⃗)^*(ϕ_i(-)_-n^ba)^†],whereμ= 0,1,2, the compact momentum k_n^ab ≡2π/NL (a-b+nN) , the 3D spatial momentum p⃗ ≡(p_1,p_2,k_n^ab) , and the frequency ω≡(^2 + (k_n^ab)^2)^1/2 (with dependence on,n,aandbimplicit). The polarization vectors{ e_μ^i(p⃗)},i = 1,2, satisfy2+1D transversality,p^μe_μ^i = 0, withp^0 ≡ω. This expansion satisfies BCs (<ref>), anti-Hermiticity and transversality ofA_μ, and the 4D free wave equation□A_μ= 0.The corresponding mode expansion for the quarks isq(t,,x_3)^aA = 1/L∑_n∈𝐙+1/2∫d^2p/(2π)^2 √(2 ω) [ e^-i (ω t - · - k_n^aA x_3) u_s(p⃗) ψ_s()_n^aA + e^i (ω t + · + k_n^aA x_3) u_-s(-p⃗)(χ_s(-)_n^aA)^†],where the quark compact momentum is k_n^aA ≡2π/NL [(a-1/2)-N/(A-1/2)+ n N] , the 3D spatial momentump⃗ ≡(p_1,p_2,k_n^aA), and the frequency ω≡(^2 + (k_n^aA)^2)^1/2 . The free particle spinorsu_s(p⃗)have helicitys=±1and satisfyγ_αp^αu_s(p⃗) = 0withp^α≡(ω,p⃗). In a chiral basis, γ_0 ≡([ - 0 1;-1 0 ] ) , γ_i ≡([ 0 σ_i; σ_i 0 ] ) , γ_5 ≡-i γ_0γ_1γ_2γ_3 = ([ 1 - 0; 0-1 ] ) , one has u_+(p⃗) = ([ ξ_+(p̂); 0 ]) and u_-(p⃗) = ([ 0; ξ_-(p̂) ]) , whereξ_±(p̂)are two-component spinors satisfyingp̂ ·σ⃗ ξ_±(p̂) = ±ξ_±(p̂)with phase conventionξ_±(p̂)^* = ±i σ_2 ξ_∓(p̂). The free particle spinors satisfyγ_5 u_s(p⃗) = s u_s(p⃗)andu_s(p⃗)^* = C u_-s(p⃗)withC ≡i γ_5 γ_2andC^†γ^αC = (γ^α)^*. The above mode expansion satisfies the boundary conditions (<ref>) and the massless Dirac equationγ^α∂_αq = 0.The coordinate space EFT operators are just 2D spatial Fourier transforms of the momentum-space mode operators,ϕ⃗()_n^ab ≡∫d^2p/(2π)^2 e^i · ϕ⃗()_n^ab ,ψ_±()_n^aA ≡∫d^2p/(2π)^2 e^i · ψ_±()_n^aA ,χ_±()_n^aA ≡∫d^2p/(2π)^2 e^i · χ_±()_n^aA .§.§ Axial U(1)_A^ Letθ= diag(θ_1,⋯,θ_). The axial transformation is standard:q(x)→ e^iγ_5 θ q(x), q(x)^aA → e^iγ_5 θ_A q(x)^aA ,withγ_5 ≡(γ_5)^†. Non-invariance under the diagonalU(1)_Aonly appears in the non-perturbative light sector.This transformation is produced byψ_± ()_n^aA → e^± i θ_A ψ_± ()_n^aA ,χ_± ()_n^aA → e^± i θ_A χ_± ()_n^aA .Building two-component operators, ψ()_n^aA ≡([ ψ_+()_n^aA; ψ_-()_n^aA ]) and χ()_n^aA ≡([ χ_+()_n^aA; χ_-()_n^aA ]) , this transformation is equivalent toψ ()_n^aA → e^i θ_A σ_3 ψ ()_n^aA ,χ ()_n^aA → e^i θ_A σ_3 χ ()_n^aA .§.§ Charge conjugation Recall thatNis assumed odd. Combine the basic charge conjugation transformation,A_μ→A_μ^*, with global color and flavor permutationsVandV_F, respectively, chosen to preserve the form (<ref>) ofΩat theℤ_Nsymmetric minimum and the quark boundary conditions,V≡δ_a+b,N+1 = δ_a̅,b , V_F≡δ_A+B,+1 = δ_A̅,B ,wherea̅ ≡N+1-a,A̅ ≡+1-A. Note that Ω^* = V ΩV^†and Ω_F^* = V_F Ω_F V_F^†. The action of charge conjugation is Ω → V Ω^* V^†≃Ω , A_μ(x)→ V A_μ(x)^* V^† , A_μ(x)^ab → (A_μ(x)^a̅,b̅)^* = -A_μ(x)^b̅,a̅ , q(x)→ C (V q(x)^* V_F^†), q(x)^aA → C (q(x)^a̅A̅)^*,This transformation is produced byϕ⃗()_n^ab → -ϕ⃗()_n^b̅ a̅ ,ψ_s()_n^aA →χ_s()_-n^a̅A̅ ,χ_s()_n^aA →ψ_s()_-n^a̅A̅ .§.§ x_3 reflection Lety ≡(x^0,x^1,x^2,L-x^3)denote the reflected coordinates. Combine the basic reflection,A_μ(x) →A_μ(y)(recallA_3 ≡0), with the global color and flavor permutationsVandV_Fdefined above. Then the action ofx_3reflection isΩ → V Ω^† V^†≃Ω , A_μ(x)→ V A_μ(y) V^† , A_μ(x)^ab → A_μ(y)^a̅b̅ , q(x)→ R_3 (V q(y) V_F^†), q(x)^aA → R_3 q(y)^a̅A̅ ,whereR_3satisfiesR_3^†γ^αR_3 = (1-2δ^α_3) γ^αand in our chiral basisR_3 = γ_5γ_3. The free particle spinors satisfyR_3 u_s(p⃗ ') = s u_-s(p⃗ )wherep⃗ '≡(p_1,p_2,-p_3).This transformation is produced byϕ⃗()_n^ab →ϕ⃗()_-n^a̅b̅ ,ψ_s()_n^aA → -sψ_-s()_-n^a̅A̅ ,χ_s()_n^aA → sχ_-s()_-n^a̅A̅ .§.§ ℤ_N center Assume here that either=0, or= N. Combine the basicℤ_Ncenter transformation,Ω→ωΩ, with global color and flavor permutationsPandP_Fchosen to preserve the form (<ref>) ofΩand the quark boundary condition,P≡δ_a,b-1 ,P_F≡δ_A,B-1 ,with color and flavor indices regarded as defined moduloN. Note that P^†ΩP = ωΩwhenΩhas the form (<ref>), and similarly P_F^†Ω_F P_F = ωΩ_F .The action of aℤ_Ncenter transformation isΩ →ωP Ω P^†≃Ω , A_μ(x)→ P A_μ(x) P^† , A_μ(x)^ab → A_μ(x)^a-1,b-1 , q(x)→ P q(x) P_F^† , q(x)^aA → q(x)^a-1,A-1This transformation is produced byϕ⃗_n^ ab()→ϕ⃗_n-δ_a+δ_b^a-1,b-1(),ψ_n^aA()→ψ_n-δ_a+δ_A^a-1, A-1(),χ_n^aA()→χ_n-δ_a+δ_A^a-1, A-1(),whereδ_a = 1ifa = 1, otherwise 0. This is equivalent to relations (<ref>) and (<ref>).JHEP | http://arxiv.org/abs/1707.08971v1 | {
"authors": [
"Kyle Aitken",
"Aleksey Cherman",
"Erich Poppitz",
"Laurence G. Yaffe"
],
"categories": [
"hep-th",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170727180010",
"title": "QCD on a small circle"
} |
Card 𝒜 #1#1#1#1 #1#1#1#1 #1{[ .#1#1#1#1 εcc1 (c) ↓ ↑BW^ BW^ CBW^ CBW^ CBWCLW_2n^ CLW_2n^ CLW_2n RLW^(2n), →∙ RLW^(2n),∙→ RBW^∙→RBW^→∙W W^ W^ ^2n, ^2n, W^2nW^(2n),W^(2n),C^2n, C^2n, C^2nC^ C^ CCyl ^ ^ ^_2n ^_2n _2n _n^ ^ PathsCircleı𝐢angle#1#1 #1#1 #1(<ref>) 1 ł⇒h h nḇ B x y G #1ℤ/#1ℤℤ/nℤℤ/2nℤ Nb Rk Ext Slice Multinomial Mat Poisson Binomial#1#1 Trace Var Cov sign Pl uniform𝒞#1#2#3#4#5[ #1:#2 ⟶#3;#4 ⟼#5 ]#1#20pt#1#2#1#1 Proof. #1 Proof of #1 . ℝ/2πℤℚ/2πℤℝ/ℤℚ/ℤ=(d)i.i.d.r.v.æa.e.#1#1#1=0pt#1#1#1#1#2 0pt#1_#2#1#2 0pt#1^#2 DA⟶ x z C K Exp#1I_t_#1mod `E𝔼`Pℙ`Rℝ`eε`oω#1 14cm #1 lemLemma[section] pro[lem]Proposition theo[lem]Theorem cor[lem]Corollary conj[lem]Conjecture defi[lem]Definition OQ[lem]Open question noteNoteexm[lem]Examplerem[lem]Remarkmetath[lem]Meta-theorem Directed, cylindric and radial Brownian webs David CoupierUniv. Lille, CNRS, UMR 8524 - Laboratoire P. Painlevé, Département GIS - Polytech'Lille, F-59000 Lille, France; E-mail :Univ. de Valenciennes, CNRS, EA 4015 - LAMAV, F-59313 Valenciennes Cedex 9, France,Jean-François MarckertCNRS UMR 5800 - LaBRI, Univ. Bordeaux, 351 cours de la Libération, 33405 Talence Cedex-France; E-mail : ,Viet Chi TranUniv. Lille, CNRS, UMR 8524 - Laboratoire P. Painlevé, UFR de Mathématiques, F-59000 Lille, France; E-mail :Accepted, Received; in original form=================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== The Brownian web (BW) is a collection of coalescing Brownian paths (W_(x,t),(x,t) ∈ `R^2) indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.Keywords :Brownian web, navigation algorithm, random spanning forests, weak convergence of stochastic processes.AMS classification : Primary60J05, 60G52, 60J65, 60D05Secondary 60G57; 60E99. Acknowledgements : This work has benefitted from the GdR GeoSto 3477. J.F.M. has been partially funded by ANR GRAAL (ANR-14-CE25-0014) and D.C. by ANR PPPP (ANR-16-CE40-0016). D.C. and V.C.T. acknowledge support from Labex CEMPI (ANR-11-LABX-0007-01).§ INTRODUCTION The Brownian web, BW in the sequel, is a much fascinating object introduced in <cit.>. It is formed by a family of coalescing Brownian trajectories (W_x,t,(x,t)∈ℝ^2), roughly speaking starting at each point (x,t) of the plane ℝ^2 (we consider only 2D objects in this paper). For (x,t)∈^2,(W_x,t(s),s≥ t) x+ (B^(x,t)_s-t, s ≥ t)where B^(x,t) is a standard Brownian motion (BM) starting at 0 and indexed by (x,t). The trajectories started from two different points of the time-space ^2 are independent Brownian motions until they intersect and coalesce. The BW appears as the continuous limit of various discrete models of coalescing random walks and navigation schemes (e.g. <cit.>). Recently, radial (2D) counterparts of these discrete directed forests have been considered and naturally, attempts have been carried to obtain invariance principles for these objects and define a “radial Brownian web” (RBW; <cit.>). Nevertheless, the rescaling needed in the BW case is somehow incompatible with a “nice Brownian limit” in the radial case. For directed forests in the plane, time is accelerated by n^2 say, while space is renormalized by n, for a scaling parameter n→+∞. In the radial case, the “space and time” parameterizations are related by the fact that the circle perimeter is proportional to its radius. This hence prevents a renormalization with different powers of n (2 and 1 for n^2 and 1/n) unless we consider only local limits. The main idea of this paper is the creation of the cylindric Brownian web (CBW) that allows to involve the angular characteristic of the radial problems, while keeping a geometry close to the plane. The usual BW is indexed by ×, where the first component is the space component. The CBW is an object indexed by the cylinder=()×where the first componentis the circle. Topologically,somehow interpolates between the plane `R× `R and the plane equipped with the polar coordinate system ()× `R_+ suitable to encode a RBW, as we will see.Similarly to (<ref>), we can define the CBW ^=(^_(x,t), (x,t)∈) as the family of coalescing trajectories(^_x,t(s),s≥ t)ł(x+ B^(x,t)_s-t,s≥ t)̊1that is, independent Brownian motions taken modulo 1 which coalesce upon intersecting. Note that the time will be flowing upwards in the graphical representations of this paper, and hence the notation with the upward arrow. Later, dual objects will be defined with their inner time running downward. Also, to distinguish between planar and cylindrical objects, cylindrical objects will be denoted with bold letters. In Section <ref>, we recall the topological framework in which the (planar) BW as introduced by Fontes et al. <cit.> is defined. Many convergence results on the plane leading to the BW can be turned into convergence results on the cylinder with the CBW as limit since the mapproj`R`R / ℤxx 1is quite easy to handle and to understand. We recall some criteria established in the literature that allow to obtain the BW as limit of discrete directed forests. Then, we extend these results to the cylinder for the CBW. We show that the CBW can arise as the limit of a cylindrical lattice web that is the analogous of the coalescing random walks introduced by Arratia <cit.>. We end the section by showing different ways to project the CBW on the radial plane to obtain radial `Brownian' webs.In Section <ref>, the properties of the CBW are investigated. We show in particular that there is almost surely (a.s.) a unique bi-infinite branch in the CBW as well as in its dual, which is a main difference with the planar BW. Starting with a discrete lattice and taking the limit, we can characterize the joint distribution of these two infinite branches as the one of a pair of reflected Brownian motions modulo 1, in the spirit of Soucaliuc et al. <cit.>. We also prove that the coalescence time between two (or more) branches admits exponential moment, when its expectation in the plane is infinite. All these behaviors are closely related to the topology of the cylinder.In Sections <ref>and <ref>, we play with the convergences to the BW in the directed plane, to the CBW in the cylinder and to the RBW in the “radial” plane. In the plane, several examples of directed forests in addition to the coalescing random walks of Arratia are known to converge to the Brownian webs, for example <cit.>. Other radial trees such as the one introduced by Coletti and Valencia <cit.> are known to converge locally to Brownian webs. We consider the corresponding cylindrical forests and show that they converge to the CBW with a proper rescaling. For example, in Section <ref>, we propose a radial forest similar to the radial forest of <cit.>, built on a sequence of circles on which a Poisson processes are thrown. When carried to the cylinder, this amounts to throwing Poisson processes with different rates on circles of various heights. We show how the rates and heights can be chosen to have the convergence of the cylindrical forest to the CBW, which is carefully established by adapting well-known criteria (e.g. <cit.>) to the cylinder. The convergence for the latter model has its own interest: as the intensity of points increases with the height in the cylinder, the convergence is obtained for the shifted forests. It is classical in these proofs that the key ingredient for checking the convergence criteria amounts in proving estimates for the tail distribution of the coalescence time between two paths. In our case, this is achieved by using the links between planar and cylindrical models, and thanks to the Skorokhod embedding theorem which connects our problem to available estimates for Brownian motions. However we have to use clever stochastic dominations as well to obtain these estimates. Projecting the cylinder on the (radial) plane then provides a radial discrete forests which converges after normalisation to the radial Brownian web. This convergence is a global convergence, whereas only local convergences are considered in <cit.>. § CYLINDRIC AND RADIAL BROWNIAN WEB In this Section we introduce the cylindric Brownian web, several natural models of radial Brownian webs together with some related elements of topology, in particular, some convergence criteria. But we start with the definition of the standard BW given in <cit.>.§.§ The standard Brownian WebFollowing Fontes & al. <cit.> (see also Sun <cit.> and Schertzer et al. <cit.>), we consider the BW as a compact randomsubset of the set of continuous trajectories started from every space-time point of ^2=[-∞,∞]^2 equipped with the following distance ρρ((x_1,t_1),(x_2,t_2))= A(x_1,t_1)-A(x_2,t_2)_∞, where the map A is given by A^2[-1,1]^2(x,t)(Φ(x,t),Ψ(t))=ł(tanh(x)/1+|t|,tanh(t))̊.For t_0∈, C[t_0] denotes the set of functions f from [t_0,+∞] tosuch that Φ(f(t),t) is continuous. Further, the set of continuous paths started from every space-time points isΠ=⋃_t_0∈ C[t_0]×{t_0}. (f,t_0)∈Π represents a path starting at (f(t_0), t_0). For (f,t_0)∈Π, we denote by f̃ the function that coincides with f on [t_0,+∞]and which is constant equals f(t_0)on [-∞,t_0). The space Π is equipped with the distance d defined byd((f_1,t_1),(f_2,t_2))=ł(sup_tł|Φ(f̃_̃1̃(t),t)-Φ(f̃_̃2̃(t),t)|̊)̊∨ |Ψ(t_1)-Ψ(t_2)|.The distance depends on the starting points of the two elements of Π, as well as their global graphs. Further, the set H of compact subsets of (Π,d) is equipped with the d_ H Hausdorff metric (induced by d), and F_ H, the associatedBorel σ-field.The BW W=(W_x,t, (x,t)∈^2) is a random variable (r.v.) taking its values in ( H, F_ H). It can be seen as a collection of coalescing Brownian trajectories indexed by ^2. Its distribution is characterized by the following theorem due to Fontes & al. <cit.>: There exists an ( H, F_ H)-valued r.v. W whose distribution is uniquely determined by the following three properties.(o) From any point (x,t)∈`R^2, there is a.s. a unique path W_x,t from (x,t),(i) For any n≥ 1, any (x_1,t_1),…,(x_n,t_n), the W_x_i,t_i's are distributed as coalescing standard Brownian motions,(ii) For any (deterministic) dense countable subset D of `R^2, a.s., W is the closure in ( H,d_ H) of (W_x,t,(x,t)∈ D). In the literature, the BW arises as the natural limit for sequences of discrete forests constructed in the plane. Let χ be a family of trajectories in H. For t>0 and t_0,a,b∈ℝ with a<b, letη_χ(t_0,t; a,b) := { f(t_0+t)|(f,s) ∈χ, f(t_0)∈[a,b] }be the number of distinct points in ℝ×{t_0+t} that are touched by paths in χ which also touch some points in [a,b]×{t_0}. We also consider the number of distinct points in [a,b]×{t_0+t} which are touched by paths of χ born before t_0:η_χ(t_0,t; a,b) := { f(t_0+t) ∈ [a,b]|(f,s) ∈χ, s≤ t_0 } .Th. 6.5. in <cit.> gives a criterion for the convergence in distribution of sequences of r.v. of ( H, F_ H) to the BW, which are variations of the criteria initially proposed by <cit.>: Let (χ^n)_n≥ 1 be a sequence of ( H, F_ H)-valued r.v. which a.s. consists of non-crossing paths. If (χ^n)_n≥ 1 satisfies conditions (I), and either (B2) or (E) below, then χ^n converges in distribution to the standard BW.(I) For any dense countable subset 𝒟 of ^2 and for any deterministic y_1,⋯,y_m∈ D, there exists paths χ^n_y_1,…χ^n_y_m of χ^n which converge in distribution as n→ +∞ to coalescing Brownian motions started at y_1,…,y_m.(B2) For any t>0, as `e→ 0^+,`e^-1lim sup_n→+∞sup_(a,t_0)∈ `R^2( η_χ^n(t_0,t;a,a+`e) ≥ 3 ) → 0 .(E) For any limiting value χ of the sequence (χ^n)_n≥ 1, and for any t >0, t_0∈, a<b∈,( η_χ(t_0,t ; a,b)) ≤(η_W(t_0,t ; a,b)) ,where W denotes the BW. In this paper we focus on forests with non-crossing paths. But there also exist in the literature convergence results without this assumption: see Th. 6.2. or 6.3. in <cit.>. For forests with non-crossing paths, the condition (I) implies the tightness of (χ^n)_n≥ 1. The conditions (B2) or (E) somehow ensure that the limit does not contain `more paths' than the BW. In the literature, proofs of (B2) and (E) are both based on an estimate of the coalescence time of two given paths. However, condition (B2) is sometimes more difficult to check. It is often verified by applying FKG positive correlation inequality <cit.>, which turns out to be difficult to verify in some models. When the forest exhibits some Markov properties, it could be easier to check (E) as it is explained in <cit.> or <cit.>, Section 6.1. Let us give some details. The condition (E) mainly follows fromlim sup_n→ +∞( η_χ^n(t_0,ε ; a,b) ) < +∞ ,for any ε>0, t_0∈ and a<b∈, which can be understood as a coming-down from infinity property. Statement (<ref>) shows that for any limiting value χ, the set of points χ(t_0 ; t_0+ε)of ×{t_0+ε} that are hit by the paths of χ(t_0)–paths of χ born before time t_0–constitutes a locally finite set.Thus, condition (I) combined with the Markov property, implies that the paths of χ starting from χ(t_0 ; t_0+ε) are distributed as coalescing Brownian motions. Hence,(η_χ(t_0)(t_0,t ; a,b)) ≤(η_W(t_0+ε,t-ε ; a,b)) =b-a/√(π (t-ε))→ b-a/√(π t) = (η_W(t_0,t ; a,b))as ε→ 0 and (E) follows. For details about the identity (<ref>) see <cit.>.§.§ The Cylindric Brownian Web We propose to define the CBW =(_x,t, (x,t)∈) on a functional space similar to H so that the characterizations of the distributions and convergences in the cylinder are direct adaptations of their counterparts in the plane (when these counterparts exist! See discussion in Section <ref>). In particular, this will ensure that the convergences in the cylinder and in the plane can be deduced from each other provided some conditions on thecorresponding discrete forests are satisfied. The closed cylinder is the compact metric space =()×, for the metricρ_O((x_1,t_1),(x_2,t_2))= d_(x_1,x_2)∨ |Ψ(t_1)-Ψ(t_2)|where d_(x,y)= min{ |x-y|, 1-|x-y|} is the usual distance in . In the sequel, we use as often as possible the same notation for the CBW as for the planar BW, with an additional index O (as for example ρ and ρ_O).For t_0∈, the set C_O[t_0] denotes the set of continuous functions f from [t_0,+∞] to , and Π_Othe set ⋃_t_0∈ C_O[t_0]×{t_0}, where (f,t_0)∈Π_O represents a path starting at (f(t_0), t_0). For (f,t_0)∈Π_O, we denote by f̃ the function that coincides with f on [t_0,+∞]and which equals to f(t_0)on [-∞,t_0). On Π_O, define a distance d_O byd_O((f_1,t_1),(f_2,t_2))=ł(sup_t d_(f̃_̃1̃(t),f̃_̃2̃(t)))̊∨ |Ψ(t_1)-Ψ(t_2)|.Further, H_O, the set of compact subsets of (Π_O,d_O) is equipped with the d_ H_O Hausdorff metric (induced by d_O), and F_ H_O, the associatedBorel σ-field. The CBW is a r.v. taking its values in ( H_O, F_ H_O), and is characterized by the following theorem (similar to the Theo. 2.1. in Fontes & al. <cit.> for planar BW). There is an ( H_O, F_ H_O)-valued r.v.whose distribution is uniquely determined by the following three properties.(o) From any point (x,t)∈, there is a.s. a unique path _x,t from (x,t),(i) for any n≥ 1, any (x_1,t_1),…,(x_n,t_n) the joint distribution ofthe _x_i,t_i's is that of coalescing standard Brownian motions modulo 1,(ii) for any (deterministic) dense countable subset D of , a.s.,is the closure in ( H_O,d_ H_O) of (W^_x,t,(x,t)∈ D). As in the planar case, the CBWadmits a dual counterpart, denoted byand called the dual Cylindric Brownian Web. For details (in the planar case) the reader may refer to Section 2.3 in <cit.>. For any t_0∈, identifying each continuous functions f∈ C_O[t_0] with its graph as a subset of , f:=-f={(-x,-t) : (x,t)∈ f} defines a continuous path running backward in time and starting at time -t_0. Following the notations used in the forward context, let us define the set Π_O of such backward continuous paths (with all possible starting time), equipped with the metric d_O (the same as d_O but on Π_O). Further, H_O denotes the set of compact subsets of (Π_O,d_O) equipped with the Hausdorff metric induced by d_O. Theorem 2.4 of <cit.> admits the following cylindric version.There exists an H_O× H_O valued r.v. (,) called the double Cylindric Brownian Web, whose distribution is uniquely determined by the two following properties:(a) and - are both distributed as the CBW.(b) A.s. no path ofcrosses any path of .Moreover, the dual CBWis a.s. determined by(and vice versa) since for any point (x,t)∈,a.s. contains a single (backward) path starting from (x,t) which is the unique path in Π_O that does not cross any path in .For all -∞≤ t≤ t'<+∞, let us denote by F^_t,t' the σ-algebra generated by the CBWbetween time t and t':ℱ^_t,t' = σ( {{^_(x,s)(s') , t<s'≤ t' } , x∈ , t<s≤ t' }) . We write ℱ^_t' instead of ℱ^_-∞,t'. The CBW is Markov with respect to the filtration (ℱ^_t)_t∈`R and satisfies the strong Markov property, meaning that for any stopping time T a.s. finite, the process{{^_(x,T+t)(T+s) , s≥ t } , x∈ , t≥ 0 }is still a CBW restricted to the semi-cylinder ^+:=()× `R^+ which is independent of ∩_t>Tℱ^_t. In the same way, we can also define the σ-algebra F^_t,t', where t≥ t', with respect to the dual CBW .The convergence criteria <cit.> or Theorem <ref> above has hence a natural counterpart on the cylinder. For a,b ∈denote by [a→ b] the interval from a to b when turning around the circle counterclockwise, and by|a→ b| its Lebesgue measure (formally: for a<b, [a→ b] = [a,b] and if a>b, [a→ b]=[a,1]∪[0,b]). For X a r.v. in H_O, denote byη_X^O(t_0,t;[a→ b]) := { f(t_0+t) |(f,s)∈ X, f(t_0) ∈ [a→ b] }be the number of distinct points in ×{t_0+t} that are touched by paths in X which also touch some points in [a→ b]×{t_0}. We also setη_X^O(t_0,t ; [a→ b]) := { f(t_0+t) ∈ [a→ b]| (f,s)∈ X, s≤ t_0 } .Here is the counterpart of Theorem <ref>in the cylinder:Let (χ^n)_n≥ 1 be a sequence of ( H_O, F_ H_O)-valued r.v. which a.s. consist of non-crossing paths. If (χ^n)_n≥ 1 satisfies conditions (IO), and either (B2O) or (EO), then χ^n converges in distribution to the CBW .(IO) For any dense countable subset 𝒟 any deterministic y_1,⋯,y_m∈ D, there exists for every n≥ 1, paths χ^n_y_1…χ^n_y_m in χ^n such that χ^n_y_1…χ^n_y_m converge in distribution as n→ +∞ to coalescing Brownian motions modulo 1 started at y_1,…,y_m.(B2O) For any t>0, as `e→ 0^+,`e^-1lim sup_n→+∞sup_(a,t_0)∈( η^O_χ^n(t_0,t;[a→ a+`e 1]) ≥ 3 ) → 0 .(EO) For any limiting value χ of the sequence (χ^n)_n≥ 1, and for any t >0, t_0∈ and a,b∈,( η^O_χ(t_0,t ; [a→ b]) ) ≤( η^O_(t_0,t ; [a → b]) ) . This section ends with a summary of the relationships between η_W, η_W, η^O_ and η^O_ where W denotes the planar BW. First, in the plane, as noticed in <cit.> Section 2, η_W(t_0,t ; a,b) and η_W(t_0,t ; a,b)+1 are identically distributed. This can be shown using duality arguments. In the cylinder the situation is a little bit different: it is not difficult to show that, for t,t_0>0 and a,b∈,η^O_(t_0,t ; [a → b]) η^O_(t_0,t ; [a → b]) + _ ,where the eventmeans that the cylindric BMs starting from (a,t_0) and (b,t_0) are allowed to coalesce before time t_0+t but not from the side [b → a] (more precisely, |_(a,t_0)(s)→_(b,t_0)(s)| stays in [0,1) for s∈[t_0,t_0+t]).Moreover, for any t,t_0>0 and a,b∈ with |a → b|<1, we will prove at the end of the current section thatη^O_(t_0,t ; [a → b]) ≤_Sη_W(t_0,t ; a,b) ,where ≤_S stands for the stochastic domination. Statement (<ref>) traduces the following natural principle: trajectories merge easier in the cylinder than in the plane. However there is no stochastic comparison between η^O_ and η_W. Indeed, the expectation of η_W(t_0,t ; a,b) tends to 0 as t→∞ thanks to identity (<ref>) whereas this does not hold in the cylinder. Theorem <ref> (below) states the a.s. existence inof a bi-infinite path. So, for any t,t_0, η^O_(t_0,t ; [0 → 1]) is larger than 1 and, by rotational invariance,(η^O_(t_0,t ; [a → b])) = |a→ b|(η^O_(t_0,t ; [0 → 1])) ≥ |a→ b| .It then remains to prove eta^O<eta. Let us focus on the planar BW W restricted to the strip ×[t_0,t_0+t]. First, by continuity of trajectories, with probability at least 1-ε, there exists δ>0 such that sup_0≤ d≤δ|W_a,t_0(t_0+d)- W_b,t_0(t_0+d)|<1 (since |a-b|<1) where W_x,t denotes the BM starting at (x,t). The coming-down from infinity property satisfied by the BW ensures that the number of remaining BMs at level ×{t_0+δ} and starting from [a,b]×{t_0} is a.s. finite. Let κ be this (random) number. When defining a realization of the BW, we need to decide, in case of coalescence of two trajectories, which one survives. In order to compute η_W(t_0,t ; a,b) we label these remaining BMs by 1,…,κ from left to right and when two of them merge, the BM having the lower label is conserved while the other one is stopped. This stopping rule allows us to determine the set of labels of remaining BMs at level ×{t_0+t}, say ℒ, whose cardinality is η_W(t_0,t ; a,b). Now, let us complete the previous stopping rule as follows: if the BM with label 2≤ j≤κ meets the path 1+W_a,t_0 between times t_0+δ and t_0+t then it stops. Although 1+W_a,t_0 does not correspond to any trajectory in the planar BW W– and then appears as artificial –, it coincides with W_a,t_0 in the cylinder and then has label 1. According to this completed rule, we obtain a new set of labels of remaining BMs at level ×{t_0+t}. It is included in ℒ and its cardinality has the same distribution than η^O_(t_0,t ; [a → b]). In conclusion the previous construction allows us to bound from above η^O_(t_0,t ; [a → b]) by η_W(t_0,t ; a,b) on an event of probability at least 1-ε, for any ε>0.§.§ The Cylindric Lattice WebAs for the BW, the CBW can be constructed as the limit of a sequence of discrete directed forests on the cylinder. For any integer n≥ 1, define the “cylindric lattice” as := {(x,t), x ∈ℤ/2nℤ, t ∈ℤ, x-t 2 = 0 },and consider (ξ(w), w ∈) a collection of i.i.d. Rademacher r.v. associated with the vertices of . The cylindric lattice web (CLW) is the collection of random walks=ł(_w,w ∈)̊indexed by the vertices of , where for w=(x,t), lcl_(x,t)(t) = x _(x,t)(s) = _(x,t)(s-1) + ξ(_(x,t)(s-1),s-1)2n, s >t.The sequence of paths (_w,w∈) is equivalent to that introduced by Arratia <cit.> in the planar case. The union of the random paths ((_(x,t)(s),s),s≥ t) for (x,t)∈, coincides with the set of edges {(w,w+(ξ(w),1)), w ∈} (see Figure <ref>).The dualofis a reversed time CLW (and shifted by 1) defined on the “dual” of := {(x,t), x ∈, t ∈ℤ, x-t2 = 1 }.is the collection of random walks =ł(_w, w ∈)̊ indexed by the vertices ofsuch that for w=(x,t)∈, and using the same family (ξ(w), w ∈) as before: lcl_(x,t)(t) = x, _(x,t)(s) = _(x,t)(s+1)-ξ(_(x,t)(s+1)-(0,1)),s),fors ≤ t.We define, for any h∈ℤ, for any direction D∈{,}, the horizontal slice by_2n^D(h)= ^D_2n∩ł( ×{h})̊,so that the random walks (_w^2n,D,w ∈_2n^D(h)) start from the points of _2n^D(h). The normalized CLW and its dual are defined as follows. For D∈{ , } and for any (x,t) in _2n^D, set ^(2n),D_(x/2n,t/n^2)(s):= 1/2n_(x,t)^2n, Dł( 4n^2 s )̊ for s≥t/n^2D=,s≤t/n^2D=. Since ^(2n),D_(x,t)(4 n^2 s ) takes its values in , 2n is the right space normalization, which implies the time normalization as usual.The pair of renormalized CLW (^(2n),,^(2n),) converges in distribution to the pair of CBW (,).Let us first prove the convergence of the marginals. Sinceandhave the same distribution (up to a reversal of time and a shift by 1).To do it, we mainly refer to the proof of the convergence towards the (planar) BW of the sequence of lattice webs (W^(2n))_n≥ 1, obtained from normalizing the random walks on the grid ={(x,t)∈^2,x-t2 =0 } similarly to (<ref>): see <cit.> for further details. As for (I) the proof of (IO) is a basic consequence of the Donsker invariance principle and is omitted here. The same coupling argument used to prove eta^O<eta leads to the following stochastic domination: for n≥ 1, t_0∈, t>0, a∈ [0,2π] and ε>0,η^O_^(2n),(t_0,t ; [a→a+ε]) ≤_S η_W^(2n)(t_0,t ; a, a+ε) .Hence condition (B2) satisfied by the rescaled (planar) lattice web W^(2n) (see Section 6 in <cit.>) implies condition (B2O) for ^(2n),. Then Theorem <ref> applies and gives the convergence of (^(2n),)_n≥ 1 to .The convergence of the marginals implies that the distributions of {(,)}_n≥ 1 form a tight sequence in the set of measures on ℋ_O×ℋ_O. It then suffices to prove that any limiting value of this sequence, say (𝒳^,𝒳^), is distributed as the double CBW (,). To do it, we check the criteria of Theorem <ref>. Item (a) has already been proved. To check (b), let us assume by contradiction that with positive probability there exists a path π_z∈𝒳^ which crosses a path π̂_ẑ∈𝒳^.By definition of (𝒳^,𝒳^), this would lead to the existence, for n large enough and with positive probability, of a path ofcrossing a path of . This is forbidden since the lattice webs have non crossing paths. §.§ Radial Brownian Webs §.§.§ The standard Radial Brownian Web and its dualOur goal is now to define a family of coalescing paths, indexed by the distances of their starting points to the origin in ℝ^2, that we will callradial Brownian web. Let us start with some topological considerations. Our strategy consists in sending the semi-cylinder ^+:=()× `R^+ onto the plane equipped with the polar coordinate system ()× `R^+ by using the mapφ_⋆× `R^+() × `R^+(x, f_⋆(t))(2π x, t) ,where f_⋆(t):=t/(4π^2). The presence of factor 1/(4π^2) will be discussed below. Let(h)={ (x,h), x ∈},be the horizontal slice at height h of . For any t>0, φ_⋆ projects (f_⋆(t)) on (0,t):=×{t}. It also identifies (0) with the origin.The map φ_⋆ induces the metric ρ_∙ on the radial plane × `R^+byρ_∙((x_1,t_1),(x_2,t_2)) := ρ_O(φ_⋆^-1(x_1,t_1),φ_⋆^-1(x_2,t_2)),for any elements (x_1,t_1),(x_2,t_2)∈ ()× `R^+. Following the beginning of Section <ref>, we can construct a measurable space ( H_∙, F_ H_∙) equipped with the distance ρ_∙. Of course, the map φ_⋆ is continuous for the induced topology, so that the image of a (weak) converging sequence by φ_⋆ is a (weak) converging sequence. We callstandard in-radial Brownian web, and denote by , the image under φ_⋆ of the dual CBWrestricted to ^+. In particular, φ_⋆ sends the trajectory _x,f_⋆(t)(s) for s going from f_⋆(t) to 0 on the path _2π x,t(s) for s going from t to 0 where_2π x,t(s) := s expł( 2iπ_x,f_⋆(t)(f_⋆(s)) )̊ .Notice that the natural time of the trajectory _2π x,t is given by the distance to the origin, since the radius satisfies:| _2π x,t(s) | = s .The families of paths (_x,f_⋆(t), (x,f_⋆(t)) ∈^+) that coalesce on the cylinder when t evolves from +∞ to 0, are then sent on radial paths(_x,t,(t exp(i x) ∈ℂ)) that coalesce when they are approaching the origin 0.This is the reason whyis said in-radial, and the notation →∙ evokes the direction of the paths, “coalescing towards the origin”.Moreover, for any 1<s≤ t, φ_⋆ sends the part of cylinder delimited by times f_⋆(s-1)=(s-1)/(4π^2) and f_⋆(s)=s/(4π^2) (i.e. with height 1/(4π^2)) to the ring centered at the origin and delimited by radii s-1 and s (i.e. with width 1). Then, on the unit time interval [s-1;s], the increment of the argument of _2π x,t, i.e.2π_x,f_⋆(t)(f_⋆(s-1)) - 2π_x,f_⋆(t)(f_⋆(s))2πis distributed according to the standard BM at time 1 taken modulo 2π. This is the reason whyis said to be standard. As a consequence, the trajectory _x,t turnsa.s. a finite number of times around the origin.As the standard BW, the CBW and the in-radial Brownian web admit special points from which may start more than one trajectory and whose set a.s. has zero Lebesgue measure. See Section 2.5 in <cit.> for details. Except from these special points, the in-radial Brownian webcan be seen as a tree made up of all the paths {_x,t(s), 0≤ s ≤ t}, (x,t)∈× `R^+, and rooted at the origin. Its vertex set is the whole plane. Th. <ref> in the sequel also ensures that this tree contains only one semi-infinite branch with probability 1.Let us denote bythe image underφ_⋆ of the CBWrestricted to ^+. We call it thestandard out-radial Brownian web. The map φ_⋆ sends the trajectory {_x,f_⋆(t)(s), s ≥ f_⋆(t)} of the cylindric BM _x,f_⋆(t) starting at (x,f_⋆(t))∈^+ on the out-radial (continuous) path {_2π x,t(s), t≥ s} where_2π x,t(s) := s expł( 2iπ_x,f_⋆(t)(f_⋆(s)) )̊ .Unlike the in-radial path _x,t, _x,t is a semi-infinite path which moves away from the origin. Finally, the out-radial Brownian webappears as the dual of the in-radial Brownian web . Indeed, the CBWsandare dual in the sense that no trajectory ofcrosses a trajectory ofwith probability 1 (see the proof of Prop. <ref>). Clearly, the map φ_⋆ preserves this non-crossing property which then holds forand . Let us recall that ^(2n), and ^(2n), denote the normalized cylindric lattice webs obtained from ^2n, and ^2n,: see (<ref>). Let us respectively denote by _2π x,t and _2π x,t the radial lattice webs obtained as images under φ_⋆ of ^(2n), and ^(2n), restricted to ^+. Using the continuity of φ_⋆, it is possible to transfer the convergence result of Prop. <ref> from the cylinder to the plane. Then, the convergence result below is a direct consequence of Prop. <ref>. The pair (,) converges in distribution to the pair of standard radial Brownian webs (,). §.§.§ Other Radial Brownian WebsIn this section we explore different radial projections of the cylindric Brownian web (,) into the plane. Let us first describe the general setting. Let f be an increasing continuous function, defining a one-to-one correspondence from an interval I⊂ `R^+ onto an interval J⊂ `R. Define the bijective map φ_f by: φ_f× J× I(θ, f(t))(2πθ, t) .As previously, × I represents a subset of `R^2 (actually a ring) parametrized by polar coordinates. The map φ_f sends the restriction of the CBWto the part of cylinder × J on a radial object defined on the ring × I, denoted by f- and also called radial Brownian web. In this construction, the function f is a winding parameter. For instance, if 1,2∈ I, the argument variation (in ) around the origin of the f-_x,2 between radii 1 and 2 (where x∈[0,2π] is the initial argument) is a centered Gaussian r.v.with variance 4π^2(f(2)-f(1)). The standard radial Brownian web introduced in the previous section corresponds to the particular case I=`R^+, J=`R^+ and f(t)=t/(4π^2), for which the argument variation of a trajectory on a ring with width c is simply a Gaussian 𝒩(0,c). Our second example of maps f allows to project the complete pair (,) parametrized byto the plane. Let us consider the bijection from I=(0,+∞) onto J=`R defined by f(t):=ln t (or any other map f sending (0,+∞) onto ℝ). Then, the radial Brownian web f-– image ofby φ_f–presents an accumulation phenomenon in the neighborhood of the origin. Indeed, the argument variation around the origin between radii ε and 1 has distribution 𝒩(0,4π^2|ln(ε)|), and thus goes to +∞ in the neighborhood of 0 (ε→ 0^+) when it stays bounded in any other bounded ring far away from 0. Our third example of map f provides a tree – given by the trajectories of f-– having many semi-infinite branches with asymptotic directions. A semi-infinite branch ζ (if it exists) of the tree f- is said to admit an asymptotic direction θ∈ whenever (z_k)→θ, for any subsequence (z_k)⊂ζ such that |z_k|→∞. To show that f- admits many semi-infinite branches, let us consider the bijection f from I=`R^+ onto J=[0;1) defined by f(t):=2/πarctan t. For a small ε∈(0,1), the map φ_f projects the thin cylinder × [1-ε;1) on the unbounded set × [tan(π(1-ε)/2);+∞). On the (small) time interval [1-ε;1), the CBMs have small fluctuations, and then the tree f- admits semi-infinite branches with asymptotic directions. The next result proposes a complete description of the semi-infinite branches of f-.The standard radial Brownian web could appear a bit impetuous to the reader: the fluctuation of the argument along a trajectory parametrized by the modulus, being a BM mod 2π, the trajectories may have important fluctuations far from the origin. The choice f(t)=ln t of Example 2 provides a radial forest where the paths look like coalescing BMs locally and far from O: between radii r and r+1, the fluctuations are of variance 1/r. This model is invariant by inversion.Consider the f-RBW for a bijection f from I=_+ into an interval J with compact closure such that f can be extended continuously to (J). With the above notations, the following statements hold.*A.s. any semi-infinite branch of f- admits an asymptotic direction.*A.s. for any θ∈, the tree f- contains (at least) one semi-infinite branch with asymptotic direction θ.*For any (deterministic) θ∈, a.s. the tree f- contains only one semi-infinite branch with asymptotic direction θ.*A.s. there exists a countable dense set 𝒟⊂ such that, for any θ∈𝒟, the tree f- contains two semi-infinite branches with asymptotic direction θ.*A.s. the tree f- does not contain three semi-infinite branches with the same asymptotic direction.The first two items generally derive from the straight property of the considered tree: see Howard & Newman <cit.>. However, in the present context, it is not necessary to use such heavy method and we will prove them directly. For the sake of simplicity, we can assume that J=[0,1). Let us first consider a semi-infinite branch ζ of f-.By construction of the f-RBW, there exists a path γ of the CBW on × J such that ζ=φ_f(γ). The path γ of the CBW on × J is a Brownian motion that can be extended by continuity to × [0,1] by (θ̅,1), say, implying that the first coordinate of ζ converges to θ̅/(2π) when the radius tends to infinity. This means that the semi-infinite branch ζ admits θ̅ as asymptotic direction. The proof of the second item is in the same spirit.The key argument for the three last statements of Th. <ref> is the following. With probability 1, for any θ∈ and x:=θ/2π, the number of CBMs ofstarting at (x,1) is equal to the number of semi-infinite branches of f- having 2π x as asymptotic direction. With Th. <ref>(o), it then follows that the number of semi-infinite branches of f- having the deterministic asymptotic direction θ∈ is a.s. equal to 1. This key argument also makes a bridge between the (random) directions in which f- admits several semi-infinite branches and the special points of . Given t∈`R, Th. 3.14 of <cit.> describes the sets of points on the real line `R×{t} from which start respectively 2 and 3 BMs. The first one is dense and countable whereas the second one is empty, with probability 1. These local results also hold for the CBW(but we do not provide proofs).Cylinders may also be sent easily on spheres, by sending the horizontal slices h∈ (a,b) of the cylinder to the horizontal slice g(h) of the sphere {(x,y,h) ∈ℝ^3: x^2+y^2+g(h)^2=1}, where -∞≤ a < b ≤ +∞, and g is an increasing and bijective function from (a,b) to (-1,1). Somehow, sending cylinders onto the plane allows to contract one slice (or one end) of the cylinder, and sending it on the sphere amounts to contracting two slices (or the two ends) of the cylinder. Again, this point of view will provide a suitable definition for the spherical Brownian web and its dual. § ELEMENTS ON CYLINDRIC LATTICE AND BROWNIAN WEBS In this section, two differences between the CBW and its plane analogous are put forward. Firstly, each of CBWandcontains a.s. exactly one bi-infinite branch; this is Th. <ref>, the main result of this section. This property is an important difference with the planar BW which admits a.s. no bi-infinite path (see e.g. <cit.> in the discrete case). The distributions of these bi-infinite paths are identified by taking the limit of their discrete counterparts on the cylindric lattice web.Secondly, the coalescence time of all the Brownian motions starting at a given slice admits exponential moments (Prop. <ref>). This is also an important difference with the planar case, where the expectation of the coalescence time of two independent Brownian motions is infinite, which comes from the fact that the hitting time τ_1 of 0 by a Brownian motion starting at 1 is known to have distribution (τ_1∈ dt)=e^-1/(2t)/√(2π t^3). §.§ The bi-infinite branch of the CBWFor any x,x'∈^D, t∈ℝ, denote byT^(x,x',t) = infł{ s>t :^_(x,t)(s) = ^_(x',t)(s) }̊T^(x,x',t) = supł{ s<t :_(x,t)(s) = _(x',t)(s) }̊the coalescence times of the cylindric Brownian motions ^_(x,t) and ^_(x',t) one the one hand, and of ^_(x,t) and ^_(x',t) on the other hand. Set for D∈{, },T^D(t)= maxł{T^D(x,x',t), (x,t),(x',t)∈(t)}̊,the coalescence time of all the Brownian motions (going upward if D= and downward if D=) starting at (t). Consider a continuous function γ : ↦.We say that γ, or rather, its graph {(γ_t,t), t ∈} is a bi-infinite path of the CBW , if there exists an increasing sequence (t_k, k ∈ℤ) such that lim_k→ -∞ t_k=-∞,lim_k→ +∞ t_k=+∞, and a sequence (x_k, k ∈ℤ) such that for any k∈ℤ,^_(x_k,t_k)(t_k+1)=x_k+1, and ^_(x_k,t_k)(s)=γ_s for s∈[t_k,t_k+1]. Similarly, we say that{(γ_t,t), t ∈} is a bi-infinite path of the CBW ^, if there exists an decreasing sequence (t_k, k ∈ℤ) such that lim_k→ -∞ t_k=+∞,lim_k→ +∞ t_k=-∞ and a sequence (x_k, k ∈ℤ) such that for any k∈ℤ, _(x_k,t_k)(t_k+1)=x_k+1, and _(x_k,t_k)(s)=γ_s for s∈[t_k+1,t_k]. With probability 1, any two branches of the CBWeventually coalesce. Furthermore, with probability 1, the CBWcontains exactly one bi-infinite branch (denoted ).A notion of semi-infinite branch is inherited from the cylinder via the map φ_⋆: The standard out-radial Brownian webpossesses a unique semi-infinite branch.The first statement is a consequence of the recurrence of the linear BM.Let us introduce some stopping times for the filtration F^. First let τ^,1=T^(0) (the coalescing time of the CBW ^ coming from (0) in the dual), and successively, going back in the past,τ^,k=T^(τ^,k-1). Since the primal and dual paths do not cross a.s., it may be checked that all primal Brownian motion ^_(x,τ^,k) for x ∈(τ^,k) have a common abscissa, say x'_k-1 at time τ^,k-1, that is in (τ^,k-1). In other words, they merge before time τ^,k-1. A simple picture shows that at x'_k-1, the dual ^ has two outgoing paths, and thus the primal ^_(x'_k-1,τ^,k-1) is a.s. a single path (see e.g. <cit.>, and use the fact that the special points of the CBW are clearly the same as those of the BW). We have treated the negative part of the bi-infinite path. The positive path is easier, since a bi-infinite path must coincide with the trajectory ^_(x'_0,0) for its part indexed by positive numbers. As a consequence, the sequence defined by : – for k≥ 0 by t_-k=τ^,k, x_-k=x'_k, – for k≥ 1 by t_k=k, x_k=W^_(x_k-1,t_k-1)(t_k) does the job if we prove that τ^,ks are finite times that go to -∞ a.s. But this is a consequence of the strong law of large numbers, since τ^,k is a sum of i.i.d. r.v. distributed asτ^,1 a.s. finite and positive (by continuity of the BM and comparison with the planar BW).Similarly, it can be proved that any two branches ofeventually coalesce and thatcontains a.s. a unique bi-infinite path that we denote 𝐂^. §.§ The bi-infinite branch of the CLW As we saw in Prop. <ref>, the CBW can be obtained as a limit of a CLW when the renormalization parameter n in the CLW tends to +∞. We first show that the CLW also has a bi-infinite path and use the explicit transition kernels for the trajectories of the CLW to obtain, by a limit theorem, the distribution of (𝐂^,𝐂^). The latter are two reflected Brownian motions, as described by <cit.>. The coalescence times of the random walks starting at height h∈ are respectively : ccl T_n^(h) = infł{t≥ h : _w(t)=_w'(t), ∀ w,w'∈^_2n(h)}̊,T_n^(h) = supł{t≤ h : _w(t)=_w'(t), ∀ w,w'∈^_2n(h)}̊.Since for any two points w,w'∈, _w and _w' eventually coalesce a.s., we have a.s., for any h,T^_n(h)<+∞,T^_n(h)>-∞. For D∈{, }, a bi-infinite path of ^D_2n is a sequence (x_i,i)_i∈, such that for all i∈, x_i-x_i-1 2n ∈{1,2n-1}.We say that ^2n,D contains a bi-infinite path ^,D if there is a bi-infinite path ^,D of ^D_2n whose edges are included in the set of edges of ^2n,D. A.s.,andeach contains a unique bi-infinite path.Take the slice h and consider T^_n(h). Since the paths fromdo not cross those of , the paths instarted from h+T^_n(h) all meet before slice h. Let (h) be their common position at height h. Let us consider the sequence (τ^,k_n, k∈) defined similarly to the one introduced in the proof of Th. <ref>: τ^,0_n=0 and for k≥ 1, τ^,k_n=T_n^(τ^,k-1_n). This sequence converges to +∞ a.s. since τ^,k_n is the sum of k independent r.v. distributed as τ^,1_n=T_n^(0). The sequence of paths γ_k=_(C^2n,(τ_n^,k),τ_n^,k) is increasing for inclusion and defines a bi-infinite path 𝐂^=lim_k→ +∞γ_k that is unique by the property (<ref>). The construction of the bi-infinite path forfollow the same lines. Let us describe more precisely the distribution of (,). Let h_1≤ h_2 be two heights. We show that (,) is distributed on a time interval [h_1,h_2], as a Markov chain with explicit transitions. For any process X=(X_i,i ∈ℤ) indexed by ℤ, and h_1 ≤ h_2, let us denoteX[h_1,h_2]:= (X_h_1, X_h_1+1, ⋯, X_h_2), X[h_2,h_1]:= (X_h_2, X_h_2-1,⋯,X_h_1).For h_1≤ h_2, we have (i)(h_1) and (h_2) are independent r.v. respectively uniformly distributed in _2n(h_1) and _2n(h_2), (ii) For any (x_1,x_2)∈_2n(h_1) ×_2n(h_2), conditionally on ((h_1),(h_2))=(x_1,x_2), ł([h_1,h_2],[h_2,h_1])̊ł(_(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_2,h_1])̊ If Pair^,(x_1,x_2,h_1,h_2) denotes the support of (_(x_1,h_1)[h_1,h_2], _(x_2,h_2)[h_1,h_2]), then for any (C_1,C_2)∈ Pair_n^,(x_1,x_2,h_1,h_2) `Pł(( _(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_1,h_2])=(C_1,C_2))̊=2^-2(h_2-h_1)+(C_1,C_2) where (C_1,C_2) is the “number of contacts" between C_1 and C_2 : (C_1,C_2)=#{i ∈ [h_1,h_2-1] : C_1(i)= C_2(i+1)}.The family ((h), h<h_1) (resp. ((h), h≥ h_2)) is a function of the Rademacher r.v. placed on ∪_h<h_1_2n(h) (resp. on ∪_h≥ h_2_2n(h)). Hence, ((h), h<h_1) and ((h), h≥ h_2) are independent, and independent of [h_1, h_2-1]. Clearly, (h_1) and (h_2) have invariant distributions by rotation, so they are uniform, and (<ref>) holds.Using the Rademacher r.v. ξ's defined at the beginning of Section <ref>, we have{(_(x_1,h_1)[h_1,h_2],_(x_2,h_2)[h_1,h_2])=(C_1,C_2)}={∀ h_1≤ i < h_2, ξ(C_1(i),i)=C_1(i+1)-C_1(i) 2nξ(C_2(i+1),i)=C_2(i+1)-C_2(i) 2n}, since the edges of the dual are determined by the edges of the primal. The number of Rademacher (ξ(w),w ∈) contributing to the above event is 2(h_2-h_1)-(C_1,C_2), hence the result. (C_1,C_2)=#{i : (C_1(i),i)= (C_2(i+1),i)} is the number of edges (ξ(u),u ∈) contributing to the definition of both (C_1,C_2). Apart these edges, each increment ofand of C^ are determined by some different Rademacher r.v. Hence 2(h_2-h_1)-(C_1,C_2) edges determine the event { (_(x_1,h_1)^[h_1,h_2],_(x_2,h_2)^[h_1,h_2])=(C_1,C_2)}. From the above Lemma, it is possible to give a representation of the vectors [h_1,h_2] and [h_2,h_1] with a Markov chain whose components both go in the same direction ↑.We haveł([h_1,h_2],[h_1,h_2])̊ (M_1[h_1,h_2],M_2[h_1,h_2])where M=(M_1,M_2) is a Markov chain whose initial distribution is uniform on _2n(h_1)×_2n(h_1), and whose transition kernel K is defined as follows:if d_ℤ/2nℤ(a,a')>1,K((a,a'),(a +2n, a' + '2n)=1/4, for any (,')∈{-1,1}^2 if d_ℤ/2nℤ(a,a')=1, {[ K((a,a+1 ),(a+1,a+2 ))= 1/2,; K((a,a+1 ),(a-1,a+2 ))= 1/4,; K((a,a+1 ),(a-1,a ))= 1/4,; K((a+1,a ),(a,a-1 ))= 1/2,;K((a+1,a ),(a+2, a-1 ))= 1/4,; K((a+1,a ),(a+2,a+1 ))= 1/4, ]. where a,a-1,a+1,a+2 are considered modulo 2n.Notice that the starting points of M is a pair of uniform points at time h_1, while for [h_1,h_2] and [h_1,h_2] the starting points were on two different slices (see Lemma <ref>). First, both distributions have same support, which is⋃_(x_1,x_2)∈^_2n(h_1)×^_2n(h_2) Pair^,(x_1,x_2,h_1,h_2),the set of pairs of non-crossing paths living on ×. By Lemma <ref>, we see that for any pair (C_1,C_2) in this support we have `P( (_h_1[h_1,h_2],_h_2([h_1,h_2])= (C_1,C_2))= 2^-2(h_2-h_1)+(C_1,C_2). The Markov kernel has been designed to satisfy the same formula.§.§ Distribution of (^, ^) In the sequel, we consider the sequence (,)_n∈ correctly renormalized and interpolated as a sequence of continuous functions. We will prove its convergence in distribution on every compact set [h_1,h_2] (with h_1<h_2) to (^,^), a pair of reflected Brownian motions modulo 1 (see Figure <ref>). This result is similar to that ofSoucaliuc et al. <cit.> introduced in the next paragraph.Let F:ℝ→[0,1] be the even, 2-periodic function defined over [0,1] by F(x)=x.Let us consider U_1 and U_2 two i.i.d uniform r.v. on [0,1], and B and B' two i.i.d. BM starting at 0 at time h_1 and independent of U_1 and U_2. Let (Y^,Y^) be the following continuous process defined for t∈ [h_1,h_2] and taking its values in ^2 (Y^, Y^)(t)=ł( U_1+B'_t/√(2)-H(t) 1, U_1+B'_t/√(2)+H(t) 1 )̊,where H(t)represents half the “distance”|Y^(t)→ Y^(t)| :H(t)=Fł(U_2+√(2)B_t)̊/2.Since F is bounded by 1, Y^ and Y^ never cross.We have the following convergences in distribution:(i) Let h_1 <h_2. Let U^_n and U^_n be two independent uniform r.v. on _2n(h_1) and _2n(h_2) respectively. Then in ([h_1,h_2],()^2): ł(_U^_n (4n^2 .) /2n, _U^_n(4n^2 .)/2n)̊ (Y^,Y^).(ii) In (,()^2):ł((4n^2 .)/2n, (4n^2 .)/2n)̊ (,)and (,)(Y^,Y^).Notice that for t=h_1,(Y^(h_1),Y^(h_1))=(U_1-F(U_2)/2 1,U_1+F(U_2)/2 1)which is indeed a pair of i.i.d. uniform r.v. on [0,1] as expected in view of Lemma <ref>(i).The remaining of this section is devoted to the proof of Theorem <ref>, which is separated into several steps. Let us start with point (i).Step 1: Tightness of ł(_U^_n(4n^2 .)/(2n), _U^_n(4n^2.)/(2n))̊ By translation invariance, we may suppose that h_1=0 and set h_2=T. The tightness of the family of distributions of ł(_U^_n(4n^2 .)/(2n), _U^_n(4n^2.)/(2n))̊ in C([0,T],()^2) follows from the tightness of its marginals that are simple well rescaled random walks on the circle. Now, our aim is to identify the limiting distribution. For that purpose, and in view of Lemmas <ref> (ii) and <ref>, we study more carefully the Markov chain (M_1,M_2). Step 2: Angle process between M_1 and M_2 Let us extend the notation [a→ b] and |a→ b| for a and b in /2n. For the Markov chain M defined in Lemma <ref>, the angle process between the two components isA(i)= |M_1(i) → M_2(i)|, i≥ 0.Of course, for any i, (M_1(i),M_2(i))= (M_1(i),M_1(i)+A(i)2n). We will focus on the asymptotics of ((M_1(i), A(i)), i≥ 0).Recall that M_1 and M_2 are simple non-independent random walks with Rademacher increments. Let us write:ccl M_1(i) = M_1(0)+∑_j=1^i R_2j-1,M_2(i) = M_2(0)+∑_j=1^i R_2jwhere (R_2i,i≥ 1) and (R_2i-1,i≥ 1) are two families of i.i.d. Rademacher r.v., the two families being possibly dependent from each other. The process A takes its values in the set of odd integers in [0,2n], and its are sums of 2 Rademacher r.v. Now, let us consider the simple random walkZ(i) = A(0) + ∑_j=1^i (-1)^jR_j=M_2(i)-M_1(i)starting from A(0). If M_1 and M_2 were allowed to cross, then A(i) would be equal to Z(2i). We have to account for the non-crossing property of the paths of . A random walk (Z_i,i≥ 0) is said to be the simple random walk reflected at 0 and 2n, and starting at someb∈0,2n if (Z_i,i≥ 0) is a Markov chain such thatrcl(Z_i+1=1|Z_i=0)= (Z_i+1=2n-1|Z_i=2n)=1 (Z_i+1=a± 1 |Z_i=a) =1/2, for anya ∈1,2n-1.For any discrete time process X, denote byΔ X_i:=X_i-X_i-1,the ith increment of X. We haveThe distribution of the process ((A(i),M_1(i)),i≥ 0) starting at (A(0),M_1(0)) where A(0)∈1,2n-1 is odd, and M_1(0) ∈0,2n-1 is even is characterized as follows:For (Z_i,i≥ 0) a simple random walk reflected at 0 and 2n, and starting from A(0), we have:(A(i),i≥ 0) (Z_2i,i≥ 0)F_2n(Z_2i,i≥ 0),where F_2n:ℤ→0,2n, the even 4n-periodic function, defined on [0,2n] by F_2n(x):= x. The random walk M_1 starting at M_1(0) admits as increments the sequence ł(Δ M_1(i),i≥ 1)̊=ł(-Δ Z_2i-1,i ≥ 0)̊,that is the opposite of the increments with odd indices of Z. Notice that the second identity in eq:tehez holds in distribution only: as defined, the reflection only modifies the increments that follow the hitting times of 0 and 2n, whereas the map F_2n turns over large part of the trajectory (Z_i,i≥ 0). Denoting by w_2n(ℓ)=⌊ℓ/(2n)⌋ the discrete “winding number” of ℓ, according to Lemma <ref>, the increments of the process M_1 under this representations areΔ M_1(ℓ) = (-1)^w(Z _2ℓ-1)ΔZ _2ℓ-1.The distance |M_1(i)→ M_2(i)| decreases when Δ M_1(i)=R_2i-1=1 increases and increases when Δ M_2(i)=R_2i=1, so that Z(2i) would be equal to A(i) if the two walks were not constrained to not cross. We would also haveΔ M_1(i) =-(Z_2i-1-Z_2i-2)=-ΔZ_2i-1.Since 0 and 2n are even, and since Z(0)=A(0) is odd, the random walk can hit 0 and 2n only after an odd number of steps. In other words, the reflection will concern only the steps with even indices. Therefore, let (Z_i,i≥ 0) be the random walk Z reflected at 0 and 2n: the odd increments of Z and of Z are the same, and the even increments correspond except when Z_2i-1∈{0,2n}, in which case the reflection implies that Z_2i=_Z_2i-1=0+(2n-1)_Z_2i-1=2n. It is easy to check from (<ref>)-(<ref>) that (Z_2i,i≥ 0) has the same distribution as the angle process (A(i), i≥ 0) started from A(0) one the one hand, and as (F_2n(Z_2i), i≥ 0) on the other hand.Finally, notice that because the odd increments are the same, (<ref>) also holds for Z. Step 3: Identification of the limitLet U_1, U_2 are two uniform r.v. on [0,1], let B and B' be two BMs, all being independent. We have in (_+,) that ł(M_1(4n^2.)/2n,A(4n^2.)/2n)̊ł( U_1+B'_.-H(.) 1, 2H(.))̊ where H(t):=F(U_2 + √(2) B_t)/2, has been defined in (<ref>).Let us first consider the angle component. Since the discrete process A(4n^2t)/(2n) is the difference between two (dependent) suitably rescaled random walks which are both tight under this rescaling, the process A(4n^2t)/2n is tight in (_+,). To characterize the limiting process, writeA(4n^2t)/2n= F_2n(Z_8n^2t)/2n= F ł( Z_8n^2t/2n)̊since for every x and every n, F_2n( 2n x ) = 2n F( x ). The central limit theorem implies the convergenceZ_8n^2t/2n U_2+ N(0, 2t) for a fixed t≥ 0. Since, the mapping g↦ (t↦ F(g(t)) is continuous on (_+,)s, the independence and stationarity of the increments of Z provide the finite dimensional convergence of the angle process in (<ref>). For the first component, we know that M_1(4n^2.)/2n converges in distribution to a BM modulo 1, but that is not independent from the limit H of A(4n^2.)/2n. The result is a consequence of the following lemma, proved in the sequel. Let B and B' be two independent BM, and let X=B+B' be the sum process. For any (b_0,x_0)∈^2, conditionally on {(X_t=x_t,t∈[0,T]) ,B_0=b_0,B'_0=x_0-b_0}, we have(B.,B'.)_[0,T]ł(b_0-x_0/2+x_./2 +B”./√(2), -b_0+x_./2-B”./√(2))̊_[0,T] for an independent BM B”.Step 4: Proof of Theorem <ref>(ii). Consider two levels h_1≤ h_2. First, remark that the restriction of (^, ^) to the compact interval [h_1,h_2] has same distribution as (_U^,_U^) on [h_1,h_2], where U^ and U^ are independent and uniformly distributed on (h_1) and (h_2) (indeed, (h_1) depends only on what happens below the level h_1 and (h_2) depends only on what happens above the level h_2. From (<ref>), itremains to prove that (_U^,_U^) on [h_1,h_2] is distributed as (Y^,Y^).For any (x,h)∈, the map Π_(x,h) :F∈ (ℋ_O,d_ℋ_O) ↦ W_(x,h)∈([h,+∞),∖) that associates to a forest the path started at (x,h) is continuous. From Prop. <ref>, we thus deduce that the Markov chain M of Lemma <ref> correctly renormalized converges on [h_1,h_2], when n→ +∞, to (_u_1-F(u_2)/2(t),_u_1+F(u_2)/2(t))_t∈ [h_1,h_2]the paths ofand its dual. We deduce that (Y^,Y^) has the same distribution. This concludes the proof of Theorem <ref>. Since we are dealing with Markov processes with stationary increments and simple scaling properties, it suffices to show that for X_1,X_2,N 3 i.d.d. N(0,1) r.v. , we have that conditionally on S=X_1+X_2,(X_1,X_2) (d)=(S/2+N/√(2),S/2-N/√(2)).This is a consequence of Cochran theorem, which gives that (X_1-S/2, X_2-S/2) is a Gaussian vector independent from S. Since X_1-S/2=(X_1-X_2)/2=-(X_2-S/2), introducing N/√(2)=X_1-S/2 finishes the proof.§.§ The coalescence times have exponential moments Th. <ref> states that the coalescence times T^(x,x',t) and T^(t) are finite a.s. Due to the compactness of the space , we can prove in fact that they admit exponential moments.(i) There exist b>0, M<+∞ such that for any x,x'∈ and any t∈`R,`E [ e^b (T^(x,x',t)-t)] < M. (ii) For any t∈ `R, the coalescence time T^(t) admits exponential moments :∃ a>0 , `E [ e^a (T^(t)-t)] < ∞ . For both assertions, by the time translation invariance of the CBW, it suffices to consider only the case t=0 .(i) We can assume that 0≤ x≤ x'< 1. We have before crossing time(^_(x',0)(t)-^_(x,0)(t), 0≤ t≤ T(x,x',0))(x'-x+ √(2)B(t) 1, 0≤ t≤ T(x,x',0)),where B is a standard usual Brownian motion. HenceT(x,x',0) has same distribution as the exit time of a linear BM B from the segment [-(x'-x),1-(x'-x)]. This exit time is known to admit exponential moments (see e.g. Revuz & Yor <cit.>). (ii) We will use a very rough estimate to prove this fact.Let for k≥ 0, A_k be the following independent events :A_k={T^(2k)≤ 2k+2}meaning that all trajectories born at height 2k have coalesce before time 2k+2. If we show that p:=P(A_k)> 0, then T(0) ≤min{ k, A_kholds} is bounded by twicea geometric r.v. p, and then has some exponential moments. So let us establish this fact.For this, we use a single argument twice. Consider Z the hitting time of two BM starting at distance 1/2 on . Clearly q:=ℙ(Z≤ 1)>0. Let us now bound P(A_0). For this consider “half of the dual CBW”(_(x,1), 0≤ x≤ 1/2) starting at (1). With probability q these trajectories merge before (0). Conditionally to this event, all primal trajectories (^_(x,0), x ∈(0)) starting at time 0 a.s. avoid the dual trajectories, and satisfy ^_(x,0)(1) ∈ (1/2,1), meaning that, with probability q at least, they will be in the half interval (1/2,1). But now, the two trajectories ^_(1/2,1) and ^_(1,1), will merge before time 2 with probability q. Conditionally to this second event, with probability ≥ 1/2, the merging time of (^_(x,1), x∈ (1/2,1)) is smaller than 1. Indeed on , by symmetry, when ^_(1/2,1) and ^_(1,1) merge, they “capture” all the trajectories starting in [1/2,1] (which will merge with them) or they capture all the trajectories starting in[0,1/2]. Since both may happen, the probability of each of this event are larger than 1/2. Hence p ≥ q^2/2 and the proof is complete.§.§ Toward explicit computations for the coalescence time distributionNotice that an approach with Karlin-McGregor type formulas can lead to explicit (but not very tractable) formulas for the distribution of the coalescing time of several Brownian motions. Let us consider 0<x_1<x_2<… <x_k<1, and denote by T_k the time of global coalescence of the k Brownian motions W_(x_1,0)^,… W_(x_k,0)^.Taking to the limit formulas obtained by Fulmek <cit.>, we can describe the distribution of the first coalescence time T^k→ k-1 between two of these paths :T^k→ k-1(x_i,1≤ i ≤ j)=min{T^(x_i,x_i+1,0), 1≤ i≤ k}with the convention that x_k+1=x_1, and where T^(x_i,x_i+1,0) is the time of coalescence of ^_(x_i,0) and ^_(x_i+1,0) as defined in (<ref>). We will omit the arguments (x_i,1≤ i ≤ j) in the notation T^k→ k-1_(x_i,1≤ i ≤ j) unless necessary. For t>0,ℙ(T^j→ j-1>t)=∫ dy_1…∫ dy_j _0<y_1<y_2<… <y_j<1[∑_i=0^j-1(σ^i) ∏_ℓ=1^j Φ_t(y_ℓ-x_σ^i(ℓ))]where σ^i denotes the rotation σ^i(ℓ)= ℓ+ijand whereΦ_t(x)=1/√(2π t)∑_m ∈exp(-(x-m)^2/2t).Explicit formulas for the Laplace transform of T^j→ j-1 are not established in general cases to our knowledge, except for the following special case when k=2 and θ<0 (see e.g. Revuz & Yor <cit.>):(e^θ T^2→ 1)=cosh(√(|θ|)1+2x_1-2 x_2/2)/cosh(√(|θ|)/2).Using that (e^θ T^j+1→ j)=1+∫_0^+∞θ e^θ t(T^j+1→ j>t)dt and the Markov property, we can finally link (<ref>) and T_k:(e^θ T_k)=∏_j=1^k-1((e^θ T^j+1→ j(_1(T_j),…_k(T_j)) | _1(T_j),…_k(T_j))),where T_j is the time of the k-jth coalescence (at which there are j Brownian motions left) and (_1(T_j),…_k(T_j)) are the values of the k coalescing Brownian motions at that time (and hence only j of these values are different). It is however difficult to work out explicit expressions from these formula.§ DIRECTED AND CYLINDRIC POISSON TREES Apart from the (planar) lattice web W^2n, defined as the collection of random walks on the grid ={(x,t)∈^2,x-t2 =0 } (see <cit.> or Figure <ref>), several discrete forests are known to converge to the planar BW; in particular the two-dimensional Poisson Tree studied by Ferrari & al. in <cit.>. In Section <ref>, a cylindric version of this forest is introduced and we state the convergence of this (continuous space) discrete forest to the CBW. See Th. <ref> below. Our proof consists in taking advantage of the local character of the assumptions (B2O) and (B2). Indeed, the cylinder locally looks like the plane and we can couple (on a small window) the directed and cylindrical Poisson trees in order to deduce (B2O) from (B2).Finally, in Section <ref>, we discuss under which assumptions, conditions (B2) and (B2O) can be deduced from each other. §.§ Convergence to the CBW Let n≥ 1 be an integer, andr>0 be a real-valued parameter. Consider a homogeneous Poisson point process (PPP in the sequel) 𝒩_λ with intensity λ >0 on the cylinderdefined in (<ref>).Let us define a directed graph with out-degree 1 having 𝒩_λ as vertex set as follows: from each vertex X=(x,t)∈𝒩_λ add an edge towards the vertex Y=(x',t')∈𝒩_λ which has the smallest time coordinate t'>t among the points of 𝒩_λ in the strip {(x”,t”) ∈ | d_`R/ℤ(t,t”) ≤ r} whered_`R/ℤ(x,x”):=min{|x-x”|, |1+x-x”|}. Let us set α(X):=Y the out-neighbor of X. Notice that even if X does not belong to 𝒩 the ancestor α(X)∈𝒩 of this point can be defined in the same way. For any element X∈, define α^0(X):=X and, by induction, α^m+1(X):=α(α^m(X)), for any m≥ 0. Hence, (α^m(X))_m≥ 0 represents the semi-infinite path starting at X. We define by _X^λ,r, the continuous function from [t;+∞) to `R/ℤ which linearly interpolates the semi-infinite path (α^m(X))_m≥ 0.The collection ^λ,r,:={_X^λ,r,, X∈𝒩_λ} is called the Cylindric Poisson Tree (CPT). This is the analogue onof the two-dimensional Poisson Tree introduced by Ferrari et al. in <cit.>. Also, ^λ,r, can be understood as a directed graph with edge set {(X,α(X)) : X ∈𝒩_λ}. Its topological structure is the same as the CBW (see Th. <ref>) or as the CLW (see Prop. <ref>). The CPT a.s. contains only one connected component, which justifies its name: it is a tree and admits only one bi-infinite path (with probability 1).Let us choose λ=n and rescale ^λ,r, into ^(n),r, defined as^(n),r, := {^n,r/n,_(x,t)(n^2 s) ; (x,t)∈𝒩_n, s≥t/n^2} .For r=1/2, the normalized CPT ^(n),r, converges in distribution to the CBW as n→+∞.As noticed in Section <ref>, only criteria (IO) and (B2O) of Th. <ref> have to be checked. The proof of (IO) is very similar to the one of (I) for the two-dimensional Poisson Tree (see Section 2.1 of <cit.>) and is omitted. The suitable value r=1/2 ensures that the limiting trajectories are coalescing standard Brownian motions.Let us now prove (B2O). By stationarity of the CPT, it suffices to prove that for all t>0,lim_`e → 0^+1/`elim sup_n→+∞(η^O_^(n),1/2,(0,t;[0→ε]) ≥ 3 ) = 0.Recall that among all the trajectories in ^(n),1/2, that intersect the arc [0→ `e] at time 0, η^O_^(n),1/2,(0,t;[0→ `e]) counts the number of distinct positions these paths occupy at time t.A first way to obtain (<ref>) consists in comparing η^O_^(n),1/2,(0,t;[0→ `e]) and η_W^(n)(0,t;0,`e), where W^(n) denotes the normalized two-dimensional Poisson tree– whose distribution converges to the usual BW, see <cit.>–by using stochastic dominations similar to (<ref>) traducing that it is easier to coalesce on the cylinder than in the plane. Since W^(n) satisfies (B2) (see Section 2.2 of <cit.>), η^O_^(n),(0,t;[0→ `e]) ≤_S η_W^(n)(0,t;0,`e) implies that ^(n),1/2, satisfies (<ref>) which achieves the proof of Th. <ref>.A second strategy is to investigate the local character of the assumptions (B2) and (B2O). Indeed, the map t↦η^O_^(n),1/2,(0,t;[0→ `e]) is a.s. non-increasing. It is then enough to prove (<ref>) for (small) 0<t≪ 1 in order to get it for any t>0. The same holds when replacing ^(n),1/2, with W^(n). Now, when t and `e are both small, the (normalized) CPT ^(n),1/2, restricted to a small window containing [0→ `e]×[0;t] behaves like the (normalized) two-dimensional Poisson tree W^(n) restricted to a window containing [0;`e]×[0;t] with high probability. As a consequence, ^(n),1/2, and W^(n) should simultaneously satisfy (B2O) and (B2).Let us write this in details. We use a coupling of the environment (the PPP) on some larger window since the trajectories of the discrete trees on a window are also determined by the environment around. Using some control of the deviations of the paths issued respectively from the intervals I^_ε=[0 →ε ]×{0} and I_ε=[0;ε]×{0}, we determine larger windows _ε^ and_ε which will determine the trajectories started from this sets to a certain time t_ε up to a negligible probability p_ε. Using the constants that emerge from this study, we thereafter design a coupling between the PPP on the cylinder and on the plane that coincides on _ε^ and_ε (up to a canonical identification). This will allow us to deduce (B2) or (B2O) from the other.To design the windows that contains all paths crossing I^_ε (or I_ε) up to time t,it suffices to follow the trajectories starting at (0,0) and (ε,0). Consider the path (X_k=(x_k,y_k), k≥ 0) started from (0,0) and consider the successive i.i.d. increments of this path denoted by (ξ^x_k,ξ^y_k)=Δ X_k. Before normalisation, (ξ^x_1, ξ^y_1) consists of two independent r.v., where ξ^x_1 is uniform on [-r,+r] with r=1/2, and ξ^y_1 has exponential distribution with parameter λ=1, sinceP(ξ^y_1 ≥ y)=`P(𝒩∩([-1/2,1/2]× [0,y])= ∅) = e^- y . Now, starting at 0, the renormalized trajectory on ^(n),1/2, is a random walk whose increments (ξ^(n),x_k, ξ^(n),y_k, k≥ 0) are i.i.d. such that nξ^(n),x_k ξ^x_1, and n^2ξ^(n),y_kξ^y_1. Let us define the number of steps for the rescaled path to hit ordinate t byτ^n_t := infł{ j≥ 1 | ∑_k=1^j ξ^(n),y_k ≥ t }̊infł{j≥ 1 | ∑_k=1^j ξ^y_k ≥ n^2t}̊ .The points {∑_k=1^j ξ^y_k, j≥ 1} form a PPP Θ on the line with intensity 1, so that τ^n_t=1+#(Θ∩[0,n^2t]) a.s. Thereforep_c,t,n := `P(τ^n_t ≥ cn^2) = `P(1+P(n^2t)≥ n^2c)where P(x) is a Poisson r.v. with parameter x. For c=2tthis probability p_2t,t,n is exponentially small in n and the event A_t,n:={τ_n^t≤ 2n^2 t} has probability exponentially close to 1. Now, on the event A_t,n, we can control the angular fluctuations of ^(n),1/2,:q_t,n:= `Pł(sup_j≤τ^n_tł|∑_k=1^j ξ^(n),x_k|̊≥ c √(t))̊≤`P(A^c_2t,n)+`Pł(sup_j≤ 2n^2 tł|∑_k=1^jξ^(n),x_k|̊≥ c √(t))̊ .Thus, consider the process defined bys_n(j/n^2) := ∑_k=1^jξ^(n),x_k 1/n∑_k=1^j ξ^x_k , j≥ 0,and interpolated in between. A simple use of Donsker theorem shows that(s_n(a))_a≥ 0ł(1/√(12)B(a))̊_a≥ 0in (_+,) where B is a Brownian motion. Since for every t, on ([0,2t],), the functional g↦max|g| is continuous, one sees thatq_t,n =`P(A^c_2t,t,n)+`Pł(sup_a≤ 2 tł|s_n(a)|̊≥ c √(t))̊`Pł(sup_a≤ 2 t |B(a)| ≥ c√(12 t))̊= `Pł(sup_a≤ 1 |B(a)| ≥ c√(6 ))̊.Take ε>0. Choose c large enough such that `Pł(sup_a≤ 1 |B(a)| ≥ c√(6))̊≤ε^2/2, and n large enough so that q_t,n≤ε^2, and t small enough so that c√(t)<1/4. We have proved that with probability larger than 1-O(ε^2), the walk hitsordinate t before its abscissa exits the window [-c√(t),c√(t)]. Since the decision sector for each step of the walker has width 2r/n, with probability more than 1-O(ε^2), the union of the decision sectors of the walk before time t are included in [-c√(t)-2r/n,c√(t)+2r/n]⊂[-1/3,1/3] for n large enough. It is now possible to produce a coupling between the PPP on the cylinder and the plane that coincides on a strip with width 2/3 : take the same PPP on the two strips (up to a canonical identification of these domains), and take an independent PPP with intensity 1 on the remaining of the cylinder or of the plane. Henceforth, any computation that depends only of such a strip in the cylinder and in the plane will give the same result. Here, we then have here, for any eventthat depends on the trajectories passing through I_ε^ or I_ε up to time t (for the constant satisfying what is said just above)`P_n^() = `P_n()+ O(ε^2),so that the inheritance of (B2) from the plane to the cylinder is guaranteed, as well as the converse.§.§ From the plane to the cylinder, and vice-versa: principlesWhen a convergence result of some sequence of coalescing processes defined on the plane to the BW has been shown, it is quite natural to think that the similar convergence holds on the cylinder too, and that the limit should be the CBW. The converse, also, should hold intuitively.The main problem one encounters when one wants to turn this intuition into a theorem, is that, in most cases the constructions we are thinking of are trees that are defined on random environments (RE) as a PPP or as lattices equipped with Rademacher r.v.. Both these models exist on the cylinder and on the plane, leading to clear local couplings of these models. But, more general RE and more general random processes exist, and it is not possible to define a “natural” model on the cylinder inherited from that of the plane. We need to concentrate on the cases where such a natural correspondence exists.A similar restriction should be done for the algorithms that build the trajectories using the RE. In the cases studied in the paper, the trajectories are made by edges, constructed by using a navigation algorithm, which decides which points to go to depending on a “decision domain” which may depend on the RE. For example, in the cylindric lattice web, the walker at position (x,t) just needs to know the Rademacher variable attached to this point, so that its decision domain is the point (x,t) itself. In the generalization ofFerrari & al. <cit.> treated at the beginning of Section <ref>, the decision domain is a rectangle[x-r,x+r]×(t,t+h] where h is smallest positive real number for which this rectangle contains a point of the point process (many examples of such navigation processes have been defined in the literature, see <cit.>). We may call such model of coalescing trajectories as coming from “navigation algorithms, with local decision domains”.There exist models of coalescing random processes of different forms, or that are not local (such as minimal spanning trees). Again, it is not likely that one may design a general theorem aiming at comparing the convergence on the cylinder with that on the plane.“For a model defined on the cylinder and on the plane on a RE” as explained inthe proof of Theorem <ref>, when a local coupling between windows (or strip) of the cylinders and of the plane exists, (B2) and (B2O)“are morally equivalent”. Informally, the 4 conditions are:1) the models are invariant by translations on respectively, the cylinder and the plane;2) there exists a coupling between both probabilistic models which allows to compare^(n), and W^(n) at the macroscopic level: on a window :=[0,A]× [0,B] for some (small) A,B>0, the environments on which are defined^(n), and W^(n) can be coupled, and, under these coupling, these REcoincide a.s.;3) the restriction of the trajectories from ^(n), and W^(n) on [0,ε]×[0,t_ε] are measurable with respect to the environment inwith probability 1-O(ε^1+a) for some a>0;4) the largest decision domain before hitting ordinate n^2t is included in a rectangle [a_n,b_n] with probability 1-O(ε^1+a) where a_n=o(1) and b_n=o(1) (for the rescaled version).§ DISCRETE CYLINDRIC AND RADIAL POISSON TREEColetti and Valencia introduce in <cit.> a family of coalescing random paths with radial behavior called the Discrete Radial Poisson Web. Precisely, a Poisson point process Θ with rate 1 on the union of circles of radius k∈∖{0}, centered at the origin, is considered. Each point of Θ in the circle of radius k is linked to the closest point in Θ in the circle of radius k-1, if any (if not, to the closest point of Θ in the first circle of radius smaller than k-1 which contains a point of Θ). They show in <cit.> that under a diffusive scaling and restricting to a very thin cone (so that the radial nature of paths disappears), this web converges to some mapping of the (standard) BW. A similar result is established in Fontes et al. <cit.> for another radial web.Our goal in this section is to establish a convergence result for an analogous of the Discrete Radial Poisson Web of Coletti and Valencia <cit.> but which holds in the whole plane. Our strategy consists in considering a cylindrical counterpart to the Discrete Radial Poisson Web and to prove its convergence to the CBW (Theorem <ref>). Thenceforth, it suffices to map the cylinder on the (radial) plane with φ_⋆ defined in (<ref>) to obtain a global convergence result for the corresponding planar radial forest. We modify a bit the model of <cit.> to make the involved normalizations more transparent and to reduce as much as possible the technical issues, while keeping at the same time the main complexity features. Consider an increasing sequence of non-negative numbers (h_k, k ∈), with h_0=0, and the associated slices of the cylinder:' = ×{h_k,k∈}=⋃_k∈(h_k) .Consider the following Poisson point process on ',Ξ=⋃_k≥ 0Ξ_k ,whereΞ_k is a PPP on (h_k) with intensity n_k>0. The sequences (h_k)_k≥ 1 and (n_k)_k ≥ 1 are the parameters of the model. Remark that the choice of n_k=n (a constant) is treated in previous sections. Here we are interested in the case where n_k, h_k→ +∞. Given Ξ, let us define the ancestor α(Z) of a point Z=(x,h_k)∈(h_k) as the closest point of Ξ_k+1 if the latter is not empty and the point (x,h_k+1) otherwise. This second alternative means that instead of moving to the closest point of the first non-empty slice with rank k'>k (as in <cit.>), one just moves vertically to the next slice.The ancestor line AL_Z of Z=(x,h_k) is the sequence (Z_j=(x_j,h_j),j≥ k) such that Z_k=Z and for j> k, Z_j+1=α(Z_j). Upon Ξ we define the Discrete Cylindric Poisson Tree𝒯 as the union of the ancestor lines of the elements of Ξ:𝒯 := ⋃_(x,h) ∈Ξ AL_(x,h) . Notice that when (x,h)∈Ξ, AL_(x,h)=𝒯_(x,h) is the path of 𝒯 started at (x,h). The notation AL_(x,h) allows to consider ancestor lines started from any points Z∈'.Contrary to Section <ref>, we do not consider a sequence of point processes parametrized by n which goes to infinity, but rather we shift the cylinder which also implies that we see more and more points. Precisely, for any k≥ j≥ 1 and any (x,h_k)∈Ξ, let AL^(j)_(x,h_k) be the ancestor line AL_(x,h_k) translated by the vector -(0,h_j). We can then associate to 𝒯, the sequence of shifted forests (𝒯^(j))_j≥ 1 by𝒯^(j) := ⋃_(x,h) ∈∪_k≥ j Ξ_k AL^(j)_(x,h) . Our purpose is to prove that:Let us consider two sequences (n_k)_k ≥ 1 and (f_k)_k ≥ 1 of positive real numbers such that lim_k→∞ f_k = 0 ,∑_k e^-n_kf_k < +∞∑_k1/n_k^2 = +∞ .Then there exists a sequence (h_k)_k ≥ 1 tending to infinity such that the sequence of shifted forests (𝒯^(j))_j≥ 1 converges in distribution to the CBW restricted to the half cylinder ^+=×_+.The map φ_⋆, as defined in def:phi, sends thehalf cylinder ^+:=()× `R^+ onto the radial plane ()× `R^+. The image of the PPP Ξ is a PPP Ξ' on the plane, which is the superposition of the Poisson point processes Ξ'_k of intensities n_k/(8π^3h_k) on the circles with radii 4π^2h_k. The image of the tree we built on the cylinder is a tree on “the radial plane”, which can in fact be directly built by adapting the navigation used in the cylinder in the plane (go to the closest point in the next circle if any, and otherwise to the point with same argument). See Figure <ref>. To get a convergence result on the radial plane, with the same flavour as that obtained in the plane, we need to discard the neighborhood of zero by a shift. The most economic way to state our results is as an immediate corollary of the previous result:Under the hypotheses of Theorem <ref>, when j→+∞, the sequence (φ_⋆(𝒯^(j)))_j≥ 1 converges in distribution to the . Let us comment on the hypothesis eq:newcond. First, it implies that n_k→∞. A consequence of Borel-Cantelli's lemma is that whenever ∑_k≥ 0e^-n_k is finite, there exists a.s. a random rank from which the Ξ_k's are non-empty. Hypothesis eq:newcond is actually slightly more demanding: the condition ∑_k 1/n_k^2=+∞ and the link between sequences (n_k)_k ≥ 1 and (f_k)_k ≥ 1 will appear in Sections <ref> and <ref>.To prove Theorem <ref>, we check the criteria of Theorem <ref>, namely (IO) and (EO). The convergence of an ancestor line of 𝒯^(j) to a BM modulo 1, when j→ +∞, is first stated in Section <ref>. In the proof, we see that the condition ∑_k 1/n_k^2=+∞ of (<ref>) is necessary. We then deduce (IO) in Section <ref>, and use at some point the second item of (<ref>). The proof of (EO) is devoted in Section <ref> and is based on a coalescence time estimate (Proposition <ref> in Section <ref>) whose proof uses the links between the cylindric and planar forests highlighted in Section <ref>. §.§ Convergence of a path to a Brownian motionLet us consider the ancestor line 𝒯_(X_j,h_j)=((X_k,h_k), k≥ j) started at a point (X_j,h_j)∈Ξ_j for j≥ 1. For k>j, this path goes to infinity by jumping from (h_k-1) to (h_k). The random increments (Δ X_k:=X_k-X_k-1,k >j) are independent. The distribution of Δ X_k is characterized for any measurable bounded map f:[-1/2,1/2]→ by,(f(Δ X_k))=e^-n_kf(0)+∫_-1/2^1/2 f(x) n_k e^-2n_k |x| dx.In other words, conditionally on the (h_k) being not empty, Δ X_k is a Laplace r.v. conditioned on having absolute value smaller than 1/2. Hence, {[ (Δ X_k) = 0; (Δ X_k) = 1/2n_k^2-e^-n_k(n_k^2+2n_k+2)/4n_k^2 =: σ_k^2; ((Δ X_k)^4) = 3/2n_k^4-e^-n_k(n_k^4+4n_k^3+12n_k^2+24n_k+24)/16n_k^4. ].As n_k→∞, the variance σ_k^2 is equivalent to 1/(2n_k^2). For the sequel, let us denote the variance of X_k-X_0 by:V_k := (X_k-X_0) = σ_1^2+⋯+ σ_k^2.The variance V_k is hence related to n_k by (<ref>). Let us now consider the time continuous interpolation of the shifted sequence (X_k,h_k-h_j)_k≥ j. For ℓ∈, we set, if h_j+t∈ [h_j+ℓ, h_j+ℓ+1),X̅_t^(j)= X_j+ ∑_k=1^ℓΔ X_j+k + (h_j+t)-h_j+ℓ/h_j+ℓ+1-h_j+ℓ Δ X_j+ℓ+1 .In order to prove the convergence of (X̅^(j))_j≥ 0 to a Brownian motion, it is natural to seth_k = V_k ,for anyk≥ 0 .Then, combining eq:displ, def:Vk, eq:hV with eq:newcond, it follows that h_k→∞ and h_k+1-h_k=σ^2_k+1→ 0, i.e. slices are getting closer and closer.The hypothesis eq:newcond is satisfied for example for n_k=k^α with 0<α<1/2 and f_k=k^α' with 0<α'<α. This entails that h_k=c_1+c_2 k^1-2α where c_1,c_2 are positive constants. Let us introduce, for the sequel,R(t) := inf{k∈, V_k≥ t}the integer index such that h_R(t)-1< t ≤ h_R(t). Note that R(h_k)=k. Under the previous notations and eq:newcond, the following convergence holds in distribution in (_+,) (X^(j)_t,t≥ 0)(B_t,t≥ 0) ,where (B_t,t≥ 0) is a standard Brownian motion taken modulo 1.We are not under the classical assumptions of Donsker theorem, since the Δ X_k's are not identically distributed and since the convergence involves a triangular array because of the shift. Because the Δ X_k's are independent, centered, with a variance in 1/n_k^2 that tends to 0, we have for all t≥ 0,lim_j→ +∞(X̅^(j)_t-X̅^(j)_0)= lim_j→ +∞∑_ℓ = j+1^R(h_j+t)(Δ X_ℓ)= lim_j→ +∞(V_R(h_j+t)-V_j)=t, implying that X̅^(j)_t-X̅^(j)_0 converges in distribution to 𝒩(0,t) by Lindeberg theorem (e.g. <cit.>). The convergence of the finite dimensional distributions is easily seen by using the independence of the Δ X_k's.The tightness is proved if (see e.g. <cit.>) for every positive ε>0 and η>0, there exists δ∈ (0,1) and j_0∈ such that for every j≥ j_0 and every t∈_+,1/δ(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|≥ε)≤η.For t∈_+ and j∈,(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t| ≥ε) =(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|^4≥ε^4) ≤ ( max_R(h_j+t)≤ℓ≤ R(h_j+t+δ) S_ℓ^4 ≥ε^4 ), where S_ℓ=h_R(h_j+t)-(h_j+t)/h_R(h_j+t)-h_R(h_j+t)-1 Δ X_R(h_j+t)+ ∑_k=R(h_j+t)+1^ℓΔ X_k.Since the Δ X_k are centered, S_ℓ defines a martingale (in ℓ≥ R(h_j+t)), and S_ℓ^4 is a submartingale. Using Doob's lemma for submartingales:ε^4 ( max_R(h_j+t)≤ℓ≤ R(h_j+t+δ) S_ℓ^4 ≥ε^4 ) ≤(S^4_R(h_j+t+δ)) ≤(∑_k=R(h_j+t)^R(h_j+t+δ) (Δ X_k)^4)+∑_k≠ℓR(h_j+t)≤ k,ℓ≤ R(h_j+t+δ)((Δ X_k)^2)((Δ X_ℓ)^2), using that the Δ X_k's are independent and centered. The last sum in the r.h.s. of (<ref>) is upper bounded by(∑_k=R(h_j+t)^R(h_j+t+δ)((Δ X_k)^2))^2that converges to δ^2 when j→ +∞. For the first term, there exists from (<ref>) a constant C such that for k large enough, ((Δ X_k)^4)≤ C (Δ X_k)^2 = C σ_k^4. Thus:ł(∑_k=R(h_j+t)^R(h_j+t+δ) (Δ X_k)^4)̊≤C M_j,t,δ∑_k=R(h_j+t)^R(h_j+t+δ)σ_k^2 ∼_j→ +∞ C M_j,t,δδwith M_j,t,δ=sup{σ_k^2, R(h_j+t)≤ k ≤ R(h_j+t+δ) }→ 0 when j→+∞.Gathering these results, we see that up to a certain constant C,1/δ(sup_t≤ s≤ t+δ| X̅^(j)_s-X̅^(j)_t|≥ε)≤C/ε^4 δ(δ^2 + CM_t_1,t,δδ), which converges to zero when δ→ 0 and j→ +∞.§.§ Coalescence time estimateIn this section, we establish a coalescence time estimate that will be useful for proving (IO) and (EO). Following the lines of Section <ref>, we can introduce a planar model corresponding to our cylindrical tree and ensuring the possibility of couplings between the cylinder and the plane. In the plane, the use of the Skorokhod embedding theorem and the results known for planar Brownian motions make it easier to obtain such estimates. We thus first introduce in Section <ref> a planar model corresponding to the forest 𝒯. We establish estimates for the coalescence time of two paths in this planar model. For this, we start with studying how the distance between the two paths evolves. The core of the proof relies on the Skorokhod embedding theorem (as in <cit.>), but with a clever preliminary stochastic domination of the distance variations. In Section <ref>, we return to the original model and deduce from the previous result estimates for the coalescence time of two paths of 𝒯^(j). §.§.§ Planar analogous We first define the planar model corresponding to our cylindrical problem. We consider the horizontal lines with ordinate (h_k)_k∈ in the upper half plane. For each k∈∈, we consider on the line L_k:=×{h_k} (or level h_k), an independent PPP Υ_k with intensity n_k. The Poisson point process on the union of the lines is denoted by Υ, similarly to (<ref>). Each point of the level h_k is linked with the closest point of the next level, namely level h_k+1. This generates a forest that we denote 𝒲, and which can be seen as the analogous of 𝒯 in the plane.For a given point Z∈∪_k∈L_k, denote by α^P(Z) the ancestor of Z for this navigation. This allows us to define, as for the cylinder, the ancestor lineAL^P_Z of any element Z∈×_+.The aim of this section is to provide an estimate on the tail distribution of the hitting time between the ancestor lines started from two points at a distance d>0 on the line L_0. Without restriction, we consider Z=(0,0) and Z'=(d,0), and denote their ancestor lines by (Z_k, k≥ 0) and (Z'_k, k≥ 0). Let us denote byD_k= Z_k'-Z_k, the distance between the two paths at level h_k. The result proved in this section is the following: Let us define by τ=inf{k∈, D_k= 0}. There exists C>0 such that for K∈∖{0},(τ>K) ≤C d/√(h_K). The remaining of the section is devoted to the proof of Prop. <ref>. In the proof, we will also need the following quantity for k≥ 1:Δ_k= D_k-D_k-1. The proof is divided into several steps. For the first step, we consider a PPP with an intensity constant and equal 1. In doing so, we introduce a sort of companion model that will help finding estimates for the planar model considered above.We will then proceed to the control of the hitting time of two ancestors lines, by using some rescaling properties.§.§.§ Step 1: Evolution of the distance in one step, when the intensity is 1Take two points Z=(0,0) and Z'=Z(d,0) at distance d in L_0. Assume that the PPP Υ_1 on L_1 has intensity 1: let (X_1,h_1)=α^P(Z) be the closest point to Z in Υ_1 and by (X_2,h_1)=α^P(Z') the closest point to Z' in Υ_1. LetD(d)=X_2-X_1=|α^P(Z')-α^P(Z)|be the “new distance”, and denote byΔ(d)=D(d)-dthe variation of the distance between the levels L_0 and L_1.The distribution μ_d of Δ(d) is the following probability measure on μ_d(du) = (d+1)e^-2dδ_-d(du)+ f_d(u) _[-d,d](u)du+de^-u-d_[d,+∞)(u)du , wheref_d(u)=-e^-2d/2 +e^-2|u|(|u|+1/2).The atom m_d= (d+1)e^-2d of μ_d at -d corresponds to case where coalescence occurs, that is α^P(Z)=α^P(Z'). Apart from this atom, μ_d is absolutely continuous with respect to the Lebesgue measure, and (Δ(d)) = 0and (D(d))=d (Δ(d)) = V(d)=1-e^-2 d+2/3 e^-2 dd^3+e^-2 dd^2.The proof is postponed to the end of the section.Notice that the distribution of Δ(d) does not depend on the height h_1 of the level L_1, and that f_d is a symmetric function.§.§.§ Step 2: The sequence (D_k,k≥ 0) is a martingale Denote by D^λ(d)=X_2-X_1=α^P(Z')-α^P(Z) the new distance if the PPP Υ_1 has intensity λ>0, and by Δ^λ(d)=D^λ(d)-d. A simple scaling argument allows one to relate the distribution of Δ^λ(d) to that of Δ^1(d)Δ(d) : Δ^λ(d)1/λΔ^1(λ d),d≥ 0,Now, since the intensity on L_k is n_k, conditional on D_k-1, ccl D_k D^n_k(D_k-1) Δ_k Δ^n_k(D_k-1)1/n_kΔ^1( D_k-1 n_k). Because of (<ref>), (D_k, k≥ 0) defines a martingale. The particular form ofμ_d makes it difficult to control the time at which it hits 0. We will dominate D_k by another martingale that is easier to handle.§.§.§ Step 3: Introduction of an auxiliary distribution μ_d We introducethe following family of distributions indexed by d>0:μ_d(du):=α_d δ_-(c_d+d)(du)+f_d(u)_[-d,d](u)du + de^-u-d_[d,+∞)(u)du+ β_d e^-u-d_[d +ν_d,+∞)(u)du. Let Δ(d) be a r.v. with distribution μ_d, and set D(d) to be the r.v. defined byΔ(d)=D(d)-d.Our strategy is as follows: we will choose carefully the functions α_d,c_d,r_d,β_d satisfying for any d>0, α_d ≤ (d+1)e^-2d, c_d ≥ 0,ν_d≥ 0,β_d ≥ 0.in such a way that for any d>0, μ_d is a probability distribution with mean 0, and which will dominate μ_d in the sense of the forthcoming Lemma <ref>. The difference between μ_d and μ_d is that the atom at -d in μ_d is replaced by an atom at -c_d-d<-d with, as a counterpart, a modification of the distribution at the right of d which is replaced by a distribution larger for the stochastic order.Proceeding like this, our idea is to bound stochastically the hitting of 0 by (D_k,k≥ 0) (the coalescing time τ) by the hitting time of (-∞,0) by an auxiliary Markov chain (D_k,k≥ 0).The measure μ_d is a density probability with mean(Δ(d))=0 iffα_d = e^-2d(1+d)(1+2d+ν_d)/1+2d+c_d+ν_dβ_d = e^ν_d(1+d)c_d/1+2d+c_d+ν_d.Compute the total mass of μ_d: ⟨μ_d,1⟩= β_d e^-2d-ν_d+α_d+1-e^-2dd-e^-2dand if the total mass is 1, the expectation of Δ(d) is : (Δ(d))=- α_d(c_d +d)+ d(d+1)e^-2d + β_d(d+ν_d+1) e^-2d-ν_d. Solving these equations in α_d and β_d provides the announced result. We hence see that we have two degrees of freedom. In the sequel, we will choose:c_d=1,ν_d=2, independent of d. This implies that:α_d=e^-2d2d^2+5d+3/2(d+2),β_d=e^2 d+1/2(d+2). From this, we can compute (Δ(d)):(Δ(d)) =1+( 4 d^4+26 d^3+66 d^2+75 d+27 ) e^-2 d/6 d+12. For the measure μ_d which we have now completely constructed, we have:Under (<ref>) and (<ref>), we have for d<d',D(d) _D(d)>0≤ _S D(d') _D(d')>0,in the sense that for all t>0, (D(d)>t)≤(D(d')>t).First, for any d', by construction of the measure μ_d',D(d')_D(d')>0≤ _S D(d')_D(d')>0.Now, recall that D(d) provides the new distance in the model of Step 1 when the intensity of the PPP on L_1 is 1 and when the starting points Z=(0,0) and Z'(d,0) are at distance d. One can follow a third point Z”(d',0). Since these paths do not cross, the distance D_1 D(d) between Z_1 and Z'_1 remains smaller than the distance D'_1=Z”_1-Z_1 D(d'). This implies that D(d)_D(d)>0≤ _SD(d')_D(d')>0 holds. This concludes the proof. §.§.§ Step 4: Introduction of an auxiliary Markov chain To dominate (D_k,k≥ 0) we introduce the Markov chain (D_k, k ≥ 0) whose distribution respects the same scaling eq:efgz as (D_k,k≥ 0): conditionally on D_k-1, we let Δ_k= D_k-D_k-1 have distributionΔ_k1/n_kΔ( D_k-1 n_k).Let us defineτ_^-=inf{k ≥ 0 : D_k≤ 0},τ_^-=inf{k ≥ 0 : D_k≤ 0} .For any d>0, if D_0=D_0=d, we haveτ_^-≤_S τ_^-.Just observe that in (<ref>), the r.v. in both sides have atoms at 0 that correspond to the entrance of D_k and D_k in (-∞,0] (in fact the hitting time of {0} for D_k and of (-∞,0) for D_k). The Markov property and (<ref>) allow to conclude. §.§.§ Step 5: Skorokhod embeddingBy the Skorokhod embedding theorem (see <cit.>), there exists a BM B started at 0 and a stopping time T_1(d) such that Δ(d)(d)= B_T_1(d). Moreover, it is possible to construct two r.v. U(d)≤ 0 and V(d)≥ 0 such that T_1(d)=inf{t≥ 0, B_t∉ [U(d),V(d)]}. U(d) and V(d) are independent from the BM B, but not independent (in general) one from the other. Since B_T_1(d)=U(d)≤ 0 or B_T_1(d)=V(d)≥ 0, U(d) and V(d) can be constructed from the distribution of B_T_1(d)(d)=Δ(d), i.e. μ_d, as follows (recall (<ref>)):*With probability 2p_d, U(d)=-V(d) and V(d) is a r.v. with density _[0,d](v) (e^-2v(v+1/2)-e^-2d/2) /p_d. We denote by A_d this event.*With probability 1-2p_d=(1+2d)e^-2d, on A^c_d, we set U(d)=-d-1. For V(d), we have two cases since the right tail of μ_d is the sum of two exponential tails, de^-d-u_[d,+∞)(u) and β_d e^-d-u_[d+ν_d,+∞)(u). Conditionally on A^c_d:∙ With probability q_d=2d/(1+2d), V(d) is a r.v. with density v+d+1/2(1+d)e^d-v_[d,+∞)(v) with respect to the Lebesgue measure. We call this event E_d.∙ With probability 1-q_d=1/(1+2d), V(d) is a r.v. with density v+d+1/3+2d e^d+2-v_[d+2,+∞)(v). This event is E^c_d∩ A^c_d.Recall from (<ref>) that μ_d admits a symmetric density f_d on [-d,d]. Thus, on the event A_d={|Δ_d|<d}, which has probability2p_d= ∫_-d^d f_d(u)du=1-e^-2d-2d e^-2d, it is sufficient to define U(d)=-V(d) with V(d) a r.v. of density f_d(v)/p_d_[0,d](v). Since the Brownian motion B started at 0 exits the symmetric interval [-V(d),V(d)] through the upper or lower bound with equal probabilities 1/2, the likelihood of B_T_1(d) for this part is as expected:2p_d (1/2f_d(v)/p_d_[0,d](v)+1/2f_d(-v)/p_d_[-d,0](v))=f_d(v)_[-d,d](v).Let us now consider A_d^c ∩ E_d. The lower bound is necessarily U(d)=-d-1, since it is the only possible value for Δ_d below -d. As for the density of V(d) conditionally to A_d^c∩ E_d, say g(v), it has to be chosen such that we recover d e^-d-u_[d,+∞)(u) once multiplied by (1-2p_d), q_d and by the probability that B exits through the upper bound V(d) rather than through the lower bound U(d)=-d-1:(1-2p_d) q_d d+1/v+d+1 g(v)=d e^-d-v_[d,+∞)(v)⇒ q_d g(v)= e^2d/1+2dv+d+1/d+1 d e^-d-v_[d,+∞)(v). Since g is a probability density, integrating over v gives q_d: q_d=2d/(1+2d). We then deduce the density of V(d) conditionally to A_d^c∩ E_d. We proceed similarly for A_d^c∩ E_d^c. By recursion, we can define for k≥ 1 the time T_k byT_k=inf{t≥ T_k-1, B_t-B_T_k-1∉ [U_k(D_k-1),V_k(D_k-1)]} where U_k(D_k-1) and V_k(D_k-1) are independent r.v. conditionally on D_k-1, such that for any D>0,U_k(D)(d)=1/n_k U(n_k D), V_k(D)(d)=1/n_k V(n_k D), where U(d) and V(d) have the law described above in the representation of T_1(d) for d>0. With this construction, we have that for k≥ 1, B_T_k(d)=D_k-d.§.§.§ Step 6: Laplace transforms of T_1(d) and T_k(d):For λ>0, there exists c_0(λ)∈ (0,1) independent of d such that0≤φ_d(λ)=(e^-λ T_1(d))≤ c_0(λ)<1. Moreover, for λ small, there exists a constant C>0 such that c_0(λ)≤ e^-Cλ. Using the Skorokhod embedding described above,φ_d(λ) =(e^-λ T_1(d) | A_d ) 2p_d + (e^-λ T_1(d) | A^c_d , E_d)(1-2p_d)q_d + (e^-λ T_1(d) | A^c_d , E_d^c)(1-2p_d)(1-q_d).Our purpose is to bound φ_d(λ) uniformly in d by a constant strictly smaller than 1. On the events, A_d and A^c_d ∩ E_d, the interval [U(d),V(d)] which defines T_1(d) has at least one extremity that gets closer and closer to zero when d tends to zero. So upperbounding the expectations in the first and second terms of the r.h.s. of (<ref>) by a constant strictly less than 1 uniformly in d is difficult. For the third term of (<ref>) however, because U(d)<-c_d=-1 and V(d)>ν_d=2, we have that(e^-λ T_1(d) | A^c_d , E_d^c)≤(e^-λ T')<1where T'=inf{t≥ 0, B_t ∉ [-1,2]}. Additionally, since (1-2p_d)(1-q_d)=e^-2d→_d→ 0 1, this shows (<ref>) withc_0(λ)=_0(e^-λ T') =cosh(√(λ/2))/cosh(3√(λ/2))<1. When λ→ 0, c_0(λ)=1-2λ+o(λ) ≤ e^-2λ which shows the second assertion with C=2.From this by using (<ref>) and the self-similarity of the standard BM started at 0,T_k-T_k-1(d)= inf{t≥ 0, B_t ∉[1/n_kU(n_k D_k-1),1/n_kV(n_k D_k-1)]} (d)= inf{t≥ 0, 1/n_k B_n_k^2 t∉[1/n_kU(n_k D_k-1),1/n_kV(n_k D_k-1)]} (d)= 1/n_k^2 T_1(n_k D_k-1). Hence it follows that(e^-λ (T_k-T_k-1) | ℱ_T_k-1)=φ_n_k D_k-1(λ/n_k^2). §.§.§ Step 7: Estimatefor the tail distribution of the coalescing timeWith the ingredients developed above, we can now follow ideas developed in <cit.> for instance. Recall that τ__-=inf{k∈, D_k≤ 0} and define θ=inf{t≥ 0, B_t=-d}. Let us consider ζ>0. Then for K∈∖{0}:(τ__->K)=(θ > T_K) ≤ (θ > ζ h_K)+(θ>T_K, T_K < ζ h_K) ≤ Cd/√(ζ h_K)+ e^λζ h_K(e^-λ∑_k=1^K (T_k-T_k-1)).For the Laplace transform in the last term, using (<ref>):(e^-λ∑_k=1^K (T_k-T_k-1)) =(_d(e^-λ∑_k=1^K (T_k-T_k-1) | ℱ_T_K-1))=(e^-λ∑_k=1^K-1 (T_k-T_k-1)(e^-λ (T_K-T_K-1) | ℱ_T_K-1)) =_d(e^-λ∑_k=1^K-1 (T_k-T_k-1)φ_n_K D_K-1(λ/n_K^2)) ≤c_0 (λ/n_K^2)_d(e^-λ∑_k=1^K-1 (T_k-T_k-1))≤∏_k=0^K-1c_0 (λ/n_k+1^2) ≤ exp(-2 λ∑_k=1^K1/n_k^2).Recall from (<ref>) that h_K=V_K∼∑_k=1^K 1/2n_k^2. Thus, from (<ref>) and (<ref>):(τ__->K)≤Cd/√(ζ h_K)+ C'exp(λ h_K(ζ-4)). Because h_K→ +∞, and because the term in the exponential is negative for ζ sufficiently small, there exists λ_0>0 and ζ_0>0 such that the r.h.s. of (<ref>) is smaller than C” d/√(ζ_0 h_K) for K large enough.This together with Proposition <ref> allow to conclude the proof of Proposition <ref>. Starting from two points Z and Z' of L_0 at distance d and denoting by τ the index of the level at which they coalesce, we have for any K∈∖{0},(τ>K)≤(τ__->K)≤Cd/√(h_K). Let us finish this subsection with the proof of Proposition <ref> that had been postponed. First, notice that X_1 has density e^-2|x|_x ∈. Then, we can compute the distribution of X_2 conditionally on X_1. In what follows, all r.v. are independent, R is a Rademacher r.v., (k) denotes an exponential r.v. with expectation 1/k.– Conditional on X_1=x_1>0, with x_1<d:∙X_2=-(d-x_1) (merge) with probability e^-2(d-x_1), ∙ with probability1-e^-2(d-x_1), X_2∼ L(R(2) | (2) <d-x_1).– Conditional on X_1=-x_1 <0, with x_1<d ∙X_2=-(d+x_1) (merge) with probability e^-2(d-x_1)-2x_1 ∙X_2∼ L(R(2) | (2) <d-x_1) with probability1-e^-2(d-x_1) ∙X_2∼ L((1)+d-x_1 | (1) <2x_1) with probabilitye^-2(d-x_1)(1-e^-2x_1) – Conditional on X_1=x_1>0, with x_1>d:∙merge with probability1– Conditional on X_1=-x_1 <0, with x_1>d ∙X_2=-(d+x_1) (merge) with probability e^-2d ∙X_2∼ L(x_1-d+ (1) | (1) <2d) with probability 1-e^-2d This yields the announced result. In particular, the two trajectories started at (0,0) and (d,0) merge at ordinate 1 with probability:`P(D(d)=0) = ∫_x=0^d e^-2x (e^-2(d-x)+e^-2(d-x)-2x)dx+∫_x=d^+∞e^-2x (1+e^-2d)dx,which is (d+1)e^-2d, as announced. §.§.§ Extension to the shifted cylinder We now conclude the section with a corollary establishing an estimate for the coalescence time in 𝒲^(j), which is the forest 𝒲 shifted by (0,-h_j) similarly to 𝒯^(j). Then, we enounce an estimate for the shifted cylindrical forest 𝒯^(j).Let d>0 and j∈. (i) Let us consider the paths in 𝒲^(j) started at (0,0) and (d,0) (if (0,h_j) and (d,h_j), these points are connected at the level j+1 to the closest point of 𝒲). Define their coalescing time as τ=inf{k≥ j, D_k=0}. There exists a constant C>0 such that for any K> j,(τ>K)≤Cd/√(h_K-h_j).This can be translated, for any t_0>0 as:(𝒲^(j)_(0,0)(t_0)≠𝒲^(j)_(d,0)(t_0))≤C d/√(h_R(h_j+t_0)-1-h_j)→_j→ +∞C d/√(t_0). (ii) Let us consider the paths of 𝒯^(j) started at (0,0) and (d,0) for d∈(0,1/2] (d=1/2 is the maximal distance in the cylinder). Then, there exists C>0 such that for any t_0>0:(𝒯^(j)_(0,0)(t_0)≠𝒯^(j)_(d,0)(t_0))≤C d/√(h_R(h_j+t_0)-1-h_j)→_j→ +∞C d/√(t_0).The proof of (i) is an adaptation of the proof Step 7 of Prop. <ref> by summing between levels L_j and L_K.Let us now consider (ii). Intuitively, the coalescence time in the cylinder is stochastically dominated by the coalescence time in the plane. But since some slices in the cylinder may contain no points of the PPP (when no line L_k in the plane is empty), and since the increments of the distance between the two paths are non standard when this distance is close to 0 and 1 (when only the case 0 matters in the plane), an additional argument is needed in the discrete case to establish the domination rigorously.Recall the model introduced in Section <ref>. We consider the Markov chain (D_k,k≥ 0) and denote by τ_d be stopping time at which the Markov chain started from d hits 0. We also introduce similarly the distance process (D̅_k,k≥ 0) in the cylinder.Now, let us define another Markov chain (D'_k,k≥ 0) with the following transitions: L(D'_k+1 | D'_k=d)=L( min(D_k+1,|1-D_k+1|) | D_k=d ). The distance D'_k somehow mimics the distance on the cylinder by considering the minimum distance between two points of the same level in the clockwise and counter clockwise senses. Let us define by τ'_d the stopping time at which (D'_k,k≥ 0) started from d hits 0. Since min(D_k+1,|1-D_k+1|)≤ D_k+1, and since τ_d ≤_S τ_d' when d<d',by using the same argument as in the proof of Lemma <ref>, we obtain by using iteratively eq:tjdktu thatτ'_d ≤_Sτ_d.To conclude, it remains to show that (D'_k,k≥ 0) coincides with (D̅_k,k≥ 0), up to a probability going to 0 in j. Since we may produce a local coupling between D'_k and D̅_k as long as D_k possesses small fluctuations, it suffices to prove that all the increments of the paths (Z̅_k, k≥ 0) and (Z̅'_k,k≥ 0) that define (D'_k,k≥ 0) in the cylinder are not 0 and smaller than 1/6 after the slice j with probability going to 1 when j→ +∞. This indeed guarantees that the cylinder effects do not prevent the coupling: no jumps “0” occur and “decision domains” do not see that the environment is a cylinder. The probability that there is no point within distance ± 1/6 for a walk is e^-n_k/3, and by Borel-Cantelli's lemma, with probabilty 1 the two walks(Z̅_k) and (Z̅'_k) will do a finite number of jumps larger than 1/6. Hence, for any `e>0, for j large enough, the distribution of (D'_k,k≥ 0) and (D̅_k,k≥ 0) coincides with probability at least 1-`e. Thus the coupling works, which allows to conclude.We have now the tools to prove the criteria of the convergence Theorem <ref>, (IO) and (EO). Both of these criteria make use of the estimates on coalescing time that we hve just established. §.§ Proof of (IO)The purpose of this section is to prove the next Proposition which implies (IO).Assume eq:newcond. Let m∈∖{0} and y_1=(x_1,t_1),…,y_m=(x_m,t_m)∈^+. For j≥ 0 and 1≤ℓ≤ m, let us denote by γ^(j)_y_ℓ the path interpolating linearly the shifted ancestor line AL^(j)_y_ℓ+(0,h_j). Then, the sequence (γ^(j)_y_1,…γ^(j)_y_m) converges in distribution, when j→ +∞, to coalescing Brownian motions modulo 1 started at y_1,… y_m. Notice that the path γ^(j)_y_ℓ starts at y_ℓ. We also recall that the ancestral line AL_y_ℓ+(0,h_j) does not necessarily starts from a point of Ξ, but links the starting point y_ℓ+(0,h_j) to the closest point of Ξ in the first non-empty slice of height greater or equal to R(t_ℓ+h_j). For the sequel, let us denote by y_j,ℓ this point. The result for m=1 is due to Lemma <ref> and the fact that y_j,1-(0,h_j) converges a.s. to y_1. The proof can be done by recursion, and we focus here on the case m=2 which can be generalized directly by following Arratia <cit.> and Ferrari, Fontes and Wu <cit.>.Let us first recall a simple fact. Let t≤ t' and a,b ∈. Two BM (^_(a,t)(s),s≥ t) and (^_(b,t')(s),s ≥ t') on the cylinder are said to be coalescing BM if ^_(a,t)(s) for t≤ s ≤ t' is a standard BM taken modulo 1, and if the two trajectories (^_(a,t)(s),s≥ t') and (^_(b,t')(s),s ≥ t') are BM till their hitting time τ. After this time, they coincide with (^_(b,t')(s), s≥τ).To prove that (γ^(j)_y_1,γ^(j)_y_2) converges to two coalescing BM, a strategy consists in decomposing the trajectories as follows, where we can assume to simplify that y_1 and y_2 are such that t_1=t_2: (a) as long as the two paths are far apart, say if d_/ł(γ^(j)_y_1(s),γ^(j)_y_2(s))̊>a^j(s) for a good sequence a^j(s)→ 0, then the next steps of these trajectories are likely to be characterized by Ξ∩ I and Ξ∩ I' for two random influence intervals I and I' that will not intersect. By the spatial properties of the PPP, it means that as long as I∩ I'=∅,the two trajectories behave as if they were constructed on different spaces, and then eventually behave as independent BM before their coalescing time (here, since the intensity is not constant, a dependence in s is needed). (b)when eq:far fails (the two paths are close) then another argument is developed to prove that the two paths will merge with a probability going to 1, within a o(1) delay. This is given by the Corollary <ref>.It remains to see in details how (a) can be handled. Let us denote by (X̅^(j)_s)_s≥ t (as in (<ref>)) the path γ^(j)_(x,t). The distribution of the increment Δ X_j+k (with the notation of section <ref>) satisfies (|Δ X_j+k|≥ r)=e^-2n_k+j r so that for r=f_j+k/2, with the f_j+k's appearing in Assumption (<ref>), the event {∀ k≥ 0, |Δ X_j+k|≤ f_j+k/2} will occur a.s. for j large enough thanks to Borel-Cantelli's lemma. We then decree that two walks are close if when they get in the slice with intensity n_k+j, their distance is smaller than f_j+k, i.e. we choose a^j(s)=f_R(h_j+s)-1. This suffices to complete the proof.§.§ Proof of (EO)We follow the strategy developed in <cit.>: we first show that the sequence (𝒯^(j))_j≥ 1 satisfies (<ref>), stated below, from which (EO) follows. The sequence (𝒯^(j))_j≥ 1 satisfies for all t_0,t>0, a>0,lim sup_j→ +∞( η_𝒯^(j)^O(t_0,t ; [0 → a]) ) < +∞ .This implies that (𝒯^(j))_j≥ 1 satisfies (EO). The rest of this section is devoted to the proof of Proposition <ref>. Let us remark that, by translation invariance and linearity of the expectation, it is enough to prove (<ref>) for small values of a. Let us start with proving that (<ref>) implies (EO). We follow the corresponding proof in the planar context which is recalled at the end of Section <ref> (see also the end of Section 6.1 in <cit.> or Section 6 of <cit.>). Except that contrary to the planar context where explicit computation is possible, namely (η_W(t_0,t ; a,b))=(b-a)/√(π t) where W is the standard BW, another argument is needed to get the limit (<ref>) in the cylinder context. We then have to show:lim_ε→ 0( η^O_(t_0+ε,t-ε ; [a → b]) ) = ( η^O_(t_0,t ; [a → b]) ) . Let us consider t,t_0>0 and a,b∈. Let us first prove that there exists ε_0∈(0,t) such that( η^O_(t_0+ε_0,t-ε_0 ; [a → b]) ) < ∞ .Without loss of generality, we still write t_0 and t instead of t_0+ε_0 and t-ε_0. The inequality (be careful to the presence or not of the hat η on η ),η^O_(t_0,t ; [0 → 1])≤ η^O_(t_0,t ; [0 → 1/2]) + η^O_(t_0,t ; [1/2 → 1])implies by rotational invariance( η^O_(t_0,t ; [a → b]) ) ≤ 2 |a→ b| ( η^O_(t_0,t ; [0 → 1/2]) ) .We finally get (<ref>) in combining the fact that η^O_(t_0,t ; [0 → 1/2]) is stochastically dominated by η_W(t_0,t ; 0,1/2)) (already stated in (<ref>)) and(η_W(t_0,t ; 0,1/2)) = 1+(η_W(t_0,t ; 0,1/2)) = 1+1/2√(π t) . With η^O_(t_0+ε,t-ε ; [a → b]) η^O_(t_0,t-ε ; [a → b]), (<ref>) andlim_ε→ 0η^O_(t_0,t-ε ; [a → b]) = η^O_(t_0,t ; [a → b]) ,the Lebesgue's dominated convergence theorem applies and leads to the searched limit (<ref>). Mainly because there is no coalescence on the arc {t_0+t}× [a → b] for the trajectories starting before t_0 with probability 1, there exists a random ε>0 such that for any 0≤ε'≤ε, η^O_(t_0,t-ε' ; [a → b]) is equal to the limit η^O_(t_0,t ; [a → b]). This proves (<ref>). 0.3cmNow, let us show that (𝒯^(j))_j≥ 1 satisfies (<ref>). The strategy to get (<ref>) can be divided into two steps. First, we bound from above the expectation in (<ref>) by twice the expected number of remaining paths γ at time t_0+t which are born before t_0 and such that γ(t_0)∈ [0→ a], i.e. 2 (η^O_𝒯^(j)(t_0,t;[0 → a])). See Lemma <ref>. Thus, using the coalescence time estimate (Corollary <ref>), we obtain an upper bound for this latter expectation when j→ +∞. This is Lemma <ref>. The various lemma on which the proof of (<ref>) is based are proved at the end of the present section. For all times t_0,t>0, for all a>0 and any j≥ 1, the following inequality holds:ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤ 2ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊ , from which we deduce that:lim sup_j→ +∞ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤ 2 lim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊, where 𝒲^(j) is the shifted forest introduced in Section <ref>. In view of (<ref>) of Lemma <ref>, we now focus on showing thatlim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊<+∞ . Let us choose r∈∖{0} (intended to be large) and m(a,r):=min{m: m≥ ar}. We consider the grid(t_0,a,r) := ł{k/r, k∈{0,… r_a}}̊×{h_R(h_j+t_0)-1-h_j} .The height h_R(h_j+t_0)-1-h_j corresponds to the (shifted) largest slice just before height h_j+t_0 (possibly h_j+t_0 itself): see Figure <ref>. Since the sequence h_k-h_k-1=σ^2_k tends to zero by (<ref>), there exists j_0 such that, for any j≥ j_0, there is a slice carrying points between t_0 and t_0+t in 𝒲^(j), i.e. t_0<h_R(h_j+t_0)-h_j<t_0+t.Let us focus on the paths starting from the points of (t_0,a,r). For 0≤ k≤ m(a,r), denote by γ_k(.) the ancestor line starting at (k/r, h_R(h_j+t_0)-1-h_j). Even if the points (k/r, h_R(h_j+t_0)-1) do not belong to the point process Υ defined in Section <ref>, they connect to the nearest point of Υ∩ L_R(h_j+t_0). So each path γ_k(.) a.s. coincides with a path of 𝒲^(j) after one step. Let us define the eventA_a,j,r := {[ 𝒲^(j)∩([0, a]×{h_R(h_j+t_0)-1-h_j}); (t_0,a,r) ]} .The event A_a,j,r is described in Figure <ref>.We claim that when the mesh 1/r of the grid (t_0,a,r) tends to 0, the probability of A_a,j,r tends to 1: For all times t_0,t>0, for all a>0 and any j≥ j_0,lim_r→ +∞(A_a,j,r^c) = 0 . The event A_a,j,r has been introduced in order to compare η_𝒲^(j)(t_0,t;0 , a) to the number of remaining paths at height t_0+t, starting from the deterministic points of (t_0,a,r). Then, the coalescence time estimate (Corollary <ref>) leads to the following bound: For all times t_0,t>0, for all a>0, there exists a constant C>0 and an integer j_1=j_1(t_0,t) (which does not depend on r) such that for any j≥ j_1 and any r∈∖{0},( η_𝒲^(j)(t_0,t ; 0,a) _A_a,j,r) ≤ 1 + 2 m(a,r) C/r √(t) ,where C is the universal constant given by Corollary <ref>. We can now conclude. Let t_0,t>0 and a>0. First, we bound (η_𝒲^(j)(t_0,t;0, a)) from above by(η_𝒲^(j)(t_0,t;0,a) _A_a,j,r) + M ( A_a,j,r^c) + ( |Υ_R(h_j+t_0)-1([0,a])| _{|Υ_R(h_j+t_0)-1([0,a])|>M}) ,for any j,M,r. Let j≥ j_0∨ j_1. Then Lemma <ref> applies and provides a bound for the first term of (<ref>). Then, take M=M_j:=2 n_R(h_j+t_0)-1 (twice the intensity of the PPP). For j large enough and with this choice of M, the third term of (<ref>) is smaller than 1. It then follows:ł(η_𝒲^(j)(t_0,t;0,a) )̊≤ 2 + 2 m(a,r) C/r √(t) + M_j ( A_a,j,r^c) .Let us point out that till now, the parameter r is totally free. So we can choose it large enough so that m(a,r)/r≤ a+1/r≤ 2a and M_j (A_a,j,r^c) ≤ 1 (by Lemma <ref>). In conclusion, for any j large enough, (η_𝒲^(j)(t_0,t;0,a)) is bounded by 3 + 4 a C/√(t). This gives (<ref>), ending the proof of Proposition <ref>.This section ends with the proofs of Lemmas <ref>, <ref> and <ref>.Let us first prove (<ref>). We denote, for an interval I of , by η^O_𝒯^(j)(t_0,t ; I ; [0 → a]) the number of paths γ∈𝒯^(j) born before t_0 and such that γ(t_0)∈ I and γ(t_0+t)∈ [0→ a]:η^O_𝒯^(j) (t_0,t ;I ; [0 → a]) := {γ(t_0+t)∈ [0→ a], γ∈𝒯^(j),γt_0, γ(t_0)∈ I} .Then, the following inequality holds almost surely:η^O_𝒯^(j)(t_0,t ; [0→ a])≤∑_k=0^[1/a]η^O_𝒯^(j)(t_0,t ; [ka → (k+1)a] ;[0→ a]),where [1/a] is the integer part of 1/a. The inequality (<ref>) is due to the fact that two paths starting from different intervals [ka → (k+1)a] and [ℓ a → (ℓ+1)a] can coalesce and give a single point in the l.h.s. while they are counted twice in the r.h.s. Notice that when a is not the inverse of an integer, the right hand side is a bit larger than what is needed, because the first and last intervals intersect. We conclude using the rotational invariance:(η^O_𝒯^(j)(t_0,t ; [0→ a]))≤ ∑_k=0^[1/a](η^O_𝒯^(j)(t_0,t ; [ka → (k+1)a] ; [0→ a]))=∑_k=0^[1/a](η^O_𝒯^(j)(t_0,t ; [0→ a] ; [-ka → -(k-1)a]))=(η^O_𝒯^(j)(t_0,t ; [0→ a])) +( η^O_𝒯^(j)(t_0,t ; [0→ a] ; [1-a[1/a]→ a]))≤ 2(η^O_𝒯^(j)(t_0,t ; [0→ a])) .since the arc [1-a[1/a]→ a] is included in [0→ a]. This proves (<ref>). We now turn to the proof of (<ref>). Having proved (<ref>), it is sufficient to show thatlim sup_j→ +∞ł(η^O_𝒯^(j)(t_0,t;[0 → a]))̊≤lim sup_j→ +∞ł(η_𝒲^(j)(t_0,t;0 , a))̊ .For this, we construct a coupling between 𝒲^(j) and 𝒯^(j). Let us introduce the following event:E_j := {[ k≥ j, (h_k) ]} .Notice that (E^c_j)≤∑_k≥ je^-n_k which converges to zero when j→ +∞, and by Borel Cantelli's lemma and (<ref>), there exists a random level J, finite a.s., such that E_J holds. Following the idea in (<ref>), we have:(η^O_𝒯^(j)(t_0,t;[0 → a])) ≤(η^O_𝒯^(j)(t_0,t;[0 → a])_E_R(h_j+t_0)-1)+ M ( E_R(h_j+t_0)-1^c) + ( |Ξ_R(h_j+t_0)-1([0,1])| _{|Ξ_R(h_j+t_0)-1([0,1])|>M}) . Choosing M=M_j=2n_R(h_j+t_0)-1, we can control the third term as in (<ref>). For this choice of M=M_j, the second term is upper bounded by 2n_R(h_j+t_0)-1∑_k≥ R(h_j+t_0)-1e^-n_k which converges to zero when j→ +∞ by (<ref>). Now for the first term in the r.h.s. of (<ref>), let us prove that(η^O_𝒯^(j)(t_0,t;[0 → a])_E_R(h_j+t_0)-1)≤(η_𝒲^(j)(t_0,t;0 , a)),in which case, taking the lim sup in (<ref>) when j→ +∞ gives (<ref>). To show (<ref>), we produce a coupling between 𝒲^(j) and 𝒯^(j) ensuring that on E_R(h_j+t_0)-1,η^O_𝒯^(j)(t_0,t;[0 → a]) ≤η_𝒲^(j)(t_0,t;0 , a)).Consider the paths of 𝒲^(j) touching [0,a]×{t_0} and that survive until level t_0+t. Let us denote by K the random number of points in [0,a]×{t_0} corresponding to these paths (K≥η_𝒲^(j)(t_0,t;[0 → a])) and let us call 0≤ a_1<… <a_K≤ a the abscissa of these points. Recall that if Z'=α(Z)∈ L_k+1 is the ancestor of Z∈ L_k, then base the isosceles triangle with apex Z and admitting Z' as other vertex contains only one atom of Ξ_k+1: Z'. This triangle is called the influence triangle of Z. Let us denote by θ_L the left border of the influence region of 𝒲^(j)_(a_1,t_0) (i.e. the union of the influence triangles of the vertices constituting 𝒲^(j)_(a_1,t_0)). We consider the point process on the cylinder consisting of the atoms of Ξ in the regionℛ_j = {(x,t)∈× [h_j+t_0,+∞), θ_L(t)≤ x<θ_L(t)+1 } ,which is a PPP conditioned on E_R(h_j+t_0)-1. On this PPP, let us construct the corresponding forest 𝒯^(j) and compute the r.v. η^O_𝒯^(j)(t_0,t;[0 → a]).Let κ∈{1,… K} be the (random) index k of the rightmost path 𝒲^(j)_(a_k,t_0) that does not intersect the border θ_L(.)+1. By construction, all the paths 𝒲^(j)_(a_k,t_0) for k∈{1,…κ} are unchanged on the cylinder (meaning that the successive ancestors of (a_k,t_0) remain the same as in the plane). Hence, among the paths started at (a_1,t_0),…,(a_κ,t_0), the remaining ones at level t_0+t are the same in the plane and in the cylinder: η^O_𝒯^(j)(t_0,t;[0 → a_κ]) and η_𝒲^(j)(t_0,t ; 0,a_κ) are equal. If κ = K then (<ref>) is proved. Else, let us consider 𝒲^(j)_(a_κ+1,t_0) and denote by Z the first vertex of this path (in the plane) such that Z∈ℛ_j and Z'=α(Z)∉ℛ_j. In other words, 𝒲^(j)_(a_κ+1,t_0) intersects θ_L+1 for the first time when going from Z to Z'. In the cylinder, the paths 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0) (if they exist) are all trapped between 𝒯^(j)_(a_κ+1,t_0) and 𝒯^(j)_(a_1,t_0). Two cases may be distinguished:∙ If 𝒯^(j)_(a_κ+1,t_0) coalesces with 𝒯^(j)_(a_1,t_0) before time t_0+t then the same holds for 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0). In this case, the contribution of 𝒯^(j)_(a_κ+1,t_0),…,𝒯^(j)_(a_K,t_0) to η^O_𝒯^(j)(t_0,t;[0 → a]) is null. So,η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ]) =η_𝒲^(j)(t_0,t ; 0,a_κ) ≤ η_𝒲^(j)(t_0,t ; 0,a) .∙ Since the influence triangle of the vertex Z overlaps the influence region of 𝒯^(j)_(a_1,t_0) then all the paths 𝒯^(j)_(a_κ+2,t_0),…,𝒯^(j)_(a_K,t_0) have to coalesce with 𝒯^(j)_(a_κ+1,t_0) or 𝒯^(j)_(a_1,t_0) before time t_0+t. Either 𝒯^(j)_(a_κ+1,t_0) coalesces with 𝒯^(j)_(a_κ,t_0) before time t_0+t and then η^O_𝒯^(j)(t_0,t;[0 → a]) is still smaller than η_𝒲^(j)(t_0,t ; 0,a) (as in the first case). Or, 𝒯^(j)_(a_κ+1,t_0) does not coalesce with 𝒯^(j)_(a_κ,t_0) before time t_0+t and η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ])+1. This also prevents the planar paths 𝒲^(j)_(a_κ+2,t_0),…,𝒲^(j)_(a_K,t_0) to coalesce with 𝒲^(j)_(a_κ+1,t_0). Their contribution to η_𝒲^(j)(t_0,t ; 0,a) is at least 1:η^O_𝒯^(j)(t_0,t;[0 → a]) = η^O_𝒯^(j)(t_0,t;[0 → a_κ]) + 1 ≤η_𝒲^(j)(t_0,t ; 0,a) .This shows (<ref>) and concludes the proof of the Lemma.Let j≥ j_0. On the event A_a,j,r^c, there exists a point of 𝒲^(j) at height h_R(h_j+t_0)-h_j, say Z, which is the ancestor of an element of 𝒲^(j)∩([0, a]×{h_R(h_j+t_0)-1-h_j}) but of none of the points of the grid Gr(t_0,a,r). This occurs only if Z belongs to the segment [0, a]×{h_R(h_j+t_0)-h_j} and is surrounded by two other points of 𝒲^(j) on the same level which are very close to it. Precisely, A_a,j,r^c implies the existence of a segment in [-1/r, a+1/r]×{h_R(h_j+t_0)-h_j} with length 2/r and containing at least 3 points of the Poisson point process Υ_R(h_j+t_0) of intensity n_R(h_j+t_0). The number of points of Υ_R(h_j+t_0) in [-1/r, a+1/r]×{h_R(h_j+t_0)-h_j} being a Poisson r.v. with parameter (a+2/r) n_R(h_j+t_0)-1, we can deduce that the minimum distance between two consecutive points of this PPP possesses a density. Thus, the probability of A_a,j,r^c tends to 0 as r→+∞.This last proof is based on the proof of Lemma 2.7 in <cit.>. Let us denote by η_r,j(t_0,t;0,a) the number of remaining paths γ_k(.), k=0,…,m(a,r), in 𝒲^(j) at time t_0+t, that started from the grid Gr(t_0,a,r). The event A_a,j,r has been introduced in order to write:η_𝒲^(j)(t_0,t ; 0,a) _A_a,j,r≤η_r,j(t_0,t;0,a) .Besides, the number of paths counted by η_r,j(t_0,t;0,a) is upper bounded by the number m(a,r)+1 of paths γ_k(.)'s starting from the grid Gr(t_0,a,r) minus the number of pairs (k,k+1) that have coalesced before height t_0+t, i.e.η_r,j(t_0,t;0,a) ≤ (m(a,r)+1) - ∑_k=0^m(a,r)-1_γ_k(t_0+t)=γ_k+1(t_0+t) .Using (<ref>), we have for sufficiently large j:( ∑_k=0^m(a,r)-1_γ_k(t)=γ_k+1(t)) = m(a,r) ( 1 - (γ_0(t_0+t)≠γ_1(t_0+t)) ) ≥m(a,r) - m(a,r) C/r √(h_R(h_j+t_0+t)-h_R(h_j+t_0)-1)≥m(a,r) - 2 m(a,r) C/r √(t) ,since h_R(h_j+t_0+t)-h_R(h_j+t_0)-1 tends to t as j→∞. Finally, (<ref>), (<ref>) and (<ref>) lead to the expected result.Just above Corollary <ref>, we observed that certain navigations on the cylinder can be sent onto navigations in the radial plane, and that both navigations are very similar in nature. In the whole paper, we often used that some structures can be transported from the plane onto the cylinder, and to the radial plane, provided the transport keeps the crucial features of the models considered. With the example adapted from the work of Coletti and Valencia <cit.>, in this Section <ref>, we illustrate how working on the cylinder allows us to state global convergence results for the radial tree correctly renormalized. However, in some cases such as the radial spanning tree (RST) of Baccelli and Bordenave <cit.>, the cylindrical forest can appear very complicated so that is can be easier to stick to the original radial problem, showing the limitations of this method.In the RST, a homogeneous PPP is given in the plane. A radial tree with vertex set the points of the PPP and rooted at the origin O is constructed. In this tree, the ancestor of a vertex x is the closest point of the PPP with smaller radius. When we send this tree and PPP in the cylinder, the circle of radius ρ is sent on the slice of height h(ρ). The resulting vertex set on the cylinder is not a homogeneous PPP. Additionally, the neighborhood of a given point becomes complicated in the cylinder, as shown in Fig. <ref>.10Arratia R. Arratia.Coalescing Brownian motions on the line.PhD thesis, University of Wisconsin, Madison, 1979.baccellibordenave F. Baccelli and C. Bordenave.The radial spanning tree of a Poisson point process.Annals of Applied Probability, 17(1):305–359, 2007.baccellicoupiertran F. Baccelli, D. Coupier, and V. Tran.Semi-infinite paths of the 2d-radial spanning tree.Advances in Applied Probability, 45(4):895–1201, 2013.berestyckigarbansen N. Berestycki, C. Garban, and A. Sen.Coalescing Brownian flows: a new approach.Annals of Probability, 43(6):3177–3215, 2015.billingsley1968 P. 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Turner.Scaling limits of anisotropic Hasting-Levitov clusters.Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 48(1):235–257, 2012.johanssonviklundsolaturner2 F. J. Viklund, A. Sola, and A. Turner.Small particle limits in a regularized Laplacian random growth model.Communication in Mathematical Physics, 334(1):331–366, 2015. ] | http://arxiv.org/abs/1707.08336v1 | {
"authors": [
"David Coupier",
"Jean-François Marckert",
"Viet Chi Tran"
],
"categories": [
"math.PR",
"60J05, 60G52, 60J65, 60D05, 60G57, 60E99"
],
"primary_category": "math.PR",
"published": "20170726094119",
"title": "Directed, cylindric and radial Brownian webs"
} |
http://arxiv.org/abs/1707.09247v2 | {
"authors": [
"J. R. Yusupov",
"D. M. Otajanov",
"V. E. Eshniyazov",
"D. U. Matrasulov"
],
"categories": [
"quant-ph",
"cond-mat.mes-hall",
"nlin.CD",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170727085802",
"title": "Classical and quantum dynamics of a kicked relativistic particle in a box"
} |
|
Poisson-Lie T-Duality in Double Field Theory Bernardo MonechiPietro Gravino Riccardo Di Clemente Vito D. P. ServedioReceived: date / Accepted: date ==================================================================================Semiconductor mode-locked diode lasers (MLLs) are compact, rugged and efficient sources of ultra-short, intense, and high repetition frequency optical pulses with many applications such as, all-optical clock recovery, Lidar, optical frequency combs, and telecommunications [1-3]. A major limitation of MLLs for most practical applications is their very high timing jitter and phase noise, as spontaneous emission noise and cavity losses make MLLs prone to broad linewidths and therefore substantial phase noise [4]. To improve timing jitter, several experimental methods such as single cavity feedback [5-8], coupled optoelectronic oscillators (OEOs) [9], injection locking [10-12] and dual-loop feedback [13-15] have been proposed and demonstrated. Of the stabilization techniques demonstrated to date, optical feedback is a promising approach in which an additional reflector creates a compound cavity with a high quality factor, with no need for an external RF or optical source. Due to the existence of the extra mirror, side-bands resonant with the round trip time of the external cavity are generated which affect the overall timing jitter and quality of the RF spectra. To overcome these issues, optoelectronic feedback [9] can also be utilized to stabilize timing jitter and to suppress cavity side-modes by conversion of the optical oscillation (using a fast photodetector) to an electrical signal used in a long feedback loop. This technique does not require an RF source, but requires optical-to-electrical conversion. Recently, a simpler dual-loop feedback technique [13-15] without optical/electrical conversion has been demonstrated to improve timing jitter of the MLLs and to filter or suppress the unwanted spurious side-bands. Dual-loop configurations proposed to date [13, 14] yields sub-kHz linewidth but produces additional noise peaks at frequencies resonant with the inverse of the delay time in the second cavity. This is undesirable in many applications where low noise and flat spectra are required, as in frequency comb generation. Recently the influence of the second feedback delay onside-mode suppression [16] and timing jitter [17] has been studied numerically. In this letter, we report experimental investigation to eliminate these adverse dynamical effects using asymmetric dual-loop feedback by appropriately choosing the length of second feedback cavity. Best side-mode suppression and lower timing jitter relative to single loop feedback is achieved with the length ratio between the two cavities ∼ 8x. It was further observed that RF linewidth and integrated timing jitter were reduced by increasing the length of the second cavity. Our findings suggest that noise stabilization and side-mode suppression depends strongly on additional feedback delay times.Devices under investigation are two-section InAs/InP quantum dash mode-locked lasers (QDash MLLs) whose active layers have 9 InAs quantum dash monolayers grown by gas source molecular beam epitaxy (GSMBE) embedded within two barriers and separate confinement heterostructure (SCH) layers (dash in a barrier structure). Both barriers and SCHs consisted of In_0.8Ga_0.2As_0.4P_0.6 quaternary materials with λ_g = 1.55 μm [18]. Total cavity length was 2030 μm with absorber lengths 240 μm (length ratio ∼ 11.8%), giving pulse repetition frequency ∼ 20.7 GHz (I_Gain = 300 mA)and average free-space output powers of a few mW. Mode-locking was obtained without reverse bias applied to the absorber section, and the heat sink temperature was fixed at 19^0C. This is a two-section device but works similarly to a single section self-mode-locked laser, since the absorber is not biased; in this case the amount of minimal residual absorption does not affect the mode-locking mechanism [12]. The absorber and gain sections were isolated by a resistance 9 kΩ. The QDash MLL was mounted p-side up on AlN submounts and copper blocks with active temperature control and electrical contacts formed by wire-bonding. A schematic for the dual-loop technique is depicted in Fig. 1. For single and dual-loop feedback configurations, a calibrated fraction of light was fed back through port 1 of an optical circulator, then injected into the laser cavity via port 2. Optical coupling loss from port 2 to port 3 was -0.64 dB. The output of the circulator was sent to a semiconductor optical amplifier (SOA) with gain of 9.8 dB, then split into two arms by a 50/50 coupler. 50% went to an RF spectrum analyzer (Keysight E-series, E4407B) via a 21 GHz photodiode and to optical spectrum analyzers (Ando AQ6317B and Advantest Q8384). The other 50% of power was split into two equal parts by a 3-dB splitter. For single loop feedback, all power passed through loop-I. For dual-loop configurations (feedback loops-I and-II) the power was split into two loops at the 3-dB splitter. Feedback strengths in both loops were controlled by variable optical attenuators and monitored using power meters. In this experimental arrangement, the length of loop-I was fixed to 160m while the length of second feedback loop was varied in three chosen lengths: 20, 53 and 80m. Polarization controllers in each loop plus one polarization controller before port 1 of the circulator ensured the light fed back through both loops matched the emitted light polarizations to maximize feedback effectiveness. In this experiment the feedback ratio into gain section was limited to ∼ -22 dB. We calculated RMS timing jitter from single sideband (SSB) phase noise spectra measured at the fundamental RF pulse repitition frequency (20.7 GHz) using:σ_RMS=1/2 π f_ML√(2 ∫_f_d^f_u L(f) df) where f_ML is the pulse repetition rate, f_u and f_d are the upper and lower integration limits. L(f) is the single sideband (SSB) phase noise spectrum, normalized to the carrier power per Hz. To measure RMS timing jitter of the laser in more detail, single-sideband (SSB) noise spectra for the fundamental harmonic repetition frequency were measured. To assess this, RF spectra at several spans around the repetition frequency were measured from small (finest) to large (coarse) resolution bandwidths. The corresponding ranges for frequency offsets were then extracted from each spectrum and superimposed to obtain SSB spectra normalized for power and per unit frequency bandwidth. The higher frequency bound was set to 100 MHz (instrument limited).To observe the RF spectrum with single-loop feedback, the length of the loop was initially set at 160m; optimally stable resonance occurred when the feedback length was fine tuned using an optical delay line (ODL-I) (which spanned 0 to 84 ps in steps of 1.67 ps). Such optimization provides a resonant condition (at delay setting=13 ps) under which the RF linewidth was reduced from 100 kHz free-running to as low as 4 kHz, with integrated timing jitter to 0.7 ps from 3.9 ps [integrated from 10 kHz - 100 MHz]. Measured phase noise traces as functions of frequency offset from fundamental mode-locked frequency and RF spectra for free running laser (green line) and single loop feedback (gray line) are given in Fig. 5 and Fig. 6, respectively. Under similar delay settings, external cavity side-modes appear in the RF spectrum with frequency spacing 1.28 MHz, the inverse of the loop round trip delay. RF spectra are shown in Fig. 2(a) (gray line) and (b) (gray line), using spans of 10 MHz and 100 MHz, respectively. Frequency resonances can be seen in both frequency spans which contribute significantly to timing jitter, particularly for the longer feedback cavities as they are closer to the main peak and are less suppressed [11]. To eliminate these fluctuations and to improve the side-mode suppression ratio (SMSR), dual-loop feedback was implemented as described in the next section.To assess the suppression of these frequency resonances, a shorter feedback cavity corresponding to half the period of noise-induced oscillations of loop-I was introduced. Feedback strengths of both cavities were equalized using variable optical attenuators (Att-I and Att-II) plus polarization controllers (PC-I and PC-II) in both loops. One optical delay (ODL-II) was adjusted to full resonance and the length of the other delay line (ODL-I) was tuned over the maximum range available 0-84 ps. When the optical delay lines ODL-I and ODL-II were fine tuned (ODL-I=15 ps and ODL-II=25 ps) so that every second mode of loop-I coincided precisely with a mode of loop-II, maximum >30 dB suppression in the first order side-mode was achieved. However, additional fluctuations (modal overlaps) appeared at frequencies resonant with the inverse of the length of second delay time which becomes the carrier signal. These noise fluctuations depends on the ratio of the loop lengths. Here these fluctuations at frequency spacing 2.60 MHz, are consistent with the length of the second feedback loop 80m. RF spectra of the asymmetric dual-loop configuration are shown in Fig. 2(a) (red line) and (b) (red line), using spans 10 MHz and 100 MHz, respectively. In this fully resonant configuration, the RF linewidth narrowed to < 1 kHz (instrument limited), with timing jitter reduced to 295 fs. Phase noise trace and RF spectra are shown in the Fig. 5 (blue line) and Fig. 6 (blue line), respectively. The RF spectra illustrated in Fig. 2(a) and (b) show that this feedback configuration is not suitable to achieve effective suppression in frequency resonances, as the second delay time will be resonant with the second mode of the first feedback loop which restrict many practical applications where flat and side-band free RF spectra are required. To improve on this situation, a different dual-loop feedback configuration with non-resonant shorter second loop (53m) was investigated, described in the next section. In the dual-loop configuration presented in this section, the length of loop-I was initially set to 160m while that of loop-II was 53m. Upon fine tuning of both optical delay lines (ODL-I=13 ps and ODL-II=15 ps), when the second delay time was resonant with the third harmonic of the first loop, suppression of the first two frequency resonances occurred, while the third harmonic (modal overlap) was unsuppressed. This harmonic was observed at frequency offset 3.9 MHz, corresponding to the 53m length of the outer feedback loop. RF spectra for the asymmetric dual-loop configuration are shown in Fig. 3(a) (red line) and (b) (red line), using spans 10 MHz and 100 MHz, respectively. In thisconfiguration, when both external cavities are fully resonant, the RF linewidth narrows to 2 kHz with integrated timing jitter 0.45 ps. Phase noise trace and RF spectra are shown in Fig. 5 (black line) and Fig. 6 (black line) respectively. These experimentally measured results show that external cavity side-modes cannot be optimally suppressed by simply choosing the second feedback delay time to be a fraction of the first. To achieve stable and flat RF spectra, we designed an asymmetric dual-loop feedback configuration for effective suppression of external cavity side-modes. Which produced flat RF spectra close to the main peak compared to conventional single- and dual-loop feedback.In this asymmetric dual-loop feedback configuration the length of loop-I was fixed (160m) andloop-II was set ∼ 8x shorter than loop-I. Fine tuning of both cavities (ODL-I=15 ps and ODL-II=21 ps) produced precise coincidence of every eighth mode of loop-I with a mode of loop-II, so that strong side-mode suppression occurred and all feedback-induced side-modes and spectral resonances were eliminated under frequency span 10 MHz. RF spectra for this dual-loop feedback configuration (red line) are shown in Fig. 4(a) and (b) with frequency spans 10 MHz and 100 MHz respectively. Furthermore, when both external cavities are fully resonant, the RF linewidth narrows to 8 kHz with integrated timing jitter 0.6 ps. The phase noise trace and RF spectra are shown in Fig. 5 (red line) and Fig. 6 (red line) respectively. In this configuration, the RF linewidth was higher than for single loop feedback, but measured timing jitter was lower; this is due to suppression of external cavity side-modes. In addition, weak modal overlap (with intensity ∼ -6 dBm) in RF spectra of dual-loop feedback was noticed at 10.2 MHz frequency spacing, consistent with our 20m outer loop; this is shown in Fig. 4(b)(red line). This behavior shows that effective suppression of external cavity side-modes and reduced timing jitter can be achieved by appropriately fine-tuning the length of the second feedback loop. It should be noted that length of loop-II (∼ 20m) is only optimal in our specific experimental setup. Further reduction in the length of second feedback loop is not possible, as the combined variable optical attenuator, optical delay line, polarization controller and 3-dB coupler have minimum length of ∼ 20m. Better suppression of external cavity side-modes could be achieved in an arrangement not subject to this limitation, such as photonic integrated circuit. Measured phase noise traces for free-running condition (green line), single-loop (gray line) and dual-loop feedback with loop-II at 20m (red line), 53m (black line) and 80m (blue line) as functions of frequency offset from the fundamental mode-locked frequency are given in Fig. 5 Comparison of RF linewidth and integrated timing jitter under stable resonant conditions, for three chosen lengths of second feedback cavity is shown in inset of Fig. 6. Measured integrated timing jitter in all dual-loop configurations was lower than for single-loop feedback. However, best suppression in external cavity side-modes was achieved with the second delay time ∼ 8x shorter than the first. Furthermore, the integrated timing jitter in this case was 16% lower than for single-loop feedback. Reduction in timing jitter occurs due to suppression of external cavity side-modes relative to single-loop feedback. In the literature [13, 14], it was experimentally observed that side-mode suppression was achieved when both feedback delays had a common multiple. This shows that effective suppression in external cavity side-modes is highly dependent on the length of the second loop. Recently, the influence of the second loop delay on suppression of external cavity side-modes [16] and timing jitter [17] was studied numerically. In this work, experimentally measured suppression of cavity side-modes and integrated timing jitter as a function of the second cavity delay correspond well with published numerical simulations [16, 17]. In summary, an asymmetric dual-loop feedback method has been demonstrated which suppress additional noise resonances found in conventional single- and dual-loop feedback schemes. These results show that dual-loop feedback with precise alignment of the second loop delay effectively suppresses external cavity side modes and produces flat RF spectra closer to the main peak. Furthermore, by increasing the length of the second loop, significant reduction in RF linewidth and integrated timing jitter was produced. Our experimental results have validated recently published numerical simulations. Using this method, stable side-band-free integrated photonic oscillators based on mode-locked lasers may be developed which are feasible and attractive for many applications in optical telecommunications, time-domain multiplexing, frequency comb generation and as synchronized pulse sources or multi-wavelength lasers for wavelength-diversity or multiplexing.Funding. The authors acknowledge financial support from Science Foundation Ireland (grant 12/IP/1658) and the European Office of Aerospace Research and Development (grant FA9550-14-1-0204).99 gallo99 A. Sano, E. Yamada, H. 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Quantum Electron. 19(4), 1101208 (2013). gallo99 E. Sooudi, G. Huyet, J. G. McInerney, F. Lelarge, K. Merghem, R. Rosales, A. Martinez, A. Ramdane, and S. P. Hegarty, “ Injection-Locking properties of InAs/InP based mode-locked quantum-dash lasers at 21 GHz,” IEEE Photonics Technol. Lett. 23(20), 1544–1546 (2011). gallo99 M. Haji, L. Hou, A. E. Kelly, J. Akbar, J. H. Marsh, J. M. Arnold, and C. N. Ironside, “ High frequency optoelectronic oscillators based on the optical feedback of semiconductor mode-locked laser diodes,” Opt. Express. 20(3), 3268–3274 (2012). gallo99 H. Asghar, E. Sooudi, P. Kumar, W. Wei, and J. G. McInerney, “ Optimum stabilization of self-mode-locked quantum dash lasers using dual optical feedback with improved tolerance against phase delay mismatch,” Opt. Express. 25(14), 15796–15805 (2017). gallo99 O. Nikiforov, L. Jaurigue, L. Drzewietzki, K. Lüdge, and S. Breuer, “ Experimental demonstration of change of dynamical properties of a passively mode-locked semiconductor laser subject to dual optical feedback by dual full delay-range tuning,” Opt. Express. 24(13), 14301–14310 (2016). gallo99L. Jaurigue,E. Schöll, and K. Lüdge, “Suppression of noise-induced modulations in multidelay systems “, PRL. 117, 154101 (2016). gallo99 http://dx.doi.org/10.14279/depositonce-5543 gallo99 F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. V. Dijk, D. Make, O. Le Gouezigou, J. Provost, F. Poingt, J. Landreau, O. Drisse, E. Derouin, B. Rousseau, F. Pommereau, and G. -H. Duan, “ Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55μm,” IEEE J. Sel. Top. Quantum Electron. 13(1), 111–124 (2007). | http://arxiv.org/abs/1707.09399v3 | {
"authors": [
"Haroon Asghar",
"John. G. McInerney"
],
"categories": [
"physics.ins-det",
"physics.optics"
],
"primary_category": "physics.ins-det",
"published": "20170726194504",
"title": "An asymmetric dual-loop feedback scheme with optimized second delay time to suppress spurious tones and timing jitter of self-mode-locked quantum dash lasers"
} |
Non-Coherent Detection for Diffusive Molecular Communications Vahid Jamali†, Nariman Farsad, Robert Schober†, and Andrea Goldsmith†Friedrich-Alexander University (FAU), Erlangen, GermanyStanford University, Stanford, California, USA This paper has been presented in part at ACM NanoCOM 2016 <cit.>.This work was supported in part by the German Science Foundations (Project SCHO 831/7-1) and the Friedrich-Alexander- University Erlangen-Nuremberg under the Emerging Fields Initiative (EFI).December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================ We study non-coherent detection schemes for molecular communication (MC) systems that do not require knowledge of the channel state information (CSI). In particular, we first derive the optimal maximum likelihood (ML) multiple-symbol (MS)detector for MC systems. As a special case of the optimal MS detector, we show that the optimal ML symbol-by-symbol (SS) detector can be equivalently written in the form of a threshold-based detector, where the optimal decision threshold is constant and depends only onthe statistics of the MC channel. The main challenge of the MS detector is the complexity associated with the calculation of the optimal detection metric. To overcome this issue, we propose an approximate MS detection metric which can be expressed in closed form. To reduce complexity even further, we develop a non-coherent decision-feedback (DF) detector and a suboptimal blind detector. Finally, we derive analytical expressions for the bit error rate (BER) of the optimal SS detector, as well as upper and lower bounds for the BER of the optimal MS detector. Simulation results confirm the analysis and reveal the effectiveness of the proposed optimal and suboptimal detection schemes compared to a benchmark scheme that assumes perfect CSI knowledge, particularly when the number of observations used for detection is sufficiently large. Molecular communications, channel state information, non-coherent detection, blind detection, and Poisson channel.§ INTRODUCTIONMolecular communication (MC) has recentlyemerged as a bio-inspired approach for synthetic communication systems having nano/micrometer scale dimensions <cit.>. Unlike conventional wireless communication systems that employ electromagnetic wavesto convey information, MC systems encode information in the number,type, or time of release of signalling molecules. Calcium signaling of neuronsand the exchange of autoinducersby bacteria in quorum sensing are among the many examples of MC in nature <cit.>.The effect of the diffusive MC channel on a concentration-based MC system is reflected in i) the probability that a molecule released by the transmitter is observed at the receiver, and ii) the expected numberof interfering molecules observedatthe receiver. Hence, i) and ii) constitute the channel state information (CSI) of the MC channel. The CSI of an MC system depends on parameters such as the diffusion coefficient of the information molecules, the velocity of the flow in the MC channel, the concentration of enzyme in the environment, the distance between the transmitter and the receiver, and the type of the adopted receiver, among other factors, see <cit.> and <cit.>. Most existing works on MC assume that the CSI isperfectly known at the receiver for reliable detection of the transmitted information bits<cit.>. In reality, the CSI is not known a priori and has to be estimated. To this end, a training-based CSI estimation framework was developed in <cit.> where several optimal and suboptimal estimators were proposed. However, we note that the CSI of MC systems may change over time due to factors such as variations in the velocity of the flow, the temperature (which leads to variations in the diffusion coefficient and the enzyme concentration), and the distance between the transmitter and the receiver (e.g. if they are suspended in a fluid) <cit.>. Therefore, the CSI acquisition has to beconducted repeatedly to keep track of CSI variations.Detection schemes requiring the acquisition of the CSI are suitable options only if the coherence time of the MC channel is sufficiently large (e.g. when the temperature, the flow, and the positions of the transmitter and the receiver change very slowly) such that the corresponding training overhead is tolerable. On the other hand, for the case when the MC channel changes rapidly,CSI estimation either entails a large overhead or results in a low CSI estimation quality. In this case, directly detecting the data symbols without spending any resources on CSI acquisition is an attractive option which is referred to as non-coherent detection.In this paper, our focus is the design of optimal and suboptimal non-coherentdetection schemes that do not require knowledge of the instantaneous CSI. In particular, we first derive the optimal maximum likelihood (ML) multiple-symbol (MS) detector. As a special case of the optimal MS detector, we show that the optimal ML symbol-by-symbol (SS) detector can be equivalently written in the form of a threshold-based detector where the optimal decision threshold is constant and depends only on the statistics of the MC channel. One of the main challenges in implementing the MS detector is the computation of the detection metrics, which requires the evaluation of expectations with respect to the probability density function (PDF) of the CSI. The PDF of the CSI depends on the considered MC environmentand a general analytical expression is not known.In practice,the PDF of the CSI for a particular MC channel can be estimated usingempirical measurements of the CSI. To obtain a closed-form expression for the MS detection metric, we approximate the PDF of the CSI by the Gamma distribution. We can show that the Gamma distribution accurately matches the exact distribution of the CSI for several examples of stochastic MC channels. Additionally, to reducecomplexity even further, we develop a non-coherent decision-feedback (DF) detector and a suboptimal blind detector. Furthermore, we derive analytical expressions for the bit error rate (BER) of the optimal SS detector, and an upper bound and a lower bound for the BER of the optimal MS detector.Our simulation resultsreveal the effectiveness of the proposed optimal and suboptimal non-coherent detectorsand show thattheir performance approaches that of a benchmark scheme that assumes perfect CSI as the number of symbols used for detection increases. In contrast to MC, for conventional wireless communication, there is a rich literature on non-coherent MS detection, see e.g. <cit.>, and the references therein. In particular, in <cit.>, MS differential detection without CSI was presentedfor radio frequency (RF) communications, and in <cit.>, non-coherent MS detection for a photon-counting receiver was studied for optical communications. We note that the detection problem and the resultingdetection strategies developed for conventional wireless communications are not applicable to the corresponding MC detection problem due to the fundamental differences in the two systems. The problem of non-coherent data detection in MC was considered before in <cit.> andheuristic low-complexity non-coherent symbol-by-symbol detectors were proposed.However, to the best of the authors' knowledge, the design ofnon-coherent MS detectors,which is considered in this paper and its conference version <cit.>, has not yet been considered in the MC literature.In addition to the detectors proposed in <cit.>, this paper derives the BER of the optimal SS detector along with lower and upper bounds for the BER of the optimal MS detector, and validates these results with simulations. Theremainderofthispaperisorganizedasfollows. In Section II, thesystem model and the CSI modelareintroduced. The proposed detectors are presented in Section III and their performance is analyzed in Section IV. Numerical results are presented in Section V, and conclusions are drawn in Section VI.Notation: We use the following notation throughout this paper: E_x{·} denotes expectation with respect to random variable (RV) x and |·| represents the cardinality of a set. Bold letters are used to denote vectors, 𝐚^T represents the transpose of vector 𝐚, and 𝐚≥0 specifies that all elements of vector 𝐚 are non-negative. Moreover, ℕ and ℝ^+ denote the sets of non-negative integernumbers and positive real numbers, respectively.⌊·⌋ and ⌈·⌉ denote the floor and ceiling functions which map a real number to the largest previous and the smallest following integer number,respectively. Γ(·) is the Gamma function and 1{·}∈{0,1} isanindicatorfunctionwhichisequalto oneiftheargumentistrueandequaltozeroifitisnot true. Moreover,Poiss(λ)denotes a Poisson RV with meanλ, Bin(n,p) denotes a binomial RV for ntrials and success probability p, 𝒩(μ,σ^2) denotes a Gaussian RV with mean μ and variance σ^2, and Gamma(α,β) denotes a Gamma distributed RV with scale parameter α and rate parameter β.§ SYSTEM MODEL In this section, we introduce the MC channel model and the CSI model used in this paper.§.§ Channel Model We consider an MC systemconsisting of a transmitter, a channel, and a receiver, see Fig. <ref>. At the beginning of each symbol interval, the transmitter releases either N^𝚃𝚇 or zero molecules corresponding to thebinary bits 1 and 0, respectively, i.e., the modulation format used is ON-OFF keying (OOK) <cit.>. In this paper, we assume that the transmitter emits only one type of molecule. The released molecules diffuse through the fluid medium between the transmitter and the receiver.The receiver counts the number of observed molecules in each symbol interval.We note that this is a rather general model for the MC receiver which includes well-known receivers such as the transparent receiver <cit.>, the absorbing receiver <cit.>, and the reactive receiver <cit.>. In particular, the number of observed molecules at the receiverin symbol interval k, denoted by r[k], is given by <cit.> r[k]=c_𝚜[k] + c_𝚗[k], where c_𝚜[k] is the number ofmolecules observed at the receiverin symbol interval k due to the release of s[k]N^𝚃𝚡 molecules by the transmitter at the beginning of symbol interval k, with s[k]∈{0,1}. We assume that the binary information bits are equiprobable, i.e., {s[k]=1}={s[k]=0}=0.5.Moreover, c_𝚗[k] is the number of interfering noise molecules comprising residual inter-symbol interference (ISI), multiuser interference (caused by other MC links), andexternal noise (originating from natural sources) observed by the receiver in symbol interval k.The MC channel is dispersive due to the diffusive propagation of the molecules <cit.>. The ISI-free communication model in (<ref>) implies thatthe symbol intervals are chosen sufficiently large such that the channel impulse response (CIR) fully decays to zero within one symbol interval. We note that enzymes<cit.> and reactive information molecules, such as acid/base molecules <cit.>, may be used to speed up the decaying of the CIR as a function of time. Nevertheless, since the length of the symbol intervals is finite, some residual ISI always exists. Throughout thispaper, we assume that the effect of the residual ISI is included in c_𝚗[k] and is sufficiently small compared to the other components of c_𝚗[k] such that c_𝚗[k] is (approximately) independent of the signal c_𝚜[k].We now describe the underlying models and associated distributions of RVs c_𝚜[k] and c_𝚗[k]. Information Molecules (c_𝚜[k]): The movements of individual molecules are assumed to be independent from each other. We further assume thatthe observations of different molecules at the receiver are independent[This assumption holds for the transparent and absorbing receiver models. For this assumption to hold for the reactive receiver model in <cit.>, the number of receptors on the surface of the receiver has to be much larger than the average number of molecules around the receiver which is a valid assumption for typical biological cells in nature, see <cit.>.]. Therefore, from a probabilistic point of view, we can assume that each molecule released by the transmitter in a given symbol interval is observed at the receiver in the same symbol interval with a certain probability, denoted by p_𝚜. Since any given molecule released by the transmitter is either observed by the receiver or not, a binary state model applies and the number of observed molecules follows the binomial distribution Bin(N^𝚃𝚡,p_𝚜). Moreover, assuming that N^𝚃𝚡 is very large while N^𝚃𝚡p_𝚜≜c̅_𝚜 is relatively small, thebinomial distribution Bin(N^𝚃𝚡,p_𝚜) converges to the Poisson distribution Poiss(c̅_𝚜) <cit.>. Another approximation of the binomial distribution Bin(N^𝚃𝚡,p_𝚜) is the Gaussian distribution 𝒩(N^𝚃𝚡p_𝚜,√(N^𝚃𝚡p_𝚜(1-p_𝚜))) which holds for very large N^𝚃𝚡 when c̅_𝚜 is relatively large<cit.>. We note that the assumptions for the Poisson approximation are more justified for MC than those of the Gaussian approximation since the number of released molecules is typically very large (on the order of hundreds or a few thousands) but, typically, only few molecules reach the receiver. Therefore, we adopt the Poisson approximation in this paper, i.e., c_𝚜[k]∼Poiss(c̅_𝚜 s[k]). The accuracy of the Poisson distribution in modeling the number ofmolecules observed at the receiver was verified in <cit.>, <cit.>and compared with the accuracy of the corresponding binomial and Gaussian models[For instance,the analytical framework developed in <cit.> can be used to show that for N^𝚃𝚡=1000, if p_𝚜≤ 0.115 holds, the Poisson distribution more accurately approximates the binomial distributionin terms of the root mean squared error (RMSE) of the cumulative distribution function (CDF), whereas, if p_𝚜>0.115 holds, the Gaussian approximation is a better fit. For MC systems, ifN^𝚃𝚡=1000 molecules are released by the transmitter, typically we expect to observe much fewer than N^𝚃𝚡p_𝚜=113 molecules at the receiver. Hence, the Poisson approximation is more accurate compared to the Gaussian approximation in this case.]. Noise Molecules (c_𝚗[k]):Noise molecules originate frominterfering natural or synthetic sources<cit.>. Suppose there are in total N^𝚗𝚘𝚒𝚜𝚎 noise molecules in the environment and they are uniformly distributed in space and time, which is a reasonable assumption if a priori knowledge about the locations and activities of the noise sources is not available. In addition,the probability that at any given time, any noise molecule is observed at the receiver is assumed to be the same for all noise molecules and is denoted by p_𝚗.Since at the sampling time, each noise molecule is either observed at the receiver or not, again, a two state model is valid and c_𝚗[k] follows a binomial distribution, i.e., c_𝚗[k]∼Bin(N^𝚗𝚘𝚒𝚜𝚎,p_𝚗). Moreover, the probability that a given noise molecule is observed at the receiver is again expected to be very small, i.e.,p_𝚗→ 0 holds. Therefore,N^𝚗𝚘𝚒𝚜𝚎→∞ has to hold such that the average number of noise molecules observed at the receiver, denoted by c̅_𝚗=N^𝚗𝚘𝚒𝚜𝚎 p_𝚗, is non-zero. Considering this, c_𝚗[k] follows also a Poisson distribution, i.e., c_𝚗[k]∼Poiss(c̅_𝚗). The channel model in (<ref>) can be generalized to the case of multiple-sample detection if the following sum detector is employed: r[k]=∑_m=1^My[k,m] =∑_m=1^Mc_𝚜[k,m] +∑_m=1^Mc_𝚗[k,m] ≜ c_𝚜[k] + c_𝚗[k], where M denotes the number of samples per symbol interval and y[k,m] is the number of molecules observed at the receiver in the m-th sample of symbol interval k. Moreover, c_𝚜[k,m] is the number ofmolecules observed at the receiver for the m-th sample of symbol interval k due to the release of s[k]N^𝚃𝚡 molecules by the transmitter at the beginning of symbol interval k,and c_𝚗[k,m] is the number of noise molecules observed at the receiver for the m-th sample of symbol interval k. Since c_𝚜[k,m] and c_𝚗[k,m] are Poisson RVs,c_𝚜[k] and c_𝚗[k] follow Poisson distributions with meansc̅_𝚜= ∑_m=1^Mc̅_𝚜^(m) and Mc̅_𝚗, respectively, where c̅_𝚜^(m)=E{c_𝚜[k,m]} and c̅_𝚗=E{c_𝚗[k,m]}. We note that the sum detector in (<ref>) includes the well-known peak value <cit.> and energy <cit.> detectors as special cases when only one sample at the peak concentration is taken and (ideally infinitely) many samples per symbol interval are taken, respectively. Unlike the conventional linear input-outputmodel for channels in wireless communications <cit.>, the channel model in (<ref>) is not linear since s[k] does not affect the observation r[k] directly but rather via the Poisson RV c_𝚜[k]. However, the expectation of the received signal is linearly dependent on the transmitted signal, i.e., r̅[k] = E{r[k]} =c̅_𝚜 s[k] + c̅_𝚗. We note that for a given s[k], in general, the actual number of molecules observed at the receiver, r[k], will differ from the expected number of observed molecules, r̅[k], due to the intrinsic noisiness of diffusion.§.§ CSI Model The state of the diffusive MC channel specified in (<ref>) can be captured by p_𝚜 and c̅_𝚗. Since we employ OOK signaling and N^𝚃𝚡 is assumed to be fixed, without loss of generality, we refer to the vector 𝐜̅=[c̅_𝚜,c̅_𝚗]^T (instead of [p_𝚜,c̅_𝚗]^T) as the CSI of the considered MC system in the remainder of this paper.Most existing detection schemes in MC assume that knowledge of the CSI is available at the receiver <cit.>. In contrast, in this paper,we directly detect a block of transmitted symbolsbased on the corresponding received observations without spending any resources on CSI acquisition at the receiver.Let 𝐬=[s[1],s[2],…,s[K]]^T and 𝐫=[r[1],r[2],…,r[K]]^T denote the vectors of the transmitted symbols and the received observations over a block of K symbol intervals, respectively. We assume that the CSI remains unchanged over one block of transmitted symbols, but may change from one block to the next (e.g., due to a change in MC channel parameters such as the temperature, the velocity of the flow, etc.). To model this, we assume thatCSI, 𝐜̅, is an RV that takes its values in each block according to PDFf_𝐜̅(𝐜̅). Furthermore, we assume that RVs c̅_𝚜 and c̅_𝚗 are independent, i.e., f_𝐜̅(𝐜̅)=f_c̅_𝚜(c̅_𝚜) f_c̅_𝚗(c̅_𝚗) where f_c̅_𝚜(c̅_𝚜) and f_c̅_𝚗(c̅_𝚗) are the marginal PDFs of c̅_𝚜 and c̅_𝚗, respectively.For future reference, in the rest of this work,we define f_𝐫(𝐫|𝐜̅,𝐬) = ∏_k f_r[k](r[k]|𝐜̅,s[k]) as the PDF of the observationvector 𝐫conditioned on both CSI 𝐜̅ andtransmitted symbol vector 𝐬,and we define f_𝐫(𝐫|𝐬) = ∏_k f_r[k](r[k]|s[k]) as the PDF of𝐫 conditionedonly on 𝐬. We note that although the proposed non-coherent detection schemes do not require knowledge of the instantaneous CSI, 𝐜̅, we assume that statistical CSI, i.e., f_𝐜̅(𝐜̅),is available for the design of the proposed non-coherent MS, SS, and DF detectors.Since obtaining the CSI statistics might be difficult for some practical systems, wealso propose a suboptimal blind detector that does not requirestatistical CSI knowledge. We note that for the conventional channel model used in wireless communications, the noise power dependslargely on the characteristics of the receiver<cit.>. For instance, in RF communications, the noise variance is determined by the receiver thermal noise <cit.>, and in optical communications, the power of the noise depends on the sensitivity of the photo-detector to the background radiation <cit.>. Therefore, when modeling these systems, it is commonly assumed that the noise power is constant. Hence, the channel gainis typically referred to as the CSI and the noise power is assumed to be fixed and known. On the other hand, the noise molecules in MC originate frominterfering natural or synthetic sources. Thus, the mean of the noise is in general dependent on the properties of the MC channel in addition to the type of the adopted receiver.Hence, it is reasonable to treat both c̅_𝚜 and c̅_𝚗as the CSI of the MC system.§ OPTIMAL AND SUBOPTIMAL NON-COHERENT DETECTIONIn this section, we first present the optimal coherent detector which provides a performance upper bound for any non-coherent detector. Subsequently, we present the proposed non-coherent detectors, namely the optimal non-coherent MS, SS, and DF detectors. Furthermore, motivated by the structure of the optimal coherent detector, we develop a simple blind detector. §.§ Coherent ML Detector As a performance upper bound, we consider the optimal detector for perfect CSI knowledge. Moreover, since the observations in different symbol intervals are independent, without loss of optimality, the considered benchmark scheme performs symbol-by-symbol detection. Thus, the optimal ML detector is given by ŝ^𝙼𝙻[k]= s[k]∈{0,1}argmaxf_r[k](r[k]|𝐜̅,s[k]) = s[k]∈{0,1}argmax( c̅_𝚜 s[k] + c̅_𝚗)^r[k]exp(- c̅_𝚜 s[k] - c̅_𝚗)/r[k]!, where f_r[k](r[k]|𝐜̅,s[k]) is the Poisson distribution function. The ML detector can be rewritten in the form ofa threshold-based detector as follows <cit.> ŝ^𝙼𝙻[k] =1, if r[k] ≥ξ^𝙼𝙻(𝐜̅) 0, otherwise where the computation of the decision threshold, ξ^𝙼𝙻(𝐜̅) = c̅_𝚜/ln(1+c̅_𝚜/c̅_𝚗), requires CSI knowledge. §.§ Non-Coherent ML DetectorWe first develop the optimal non-coherent MS and SS detectors. Following this development, we propose suboptimal decision metrics to cope with the complexity of evaluating the optimal MS and SS metrics for this detector.§.§.§ Optimal MS DetectorWe first formulate the ML problem for non-coherent detection of a block of symbols. Note that, in this case, joint detection ofmultiple symbolsis beneficial, despite the fact that the symbols are independent, since observing multiple symbols provides more information about the MC channel than observing just one symbol. The optimal ML MS detector is mathematically given by 𝐬̂^𝙼𝚂 = 𝐬∈𝒜argmaxf_𝐫(𝐫|𝐬)= 𝐬∈𝒜argmax∫_𝐜̅≥0 f_𝐫(𝐫|𝐜̅,𝐬)f_𝐜̅(𝐜̅) d𝐜̅ = 𝐬∈𝒜argmax∫_c̅_𝚜≥ 0∫_c̅_𝚗≥ 0∏_k=1^K( c̅_𝚜 s[k] + c̅_𝚗)^r[k]exp(- c̅_𝚜 s[k] - c̅_𝚗)/r[k]!× f_c̅_𝚜(c̅_𝚜)f_c̅_𝚗(c̅_𝚗) dc̅_𝚜dc̅_𝚗, where 𝒜 is the set of all 2^K possiblebinary sequences of length K. In (<ref>), we employ the multivariate Poisson distribution function f_𝐫(𝐫|𝐜̅,𝐬) andexploit the fact that the observations in different symbol intervals are independent and that RVs c̅_𝚜 and c̅_𝚗 are independent. Before presenting the optimal MS detector as a solution of (<ref>) in the following theorem, we introduce some auxiliary variables. For a given hypothetical sequence 𝐬, let𝒦_1 and 𝒦_0 denote the sets of indices k for which s[k]=1 and s[k]=0 holds, respectively. Moreover, we define 𝐧=[n_1,n_0]^T where n_1 and n_0 are the number of ones and zeros in the given hypothetical sequence 𝐬. Additionally, for a given observation vector 𝐫, we define𝐍=[N_1,N_0]^T where N_1=∑_k∈𝒦_1 r[k] and N_0=∑_k∈𝒦_0 r[k].The optimal non-coherent MS detector as the solution to (<ref>) selects a sequence whose “1" elementscorrespond to the ζ^𝙼𝚂(𝐫) largest elements of 𝐫. Moreover, the optimal thresholdζ^𝙼𝚂(𝐫) is obtained as ζ^𝙼𝚂(𝐫) =n_1 ∈{0,1,…,K}argmaxΛ^𝙼𝚂(𝐬,𝐫), where Λ^𝙼𝚂(𝐬,𝐫) is the MS detection metric given by Λ^𝙼𝚂(𝐬,𝐫) = E_𝐜̅{( c̅_𝚜 + c̅_𝚗)^N_1c̅_𝚗^N_0e^-n_1c̅_𝚜- K c̅_𝚗}=∑_i=0^N_1N_1iE_c̅_𝚜{c̅_𝚜^N_1-i e^- n_1 c̅_𝚜}E_c̅_𝚗{c̅_𝚗^N_0+i e^- K c̅_𝚗}. Furthermore, as K→∞, we obtainζ^𝙼𝚂(𝐫)→ K×{s[k]=1}=K/2.The proof is provided in Appendix <ref>. The complexity of the proposedMS detector in Theorem <ref> is significantly smaller than that of the full search required in (<ref>). In particular, the complexity of the full search in (<ref>) grows exponentially in K, i.e., |𝒜|=2^K, whereas the complexity of the search in Theorem <ref> is linearly increasing in K, i.e., there are K+1 possibilities.Furthermore, for each search step, we have to calculate the metric Λ^𝙼𝚂(𝐬,𝐫) which is a function of the statistical CSI, but not of the instantaneous CSI, sincec̅_𝚜 and c̅_𝚗 are averaged out in Λ^𝙼𝚂(𝐬,𝐫), cf. (<ref>).Moreover, all expectations in the MS detection metric are in the form of E_x{x^ae^-bx} where x∈{c̅_𝚜,c̅_𝚗} and a and b are constants.Therefore, if these expectations for all required a and b can be computed offline, they can be stored at the receiver and used for online data detection. As a special case of the optimal MS detector in Theorem <ref>, in the following corollary we present the optimal SS detector which, unlike the general form in Theorem <ref>, lends itself to a simple threshold-based detection. The optimal non-coherent SS detector as the solution to (<ref>) for K=1 can be written in the form of the following threshold-based detector: lllŝ^𝚂𝚂[k] =1, if r[k] ≥ξ^𝚂𝚂 0, otherwisewhere threshold ξ^𝚂𝚂is given bylll ξ^𝚂𝚂= ⌈ξ∈ℝ^+ | E_𝐜̅{( c̅_𝚜 + c̅_𝚗 )^ξ e^-c̅_𝚜- c̅_𝚗 } = E_c̅_𝚗{ c̅_𝚗^ξ e^ - c̅_𝚗 }⌉ =min{ ξ∈ℕ | ∑_i = 0^ξξiE_c̅_𝚜{ c̅_𝚜^ξ-ie^-c̅_𝚜 }E_c̅_𝚗{ c̅_𝚗^ie^-c̅_𝚗 } > E_c̅_𝚗{ c̅_𝚗^ξ e^ - c̅_𝚗} }.The proof is provided in Appendix <ref>. Corollary <ref> reveals that, similar to the coherent ML detector under perfect CSI knowledge in (<ref>), the optimal non-coherent SS detector has a threshold-based structure. However, the threshold for the optimal SS detector depends only on the statistics of the MC channel. Therefore, ξ^𝚂𝚂 remains fixed as long as the statistics of the MC channel do not change. Hence, ξ^𝚂𝚂 can be obtained offline once and thereafter used for online detection. Consequently, the optimal SS detector in Corollary <ref> is considerably less complex than the equivalent detector obtained from Theorem <ref> for K=1, which requires the online computation of threshold ζ^𝙼𝚂(r[k])or, equivalently, detection metrics Λ^𝙼𝚂(s[k]=1,r[k]) and Λ^𝙼𝚂(s[k]=0,r[k]) in each symbol interval.§.§.§ Suboptimal ML Metric The main challenge of the optimal MS detectoris thecalculation of the detection metric in (<ref>). In particular, the detection metric in (<ref>) involves expectations which require the PDF of the CSI, i.e., f_x(x),x∈{c̅_𝚜,c̅_𝚗}. These PDFsdepend on the considered MC environmentandgeneral analytical expressions for f_x(x),x∈{c̅_𝚜,c̅_𝚗}, are not known.In practice, for a particular MC channel, these PDFs can be obtained usingempirical measurements of c̅_𝚜 and c̅_𝚗. However, the empirical PDFs might not lend themselves to a simple analytical form. Therefore, one convenient approach for obtaining a mathematical expression for the detection metric in(<ref>) is to assume a particular parametric model for f_x(x) and to adjust the parameters of the model to match the simulation/experimental data. In general, it is desirable that a parametric PDF model, f_x^𝚙𝚊𝚛𝚊(x), meets the following criteria: * f_x^𝚙𝚊𝚛𝚊(x) should meet any inherent constraints of the RV x. Such constraints might include that x is non-negative or non-positive, or that it is discrete, continuous, or mixed. * f_x^𝚙𝚊𝚛𝚊(x) should facilitate analysis and ideally lead to closed-form expressions for metric associated with x under consideration, which in this case is the detection metric (<ref>). * f_x^𝚙𝚊𝚛𝚊(x) should be flexible enough to provide an accurate approximation of the exact distribution for a wide range of system parameters.For the problem at hand, the PDF of the CSI has to be supported only over the non-negative range, i.e., f_x(x)=0 for x<0, sincec̅_𝚜 and c̅_𝚗 assume only non-negative values. In addition, for the purposes of this paper, the adopted PDF f^𝚙𝚊𝚛𝚊_x(x) should lead to a sufficiently simple detection metric (<ref>). We have investigated several well-knowndistributions as parametric models for the PDF of the CSI, including the Nakagami <cit.>, inverse Gaussian <cit.>, Levy<cit.>, and Gamma distributions <cit.>. We found that the Gamma distribution, f^𝚐𝚊𝚖𝚖𝚊_x(x) = β^α x^α-1 e^-β x/Γ(α), if x≥ 0 0,otherwise with parameters α,β>0 <cit.>, is the best fit to the PDF of the CSI, f_x(x),x∈{c̅_𝚜,c̅_𝚗}, for different system realizations. Moreover, it leads to a simple detection metric.We emphasize that an experimentally verified stochastic channel model for MC systems has not been reported yet. Despite this shortcoming, our motivation for adopting the Gamma distribution is as follows. First, for the stochastic MC channel used for simulation in Section V,the Gamma distribution can accurately model the randomness of the CSI introduced by random variations of the underlying MC channel parameters, e.g., the flow velocity, the enzyme concentration, the diffusion coefficient, etc. In fact, in Section V, we show that the BERs of the optimal MS detector using the exact PDF of the CSI and the Gamma distribution perfectly match. Secondly, for the Gamma distribution, the ML decision metrics can be calculated in closed form. In particular, the expectations in the detection metric in(<ref>) areof the formE_x{x^ae^-bx} where x∈{c̅_𝚜,c̅_𝚗} and a and b are constants. Therefore, using the Gamma distribution,E_x{x^ae^-bx} can be expressed as E_x{x^ae^-bx}= ∫_x=0^∞x^ae^-bx×β^α x^α-1 e^-β x/Γ(α)dx=β^α/Γ(α)∫_x=0^∞ x^a+α-1e^-(b+β)xdx=β^αΓ(a+α)/Γ(α) (b+β)^a+α=1∫_x=0^∞(b+β)^a+α x^a+α-1e^-(b+β)x/Γ(a+α)dx = Γ(a+α) β^α/Γ(α) (b+β)^a+α.The parameters α and β have to be properlychosen such that the resulting Gamma distribution well approximates the exact distribution or, if only the measurement data is available, the histogram of the measurement data. To this end, we adopt the weighted mean square error as a criterion to be minimized for the optimal choice of α and β. In particular, the optimal α^* and β^* are obtained as (α^*,β^*) =α,β>0argmin∫_x=0^∞ w(x) |f_x(x)-f_x^𝚐𝚊𝚖𝚖𝚊(x)|^2 dx, where w(x)≥ 0,∀ x, is an appropriately chosen weight function that can be used to prioritize the accuracy of the approximation in a desired range of x. Using the Gamma distribution with the optimized parameters, the detection metrics required for the MS and SS detectors are given in closed form based on (<ref>). Since the feasible sets of α and β are semi-infinite, i.e., α,β∈(0,+∞), performing a full search to find the optimal α^* and β^* is not possible. To overcome this challenge, we first note that for a Gamma distribution with mean μ_x and variance σ_x^2, the parameters (α,β) are uniquely obtained as(α,β) = (μ_x^2/σ_x^2,μ_x/σ_x^2). Since the optimal parameters (α^*,β^*) from (<ref>) are expected to lead to a Gamma distribution that has a mean and a variance that are close to those of the exact distribution, we can efficiently bound the search for the optimal values to intervals α∈ [(1-δ)μ̅_x^2/σ̅_x^2,(1+δ)μ̅_x^2/σ̅_x^2] and β∈ [(1-δ)μ̅_x/σ̅_x,(1+δ)μ̅_x/σ̅_x], where μ̅_x and σ̅_x^2 are the mean and the variance of the exact distribution, respectively, and δ≥ 0 determines how large the search intervals are. §.§ Non-Coherent DF Detector As will be shown in Section V, the proposed optimal MS detector significantly outperforms the optimal SS detector, particularly when the number of jointly detected symbols is large. However, the gain obtained by the optimal MS detector in this case comes at the expense of a large detection delay. In particular, we have to wait until all K symbols in one detection window arrive before detection can be performed. In order to mitigate this problem and to further reduce complexity while still exploiting the benefits of MS detection, we consider a DF detector in the following. In particular, the DF detectorinstantly detects the received symbols while exploiting the memory of the previously detected symbols assuming that these symbols were correctly detected. To characterize the DF detector, let us first define 𝐫_k=[r[k-1],…,r[k-K+1]]^T and 𝐬̂_k=[ŝ[k-1],ŝ[k-2],…,ŝ[k-K+1]]^T as the observation vector and the detected symbol vector for the previous K-1 symbol intervals.TheDF detector is derived in a similar manner as the optimal MS detector, cf. Appendix <ref>. In particular, given the K-1 previously detected symbols in 𝐬̂_k, weevaluate the detection metric in (<ref>) for the two possibilities s[k]=1 and s[k]=0 which leads tothe detection metrics Λ^𝙼𝚂(𝐬_1,𝐫) and Λ^𝙼𝚂(𝐬_0,𝐫), respectively, where𝐬_0=[0,𝐬̂_k^T]^T, 𝐬_1=[1,𝐬̂_k^T]^T, and 𝐫=[r[k],𝐫_k^T]^T. This leads to the following decision rule.Proposed DF Detector: The non-coherent DF detector for the MC system considered in this paper is given by lllŝ^𝙳𝙵[k] =1, ifΛ^𝙼𝚂(𝐬_1,𝐫) ≥Λ^𝙼𝚂(𝐬_0,𝐫) 0, otherwiseSince the above DF detector uses the MS detection metric in (<ref>), we can also employ (<ref>) to obtain an approximate metric based on the Gamma distribution proposed in Subsection III-A.Given 𝐫_k and 𝐬̂_k, the ratio Λ^𝙼𝚂(𝐬_1,𝐫) / Λ^𝙼𝚂(𝐬_0,𝐫) is a monotonically increasing function of r[k]. Since Λ^𝙼𝚂(𝐬_1,𝐫) < Λ^𝙼𝚂(𝐬_0,𝐫) holds forr[k]=0, there exists a unique threshold for r[k], denoted by ξ^𝙳𝙵(𝐬̂_k,𝐫_k), above which Λ^𝙼𝚂(𝐬_1,𝐫) > Λ^𝙼𝚂(𝐬_0,𝐫) holds. Hence, the non-coherent DF detector canbe equivalently written in the following threshold-based formlllŝ^𝙳𝙵[k] =1, ifr[k]≥ξ^𝙳𝙵(𝐬̂_k,𝐫_k) 0, otherwiseThe adaptive DF detection threshold ξ^𝙳𝙵(𝐬̂_k,𝐫_k) is given bylllξ^𝙳𝙵(𝐬̂_k,𝐫_k) =min{ξ∈ℕ| Λ^𝙼𝚂(𝐬_1,𝐫^ξ) > Λ^𝙼𝚂(𝐬_0,𝐫^ξ)}, where 𝐫^ξ=[ξ,𝐫_k^T]^T. We will employ the threshold detection form of the proposed DF detector in (<ref>) for the performance analysis provided in Section IV-C.§.§ Suboptimal Detector Based on Blind CSI Estimation The non-coherent detectors developed so far require statisticalknowledge of the CSI which might be difficult to acquire for someMC systems deployed in practice. Therefore, in the following, wepropose a suboptimal detector that does not needstatistical CSI knowledge. The main idea behind the simple detector, which we propose in this subsection, is to first estimate the CSI based on the symbols received in the considered detection window in order to approximate the optimal ML threshold which is denoted by ξ^𝙼𝙻_𝙱𝙻. Subsequently, symbol-by-symbol detection can be performed based on the approximated threshold ξ^𝙼𝙻_𝙱𝙻. We note that the channel estimator is blind in the sense that a training sequence is not used. For a given observation block 𝐫, let𝒦_1 (𝒦_0) denote the sets of indices k for the ⌈K/2⌉-th largest (⌊K/2⌋-th smallest) r[k]in the block. The proposed suboptimal detector is formally presented in the following. Proposed Blind ML-Based Detector: Theproposed blind detector forON-OFF keying modulation in diffusive MC is given by ŝ^𝙼𝙻_𝙱𝙻[k] =1, if r[k] ≥ξ^𝙼𝙻_𝙱𝙻(𝐫) 0, otherwise whereξ^𝙼𝙻_𝙱𝙻(𝐫) = ĉ̅̂_𝚜/ln(1+ĉ̅̂_𝚜/ĉ̅̂_𝚗) is the detection threshold. Hereby, the CSI estimates ĉ̅̂_𝚜 and ĉ̅̂_𝚗 are obtained asĉ̅̂_𝚜 = 1/⌈K/2⌉∑_k∈𝒦_1 [r[k] - ĉ̅̂_𝚗 ] ĉ̅̂_𝚗 = 1/⌊K/2⌋∑_k∈𝒦_0 r[k].The CSI estimates ĉ̅̂_𝚜 and ĉ̅̂_𝚗 in (<ref>) correspond to a simple averaging over the expected positions of s[k]=1 and s[k]=0, respectively. Since the noise molecules are always present whereas the signal molecules are present only when s[k]=1 holds, the expected number of molecules observed at the receiver in positions with s[k]=1 islarger than that in positions with s[k]=0.Therefore, we first compute ĉ̅̂_𝚗 directly from (<ref>) and thenĉ̅̂_𝚜 from (<ref>). Having the estimated CSI (ĉ̅̂_𝚜,ĉ̅̂_𝚗), we can compute the ML threshold ξ^𝙼𝙻_𝙱𝙻 and perform detection based on (<ref>). We note that although s[k]=1 and s[k]=0 are equiprobable, the number of ones and zeros in each detection window may not be exactly K/2. Therefore, assuming that the ⌈K/2⌉-thlargest elements (⌊K/2⌋-th smallest elements) of 𝐫 correspond to the positions of s[k]=1 (s[k]=0) leads to an inherent CSI estimation error. § PERFORMANCE ANALYSIS In this section, we derive exact expressions andperformance bounds for the BERs of the proposed detection schemes. In particular, the average BER, denoted by P̅_e, can be written mathematically aslllP̅_e = E_𝐜̅{P_e(𝐜̅)} = ∫_c̅_𝚜≥0 ∫_c̅_𝚗≥0 {ŝ[k]≠s[k]|𝐜̅}f_c̅_𝚜(c̅_𝚜)f_c̅_𝚗(c̅_𝚗) dc̅_𝚜dc̅_𝚗, where P_e(𝐜̅)≜{ŝ[k]≠ s[k]|𝐜̅} is theBER conditioned on a particular state of the MC channel characterized by the CSI 𝐜̅. In the following, we derive the conditional BER of the SS detector and conditional bounds for the MS detector. §.§ BER of the Optimal SS Detector We first derive the exact expression for the BER of the optimal SS detector. In particular, for a given 𝐜̅, the BER of the optimal SS detector is obtained as rllP_e^𝚂𝚂(𝐜̅) ={ŝ[k]=0|𝐜̅,s[k]=1}{s[k]=1}+ {ŝ[k]=1|𝐜̅,s[k]=0}{s[k]=0} =1/2 {r[k] < ξ^𝚂𝚂|𝐜̅,s[k]=1} +1/2 {r[k]≥ξ^𝚂𝚂|𝐜̅,s[k]=0}(a)=1/2 ∑_r[k]=0^⌊ξ^𝚂𝚂⌋( c̅_𝚜 + c̅_𝚗 )^r[k] exp(- c̅_𝚜 - c̅_𝚗 )/r[k]! + 1/2 ∑_r[k]=⌈ξ^𝚂𝚂⌉^∞ c̅_𝚗^r[k] exp(- c̅_𝚗 )/r[k]! (b)=1/2 + 1/2 [ Q(⌈ξ^𝚂𝚂 ⌉, c̅_𝚜 + c̅_𝚗 ) -Q(⌈ξ^𝚂𝚂 ⌉, c̅_𝚗 ) ], where Q(x,y)=Γ(x,y)/Γ(x) is the regularized Gamma function and Γ(·,·) is the upper incomplete Gamma function <cit.>. For equality (a), we used the fact that r[k] conditioned on 𝐜̅ and s[k] follows a Poisson distribution, and for (b), we exploited the expression for the CDF of a Poisson RV with mean λ given by F_r[k](ξ)={r[k]≤ξ}=Q(⌊ξ+1⌋,λ) which leads to {r[k]<ξ}=Q(⌈ξ⌉,λ) <cit.>. The BER expression in (<ref>) can be also used to compute the BER of the benchmark scheme with perfect CSI knowledge given in Subsection III.A after replacing ξ^𝚂𝚂 with ξ^𝙼𝙻(𝐜̅). §.§ BER Upper Bound for the Optimal MSDetector Unfortunately, the derivation of an exact expression for the BER of the optimal MS detector is a difficult, if not impossible,task due to the multi-dimensional structure of the statistics of the observation vector. Hence, in this subsection, we use the union bound to arrive at an upper bound on the BER of the optimal MS detector. The pairwise error probability (PEP), denoted by P(𝐬→𝐬̂), is defined as the probability that, assuming 𝐬 is transmitted, 𝐬̂ is detected. We note that, due to the structure of the optimal MS detector, the error probabilities for alltransmitted sequences which have the same number of ones are identical. Hence, without loss of generality, we consider only K+1 sequences 𝐬 whose first n_1∈{0,…,K}, bits are ones and which are collected in set 𝒜̃. Using the PEP, the BER is upper bounded based on the union bound as follows rllP_e^𝙼𝚂(𝐜̅) ≤1/K ∑_𝐬∈𝒜̂ ∑_𝐬̂∈{𝒜\𝐬} h(𝐬,𝐬̂) P^𝙼𝚂(𝐬→𝐬̂) {𝐬}, where h(𝐬,𝐬̂) denotes the Hamming distance between 𝐬 and 𝐬̂ and {𝐬}=1/2^KK n_1 where n_1 is the number of ones in 𝐬. We note that in order to calculate the PEP for a given 𝐬 and 𝐬̂, a K-dimensional summation with respect to the different possibilities for the observation vector 𝐫 is needed, which is a computationally challenging task.However, as can be seen from (<ref>), the detection metric is a functionsolely of n_1, N_1, and N=N_1+N_0. Thus, we introduce the new notation Λ^𝙼𝚂(n_1,N_1,N) to indicate this. Let Λ^𝙼𝚂(n_1,N_1,N) and Λ^𝙼𝚂(n̂_1,N̂_1,N) be the detection metrics for hypothesis sequences 𝐬 and 𝐬̂, respectively, for agiven 𝐫. We note that, given 𝐬 and 𝐬̂, the values of n_1 and n̂_1 are known, respectively, and for a given 𝐫, the value of N is the same for both 𝐬 and 𝐬̂. Hence, unlike the expression in (<ref>), which requires a K-dimensional summation with respect to the elements of 𝐫,it suffices to consider a three-dimensional summation with respect to N_1, N̂_1, and N. In particular,in order to calculate P^𝙼𝚂(𝐬→𝐬̂), we have to find the probabilities ofthe observation events (N_1,N̂_1,N), denoted by { N_1,N̂_1,N | 𝐜̅}, for which Λ^𝙼𝚂(n_1,N_1,N)<Λ^𝙼𝚂(n̂_1,N̂_1,N) holds. This leads to rll P^𝙼𝚂(𝐬→𝐬̂)= {Λ^𝙼𝚂(𝐬,𝐫)<Λ^𝙼𝚂(𝐬̂,𝐫)}=∑_N∑_N_1∑_N̂_11{Λ^𝙼𝚂(n_1,N_1,N)<Λ^𝙼𝚂(n̂_1,N̂_1,N)} { N_1,N̂_1,N | 𝐜̅}. Conditioned on 𝐜̅, variables N_1, N̂_1, and N are correlated Poisson RVs. Hence, in order to compute { N_1,N̂_1,N| 𝐜̅}, we divide the observation vector 𝐫 into the following four partitions:𝒮_1: Positions of 𝐫 where the corresponding elements of both 𝐬 and 𝐬̂ are one.𝒮_2: Positions of 𝐫 where the corresponding elements of 𝐬 and 𝐬̂ are one and zero, respectively.𝒮_3: Positions of 𝐫 where the corresponding elements of 𝐬 and 𝐬̂ are zero and one, respectively.𝒮_4: Positions of 𝐫 where the corresponding elements of both 𝐬 and 𝐬̂ are zero.The sum of observed molecules in the positions specified by 𝒮_1, 𝒮_2, 𝒮_3, and 𝒮_4 are denoted by M_1, M_2, M_3, and M_4, respectively. It follows that, conditioned on 𝐜̅, variables M_1, M_2, M_3, and M_4 are independent Poisson RVs with means λ_1=υ(c̅_𝚜 + c̅_𝚗), λ_2=(n_1-υ)(c̅_𝚜 + c̅_𝚗), λ_3=(n̂_1-υ)c̅_𝚗, and λ_4=(K+υ-n_1-n̂_1)c̅_𝚗, respectively, where υ=|𝒮_1|. Moreover, we have N_1=M_1+M_2, N̂_1=M_1+M_3, and N=M_1+M_2+M_3+M_4.Using these results, we obtain lll { N_1,N̂_1,N | 𝐜̅} =∑_i=0^min{N_1,N̂_1} { M_1=i | 𝐜̅} {M_2=N_1-i | 𝐜̅}×{M_3=N̂_1-i| 𝐜̅} {M_4=N_1+N̂_1-i| 𝐜̅}= ∑_i=0^min{N_1,N̂_1} λ_1^ie^-iλ_1/i! ×λ_2^N_1-ie^-(N_1-i)λ_2/(N_1-i)! ×λ_3^N̂_1-ie^-(N̂_1-i)λ_3/(N̂_1-i)! ×λ_4^N_1+N̂_1-ie^-(N_1+N̂_1-i)λ_4/(N_1+N̂_1-i)!. Using (<ref>) and (<ref>)-(<ref>), the average PEP can be analytically computed.In Section V, we employ (<ref>) as an upper bound for the BER of the optimal MS detector and verify its tightness by comparing it with the exact BER obtained via simulation. We note that the computation of the average PEP in an analytical form based on (<ref>) and (<ref>)-(<ref>) is a computationally challenging task due to the numerical integration required for evaluating (<ref>). In order to reduce the complexity of the evaluation of the average PEP, one can employ a hybrid approach where, for a given CSI 𝐜̅, the PEP bound in (<ref>) is calculated numerically and the expectation in (<ref>) is performed via Monte Carlo simulation. §.§ BER Lower Bound for the MS/DF Detectors In this subsection, we derive the BER of a genie-aided DF detector, i.e., we assume error-free decision feedback, whichconstitutesa lower bound on the BERs of the MS/DF detectors. Similar to the MS detection metric,the DF detection threshold ξ^𝙳𝙵(𝐬̂_k,𝐫_k) is also a function of only n_k,1, N_k,1, N_k,0, i.e., we may use the notation ξ^𝙳𝙵(n_k,1,N_k,1,N_k,0). Exploiting this observation, the BER of the genie-aided DF detector conditioned on 𝐜̅ is written asrllP_e^𝙳𝙵(𝐜̅) = ∑_N_k,0∑_N_k,1∑_n_k,1{ŝ[k]≠s[k]|𝐜̅,n_k,1,N_k,1,N_k,0}{n_k,1,N_k,1,N_k,0|𝐜̅} = ∑_N_k,0∑_N_k,1∑_n_k,1 {ŝ[k]≠s[k]|𝐜̅,n_k,1,N_k,1,N_k,0} {N_k,1,N_k,0|𝐜̅,n_k,1}{n_k,1|𝐜̅}. In (<ref>), n_k,1denotes the number of ones in the K-1 previously detected symbols and is a binomial RV. Moreover, N_k,1 and N_k,0 conditioned on 𝐜̅ and n_k,1 are independent Poisson RVs with means n_k,1(c̅_𝚜+c̅_𝚗) and (K-n_k,1)c̅_𝚗, respectively. Therefore, {n_k,1|𝐜̅},{N_k,1|𝐜̅,n_k,1}, and {N_k,0|𝐜̅,n_k,1} can be computed as rll{n_k,1|𝐜̅}=1/2^K-1 K-1 n_k,1 {N_k,1|𝐜̅,n_k,1}= n_k,1^N_k,1(c̅_𝚜+c̅_𝚗)^N_k,1 e^-n_k,1(c̅_𝚜+c̅_𝚗)/N_k,1!{N_k,0|𝐜̅,n_k,1}= (K-n_k,1)^N_k,0c̅_𝚗^N_k,0 e^-(K-n_k,1)c̅_𝚗/N_k,0!.Furthermore, {ŝ[k]≠ s[k]|𝐜̅,n_k,1,N_k,1,N_k,0} for the genie-aided DF detector is given by lll{ŝ[k]≠s[k]|𝐜̅,n_k,1,N_k,1,N_k,0} =1/2 {r[k] < ξ^𝙳𝙵(n_k,1,N_k,1,N_k,0)|𝐜̅,s[k]=1}+1/2 {r[k]≥ξ^𝙳𝙵(n_k,1,N_k,1,N_k,0)|𝐜̅,s[k]=0}= 1/2 + 1/2 [ Q(⌈ξ^𝙳𝙵(n_k,1,N_k,1,N_k,0) ⌉, c̅_𝚜 + c̅_𝚗 ) -Q(⌈ξ^𝙳𝙵(n_k,1,N_k,1,N_k,0) ⌉, c̅_𝚗 ) ]. § NUMERICAL RESULTS In this section, we first present the stochastic MC channel model and the adopted system parameters used for simulation. Subsequently, we evaluate the performance of theproposed non-coherent detectors where we employ the optimal coherent detector as a benchmark scheme.§.§ Stochastic MC Channel Model In this section, we present the stochastic MC channel model used for the simulation results. However, we emphasize that the proposed non-coherent detection framework is valid for any given f_c̅_𝚜(c̅_𝚜) and f_c̅_𝚗(c̅_𝚗) and is not limited to the stochastic MC channel model used here for simulation.Let us assume a point source with impulsive molecule release, a fully transparent spherical receiver with volume V^𝚁𝚇,and an unbounded environment with diffusion coefficient D. Moreover, we denote the distance between the transmitter and the receiver by d. In addition, we assume thatthere is steady uniform flow (or drift)with parallel and perpendicularvelocity components, denoted by v_∥ and v_⊥, respectively, with respect to the direction from the transmitter to the receiver. Furthermore, the signaling molecules may react withenzyme molecules, which are present in the MC environment, and degrade into a form that cannot be detected by the receiver. We assume a uniform and constant concentration of the enzyme, denoted by c̅_𝚎, and a first order reaction mechanism between the signaling and enzyme molecules with constant reaction rate κ <cit.>.Based on the aforementioned assumptions,the expected number of molecules observed at the receiver as a function of time, denoted by c̅_𝚜(t), is given by <cit.>c̅_𝚜(t) = N^𝚃𝚇V^𝚁𝚇/(4π D t)^3/2exp(-κc̅_𝚎t-(d-v_∥t)^2+(v_⊥t)^2/4Dt). Furthermore, assuming a peak observation detector, the sample time from the beginning of each symbol interval is chosen as t^𝚖𝚊𝚡=t>0argmaxc̅_𝚜(t) which leads to c̅_𝚜=t>0maxc̅_𝚜(t). The channel parameters undergo random variations that lead to random variations in c̅_𝚜. For instance, the flow velocity components, v_∥ and v_⊥, may vary over time or the diffusion coefficient, D, and enzyme concentration, c̅_𝚎, may change due to variations in the environment temperature. To capture these effects,we assume that the channel parametersin each detection window are realizations of RVs according toz=z^𝚍𝚎𝚏(1+σ_z𝒩(0,1)),z∈{D,v_∥,v_⊥,c̅_𝚎} where z^𝚍𝚎𝚏 denotes the mean value of parameter z and σ_zz^𝚍𝚎𝚏 is its standard deviation, which determines how much the parameter may deviate from the mean[We note that a Gaussian RV may assume negative values whereas D and c̅_𝚎are non-negative parameters. Therefore, we assume small values for σ_D and σ_c̅_𝚎,and omit those realizations for which z∈{D,c̅_𝚎} is negative.]. As σ_z→ 0,∀ z, the respective MC channel becomes deterministic, and for largeσ_z, the corresponding MC channel is highly stochastic. Substituting the Gaussian RVs z∈{D,v_∥,v_⊥,c̅_𝚎} into (<ref>) may not lead to a closed-form analytical expression for f_c̅_𝚜(c̅_𝚜). Therefore, we employ Monte Carlo simulation to determine a histogram for c̅_𝚜 as the “true" distribution, f_c̅_𝚜(c̅_𝚜), and to verify the effectiveness of the proposed Gamma distribution to approximate the true distribution. For the results provided in this section,we assume that the variation of themean of the noise c̅_𝚗 is modeled similarlyto the variation of the mean of the signal c̅_𝚜. In particular, we choose c̅_𝚗=𝚂𝙽𝚁^-1 X where X is an RV whose PDF is identical to that of c̅_𝚜 and 𝚂𝙽𝚁 is a constant analogous to the signal-to-noise ratio (SNR) in conventional wireless systems. Furthermore, if c̅_𝚜∼Gamma(α,β) holds, we obtain c̅_𝚗∼Gamma(α,𝚂𝙽𝚁β) <cit.>.§.§ Simulation Parameters, DF Window Size, and Benchmark Scheme Simulation Parameters: The default values of the channel parameters are given in Table I.Moreover, we consider the following three scenarios for the stochastic MC channel: Scenario 1: (σ_D,σ_v_∥,σ_v_⊥,σ_c̅_𝚎)=(0.1,0.5,0.5,0.1), Scenario 2: (σ_D,σ_v_∥,σ_v_⊥,σ_c̅_𝚎)=(0.2,1,1.5,0.1), and Scenario 3: (σ_D,σ_v_∥,σ_v_⊥,σ_c̅_𝚎)=(0.1,1.5,0.5,0.2).Each of these scenarios leads to a pair of PDFs forc̅_𝚜 and c̅_𝚗. In particular, we choose c̅_𝚗 such that for the default parameters in Table I and N^𝚃𝚡=10^4, we obtain E{c̅_𝚗}=E{c̅_𝚜}, i.e., 𝚂𝙽𝚁=1 holds for this case. To obtain different SNRs, we change the number of molecules released by the transmitter. We used Monte Carlo simulation to obtain simulation results. Thereby, we first generated N=10^6 realizations of the CSI vector 𝐜̅. Then, for each 𝐜̅, we generated the number ofmolecules observed at the receiver in each symbol interval based on the channel model in (<ref>) where RVs c_𝚜[k] and c_𝚗[k] are Poisson distributed with means c̅_𝚜 and c̅_𝚗, respectively. For the analytical results, we numerically evaluate the expression in (<ref>). DF Window Size: In order to employ the proposed DF detector with window size K, knowledge of the K-1 previously detected symbols is required. However, at the beginning of each transmission interval, i.e., for symbol intervals k<K, such knowledge is not available. Therefore, for the results presented in this section, we assume that the DF detector detects the first symbol without knowledge of the previous symbols, i.e., with window size one. Then, it detects the second symbol based on the knowledge of the detected symbol in the first symbol interval, i.e., with window size two. We continue this process until symbol interval K where, for all the following symbols k≥ K, the previous K-1detected symbols are available and a fixed window size of K can be used for DF detection. Because of the overlapping detection windows, we implicitly have to assume that the CSI is constant for B≥ K symbol intervals such that a DF detector with window size K can make B-K+1 symbol decisions. For the results shown in this section, we consider the BER for only the B-K+1 symbol intervals for which the DF detector actually employs a detection window of size K. Fig. <ref> schematically shows the different detection window sizes used forDF detection assuming K=5 and B=10. Benchmark Scheme: We note that the non-coherent detector in <cit.> was designed assuming an additive white Gaussian noise (AWGN) channel model. As discussed in Subsection II-A, the Poisson distribution for the observed number of molecules at the receiver is a more accurate model for the diffusive MC channel than the Gaussian distribution <cit.>.Additionally, the signal model assumed in <cit.> is different from that considered in this paper. Therefore, we have not included the detection scheme in <cit.> as a benchmark in this section as a direct comparison would not be fair. Instead we use the optimal coherent detector, cf. Subsection III.A, as a benchmark scheme which in fact constitutes a performance upper bound for any non-coherent detector.§.§ Performance Evaluation In this subsection, we first verify the accuracy of the proposed Gamma distribution, cf. Subsection III-B, in Fig. <ref>, and our analytical derivations, cf. Section IV, in Fig. <ref>. Subsequently,in Figs. <ref>-<ref>, we evaluate the performance of the proposed non-coherent detectors for different system parameters.Fig. <ref> shows the histogram of the CSI, c̅_𝚜, obtained by Monte Carlo simulation, and the corresponding Gamma PDF approximation for the three considered stochastic scenarios. Additionally, the result for the case when all the underlying channel parameters in (<ref>) assume their nominal values given in Table I, i.e., when the channel is deterministic, are shown. The optimal parameters of the Gamma distribution are also shown in Fig. <ref>and found using the search procedure presented in Subsection III-A and Remark <ref> with w(x)=1,∀ x, and δ=0.5. For Scenario 3,Nakagami, inverse Gaussian, and Levy distribution approximations are also plotted with their parameters optimized based on a similar search procedure as proposed in Subsection III-A for the Gamma PDF. We visually observe a very close match between the histogram (exact PDF) and the Gamma PDF approximation for all three scenarios. Moreover, Fig. <ref> suggests that for Scenario 3, the Gamma distribution is a better match to the exact distribution than the Nakagami, inverse Gaussian, and Levy distributions. Note that although theNakagami distribution seems to also accurately match the exact distribution in Fig. <ref>, it does not lead to a closed-form expression for the ML detection metric, and hence, at leastfor the purpose of this paper, the Nakagami distribution is not a good option. We note that the variances of the channel parameters (σ_D,σ_v_∥,σ_v_⊥,σ_c̅_𝚎) are larger for Scenario 3 compared to Scenario 1, i.e., the underlying channel parameters for Scenario 3 are more random compared to Scenario 1. From Fig. <ref>, we observe that as the randomness in the MC channel increases, i.e., from Scenario 1 to Scenario 3, the mean of the CSI decreases and its variance increases.In Fig. <ref>, we verify the simulation results using the performance analysis developed in Section IV. In particular, we show the BER versus the SNR in dB for Scenario 1 and K=10. For the symbol-by-symbol detectors, i.e., the optimal coherent ML (C-ML) detector and the optimal non-coherent SS (NC-SS) detector, we observe that the analytical results obtained from (<ref>) match perfectly with the simulation results. The accuracy of the proposed Gamma distribution to model the considered stochastic MC channel was verified in Fig. <ref> by showing the true and approximated PDFs. In Fig. <ref>, we verify the Gamma approximation in terms of the resulting BER performance. In particular, we show results where the optimal MS detection metric is obtained via expectation over 10^6 realization of the true CSI, i.e., Monte Carlo simulation, and analytically using the proposed Gamma distribution, cf. (<ref>). From Fig. <ref>, we observethat the BERs of the non-coherent MS detectors which employ the optimal metric (NC-MS-OM) and approximated metric (NC-MS-AM)are almost identical. Furthermore,we show the union bound (UB) in (<ref>) and the BER of the genie-aided DF (GA-DF) detector in (<ref>) as an upper bound and a lower bound for the BER of the optimal MS detector, respectively. Note that the union bound becomes a tight upper bound for large SNRs whereas the BER of the genie-aided DF detector is a tight lower bound for all SNR values considered in Fig. <ref>. In Fig. <ref>, we show the BER versus κ=B/K, see Subsection V.B, for the three considered scenarios, K=5, and 𝚂𝙽𝚁=10 dB. Note that the actual CSI in (<ref>) is used for simulation whereas the approximated Gamma distribution is employed for calculation of the detection metrics for the proposed MS and DF detectors by using (<ref>), where the corresponding curves in Fig. <ref> are denoted by NC-MS-AM and NC-DF-AM, respectively. We observe that the optimal MS detector with detection window size B outperforms the DF detector for all κ≥ 1 and approaches the performance of the coherent ML detector as κ→∞. For small κ, the BER of the DF detector is higher than the BER of the optimal MS detector with the same detection window size K, however, as κ increases, the DF detector outperforms the optimal MS detector with detection window size K by a small margin. This is due to the fact that the DF detector implicitly exploits the property that the CSI is fixed for B≥ K symbol intervals whereas the decision of the optimal MS detector is independent for each detection window size K. Moreover, from Fig. <ref>, we observe that the BER increases for all considered detectors as the MC channel becomes more stochastic, i.e., from Scenario 1 to Scenario 2 to Scenario 3. In Fig. <ref>, we show the BER versus the detection window size K for Scenario 2 and 𝚂𝙽𝚁∈{10,20} dB.We assume κ=1 for the DF detector. Note thatthe BERs ofthe proposed detectors decrease and finally converge to thelower bound provided by the coherent ML detector as K→∞. Furthermore, the gap between the BER of the optimal non-coherent MS detector and the coherent ML detector is small for K=20, which reveals the effectiveness of the optimalMS detector, although no resources are spent for trainingand CSI acquisition.Moreover, the gap between the BER of the proposed optimal MS detector and the proposed suboptimal non-coherent blind (NC-BL) detector decreases forlarger values of K, which confirms the effectiveness of the proposed suboptimal blind detector for large detection window sizes. Furthermore, as expected, the performance of all detection schemes in Fig. <ref> is better for 𝚂𝙽𝚁=20 dB compared to 𝚂𝙽𝚁=10 dB.In Fig. <ref>, we plot the BER versus the SNR in dB for Scenario 2 and K∈{5,10}. In this figure, we observe that as the SNR increases, the BER improves for all considered detectors. We note that as 𝚂𝙽𝚁→∞, the BER of the proposed blind detector saturates to a certain error floor. This is due to the fact that for the blind CSI estimator in (<ref>), we assume that the percentages of ones and zeros in a given detection window are exactly 50%, which is not always true, especially for small values of K. This introduces an inherent CSI estimation error and leads to the aforementioned error floor for the proposed blind detector. In contrast, none of the other detectors has a BER error floor. Note that as K increases, the BER of the optimal non-coherent MS detector decreases and approaches the BER of the coherent ML detector.We can also observe from Fig. <ref> that the optimal MS detector outperforms the proposed suboptimal blind detector, particularly for small K, but the gap between the BERs of these two detectors decreases as K increases. Recall that we assume ISI-free transmission in thesystem model adopted in this paper. However, although the ISI may be considerably reduced using, e.g., the methods in <cit.> and <cit.>, some residual ISI always exists as the length of the symbol intervals is finite. Therefore, in the system model, we assumed that the effect of the residual ISI is included in c_𝚗[k] and is sufficiently small compared to the other components in c_𝚗[k] such that c_𝚗[k] is (approximately) independent of the signal component c_𝚜[k].In Fig. <ref>we study the BER performance loss if this assumption does not hold. To this end, we consider a finite symbol interval length, denoted by T_𝚜𝚢𝚖𝚋, and assume that the contribution of the ISI from the previous symbol intervals is present. The detectors treat the ISI as noise, i.e., c̅_𝚗=c̅_𝚗^𝚎𝚡𝚝+c̅_𝚗^𝙸𝚂𝙸 where c̅_𝚗^𝚎𝚡𝚝 is the mean of the external noise and c̅_𝚗^𝙸𝚂𝙸 is the expected ISI given by lllc̅_𝚗^𝙸𝚂𝙸 = E{∑_l=2^∞s[k-l+1]c̅_𝚜,l} = 1/2∑_l=2^∞c̅_𝚜,l, where c̅_𝚜,l is the l-th channel tap, i.e., if the transmitter releases N^𝚃𝚇 molecules in symbol interval k, c̅_𝚜,l is the expected number of molecules arriving at the destination in symbol interval k+l-1. Therefore, c̅_𝚜,1 is the channel tap for the desired signal and c̅_𝚜,l,l≥ 2, are channel taps that create ISI.In Fig. <ref>, we show the BER versus 𝚂𝙽𝚁^𝚎𝚡𝚝=c̅_𝚜,1/c̅_𝚗^𝚎𝚡𝚝 in dB for Scenario 3, K=10, and symbol intervals T_𝚜𝚢𝚖𝚋∈{2,5,10}× T_𝚖𝚊𝚡 where T_𝚖𝚊𝚡=t>0argmaxc̅_𝚜(t) and c̅_𝚜(t) is obtained from (<ref>) assuming the nominal values of the MC system parameters in Table I. We observe from Fig. <ref> that as 𝚂𝙽𝚁^𝚎𝚡𝚝→∞, the BERs of all detectors saturate to a BER floor if ISI is present. Moreover, due to the severe mismatch between the system model assumed for development of the considered detectors and the actual simulated MC channel for the strong ISI case, the relative performance of these detectors is not obvious. In fact, we observe from Fig. <ref> thatfor the considered system parameters and T_𝚜𝚢𝚖𝚋=2 T_𝚖𝚊𝚡, the performance of the coherent ML detector considerably deterioratescompared to that of the MS detector and, similarly, the performance of the MS detector degrades compared to that of the blind detector. This suggests that the coherent ML detector is more sensitive to the considered system model mismatch compared to the MS detector and, similarly, theMS detector is more sensitive compared to the blind detector. Finally, Fig. <ref> shows that as the ISI becomes weaker, i.e., as T_𝚜𝚢𝚖𝚋 increases, the BERs of all detectors approach the respective BERs for the ISI-free case[We note that optimal non-coherent detection for the case when strong ISI is present is an important research problem which is beyond the scope of this paper and left for future work.].§ CONCLUSIONS We have derived the optimal non-coherentMS and SS detectors as well as a non-coherent DF detectorwhich do not require instantaneous CSI knowledge. As compared to the coherent detectors previously studied, the proposed non-coherent detectors may be preferable in practical scenariossince the complexity and the overhead associated with CSI acquisition are avoided.In order to further reduce the complexity of our detectors, we proposed an approximate detection metric and a low-complexity suboptimal blind detector. We further derived an analytical expression for the BER of the optimal SS detector, and a lower bound and an upper boundfor the BER of the optimal MS detector. Simulation results confirmed the analysis and showed that the proposed optimal MS detector outperforms the suboptimal blind detector, particularly for small detection window sizes. However, as the size of the detection window increases, the performances of the proposed optimal and suboptimal detectors converge to that of the benchmark coherent ML detector which requires perfect CSI. This demonstrates the effectiveness of the proposed detection schemes. § PROOF OF THEOREM <REF>Using sets 𝒦_1 and 𝒦_0, (<ref>) can be simplified to 𝐬^𝙼𝚂 =𝐬∈𝒜argmax∫_c̅_𝚜≥ 0∫_c̅_𝚗≥ 0 ( c̅_𝚜 + c̅_𝚗)^∑_k∈𝒦_1 r[k]c̅_𝚗^∑_k∈𝒦_0 r[k]×exp(-|𝒦_1|c̅_𝚜- K c̅_𝚗) f_(c̅_𝚜,c̅_𝚗)(c̅_𝚜,c̅_𝚗) dc̅_𝚜dc̅_𝚗, where the term r[k]! in (<ref>) is removed in (<ref>) since it does not affect the MS detection result. Moreover, the optimal MS detector can be written equivalently inexpectation form as 𝐬^𝙼𝚂 = 𝐬∈𝒜argmaxE_𝐜̅{A(𝐬,𝐫)( c̅_𝚜 + c̅_𝚗)^∑_k∈𝒦_1 r[k]c̅_𝚗^∑_k∈𝒦_0 r[k] B(𝐬,𝐫)exp(-|𝒦_1|c̅_𝚜- K c̅_𝚗)}. We can conclude the following properties from the MS detection metric Λ^𝙼𝚂(𝐬,𝐫)=E_(c̅_𝚜, c̅_𝚗){A(𝐬,𝐫) B(𝐬,𝐫)}. First, with respect to the data sequence 𝐬, term B(𝐬,𝐫) is only a function of the number of ones in 𝐬. Second, for a given number of ones in 𝐬, term A(𝐬,𝐫) is maximized if the ones in 𝐬 correspond to the largestelements of the observation vector 𝐫. Note that these two properties hold for any given CSI 𝐜̅. Hence, they also hold after the expectation operation with respect to 𝐜̅, i.e., E_𝐜̅{A(𝐬,𝐫) B(𝐬,𝐫)}. Therefore, we can avoid searching over all 𝐬∈𝒜, and instead, find the optimal threshold n_1 ∈{0,1,…,K} which sets the elements of 𝐬 corresponding to the n_1 largest elements of 𝐫 to one and the remaining elementsto zero. Moreover, we can further simplify the MS detection metric in (<ref>) asΛ^𝙼𝚂(𝐬,𝐫) (a)=E_(c̅_𝚜,c̅_𝚗){[∑_i=0^N_1N_1ic̅_𝚜^N_1-ic̅_𝚗^i ]c̅_𝚗^N_0exp(- |𝒦_1|c̅_𝚜- K c̅_𝚗) }(b)=∑_i=0^N_1N_1iE_c̅_𝚜{c̅_𝚜^N_1-i e^- n_1 c̅_𝚜}E_c̅_𝚗{c̅_𝚗^N_0+i e^- K c̅_𝚗} , where in equality (a), we employ the Binomial expansion, i.e., (x+y)^n = ∑_i = 0^nni x^n-i y^i with ni=n!/i!(n-i)!,and inequality (b),we use n_1=|𝒦_1| and the assumption thatRVs c̅_𝚜 and c̅_𝚗 are independent. Furthermore, recalling that{s[k]=1}={s[k]=0}=0.5 holds, the optimal threshold, denoted by ζ^𝙼𝚂(𝐫), approaches K/2 as K→∞. This concludes the proof. § PROOF OF COROLLARY <REF>For the special case of the symbol-by-symbol detection, we obtain two detection metrics from (<ref>) corresponding to s[k]=1 and s[k]=0, respectively, i.e.,lllΛ(s[k]) =E_(c̅_𝚜,c̅_𝚗){ ( c̅_𝚜 + c̅_𝚗 )^r[k] e^- (c̅_𝚜 + c̅_𝚗 )}, if s[k] = 1 E_c̅_𝚗{ c̅_𝚗^r[k] e^ - c̅_𝚗}, otherwiseNow, using Lemma <ref>, we first show that Λ(s[k]=1)/Λ(s[k]=0) is a monotonically increasing function of r[k]. Moreover, for r[k]=0 and r[k]→∞, we obtain that Λ(s[k]=1)<Λ(s[k]=0) and Λ(s[k]=1)>Λ(s[k]=0) hold, respectively. Therefore, there exists a unique threshold for r[k] below which Λ(s[k]=1)<Λ(s[k]=0) holds.If f_n(x)/g_m(x) is a monotonically increasing function of x for all possible pairs of (m,n), then function ∑_nf_n(x)/∑_m g_m(x) is also monotonically increasing in x. Please refer to Appendix <ref>.In order to apply the result of Lemma <ref> to Λ(s[k]=1)/Λ(s[k]=0), we first note that the expectation terms E_x{f(x)} can be written in summation form as ∑_n {x_n} f(x_n) by discretizing the domain of x into a set {x_1,x_2,…}.Therefore, we have Λ(s[k]=1)≜∑_n f_n(r[k]) = ∑_n {c̅_𝚜}{c̅_𝚗} ( c̅_𝚜 + c̅_𝚗)^r[k] e^- (c̅_𝚜 + c̅_𝚗 ) and Λ(s[k]=0)≜∑_m g_m(r[k]) = ∑_n {c̅_𝚗}c̅_𝚗^r[k] e^-c̅_𝚗. Now, we have to show that f_n(r[k])/g_m(r[k]) = {c̅_𝚜}{c̅_𝚗}( c̅_𝚜 + c̅_𝚗)^r[k] e^- (c̅_𝚜 + c̅_𝚗 ) /{c̅_𝚗}c̅_𝚗^r[k] e^-c̅_𝚗 = {c̅_𝚜}(1+c̅_𝚜/c̅_𝚗)^r[k] e^-c̅_𝚜 is a monotonically increasing function of r[k] for all c̅_𝚜 and c̅_𝚗 which straightforwardly holds.To further simplify the SS detection metric under hypothesis s[k]=1, we employ the Binomial expansion which leads to lllΛ(s[k]=1) = ∑_i = 0^r[k]r[k] iE_c̅_𝚜{ c̅_𝚜^r[k]-ie^-c̅_𝚜 }E_c̅_𝚗{ c̅_𝚗^ie^-c̅_𝚗 }. The above results are conciselystated in Corollary <ref> which concludes the proof.§ PROOF OF LEMMA <REF>If f_n(x)/g_m(x) is a monotonically increasing function of x, its derivative with respect to x has to be positive, i.e.,lll∂/∂x (f_n(x)/g_m(x)) = f'_n(x)g_m(x)-f_n(x)g'_m(x)/(g_m(x))^2 > 0,∀n,m. In other words, f'_n(x)g_m(x)-f_n(x)g'_m(x)> 0 has to hold for ∀ n,m. Using this result, the sufficient condition for ∑_nf_n(x)/∑_m g_m(x) to be a monotonically increasing function of x can be shown aslll ∂/∂x (∑_n f_n(x)/∑_m g_m(x)) = ∑_n f'_n(x)×∑_m g_m(x) - ∑_n f_n(x)×∑_m g'_m(x) /(∑_m g_m(x))^2 (a)= ∑_n∑_m f'_n(x) g_m(x) - ∑_n ∑_m f_n(x) g'_m(x) /(∑_m g_m(x))^2 = ∑_n∑_m [f'_n(x) g_m(x) -f_n(x) g'_m(x)] /(∑_m g_m(x))^2 (b)> 0, where equality (a) follows the sum-product rule, i.e., ∑_n x_n ∑_m y_m = ∑_n∑_m x_n y_m, and inequality (b) follows from (<ref>). This completes the proof. IEEEtran | http://arxiv.org/abs/1707.08926v1 | {
"authors": [
"Vahid Jamali",
"Nariman Farsad",
"Robert Schober",
"Andrea Goldsmith"
],
"categories": [
"cs.IT",
"math.IT"
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"primary_category": "cs.IT",
"published": "20170727163407",
"title": "Non-Coherent Detection for Diffusive Molecular Communications"
} |
[pages=1-last]tse.pdf | http://arxiv.org/abs/1707.08411v2 | {
"authors": [
"Maurice H. ter Beek",
"Axel Legay",
"Alberto Lluch Lafuente",
"Andrea Vandin"
],
"categories": [
"cs.SE"
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"primary_category": "cs.SE",
"published": "20170726124812",
"title": "A framework for quantitative modeling and analysis of highly (re)configurable systems"
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Sequential design of experiments to estimate a probability of exceeding a threshold in a multi-fidelity stochastic simulatorRémi STROH*lne, Trappes, France - [email protected], CentraleSupélec, Univ. Paris-Sud, cnrs, Université Paris-Saclay, Gif-sur-Yvette, France Séverine DEMEYER, Nicolas FISCHERlne, Trappes, France - [email protected] Julien BECT, Emmanuel VAZQUEZl2s, CentraleSupélec, Univ. Paris-Sud, cnrs, Université Paris-Saclay, Gif-sur-Yvette, France - [email protected] In this article, we consider a stochastic numerical simulator to assess the impact of some factors on a phenomenon. The simulator is seen as a black box with inputs and outputs.The quality of a simulation, hereafter referred to as fidelity, is assumed to be tunable by means of an additional input of the simulator (e.g., a mesh size parameter): high-fidelity simulations provide more accurate results, but are time-consuming. Using a limited computation-time budget, we want to estimate, for any value of the physical inputs, the probability that a certain scalar output of the simulator will exceed a given critical threshold at the highest fidelity level. The problem is addressed in a Bayesian framework, using a Gaussian process model of the multi-fidelity simulator. We consider a Bayesian estimator of the probability, together with an associated measure of uncertainty, and propose a new multi-fidelity sequential design strategy, called msur, to select the value of physical inputs and the fidelity level of new simulations. The msur strategy is tested on an example. Keywords: Multi-fidelity; Sequential design; Bayesian analysis; Gaussian process.§ INTRODUCTION The objective of this article is to propose a new Bayesian algorithm for sequential design of experiments in the context of multi-fidelity stochastic numerical simulators.A numerical simulator is a computer program modeling a physical phenomenon or a system. When the simulator is deterministic, running the computer program twice using the same inputs yields the same outputs.In this case, the simulator can be viewed formally as a function. When the simulator is stochastic, running twice the simulator with the same inputs does not return the same output.Moreover, we assume that the simulator has a particular input called fidelity parameter—e.g., the mesh size of a finite-difference partial differential equation solver—that controls a trade-off between quality of simulation and computation time.A high-fidelity simulation provides an accurate result, but is time-consuming.Let t ∈T be the fidelity input of the simulator, and x ∈X the vector of all other inputs.At fixed (x,t), the output of the simulator Z follows a probability distribution ^Z_x,t.We denote bythe value of the fidelity input associated to the highest available level of fidelity:this is the level of interest for the user of the simulator.In this article, we focus on comparing the output Z to a critical threshold —typically, a level of output that should not be exceeded by the physical phenomenon or the system under study.This comparison can be studied from two points of view: either locally, by considering the function that gives for each point x the probability of exceeding the threshold at this point,(x) = ^Z_x, (Z > ), x∈X,or globally, by computing the global probabilityof exceeding the threshold, which may be written as= ∫_X(x) (),whereis a probability distribution which models uncertainty about the value of the input factors.Our objective is to estimate these quantities by observing the stochastic outcomes z_1, z_2… of simulations at points (x_1,t_1),(x_2, t_2)…The estimators are built using a Bayesian model of the simulator.More precisely, we assume a prior model about ^Z_x, t and compute the posterior distribution of the quantity of interest (eitheror ) given χ_n = (x_i,t_i;z_i)_i≤ n.In this article, we suggest a new sequential design algorithm to select points (x_1,t_1),(x_2, t_2)… in order to obtain a fast reduction of the uncertainty about or .Our methods deals both with the stochastic nature of the simulator and the tunable simulation cost.The paper is organized as follows.Section <ref> sets the Bayesian framework that we consider for stochastic multi-fidelity simulators.Section <ref> describes our sequential algorithm and our new sampling criterion called msur.Section <ref> illustrates the method on a test problem, and Section <ref> concludes the paper. § BAYESIAN FRAMEWORKFollowing our previous work on multi-fidelity stochastic simulators <cit.>, assume that the output Z of a simulation at (x,t) follows a normal distribution:Z |ξ, λ ∼ (ξ (x,t), λ(x,t)) .Assume moreover that conditional on ξ and λ, all runs of the simulator are independent.Of course, the choice of a normal distribution here is a simplifying hypothesis, that must be verified in practice for each particular simulator [ Other (possibly non-parametric) families of distributions could be considered as well.Note, however, that the convenient conjugacy property of the Gaussian process prior with respect to the Gaussian likelihood would be lost by doing so.].Assume a Gaussian process prior for the mean function ξ:ξ| m, k∼ (m,k((x,t), (x',t'))), m∼ 𝖴(R),where m is the (constant) mean of the Gaussian process, k its covariance function, and 𝖴(R) the (improper) uniform distribution on R.For the sake of simplicity, the functions λ and k are assumed to be known.Under this prior, the posterior distribution of ξ conditionally to the observations χ_n is Gaussian:ξ|χ_n∼ (m_n(x,t),k_n((x,t), (x',t'))),where m_n and k_n are given by the kriging formulas <cit.>.Since Z has a Gaussian distribution conditional on ξ and λ, for all x∈X, the probability of exceeding a critical value can be written as(x) = Φ(ξ(x, ) - /√(λ(x, ))),with Φ the cumulative distribution function of the normal distribution.Then, denoting by _n = (·|χ_n) and _n = ( ·|χ_n) the posterior mean and variance operators, we have: _n((x)) = Φ(u_n(x) ), _n((x)) = Φ_2(u_n(x), u_n(x); r_n(x, x)) - Φ(u_n(x) )^2,where Φ_2( ·, · ;r) stands for the bivariate normal distribution function with correlation r,V_n(x,t) = λ(x,t) + k_n((x,t), (x,t)),u_n(x) = m_n(x, ) - /√(V_n(x, )), andr_n(x,x') = k_n((x, ),(x', ))/√(V_n(x, ) V_n(x', ) ). § PROPOSED ALGORITHM §.§ A SUR criterion to estimate a probability of exceeding a threshold in the case of stochastic outputs The goal of sequential design of experiments is to select a simulation point (x_n+1, t_n+1), using the result of the previous observations χ_n, to obtain a good improvement of our knowledge about a given quantity of interest.Our algorithm is based on the idea of sur strategies <cit.>.The main idea of SUR strategies is to define a measure of uncertainty _n about the estimator of a quantity of interest.Then, assuming that there is no fidelity parameter t, the next observation point is selected in order to minimize the expected uncertainty using this new observation, whose outcome is random:x_n+1 = _x ∈X _n(_n+1| X_n+1 = x). Several measures of uncertainty _n have been proposed in the literature <cit.> for the problem of estimating the volume or contour of an excursion set of a deterministic function.In our framework, however, we deal with a stochastic simulator, and we propose a new measure of uncertainty for this case.Let L be the L^2 loss function(f, g) = f - g^2_ = ∫_X(f(x) - g(x))^2 μ(),where μ denotes a positive measure on X.We suggest to measure uncertainty by the L^2 loss incurred when estimatingby _n = _n():_n = _n( (_n, )) = ∫_X_n( (x) ) μ().Note that in the case where the goal is to estimate , _n provides an upper bound on the posterior variance for any that admits a density g with respect to μ:_n() ≤ G _n, where G = ∫ g^2 and g =/.Consequently, a sequential design strategy that aims to reduce _n will be useful not only to estimate itself but also to estimate for a large class of probability distributions .Let J_n(x,t) = _n(_n+1| X_n+1 = x, T_n+1 = t). Since the output of the simulator has a Gaussian distribution, J_n(x, t) can be written asJ_n(x,t) = ∫_X( Φ_2(u_n(y), u_n(y); r_n(y, y)) - Φ_2(u_n(y), u_n(y); k_n((y, ), (x,t))^2/V_n(y, )V_n(x,t)) ) μ (). §.§ Dealing with tunable fidelity The sur strategy (<ref>) is relevant when there is no fidelity parameter t.In a multi-fidelity context, however, the simulator becomes more and more time-consuming as the fidelity parameters comes closer to .We propose a new sequential design algorithm which takes into account the variable simulation cost.Let C(x,t) > 0 be the cost of observing the simulator at (x,t).The idea is to balance the benefit of an observation against its cost.For us, the benefit of an observation is the reduction of uncertainty.Hence, we propose the following sequential strategy called msur:(x_n+1; t_n+1) = _(x, t) ∈X×T _n - J_n(x, t)/C(x, t).This idea is similar to the one proposed by <cit.> in the context of optimization, where the expected improvement is divided by the cost of an observation. Likewise, for the purpose of providing a surrogate model of the simulator, <cit.> suggested to choose the new level of fidelity by comparing the cost of the level and the reward in terms of reduction of uncertainty.Note that in many applications, the cost C depends only on the level t: C(x,t) = C(t).In this case, (<ref>) can be divided into two steps: first, select the input x at each level that minimizes the sur criterion; second, select the level t that maximizes the msur criterion.Our sequential design algorithm can thus be rewritten in this case as{x̃(t) =_x ∈XJ_n(x, t), t_n+1= _t ∈T _n- J_n(x̃(t), t)/C(t), x_n+1= x̃(t_n+1). . § ILLUSTRATIONWe illustrate our algorithm on a two-dimensional example inspired by <cit.>.We consider a damped harmonic oscillator subject torandom forcing, described by the stochastic differential equationẌ(u) + 2ζω_0Ẋ(u) + ω_0^2X(u) = W(u),X(0) = 0, Ẋ(0) = 0, u ∈[0; = 30],where W is a Gaussian white noise (with spectral density equal to one).We compute an approximation (X_n)_n=0,…, ⌊/⌋ of X by finite differences with a time step .We use an explicit exponential Euler scheme <cit.>. The output of our simulator is defined as[ Z:[0; 30] ×[0; 1] ×]0; 1]→R;(ω_0, ζ, )↦ log(max{X_n, n ≤⌊/⌋}). ] The pair (ω_0, ζ) corresponds to the input vector x and the step timeplays the role of the fidelity parameter. The critical thresholdis set to -3.The highest level of fidelity is set to = 0.01.The computational cost is empirically linear with respect to the fidelity level: C() = a/ + b, with a = 0.0098 and b = 0.02 (coefficients chosen to have C( ) = 1).Figure <ref> represents the contour plots of the mean function, the standard deviation and the probability function at three different levels of fidelity.We compute a reference valueofon a 100× 100 regular grid at the highest level of fidelity using 10^4 simulations of Z (see Figure <ref>).We suppose that the input distributionis the uniform distribution on [0; 30]×[0; 1] and let μ =.The mean value = 83.3% ofon the grid serves as a reference value for .We use the Bayesian model of Section <ref> with fixed hyper-parameters, and ten fixed levels of fidelity: = 1, 0.5, 0.33, 0.25, 0.2, 0.17, 0.1, 0.05, 0.02 and 0.01.The initial design (initial simulations before applying our msur strategy) consists of 180× 60× 20 × 10 × 5 nested observations on the five lowest fidelity levels (180 on the level = 1, 60 on the level = 0.5, …). The initial design is set using the algorithm of <cit.>.In our experiments, we compare two strategies: five sl strategies and the msur strategy (<ref>).For sl strategies, all new points are sequentially selected on a fixed level, using then sur strategy (<ref>).Each sl strategy corresponds to one level of fidelity.Each time a new observation point must be selected, we choose the point that achieves the best value of J_n among 500 candidate points per level drawn from .We allocate a simulation-time budget of 20 for each strategy.All experiments are repeated 12 times.Integrals are approximated by a sum on the 100× 100 regular grid.The strategies are compared based on the mean square error between estimations and references.Results are presented on Figure <ref>. Figure <ref> represents the mean square error of the estimation _n of , and Figure <ref> represents the mean square error of the estimation _n of , as a function of the computational cost.Each curve corresponds to an average over 12 experiments with the same method of sequential design.The plain red curve corresponds to the msur strategy.The dotted crossed blue-green curves correspond to sl strategies (one curve per level). We can see that the best level is achieved at = 0.05.The lower fidelity levels are too biased to estimate the probabilities correctly, and the upper fidelity levels are too expensive to make it possible to carry out enough observations in order to estimate the probabilities accurately.Moreover, we can see that the msur strategy is as good as the sl strategy at = 0.05.Consequently, with the msur strategy, one does not need to know which level yields the best trade-off betweenaccuracy and computational cost. § CONCLUSIONThis article makes two contributions.First, we suggest a new sur criterion to estimate a probability of exceeding a threshold in the case of a stochastic simulator.Second, we construct a sequential strategy called msur as an adaptation of the sur strategies to deal with multi-fidelity simulators.Our first results are promising, because the msur strategy succeeds to get the better of all single-level sur strategies without knowing which level provides the best compromise between speed and accuracy. apalike | http://arxiv.org/abs/1707.08384v1 | {
"authors": [
"Rémi Stroh",
"Séverine Demeyer",
"Nicolas Fischer",
"Julien Bect",
"Emmanuel Vazquez"
],
"categories": [
"stat.CO",
"stat.ME",
"stat.ML"
],
"primary_category": "stat.CO",
"published": "20170726113558",
"title": "Sequential design of experiments to estimate a probability of exceeding a threshold in a multi-fidelity stochastic simulator"
} |
Department of Physics, National Technical University of Athens, GR-15780 Athens, Greece Materials Science Department, University of Patras, Rio GR-26504, Greece Crete Center for Quantum Complexity and Nanotechnology (CCQCN), Physics Department, University of Crete GR-71003 Heraklion, GreeceInstitute of Electronic Structure and Laser, FORTH, Heraklion, Greece Department of Physics, University of South Florida, Tampa, Florida 33620, USATheoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vass. Constantinou 48, GR-11635 Athens, GreeceWe present a simple torsional potential for graphene to accurately describe its out-of-plane deformations. The parameters of the potential are derived through appropriate fitting with suitable DFT calculations regarding the deformation energy of graphene sheets folded around two different folding axes, along an armchair or along a zig-zag direction. Removing the energetic contribution of bending angles, using a previously introduced angle bending potential, we isolate the purely torsional deformation energy, which is then fitted to simple torsional force fields. The presented out-of-plane torsional potential can accurately fit the deformation energy for relatively large torsional angles up to 0.5 rad. To test our proposed potential, we apply it to the problem of the vertical displacement of a single carbon atom out of the graphene plane and compare the obtained deformation energy with corresponding DFT calculations. The dependence of the deformation energy on the vertical displacement of the pulled carbon atom is indistinguishable in these two cases, for displacements up to about 0.5 Å. The presented potential is applicable to other sp^2 carbon structures.A Torsional potential for graphene derived from fitting to DFT results Nektarios N. Lathiotakis December 30, 2023 ======================================================================§ INTRODUCTIONFollowing the isolation of single layer graphene <cit.> an enormous research effort has been devoted to the study of this two-dimensional material and its properties <cit.>. Potential applications have been explored in electronics <cit.>, opto-electronics <cit.>, gas filtering <cit.>, energy storage <cit.>, uses related to its unique mechanicalproperties <cit.>, etc.Many empirical force fields have been used in atomistic simulations, calculating various structural, mechanical or phonon properties of graphene <cit.>. Besides the older, well known Tersoff <cit.> and Brenner <cit.> potentials, more accurate force fields have been introduced the last two decades. For example, optimized parameter sets for the latter potentials, providing better description of structural and phonon properties of graphene are presented in Ref. <cit.>. LCBOP <cit.> and AIREBO <cit.> are efficient potentials that have been widely applied in many calculations. Other potentials leading to good predictions of elastic and thermal properties of graphene have been also discussed <cit.>.More recently, we have presented simple analytical expressions for the accurate description of bond stretching and angle bending potentials of graphene <cit.>. These potentials are derived by fitting analytical functions to the deformation energy of proper distortions of graphene, obtained through accurate calculations from first principles' methods (DFT). The presented force field is applicable only to distortions restricted within the plane of graphene. These in-plane potentials can accurately describe elastic properties and the mechanical response of graphene in various extensional loads <cit.>. In this work, using similar ideas and methods, we extend this force field with torsional energy terms, in order to be able to describe out-of-plane distortions in graphene.The basic motivation is to provide a simple and computational efficient classical potential which can be used for accurate large-scale atomistic calculations. The torsional potential presented here is also capable to describe other non-planar sp^2 carbon systems, like fullerenes and carbon nanotubes <cit.>.In the present work, we describe in detail the procedure followed and the necessary analytical calculations in order to fit the proposed torsional potential to ab-initio data.The full potential is then tested in the case of the deformation energy due to the vertical displacement of a C atom outside graphene's plane. A more comprehensive benchmark study for fullerenes, nanotubes and graphene's phonons is presented elsewhere <cit.>. We have considered two types of folding of graphene sheets around different axes (either an armchair or a zig-zag one). The corresponding deformation energies are calculated using DFT methods. Following the removal of the contribution of angle bending terms in the total deformation energy, we isolated the pure torsional energy. Then, the analytic modeling of this energy in terms ofindividual torsional contributions, leads to a fitting procedure providing the optimal parameters of the out-of-plane torsional energy.This paper is organized as follows: In Sect. <ref> we describe the structures and methodology adopted for the DFT simulations that was used to obtain the deformation energies. Then in Sect. <ref>, we present the analytic work for removing the angle bending contributions from the deformation energies.The analytic expressions of the torsional energy terms in terms of the folding angles are provided in Sec. <ref>. The fitting of the torsional terms of bothModel 1 and 2 is described in Sec. <ref> completing the presentation of the derivation of the new torsional force fields. Then, in Sec. <ref>, we present a test case, the deformation energydue to the vertical displacement of a carbon atom outside graphene's plane, as afirst application of the proposed scheme and compare the prediction of Models 1 and 2 with DFT results.Finally, a summary and conclusions are given in Sect. <ref>.§ STRUCTURES AND DFT CALCULATIONS§.§ Torsion angles in graphene In Fig. <ref>, we show a part ofthe honeycombstructure of graphene and a few carbon-atom positions labeled as i, j, k, l, m.The quadruple (i, j, k, l), with 3 of these positions belonging to the same hexagonal ring and one to anadjacent iscustomary called “trans" while the quadruple(i, j, k, m), with all belonging to the same ring, “cis". In the case that the structure is distorted and the atoms in the quadruple are no longerco-planar, we can define a torsion angle, which we label as (i-j-k-l), as the dihedral angle of the planes i-j-k, j-k-l. The torsion angles can then be classifiedas “trans" or “cis" accordingly.The dihedral angle between two planes, e.g. i-j-k and j-k-l can be defined as the angle between the vectors perpendicular to the planes. We assumed that the perpendicular vectors are pointinginwards for clockwise triples (like i-j-k) or outwards for anti-clockwise triples (like j-k-l).Under this assumption, torsional angles are in the range [0,π] with “cis" angles smaller than π/2 and “trans" angles larger than π/2.§.§ DFT results DFT calculations were performed for two distorted graphene structures, one that graphene is folded around an armchair axis and one around a zig-zag. These structures are shown in Fig. <ref> where we label all atoms relevant to the present discussion. They are periodic along the folding axis while on the vertical they are not. Thefolding angle around either the armchair or zig-zag axis is denoted by ϕ.Withsymbol θ, we denote “usual” angles between carbon bonds (bending angles) and with ω torsion angles as definedin the previous subsection.All calculations were performed with Quantum-Espresso periodic-DFT code <cit.>, with the same pseudopotential <cit.> as in Ref. kalosakas. The wave-function and density plane-wave cutoffs were chosen 40 Ry and 400 Ry respectively. The unit cell we chose is minimal in the periodic direction (that of the folding axis) while, in the vertical, it is appropriately large to avoid edge-effects.Thus, the simulated structures are nanoribbons that are folded around their middle line direction.In the case of the armchair folding (Fig. <ref>(top)), the vertical unit cell direction is such thatneighbors up to the 5th in the vertical direction were included. In the case of the zig-zag folding that size is long enough to include up to 8th neighbors. Thus, the unit cells contain 22 and 18 atoms for the armchair and the zig-zag folding, respectively. In the reciprocal space, we used a mesh of 1×24×1, i.e. 24 points were assumed along the bending direction that the structure is periodic.In Fig. <ref>, we show with filled circles the total deformation energy per unit cell along the folding direction, as calculated by DFT, as a function of the folding angle ϕ, for both armchairand zig-zag folding actions, E_d^ (a) and E_d^ (z), respectively. The total deformation energy per unit-cell is taken as the energy differencebetween the folded structure and the not folded one (ϕ=0).Apparently, the two structure distortions due to the considered foldings are complex and consist of several individual angle-bending andtorsional deformations. Note that bond lengths are not altered so there is no bond-stretching contributionin the total deformation energy. As we see in Fig. <ref>, the contribution from angle bendingis significant for ϕ larger than 0.2 rad. In order to perform a fitting for the torsional terms alone, we first needto exclude angle-bending contributions from the total deformation energy. In order to do so, (i) we identify allangle-bending terms and express analytically their corresponding bending angles θ in terms of ϕ and then(ii) we remove the angle-bending terms using the analytic terms in the Ref. <cit.>.The residual, torsional energy per unit cell, when the contribution from angle-bending is subtracted, as a function of the out-of-plane folding angle ϕ is shownin Fig. <ref> (diamonds), for the two folding directions. In order to fit an analytic expression tothe torsional terms, we also have to (i) identify all the individual torsional terms that contributefor each of the zig-zag and armchair cases, and subsequently (ii) express the corresponding torsion angles as functions of the folding angle, ϕ.These steps are described below where we provide all necessary analytical expressions.§ REMOVING THE ANGLE-BENDING TERMS §.§ For the folding around armchair axis The folding around the armchair direction (Fig. <ref>, top)alterstwo bond angles per unit cell, (32̂4),(61̂5), which are equal. One can show thatthese angles, in terms of ϕ, are given byθ^ (a)=2arcsin(√(3/8)√(cosϕ+1)) The angle-bending energy that one needs to remove from the total energy isU_b^ (a) = 2 V_b ( θ^ (a)(ϕ) ) ,where V_b(θ) is the analytical expression for the angle bending given in Ref. <cit.>, i.e.V_b(θ) = k/2( θ-2π/3)^2 - k^'/3( θ-2π/3)^3 ,with k=7.0 eV/ rad^2 and k^'=4 eV/ rad^3.Removing these terms from the total deformation energies, E_d^ (a) we find thetotal torsional energyE_t^(a) = E_d^ (a) - U_b^ (a) ,shown in Fig. <ref>. §.§ For the folding around zig-zag axis Similarly, the folding around the zig-zag direction (Fig. <ref>, (bottom))affects two angles per unit cell, (21̂6),(21̂5), that are also equal.In terms of ϕ, these angles are given byθ^ (z)=2arcsin(1/2√(2+cosϕ)) Again, the angle-bending energy that one needs to remove from the total energy isU_b^ (z) = 2 V_b ( θ^ (z)(ϕ) ) ,we remove these terms from the total deformation energy and we find the total torsional energy,E_t^(z) = E_d^ (z) - U_b^ (z) ,shown in Fig. <ref>. § ANALYTICAL EXPRESSIONS FOR THE TORSIONAL TERMSHere we provide analytical expressions, U_t^ (a)(ϕ), U_t^ (z)(ϕ), for the total torsionalenergies, as functions of ϕ that will contain parameters to be fitted so that these expressionsreproduce as close as possible the E_t^ (a), E_t^ (z) points shown in Fig. <ref>. To arrive to such analytical expressions we need first to identify all altered torsion angles (per unit cell)and express them in terms of the folding angle ϕ. Then, U_t^ (a)(ϕ), U_t^ (z)(ϕ) will be just the sum of all individual torsional terms that correspond to these altered torsional angles.Regarding the individual torsional energy term, V_t(ω), two different functional forms would be considered. The most frequently used formula, referred as Model 1 here, is V_t(ω)= 1/2 V_1 [ 1 + cosω] + 1/2 V_2 [ 1 - cos (2 ω ) ]. An alternative model that we considered, which we call Model 2, assumes a different fitting formula for cis or trans dihedral angles ω[ V_t^ (cis)(ω) = K_ cis[ 1 - cos (2 ω ) ] ,; V_t^ (trans)(ω) = K_ trans[ 1 - cos (2 ω ) ] , ] where either the first or the second expression is used for cis or trans torsion angles, respectively. Below we use both Models 1 and 2 to fit their parameters to the obtained DFT results.§.§ For the folding around armchair axisInspecting the Fig. <ref> (top) we identify the following torsion (dihedral) angles per unit cell that are altered by folding along the armchair axis ∙2 trans dihedral angles, (5-1-2-3), (4-2-1-6), withω_1^ (a)(ϕ)=arccos(-cosϕ) ∙ 4 cis dihedral angles, (11-4-2-3), (14-5-1-6), (7-3-2-4), (10-6-1-5), withω_2^ (a)(ϕ)=arccos(√(3)1+cosϕ/√(9sin^2ϕ+6(1+cosϕ))) ∙ 4 trans dihedral angles, (12-4-2-3), (13-5-1-6), (8-3-2-4), (9-6-1-5), withω_3^(a)(ϕ)=arccos(-√(3)1+cosϕ/√(9sin^2ϕ+6(1+cosϕ)))Through the angle expressions given above, the total torsional energy, U_t^ (a) within the Model 1,becomes an analytic function of ϕ:U_t^ (a)(ϕ) = 2 V_t (ω_1^ (a)(ϕ)) + 4 V_t ( ω_2^(a)(ϕ) ) +4 V_t ( ω_3^(a)(ϕ) )where V_t(ω) is the individual torsional term Eq. (<ref>).For the Model 2, the corresponding expression of the total torsional energy U_t^ (a) isU_t^ (a)(ϕ)=2 V_t^ (trans) (ω_1^(a)(ϕ)) + 4 V_t^ (cis) ( ω_2^(a)(ϕ) ) +4 V_t^ (trans) ( ω_3^(a)(ϕ) )with V_t^ (cis) and V_t^ (trans) given by Eq. (<ref>).§.§ For the folding around zig-zag axisInspecting the Fig. <ref> (bottom) we identify the following dihedral angles per unit cell that are affected ∙2 cis dihedral angles, (3-2-1-6), (4-2-1-5), withω_1^ (z)(ϕ)=arccos(√(3/sin^2ϕ+3))∙ 2 trans dihedral angles, (4-2-1-6), (3-2-1-5), withω_2^ (z)(ϕ)=arccos(-√(3/sin^2ϕ+3))∙ 2 cis dihedral angles, (2-1-5-13), (2-1-6-9), withω_3^ (z)(ϕ)=arccos(√(3/sin^2ϕ+3)cosϕ)∙ 2 trans dihedral angles, (2-1-5-14), (2-1-6-10), withω_4^ (z)(ϕ)=arccos(-√(3/sin^2ϕ+3)cosϕ)And the total torsional energy is given byU_t^ (z)(ϕ) = 2 V_t (ω_1^ (z)(ϕ)) + 2 V_t ( ω_2^ (z)(ϕ) ) +2 V_t ( ω_3^ (z)(ϕ)) + 2 V_t (ω_4^ (z)(ϕ)) ,where V_t is given by Eq. (<ref>) for Model 1. In case of Model 2, the above formula becomes U_t^ (z)(ϕ)= 2 V_t^ (cis) (ω_1^ (z)(ϕ)) + 2 V_t^ (trans) ( ω_2^ (z)(ϕ) ) +2 V_t^ (cis) ( ω_3^ (z)(ϕ)) +2 V_t^ (trans) (ω_4^ (z)(ϕ)). § FITTING PROCEDUREThe total torsional energy data, (ϕ_i, E_t,i^ (a)) and (ϕ_i, E_t,i^ (z)), shown in red and black diamonds in the Fig. <ref>, and the analytical (to be fitted)expressions U_ t^ (a), U_ t^ (z) given in the Eqs. (<ref>)and (<ref>) for the Model 1 (or the Eqs. (<ref>) and (<ref>) for the Model 2) can be used to obtain the optimal parameters V_1 and V_2 (or K_ cis and K_ trans) of theindividual torsional terms so that U_ t^ (a), U_ t^ (z) reproduce thedependence of E_t^ (a)(ϕ) and E_t^ (z)(ϕ) as close as possible.For this purpose, adopting a standard procedure, we minimize an objective functionO(V_1,V_2) which is the equal-weighted sum of the square differences,O(V_1,V_2)= ∑_i=1^ϕ_i < ϕ_ max[ E_t,i^ (a) - U_ t^ (a)(ϕ_i)]^2 + ∑_i=1^ϕ_i < ϕ_ max[E_t,i^ (z) - U_ t^ (z)(ϕ_i)]^2.The sums in the above expression runs over all i for which ϕ_i is smaller than anupper-limit angle ϕ_ max.For the Model 1, the fitted total torsional energiesU_ t^ (a)(ϕ) and U_ t^ (z)(ϕ) given in the Eqs. (<ref>) and (<ref>) depend on V_1 and V_2 through the dependence of the individual terms V_t of the Eq. (<ref>). In the case of Model 2, we are optimizing K_ cis and K_ trans parameters, and the expressions(<ref>) and (<ref>) are used instead and the individual termsV_t are given by the Eq. (<ref>).The choice of ϕ_ max is expected to affect the quality of fitting for small and large ϕ. We are interested in seeing whether the fitting parameters depend on ϕ_ max and, if so, at what extend.§.§ Model 1: fitting Results for V_1, V_2 We performed fitting of V_1, V_2 of Eq. (<ref>), for three differentϕ_ max values: 10^∘, 20^∘ and 30^∘. The optimal parameters V_1 and V_2 given in table <ref>.In Fig. <ref>, we show, the total torsional energies, E_t^ (a) andE_t^ (z) and the fitted lines for these three values of ϕ_ max.As we see, the fitting is better for small values of ϕ and deteriorates as ϕ increases. In all cases, it reproduces the armchair data in closer agreement than the zig-zag, i.e. for a larger range of ϕ. For ϕ_ max=10^∘, there is a satisfactory agreement for values ofϕ up to 0.55 rad for the armchair case and 0.4 rad for the zig-zag case. For ϕ_ max=20^∘, the range of satisfactory agreement increases roughly up to 0.65 radand 0.45 rad for the armchair and zig-zag cases respectively. Finally forϕ_ max=30^∘,the agreement range increases further up to 0.7 rad and 0.5 rad, respectively. Although byincreasing ϕ_ max, the range of satisfactory agreement also increases, this is at the cost of the agreement for smaller angles. As the code is trying to fit better at larger values of ϕ the quality for smaller angles deteriorates. This deterioration, however, is rathersmall as we observe in the insets of the Figs. <ref>(a), (b), (c) where we zoom in thatregion. However, in general, for angles ϕ up to 0.4 rad (22^∘) corresponding to energies 0.2 to 0.3 eV, all fittings are satisfactory. On the other hand, as we see in Table <ref>, the fitted values of V_1 and V_2 are not so sensitive to the value of ϕ_ max: V_1 remains close to 0 while V_2 is in the range 0.22-0.23 eV.In addition, the large value of V_2, i.e. 0.23 eV, obtained for ϕ_ max=20^∘,30^∘performs better for larger angles, up to 0.5 rad (∼30^∘), corresponding to energies of0.4-0.5 eV, while on the other hand the fitted results in the region of small ϕ remainsatisfactory. These considerations suggest that it is quite reasonable to adopt as optimal V_1=0 and V_2=0.23 eV and our proposed torsional potential has the simple formV_t(ω)=1/2V_2(1-cos(2ω)), V_2=0.23eV .This potential is shown in Fig. <ref> together with E_t^ (a) and E_t^ (z).§.§ Model 2: fitting results for K_ cis, K_ trans As in the previous section, we performed the fitting of K_ cis and K_ trans of Eq. (<ref>) for the same values of ϕ_ max, i.e. 10^∘, 20^∘ and 30^∘. The optimal parameters K_ cis and K_ trans for each of these cases are given inTable <ref>. In Fig. <ref>, we show the fitting lines for the armchair and zig-zag folding casesfor all three values of ϕ_ max compared to the data points for the total torsional energy per unit cell, E_t^ (a) andE_t^ (z). The fitting quality is quite similar to that of the previous section. Again,for ϕ_ max=20^∘ and 30^∘, the quality improves for largervalues of ϕ and at the same time the fit for smaller ϕ does not deteriorate substantially. Thus, we propose a rounded optimal setK_ cis=0.14 eV and K_ trans=0.10 eV which is close to the values obtained forϕ_ max=20^∘ and 30^∘. In Fig. <ref>, we show the torsional energyobtained with Model 2 and these values for K_ cis and K_ trans compared with the data E_t^ (a) and E_t^ (z).To summarize, for the Model 2, we propose[V_t^ (cis)(ω) = K_ cis[ 1 - cos (2 ω ) ],K_ cis=0.14,; V_t^ (trans)(ω) = K_ trans[ 1 - cos (2 ω ) ],K_ trans=0.1, ] where either the first or the second expression is used depending on whether the torsional angle ω is cis or trans. §.§ Comparison of the two modelsIn Fig. <ref> we show the torsional energies per unit cell, for both fitting forms of Models 1 and 2, for thecase of the optimal parameters we arrived at.We notice that the two models are of the same quality. They almost coincide for the armchair case, while for the zig-zag, the Model 2 is slightly better for larger ϕ's and the Model 1 marginally better for smaller. The differences however are not significant for ϕ's up to 0.5 rad. Note that the obtained optimal parameters of Model 2, K_ cis=0.14 eV and K_ trans=0.10 eV, are close to each other, indicating that a single parameter with value the average of themwould offer a reasonable description. Moreover, this average value is almost equal to V_2 /2. Thus, it is rather unnecessary to assume different parameters for cis and trans dihedral anglesand, to keep things as simple as possible, the simple form of the Eq. (<ref>), is quite sufficient to describe all torsional distortions. Therefore, the Model 1 of Eq. (<ref>) is our proposed one, containing a single parameter V_2. We should mention that our modeling describes accurately the energy of torsional angles ϕ up to 0.5 rad which is already a sufficiently large value, corresponding to rather unphysical structuraldeformations.§ APPLICATION TO THE VERTICAL DISPLACEMENT OF A CARBON ATOM IN GRAPHENE In order to test the accuracy of the proposed parameters for the torsional terms,we consider the deformation energy of graphene due to a vertical, out-of-plane displacementof a single carbon atom. We consider that apart from the vertically displaced atom, all other atoms remain fixed at their equilibrium positions within graphene's plane. The task is to comparethe deformation energy obtained by the present potential, along with the in-plane force field of Ref. <cit.>, with that obtained by DFT calculations (using the same method that was used to produce the data discussed above). The process of moving a carbon atom vertically outside graphene's plane is described by a deformationenergy consisting of all kinds of individual terms, i.e. bond-stretching,angle-bending and of course torsional terms. Concerning the bond-stretching terms, the vertical movement of a carbon atom at a displacement zover the plane, alters only the three bonds of that atom (see Fig. <ref>). If the lengthof these bonds at z=0 is d their altered length d' becomes d'(z)=√(d^2+z^2) At the same time, two different kinds of angle-bending terms appear corresponding to: (i) the three angles, θ_1, between the atom's bonds (marked in red in Fig. <ref>) and (ii) the six angles, θ_2 between these bonds and the bonds marked in blue in Fig. <ref>. These angles can be expressed in terms of the displacement z asθ_1(z)=arccos(2z^2-d^2/2(d^2+z^2))and θ_2(z)=arccos(-d/2√(d^2+z^2)) Finally, several torsional terms also contribute. There are rotations around the 3 bondsof the displaced atom and its first neighbors (marked in red in Fig. <ref>) as well asrotations around the 6 bonds of the first neighbors and the second neighbors. These rotations correspond to the following torsional angles * 6 cis dihedral angles around the bonds marked in red in Fig. <ref> given byω_ cis^(1)(z)=arccos(3/4/√((z/d)^2+ 3/4)√(3(z/d)^2+3/4))* 6 trans dihedral angles around the bonds marked in red in Fig. <ref> given byω_ trans^(1)(z)=arccos(-3/2(z/d)^2- 3/4/√((z/d)^2+3/4)√(3(z/d)^2+3/4)) * 6 cis dihedral angles around the bonds marked in blue in Fig. <ref> given byω^(2)_ cis(z)=arccos(1/√(4/3(z/d)^2+1)) * 6 trans dihedral angles around the bonds marked in blue in Fig. <ref> given by ω^(2)_ trans(z)=arccos(-1/√(4/3(z/d)^2+1)) Then, the deformation energy is given byE_d(z)=3 V_s (d'(z)) + 3 V_b (θ_1(z)) + 6 V_b (θ_2(z))+6 V_t(ω_ cis^(1)(z)) + 6 V_t(ω_ trans^(1)(z)) +6 V_t(ω_ cis^(2)(z)) + 6 V_t(ω_ trans^(2)(z)) ,where V_s (d), V_b (θ) are respectively the bond stretching and angle bending termsgiven in Ref. <cit.>. V_t is the individual torsional term of Eq. (<ref>) for Model 1, while for Model 2 it should be replaced by V_t^ (cis) or V_t^ (trans) of Eq. (<ref>) for the two ω_ cis and the two ω_ trans respectively. In Eq. (<ref>), the deformation energy, E_d, becomes an analytic function of z through the explicit dependence on z of the bonds d', the angles θ_1, θ_2 and the dihedral angles ω_ cis^(1), ω_ trans^(1), ω_ cis^(2), ω_ trans^(2). In Fig. <ref>,we show E_d in comparison with the corresponding DFT results.As we see, both models perform equally well and the error does not exceed 0.05 eV for thedeformation range shown.There are small differences in theiragreement with DFT, for example Model 2 seems slightly better for indermediate-sizedisplacements while Model 1 for larger ones, however, these small differences areinsignificant validating our preference for Model 1 on the basis of its simplicity. § CONCLUSION In summary, we present a simple torsional force field for graphene and other sp^2 carbon nanostructures. To obtain this potential we performed two sets of DFT calculations by folding two different graphene nanoribbonstructures around their middle line. The first set of calculations concern the folding of an armchair nanoribbonaround its middle line, an armchair direction. The second concerns the folding of a zig-zag nanoribbon around itsmiddle line which is a zig-zag direction. From the deformation energies we isolated the “pure” torsional contribution by removing angle bending terms with the use of our previously proposed angle bending terms<cit.>. The purified torsional deformation energy was then fit two different analytic forms with two parameters each that we call Models 1 and 2. The first (Model 1) is that of Eq. (<ref>) and does not distinguish torsional angles, while the second (Model 2) of Eq. (<ref>) treats differently “cis” and “trans” torsional angles. We found thatthe form of Model 1 reduces to one parameter form, see Eq. (<ref>), which was found to be an average of the “cis”and “trans” terms of Model 2 which differ very little from each other, see Eq. (<ref>). That suggests that theuse of two different terms is redundant and the single term of Model 1 suffices at a reasonable level or accuracy. We foundthat both models reproduce accurately the torsional deformation energy of graphene nanoribbons due to the folding weconsidered up to θ_ max of the order of 30^o (≈ 0.5 rad).As an additional validation test we considered the case of the deformation energy due to the vertical displacement of asingle C atom outside of graphene's plane. For this task we used the torsional terms of either Model 1 or 2 combined withthe bond stretching and angle bending terms of Ref. kalosakas. For all terms in this formula we provideanalytic expressions in terms of the displacement z. We found that both models perform equally well in this case with errors not exceeding 0.05 eV for a relatively large range of z, up to 0.4-0.5 Å. The good performance of both Models in this case validates our choice for the simpler Model 1. The torsional force field presented here, in combination with the bond stretching and angle bending terms of Ref. kalosakas, which were also fitted to DFT results using the same density functional approximation provide a complete, accurate, but simple in form atomistic potential, which is computationally efficient due to its simplicity. Thus, we expect thatit will be proven a very useful tool for large scale atomistic simulation of graphene and other sp^2 nanostructures. Acknowledgements: We acknowledge helpful discussions with E. N. Koukaras. The research leading to the present results has received funding from Thales project “GRAPHENECOMP”, co-financed by the European Union (ESF) and the Greek Ministry of Education(through ΕΣΠΑ program). 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Galiotis, K. Papagelis, N. N. Lathiotakis, preprint (2017). QE-2009 P. Giannozzi et al, J. Phys.: Cond. Matt. 21, 395502 (2009).pseudo A. Dal Corso, http://www.quantum-espresso.org/wp-content/uploads/upf_files/C.pbe-rrkjus.UPF | http://arxiv.org/abs/1707.09059v1 | {
"authors": [
"Georgios D. Chatzidakis",
"George Kalosakas",
"Zacharias G. Fthenakis",
"Nektarios N. Lathiotakis"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170727221407",
"title": "A Torsional potential for graphene derived from fitting to DFT results"
} |
1]Carlo Ciliberto 1]Mark Herbster 2,3]Alessandro Davide Ialongo 1,4]Massimiliano Pontil 1,5]Andrea Rocchetto Corresponding author: [email protected] 1,6]Simone Severini 1,5,7]Leonard Wossnig[1]Department of Computer Science, University College London [2]Department of Engineering, University of Cambridge [3]Max Planck Institute for Intelligent Systems, Tübingen [4]Computational Statistics and Machine Learning - Istituto Italiano di Tecnologia [5]Department of Materials, University of Oxford [6]Institute of Natural Sciences, Shanghai Jiao Tong University [7]Theoretische Physik, ETH Zürich Spectral sequences of Type Ia supernovae. I. Connecting normal and sub-luminous SN Ia and the presence of unburned carbon [=========================================================================================================================Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed. § INTRODUCTION In the last twenty years, thanks to increased computational power and the availability of vast amounts of data, machine learning (ML) algorithms have achieved remarkable successes in tasks ranging from computer vision <cit.> to playing complex games such as Go <cit.>. However, this revolution is beginning to face increasing challenges. With the size of datasets constantly growing and Moore's law coming to an end <cit.>, we might soon reach a point where the current computational tools will no longer be sufficient. Although tailored hardware architectures, like graphics processing units (GPUs) and tensor processing units (TPUs), can significantly improve performance, they might not offer a structural solution to the problem. Quantum computation is a computational paradigm based on the laws of quantum mechanics. By carefully exploiting quantum effects like interference or (potentially) entanglement, quantum computers can efficiently solve selected problems <cit.> that are believed to be hard for classical machines. This review covers the intersection of machine learning and quantum computation, also known as quantum machine learning (QML). The term quantum machine learning has been used to denote different lines of research such as using machine learning techniques to analyse the output of quantum processes or the design of classical machine learning algorithms inspired by quantum structures. For the purpose of this review we refer to QML solely to describe learning models that make use of quantum resources. The goal of this review is to summarise the major advances in QML for a mixed audience of experts in machine learning and quantum computation and serve as a bridge between the two communities. Most problems will be analysed under the lens of computational complexity, possibly, a unifying language for both communities. We do not aim for completeness but rather discuss only the most relevant results in quantum algorithms for learning. For the interested reader there is now a number of resources covering quantum machine learning in the broader sense of the term <cit.>. For an introduction to quantum algorithms we refer to the reviews of Montanaro <cit.> and Bacon <cit.>, while for machine learning to the books by Bishop <cit.> and Murphy <cit.>.Why should a machine learning expert be interested in quantum computation? And why are we expecting quantum computers to be useful in machine learning? We can offer two reasons. First, with an ever growing amount of data, current machine learning systems are rapidly approaching the limits of classical computational models. In this sense, quantum algorithms offer faster solutions to process information for selected classes of problems. Second, results in quantum learning theory point, under certain assumptions, to a provable separation between classical and quantum learnability. This implies that hard classical problems might benefit significantly from the adoption of quantum-based computational paradigms. But optimism should come with a dose of scepticism. The known quantum algorithms for machine learning problems suffer from a number of caveats that limit their practical applicability and, to date, it is not yet possible to conclude that quantum methods will have a significant impact in machine learning. We will cover these caveats in detail and discuss how classical algorithms perform in light of the same assumptions. Quantum computation is a rapidly evolving field but the overarching question remains: when will we have a quantum computer? Although it is not within the scope of this review to present a time-line for quantum computation it is worth noting that in the last years the worldwide effort to build a quantum computer has gained considerable momentum thanks to the support of governments, corporations and academic institutions. It is now the general consensus that general purpose quantum computation is within a 15 years time-line <cit.>. The review is structured as follow. We start with Section <ref> by providing some essential concepts in quantum computation for the reader with no prior knowledge of the field. In Section <ref> we introduce the standard models of learning, their major challenges and how they can be addressed using quantum computation. Section <ref> surveys results in quantum learning theory that justify why we expect quantum computation to help in selected learning problems. We proceed in Section <ref> by discussing how to access data with a quantum computer and how these models compare with parallel architectures. We continue by presenting different computational and mathematical techniques, that find widespread application in machine learning, and can be accelerated with a quantum computer. More specifically, we survey quantum subroutines to speedup linear algebra (Section <ref>), sampling (Section <ref>), and optimisation problems (Section <ref>). For each section, we discuss the asymptotic scaling of the classical and quantum subroutine and present some learning applications. The following section is dedicated to quantum neural networks (Section <ref>). Even if neural networks are not a mathematical technique on their own, they are surveyed in a dedicated section due to their prominence in modern machine learning. The last two sections cover two promising applications of quantum computation in machine learning. In Section <ref> we consider the case of learning under noise while in Section <ref> we discuss computationally hard problems in machine learning. We conclude with an outlook section.§ ESSENTIAL QUANTUM COMPUTATIONQuantum computing focuses on studying the problem of storing, processing and transferring information encoded in quantum mechanical systems. This mode of information is consequently called quantum information. The book by Nielsen and Chuang <cit.> is a standard introduction to the field. Loosely speaking, quantum computational models propose a probabilistic version of (time) reversible computation, i.e. computation in which the output is in one-to-one correspondence with the input. According to quantum theory, physical states are mathematically represented by density matrices, which are trace-one, positive-semidefinite matrices that generalise the concept of probability distributions. The logical states used by a quantum computational model are then identified with the physical states of the quantum system that implements it.A computation is executed reversibly by applying a sequence of unitary matrices to an initialised state. A probabilistic output is obtained according to the distribution encoded by the final density matrix. In this framework, the fundamental unit of quantum information is the state of any quantum system with two degrees of freedom distinguishable by an observer, which then coincide with the usual logical values 0 and 1. This is called qubit and, for our purposes, it is a vector ψ = α_0 e_0 + α_1 e_1, where α_0, α_1 ∈ℂ, |α_0|^2 + |α_1|^2 = 1 and, e_i denotes the i-th standard basis vector. The values of a distribution on 0 and 1 are given by {|α_1|^2,|α_2|^2}. It is a basic fact that the information content of a qubit is equivalent to a single bit. Registers of multiple qubits are assembled with the use of a tensor product. Unitary matrices acting on a small number of qubits can then be interpreted as a generalisation of logic gates. The induced dynamics is responsible for interference, a key ingredient of quantum computation. By exploiting interference effects, quantum computers are able to simultaneously evaluate a function on every point of its domain. Although the result of this computation is not immediately accessible to a classical observer, the possibility of using quantum dynamics to increase the probability of determining a given property of the function is promised to allow a quantum computer to solve some computational problems exponentially faster than classical devices. Although the true roots, and extents, of quantum speedups are still unclear, it is believed that structure, certain symmetries and non-classical correlations play an important role in the observed advantages <cit.>.In the context of the analysis of classical data, we can exploit the encoding of quantum information to efficiently represent classical probability distributions with exponentially many points. For instance, when v = (v_1, …, v_2^n) is a probability vector of size 2^n, we can write an n-qubit state (register), ψ = ∑_i=1 ^2^n√(v_i) e_i. Finally, when we use the term qubits we always refer to idealised, error free, objects. In practice quantum states are extremely fragile and require extensive error correction to be shielded from the effects of noise. The theory of error correction guarantees that if the physical errors are below a certain threshold it is possible to correct the system efficiently. The theory of error correction is reviewed by Preskill <cit.>. We further discuss the types of error affecting quantum systems in Section <ref>. §.§ Comparing the performance of classical and quantum algorithms Computational complexity studies the resources needed to perform a given computational task. One of the major goals of this theory is to classify problems according to their time complexity, which roughly corresponds to the number of elementary steps necessary to solve the problem as a function of the size of the input. The books by Papadimitrou <cit.> and Arora and Barak <cit.> provide extensive introductions.We define quantum speedup as the advantage in runtime obtained by a quantum algorithm over the classical methods for the same task. We quantify the runtime with the asymptotic scaling of the number of elementary operations used by the algorithm with respect to the size of the input. To compare the performance of algorithms, we use the computer science notation 𝒪(f(n)) indicating that the asymptotic scaling of the algorithm is upper-bounded by a function f(n) of the number n of parameters characterizing the problem, i.e. the size of the input. The notation 𝒪̃(f(n)) ignores logarithmic factors.In computational complexity theory it is customary to analyse algorithms with respect to the number of queries to some efficiently implementable oracles, which can be either classical or quantum. This approach to analysing algorithms is called the query model. In the query model an algorithm is said to be efficient if it queries the oracle a polynomial number of times. Throughout this review many speedups are obtained in the query model. A standard oracle for QML assumes that the classical datasets can be efficiently produced in quantum superposition. The QRAM discussed in Section <ref> is a possible implementation of this oracle. § SETTING THE PROBLEM: PERSPECTIVES IN MACHINE LEARNING The term machine learning refers to a variety of statistical methods to analyse data. The prototypical goal is to infer, from a finite number of observations (the training data), the future behaviour of an unknown and possibly non-deterministic process (such as the dynamics of the stock market or the activations patterns in the human brain). In the last four decades, the field of machine learning has grown to such an extent that even providing a brief overview of the most prominent ideas and frameworks would require a review on its own. To this end, for the purpose of this review, we mainly focus on one of the most well-established and mature areas of research, namely supervised learning from the perspective of learning theory. To the reader interested in a more satisfying overview of machine learning we recommend, for instance, <cit.>. Learning theory aims to place the problem of learning from data on solid mathematical foundations. Typical questions that one asks in this setting are: how many examples are required to learn a given function? How much computational resources are required to perform a learning task? Depending on a number of assumptions about the data access model and on the goal of learning, it is possible to define different learning models. Two prominent ones are the Probably Approximately Correct (PAC) framework developed by Valiant <cit.> and the Statistical Learning Theory by Vapnik <cit.>. Here, a learner seeks to approximate an unknown function based on a training set of input-output pairs. Examples in the training set are assumed to be drawn from an unknown probability distribution and predictions are tested on points drawn from the same distribution. PAC and statistical learning theory model the efficiency of an estimator with two quantities: the sample complexity and the time complexity. The sample complexity is the minimum number of examples required to learn a function up to some approximation parameters and it is directly related to the capacity of the hypotheses space and the regularity of the data distribution; the time complexity corresponds to the runtime of the best learning algorithm. A learning algorithm is said to be efficient if its runtime is polynomial in the dimension of the elements of the domain of the function and inverse polynomial in the error parameters. In these settings the goal is to find a model that fits well a set of training examples but that, more importantly, guarantees good prediction performance on new observations. This latter property, also known as generalisation capability of the learned model, is a key aspect separating machine learning from the standard optimisation literature. Indeed, while data fitting is often approached as an optimisation problem in practice, the focus of machine learning is to design statistical estimators able to “fit” well future examples. This question is typically addressed with so-called regularisation techniques, which essentially limit the expressive power of the learned estimator in order to avoid overfitting the training dataset. A variety of regularisation strategies have been proposed in the literature, each adopting a different perspective on the problem (see <cit.> for an introduction on the main ideas). Among the most well-established approaches it is worth mentioning those that directly impose constraints on the hypotheses class of candidate predictors (either in the form of hard constraints or as a penalty term on the model parameters, such as in Tikhonov regularisation) or those that introduce the regularisation effect by “injecting” noise in the problem (see Section <ref>). These ideas have led to popular machine learning approaches currently used in practice, such as Regularised Least Squares <cit.>, Gaussian Process (GP) Regression and Classification <cit.>, Logistic Regression <cit.>, and Support Vector Machines (SVM) <cit.> to name a few. From a computational perspective, regularisation-based methods leverage on optimisation techniques to find a solution for the learning problem and typically consist of a sequence of standard linear algebra operations such as matrix multiplication and inversion. In particular, most classical algorithms, such as GP or SVM, require a number of operations comparable to that of inverting a square matrix that has size equal to the number N of examples in the training set. This leads, in general, to a time complexity of 𝒪(N^3) which can be improved depending on the sparsity and the conditioning of the specific optimisation problem, see Section <ref>. However, as the size of modern datasets increases, the above methods are approaching the limits of their practical applicability.Recently, alternative regularisation strategies have been proposed to reduce the computational costs of learning. Instead of considering the optimisation problem as a separate process from the statistical one, these methods hinge on the intuition that reducing the computational burden of the learning algorithm can be interpreted as a form of regularisation on its own. For instance, early stopping approaches perform only a limited number of steps of an iterative optimisation algorithm (such as gradient descent) to avoid overfitting the training set. This strategy clearly entails less operations (less number of steps) but can be shown theoretically to lead to same generalisation performance of approaches such as Tikhonov regularisation <cit.>. A different approach, also known as divide and conquer, is based on the idea of distributing portions of the training data onto separate machines, each solving a smaller learning problem, and then combining individual predictors into a joint one. This strategy benefits computationally from both the parallelisation and reduced dimension of distributed datasets and it has been shown to achieve the same statistical guarantees of classical methods under suitable partitions of the training data <cit.>. A third approach that has recently received significant attention from the machine learning community is based on the idea of constraining the learning problem to a small set of candidate predictors, obtained by randomly sampling directions in a larger, universal hypotheses space (namely a space dense in the space of continuous function). Depending on how such sampling is performed, different methods have been proposed, the most well-known being random features <cit.> and Nystrom approaches <cit.>. The smaller dimensionality of the hypotheses space automatically provides an improvement in computational complexity. It has been recently shown that it is possible to obtain equivalent generalisation performance to classical methods also in these settings <cit.>. For all these methods, training times can be typically reduced from the 𝒪(N^3) of standard approaches to 𝒪 (N^2) while keeping the statistical performance of the learned estimator essentially unaltered.Because the size of modern datasets is constantly increasing, time complexities in the order of 𝒪 (N^2) might still be too demanding for practical applications. In this regard, quantum computation could offer the potential to further improve the efficiency of such methods allowing them to scale up significantly. Indeed, through a number of quantum algorithms for linear algebra, sampling and optimisation techniques, we could in principle obtain up to exponential speedups over classical methods. However, as it will be discussed in Section <ref>, current QML methods require fast memory access and particular data structures that might limit their applicability in real settings. Nevertheless, as we will discuss in the following section, a number of results in quantum learning theory point, under specific assumptions, to a clear separation between classical and quantum learning paradigms in some specific settings. § “CAN WE DO BETTER?”: INSIGHTS FROM QUANTUM LEARNING THEORYLearning theorists have been interested to study how quantum resources can affect the efficiency of a learner since the 90's. Although different learning models have been translated into the quantum realm, here we focus on the quantum version of the PAC model. The reason of this choice is that in this model we have results for both the sample and the time complexity. For an extensive overview of the known results in quantum learning theory we refer the reader to the review by Arunachalam and de Wolf <cit.>.The quantum PAC model has been introduced in <cit.>. Here it is assumed that the learner has access to a quantum computer and to an oracle that returns the training set in quantum superposition.In terms of sample complexity, it has been shown in a series of papers, which constantly improved the bounds until reaching provable optimality <cit.>, that the quantum PAC model under an unknown distribution and standard PAC are equivalent up to constant factors. This implies that, in general, quantum mechanics does not help to reduce the amount of data required to perform a learning task. However, if one considers a different learning model, like the exact learning framework developed by Angluin <cit.>, it is possible to prove that quantum learners can be polynomially more efficient than classical in terms of number of queries to the data oracle <cit.>.Although quantum and classical examples are equivalent up to constant factors when learning under general distributions, the quantum PAC model can offer advantages over its classical counterpart in terms of time complexity. One of the central problems studied in the classical literature is the learnability of disjunctive normal forms (DNFs). To date, the time complexity of the best algorithm for learning DNFs under an unknown distribution is exponential <cit.>. A number of assumptions can be made to relax the hardness of the problem. For instance, if the learner is provided with examples drawn from the uniform distribution then the runtime of the best learner becomes quasipolynomial <cit.>. For the sake of completeness we note that the methods presented in Section <ref>, like SVMs, have been shown to not being able to learn efficiently DNF formulas <cit.>. The learnability of DNF formulas has also been studied in the quantum PAC model <cit.>. Here DNFs have been shown to be efficiently learnable under the uniform distribution. This quantum speedup is obtained through an efficient algorithm <cit.> that allows to sample exponentially faster from the probability distribution described by the coefficients of a boolean Fourier transform. Interestingly DNF formulas can be shown to be efficiently learnable under noise. We will return to this point in Section <ref>. Another case where it is believed that learning can be performed efficiently only when the learner has access to quantum resources is based on a class of functions developed by Kearns and Valiant <cit.>. This class is provably hard to learn under the assumption that factoring Blum integers is also hard (an assumption widely believed to be true; for a brief introduction to the concept of hardness in computational complexity see Section <ref>). Servedio and Gotler <cit.>noted that thanks to Shor'squantum factoring algorithm <cit.> this class of functions can be learned efficiently in the quantum PAC model. The results coming from the quantum learning theory literature show that by carefully exploiting quantum mechanical effects, depending on the type of learning model considered, it is possible to have a better generalisation error (i.e. we can learn with less examples) or we can learn functions that would otherwise be hard for classical learners.§ DATA ACCESS, COMMUNICATION AND PARALLELISMOne of the roots of the speedups theorised in quantum computation is the ability to process information in quantum superposition <cit.>. Because machine learning is ultimately about analysing vast amounts of data it is important to address the question of how data is turned into quantum superposition. We distinguish between two types of algorithms: those that operate on quantum data (i.e. data that is output of a quantum process, for example, a quantum chemistry problem) and those that seek to process data stored in a classical memory. The first case is ideal for QML. The data is ready to be analysed and we do not have to spend computational resources to convert the data into quantum form. The second case is more elaborate as it requires a procedure that encodes the classical information into a quantum state. As we will see the computational cost of this operation is particularly relevant to determine whether we can obtain quantum speedups in machine learning for classical data. Let us assume that one wants to process N d-dimensional classical vectors with a quantum algorithm. The quantum random access memory (QRAM) <cit.> isa quantum device that can encode in superposition N d-dimensional vectors into log(Nd) qubits in 𝒪(log(Nd)) time by making use of the so called “bucket-brigade” architecture.The idea is to use a tree-structure where the Nd leaves contain the entries of the N vectors in ℝ^d. The QRAM, with a runtime oflog(Nd), can return a classical vector in quantum superposition efficiently. However, the number of physical resources it requires scales as Nd. As we will see this exponential scaling (with respect to the number of qubits) has been used to question whether the QRAM can be built in an experimental setting or whether it can provide a genuine computational advantage <cit.>. Fundamentally the issue can be related to whether the exponential number of components needs to be continuously “active”. The proponents of the QRAM <cit.> claim that only log(Nd) components need to be active while the others can be considered as “non-active” and error free. Whether this assumption holds in an experimental setting is unclear <cit.>. We now proceed to discuss its implications. The first issue that appears with QRAM is whether all the components require to be error corrected (we briefly discuss errors in quantum computation in Section <ref>). Indeed, if the exponential physical resources required full error correction then it would be impractical to build the device in an experimental setting. Arunachalam et al. <cit.> addressed this question and showed that, provided a certain error model, algorithms that require to query the memory a polynomial number of times (like the quantum linear system algorithm presented in Section <ref>) might not require fault-tolerant components. However, for superpolynomial query algorithms, like Grover's search <cit.> (a subroutine required, for example, in some of the quantum methods for training restricted Boltzmann machines discussed in Section <ref>) the QRAM requires error-corrected components. A second problem related to the exponential number of resources in an active memory has been raised by Aaronson <cit.> and by Steiger and Troyer <cit.>. The authors argue that the only fair comparison of a system which requires an exponential number of resources is with a parallel architecture with a similar amount of processors. In this case many linear algebra routines, including solving linear systems and singular value decomposition, can be solved in logarithmic time <cit.>). A third caveats of the QRAM is the requirement of having data distributed in a relatively uniform manner over the quantum register. As pointed out in <cit.>, if that was not the case the QRAM would violate the search lower bounds proved in <cit.>. In the case of non-uniformly distributed data, the QRAM is no longer efficient and take √(N) to turn the classical dataset into quantum superposition. As a last comment on the QRAM, the possibility of loading the data in logarithmic time, when the size of the data is considerable, can be controversial due to speed of communication arguments. In fact, as suggested in <cit.>, latency can play a role in big memory structures. In particular, a lower bound on the distance which the information has to travel implies a lower bound on latency, due to considerations on the limits set by the speed of light. In a three dimensional space these are given by √(Nd). In practice these considerations will only dominate if the amount of memory is extremely large but, because in quantum machine learning we aim at datasets that surpass the current capability of classical computers, this bound is a potential caveat. In conclusion, the QRAM allows to upload data efficiently but might be hard to implement experimentally or might not allow a genuine quantum advantage if we take into account all the required resources. Noticeably the fast data access guaranteed by the QRAM is only required for QLM algorithm that run in sublinear time. Although many known QML algorithms run in sublinear time, quantum learning theory suggest that for some classically hard problems quantum resources might give exponential advantages. In this case, a memory structure that can prepare a quantum superposition in polynomial time (i.e. in 𝒪(Nd)) can still be sufficient to maintain a quantum speedup compared to the classical runtime. We will discuss hard classical learning problem in Section <ref>.Finally, we note that, although the QRAM, due to its generality, is the most widely used memory structure in QML algorithms other protocols to encode classical data in superposition exist. For example, a technique developed by Grover and Rudolph allows one to generate a quantum superposition that encodes an approximate version of a classical probability distribution provided its density is efficiently integrable <cit.>.§ FAST LINEAR ALGEBRA WITH QUANTUM MECHANICSA significant number of methods in the quantum machine learning literature is based on fast quantum algorithms for linear algebra. In this section we present the two main quantum subroutines for linear algebra: a quantum algorithm for matrix inversion and a quantum algorithm for singular value decomposition. We summarise the major applications of these techniques to machine learning problems and how they compare with classical an parallel implementations in Table <ref>. §.§ Fast matrix inversion: the quantum linear system algorithm Solving linear systems of equation is an ubiquitous problem in machine learning. As discussed in Section <ref>, many learning problems, like Gaussian Processes or SVMs, require the inversion of a matrix. For a system of linear equations A x = b with A ∈ℝ^N × N and x,b ∈ℝ^N, the best classical algorithm has a runtime of N^2.373 <cit.>. However, due to a large pre-factor, the algorithm is not used in practice. Standard methods, for example, based on QR-factorisation take N^3 steps <cit.>. The quantum linear system algorithm (QLSA) <cit.>, also known as HHL after the three authors Harrow, Hassidim, and Lloyd, promises to solve the problem in log (N) κ^2 s^2 /ϵ, where κ is the condition number (defined to be the ratio of the largest to the smallest eigenvalue), s is the sparsity or the maximum number of non-zero entries in a row and column of A and ϵ is the precision to which the solution is approximated. The precision is defined as the distance of the solution vector a to the true result a, which is given by ||a -a || = √(1-2 Re⟨ a,a ⟩)≤ϵ. Ambainis <cit.> and Childs et al. <cit.> improved the runtime dependency of the algorithm in κ and s to linear and the dependency in ϵ to poly-logarithmic. Although the QLSA algorithm solves matrix inversion in logarithmic time a number of caveats might limit its applicability to practical problems <cit.>. First, the QLSA algorithm requires the matrix A to be sparse. Second, the classical data must be loaded in quantum superposition in logarithmic time. Third, the output of the algorithm is not x itself but a quantum state that encodes the entries of x in superposition. Fourth, the condition number must scale at most sublinearly with N. An interesting problem that satisfies this requirements is discussed in <cit.>.Recently, <cit.> addressed the first caveat. By using a quantum walk based approach the authors derived an algorithm that scales as κ^2 ||A||_F log N/ϵ and can also be applied to dense matrices (however, in this case, the speedup is only quadratic since ||A||_F=√(N)). This result has been improvedin <cit.>, currently the best known lower bound for matrices with this property. The second caveat inherits the same issues of the QRAM discussed in Section <ref>: it is an open question whether we can practically load classical data in quantum superposition in logarithmic time.The third caveat is a common pitfall of quantum algorithms. As pointed out by Childs <cit.> and Aaronson <cit.>, in order toretrieve classical information from the quantum state, we need at least a number of measurements that is proportional to N. This would destroy every exponential speedup. One way forward is to use the QLSA algorithm only to compute certain features of the classical vector, which can be extracted efficiently using quantum mechanics, for example, the expected value x^T A x of a matrix A. A number of other possible applications is discussed in <cit.>. It is then natural to question how quantum algorithms compare to their classical analogues after all the caveats have been taken into account. For example, it is possible to show that calculating an expectation value of the form x^T A x can be done in time linear in the sparsity of the matrix A, using classical sampling methods. Furthermore, conjugated gradient descent can obtain the full solution of the linear system (also when A is not sparse) in linear time in the dimensionality and, in most cases, the sparsity of A <cit.>. We present a general comparison of the asymptotic scalings of classical, quantum and parallel algorithms for linear algebra and their major applications in machine learning in Table <ref>.1.5 Comparing algorithms based on their worst case running time may not be the right approach when considering their practical applicability, as it is commonly done in machine learning. Indeed, despite its worst case running time, an algorithm solving a given problem will often terminate much faster: average-case complexity can be much lower than worst case. Furthermore, smoothed analysis <cit.> provides a framework for studying the time performance of an algorithm in the presence of realistic input noise distributions. This gives another way to quantify the complexity of algorithms. To date, no quantum algorithm has been analysed in the smoothed analysis framework.Statistical considerations can also lead to interesting insights on the computational hardness of a learning problem. Kernel regularized least squares provide a good example. Under standard technical assumptions on the target function of the learning problem, computational regularization methods for kernel regularized least squares <cit.> (see Section <ref>) achieve the optimal learning rate of ϵ = N^-1/2 while requiring only N^2 operations. With optimal learning rates we mean that any learning algorithm cannot achieve better prediction performance (uniformly) on the class of problems considered. Interestingly, such assumptions also allow us to derive estimates for the condition number of the kernel matrix to be of order κ = N^1/2 <cit.>. The corresponding quantum scaling for the inversion of the kernel matrix is N^2 and it is therefore comparable to that of computational regularization methods implementable on classical machines (which, in addition, provide the full solution vector). Finally, it is worth comparing the QLSA to classical parallel methods for matrix inversion. In the parallel model of computation <cit.>, inverting an N× N matrix takes log^2(N) computational steps using a number of processors which is of order poly(N) (a crude upper bound of N^4 is given by <cit.>). Although the parallel model of computation does not resemble the actual behaviour of parallel machines, it can be a fair comparison considering that quantum computers might also face connectivity issues and hence communication overheads among the qubits. Inparticular when exponentially large amounts of error-corrected qubits are required, as with the QRAM, it is likely that latency issues arise. To conclude, the QLSA is a logarithmic time quantum algorithm for matrix inversion, a task arising in many learning problems. However, a number of caveats that include the requirement of a logarithmic access time to the memory and the impossibility of retrieving the solution vector with one measurement, lead to question whether classical or parallel algorithms that make use of the same assumptions obtain similar, or better, runtimes. In this respect, experimental implementations will greatly contribute to asses the true potential of these methods in realistic scenarios. §.§ Quantum singular values estimation The singular value decomposition (SVD) of a M × N, rank r matrix A is a factorisation of the formA = U Σ V^†, where U and V are, respectively, M × M and N × N unitary matrices and Σ is a M × N diagonal matrixwith r positive entries σ_1 , …, σ_r which are called the singular values of A.Singular value estimation is a fundamental tool in many computational problems and applications ranging from matrix inversion for linear regression to matrix approximation <cit.>. It is also of particular interest for problems of dimensionality reduction like principal component analysis (PCA) <cit.>. Classically, finding such a decomposition is computationally expensive, and for M > N it takes M N^2 <cit.>.Prakash and Kerenidis <cit.> introduced the quantum singular value estimation (QSVE) algorithm, based on Szegedy's work on quantum walks <cit.>, which runs in time ||A||_F logMN/ϵ. Their algorithm returns an estimate of the singular values σ̃_̃ĩ such that | σ_i - σ̃_i| ≤ϵ. As for the QLSA, the QSVE algorithm outputs the singular valuesin quantum superposition. As such, in order to read out all the r-values, the algorithm must be run Nlog N, times thus destroying any exponential speedup. However, it is still possible to construct useful applications of the QSVE algorithm. For example, <cit.> proposed a recommendation system which runs in poly(r)polylog(MN) (assuming a good r-rank approximation of the preference matrix).We note that the QSVE algorithm requires an oracle that can prepare quantum states that encode the rows and the columns of the matrix A in polylogarithmic time. It is possible to implement this oracle with the QRAM, and hence it will inherit the caveats discussed in Section <ref>. An alternative method for quantum singular value estimation, has been proposed by Lloyd, Mohseni, and Rebentrost <cit.>. The scaling of this algorithm is quadratically worse in ϵ but the requirements on the memory structure are less stringent than in <cit.>. This is advantageous in some applications, like analysing the principal components of kernel matrices <cit.>.§ QUANTUM METHODS FOR SAMPLINGMany learning problems of practical interest, as, for example, exact inference in graphical models, are intractable with exact methods. We discuss in detail hard learning problems in Section <ref>. Sampling methods are a common technique to compute approximations to these intractable quantities <cit.>. There is a rich literature on sampling methods <cit.>. The most commonly used ones are Monte Carlo methods and in particular the MCMC. The quantum algorithms discussed in this section are devoted to speed up MCMC methods. MCMC methods <cit.>, like Gibbs Sampling or the Metropolis algorithm, allow to sample from a probability distribution Π defined over a state space using a Markov chain that after a number of steps converges to the desired distribution (in practice one will only reach a distribution which is ϵ-close). The number of steps τ required to converge to Π is referred to as the mixing time. Estimating the mixing time can be reduced to bounding the spectral gap δ, which is the distance between the largest and the second largest eigenvalue of a stochastic map that evolves the Markov chain. The mixing time satisfies the inequality τ≥1/2 δlog(2 ϵ)^-1 and it is possible to show <cit.> that for the classical MCMC algorithm, τ is of the order 1/(δlog(1/Π^*)), where Π^* is the minimum value of Π. Recently, there has been a significant interest in quantum algorithms that allow to speedup the simulations of the stochastic processes used in MCMC. A common feature of these algorithms is a quadratic speedup in terms of spectral gap, inverse temperature, desired precision or the hitting time. Advances in this field include algorithms for thermal Gibbs state preparation <cit.> which provide polynomial speedups in various parameters, such as the spectral gap. Other methods have introduced the concept of quantum hitting time of a quantum walk <cit.>. In this framework it is possible to obtain an polynomial speedup with respect to most classical variants (this can be exponential for the hitting time). A number of other algorithms accelerate classical Monte Carlo methods applied to the estimation of quantities such as expectation values and partition functions, which play a major role in physics <cit.>. § QUANTUM OPTIMISATIONAs discussed in Section <ref>, optimisation methods are a fundamental building block of many machine learning algorithms. Quantum computation provides tools to solve two broad classes of optimisation problem: semidefinite programming and constraint satisfaction problems. §.§ Quantum algorithms for semidefinite programming Semidefinite programming <cit.> is a frameworkfor solving certain types of convex optimisation problems. Semidefinite programs find widespread applications in ML <cit.>. In a semidefinite program (SDP) the objective is to minimise a linear function of a N × N positive semidefinite matrix X over an affine space defined by a set of m constraints. The best known classical SDP-solvers <cit.> runs in time 𝒪(m(m^2+n^ω + mns)log^O(1)(mnR/ϵ)), where ϵ is an approximation parameter, ω∈ [2,2.373) is the optimal exponent for matrix multiplication, s is the sparsity of A, and R is an upper bound on the trace of an optimal X. Based on a classical algorithm to solve SDPs by Arora and Kale <cit.>, that has a runtime of 𝒪̃ (nms(Rr / ϵ)^4+ns(Rr/ ϵ)^7), where r is an upper bound on the sum of the entries of the optimal solution to the dual problem, in 2016 Brandão and Svore <cit.> developed a quantum algorithm for semidefinite programs that is quadratically faster in m and n. The dependence on the error parameters of this result has been improved in <cit.>. In this work the authors obtain a final scaling of 𝒪(√(mn)s^2 (Rr/ϵ)^8). The main problem of these quantum algorithms is that the dependence on R,r,s and 1/ϵ is considerably worse than in <cit.>. This quantum algorithm thus provides a speed-up only in situations where R,r,s,1/ϵ are fairly small compared to mn and, to date, it is unclear if there are interesting examples of SDPs with these features (for more details see <cit.>). §.§ Quantum algorithms for constraint satisfaction problems In a constraint satisfaction problem (CSP), we are given a set of variables, a collection of constraints, and a list of possible assignments to each variable <cit.>. The task is to find values of the variables that satisfy every constraint. This setting prompts to exact and approximate cases. For many families of CSPs efficient algorithms are unlikely to exist. Two quantum algorithms are known for CSPs: the quantum approximate optimisation algorithm and the quantum adiabatic algorithm. Due to its generality and a profoundly different way of exploiting quantum evolution, the latter algorithm is also regarded as an independent computational model called adiabatic quantum computation (AQC). We will provide a brief introduction to AQC in the following paragraphs. §.§.§ The quantum approximate optimisation algorithm The quantum approximate optimisation algorithm (QAOA) developed in 2014 by Farhi, Goldstone and Gutman is a quantum method to approximate CSPs <cit.>. The algorithm depends on an integer parameter p ≥ 1 and the approximation improves as p increases. For small values of p the QAOA algorithm can be implemented on a shallow circuit. As argued <cit.> this feature makes the QAOA algorithm a good candidate for first generation quantum hardware. For certain combinatorial optimisation problems the QAOA algorithm can give approximation ratios that are better than what can be achieved by random sampling <cit.> but worse than the best classical solvers. In specific instances of MAX-kXOR the QAOA algorithm with p = 1 was believed to outperform the best classical solver <cit.>. This sparked further research in the classical community and Barak et al. designed a classical algorithm able to outperform the quantum scaling <cit.>.§.§.§ The quantum adiabatic algorithm The quantum adiabatic algorithm (QAA) <cit.> is an optimisation method that operates in the adiabatic model of quantum computation. The QAA can be thought of as a quantum analogue of simulated annealing <cit.>. The algorithm encodes the solution to a computational problem in the unknown ground state of a quantum system (usually an Ising spin glass Hamiltonian). By starting off in the ground state of a known and easy to implement Hamiltonian, the QAA exploits a slow, time-dependent, Hamiltonian dynamics to obtain the solution to the problem. If the evolution is slow enough, the quantum adiabatic theorem <cit.> guarantees that the system will reach the desired ground state. If the energy barriers have specific configurations (e.g. tall and narrow) and the energy gap between the ground state and the first excited state remains large enough, the algorithm can obtain significant speedups over classical simulated annealing <cit.>.Although QAA and AQC are usually considered synonyms in the literature we shall keep the two concepts distinct as to mark the difference between the computational model and the algorithm. Another name which is frequently used in the literature as synonym of QAA and AQC is quantum annealing (QA). Although there is not a clear consensus in the literature over the differences between these three concepts, we refer to QA only when the adiabatic evolution occurs at non-zero temperature. Aharonov et al. <cit.> showed that AQC is universal for quantum computation, i.e. it is capable of solving any computational problem that can be solved by a quantum computer. Although it is clearly possible to encode -hard problems <cit.>, quantum mechanics is not expected to solve these in polynomial time (however the scaling constants of the quantum algorithm might be smaller). Finally, it is important to note that the adiabatic algorithm lacks worst case upper bounds on its runtime. Its performance has been analysed with numerical experiments <cit.>. However, these are limited to small size systems and only running the algorithm on actual hardware will be able to determine the strength of this approach. § QUANTUM NEURAL NETWORKSThe term artificial neural network (ANN) denotes a variety of models which have been widely applied in classification, regression, compression, generative modelling and statistical inference. Their unifying characteristic is the alternation of linear operations with, usually preselected, non-linear transformations (e.g. sigmoid functions) in a potentially hierarchical fashion.While in the last decade neural networks have proved successful in many applications, fundamental questions concerning their success remain largely unanswered: are there any formal guarantees concerning their optimisation and the predictions they return? How do they achieve good generalisation performance despite the capacity to completely overfit the training data? Artificial neural networks have been extensively studied in the QML literature. The major research trends have focused on accelerating the training of classical models and on the development of networks where all the constituent elements, from the single neurons to the training algorithms, are executed on a quantum computer (a so called quantum neural network). The first works on quantum neural networks appeared in the 90's <cit.> and a number of papers have been published on the topic. However, it is worth noticing that the field has not reached a level of scientific maturity comparable to the other areas of QML discussed in this review. Possible reasons for the difficulties encountered in making progress in this area can be traced to the inherent differences between the linearity of quantum mechanics and the critical role played by non-linear elements in ANNs or the fast developments occurring in the field of classical ANNs. The literature on accelerated training of NNs using quantum resources has mainly focused on restricted Boltzmann machines (RBMs). RBMs <cit.> are generative models (i.e. models that allow to generate new observational data based on prior observations) that are particularly apt to be studied from a quantum perspective due to their strong connections with the Ising model. It has been shown that computing the log-likelihood and sampling from an RBM is computationally hard <cit.>. Markov Chain Monte Carlo (MCMC) methods are the standard techniques used to overcome these difficulties. Nonetheless, even with MCMC the cost of drawing samples can be high <cit.> for models with several neurons. Quantum resources can help to reduce the training cost.There are two main classes of quantum techniques to train RBMs. The first one is based on methods from quantum linear algebra (discussed in Section <ref>) and quantum sampling (discussed in Section <ref>). Wiebe, Kapoor and Svore <cit.> developed two algorithms to efficiently train a RBM based on amplitude amplification <cit.> and quantum Gibbs sampling. These obtain a quadratic improvement in the number of examples required to train the RBM but, the scaling of the algorithm is quadratically worse in the number of edges than contrastive divergence <cit.>. A further advantage of the approach proposed in <cit.> is that it can be used to train full Boltzmann machines (a classical version of this algorithm has also been proposed <cit.>). A full BM is a type of Boltzmann machine where the neurons correspond to the nodes of a complete graph (i.e. they are fully connected). Although full BMs have an higher number of parameters with respect to RBMs, they are not used in practice due to the high computational cost of training and, to date, the true potential of large scale, full BMs is not known. The second direction to training RBMs is based on quantum annealing, a model of quantum computation that encodes problems in the energy function of an Ising model (quantum annealing will be discussed in Section <ref>). Specifically, <cit.> make use of the spin configurations generated by a quantum annealer to draw Gibbs samples that can be then used to train a RBM. These types of physical implementations of RBMs present several challenges. Benedetti and co-authors <cit.> pointed out the difficulties in determining the effective temperature of the physical machine. In order to overcome this problem they introduced an algorithm to estimate the effective temperature and benchmarked the performance of a physical device on some simple problems. A second critical analysis of quantum training of RBMs was conducted by Dumoulin et al. <cit.>. Here the authors showed with numerical models how the limitations that the first generations quantum machines are likely to have in terms of noise, connectivity and parameters tuning, severely limit the applicability of quantum methods. An hybrid approach between training ANNs and a fully quantum neural network is the quantum Boltzmann machine proposed by Amin et al. <cit.>. In this model the standard RBM energy function gains a purely quantum term (i.e. off diagonal) that, according to the authors, allows to model a richer class of problems (i.e. problems that would otherwise be hard to model classically like quantum states). Whether these models can provide any advantage for classical tasks is unknown. Kieferova and Wiebe <cit.> suggest quantum Boltzmann machines could provide advantages for tasks like reconstructing the density matrix of a quantum state from a set of measurements (this operation is known in the quantum information literature as quantum state tomography). Although there is no consensus on the defining features of a quantum artificial neural network, the last two decades have seen a variety of works that attempted to build networks whose elements and updating rules are based solely on the laws of quantum mechanics. The review by Schuld, Sinayisky and Petruccione <cit.> provides a critical overview of the different strategies employed to build a quantum ANN and highlights how most of the approaches do not meet the requirements of what can be reasonably defined as a quantum ANN. In particular, most of the papers surveyed by Schuld et al. failed to reproduce basic features of ANNs (for example, the the attractor dynamics in Hopfield networks). On the other side, it can be argued that the single greatest challenge to a quantum ANN is that the quantum mechanics is linear but ANNs require non linearities <cit.>. Recently, two similar proposals <cit.> have overcome the problem of modelling non-linearities by using measurements and introducing a number of overhead qubits in the input and output of each node of the network. However these models still lack some important features of a fully quantum ANN. For example, the models parameters remain classical, and it is not possible to prove that the models can converge with a polynomial number of iterations. The authors of the papers acknowledge that, in their present forms, the most likely applications of these models appear to be learning quantum objects rather than enhancing the learning of classical data. Finally, we note that, to date, there are no attempts to model non linearities directly on the amplitudes. § LEARNING WITH NOISENoise can play different, potentially beneficial, roles in learning problems. In a classical setting, it has been shown that noise can alleviate two of the most common model-fitting issues: local optima and generalisation performance. Perturbing gradients can help with the former by “jumping out” of local optima, whereas perturbing training inputs or outputs can improve the latter. The possibility of exploiting advantageously the effects of noise is particularly interesting in the context of quantum computation. Early quantum computers are expected to have too few qubits to implement full error correction and the community is actively looking for problems where noise not only does not destroy the computation but can play a beneficial role.The analysis of noisy learning problem from a quantum perspective becomes particularly promising in selected cases. As we will discuss in this section, quantum resources allow to solve efficiently noisy learning problems that would be otherwise classically hard. Although few results are known in this area, further research in this direction might provide new cases of a separation between the classical and quantum case in a learning setting. The goal of this section is to inspire future research aimed at understanding how quantum learners behave in noisy settings. We begin by reviewing for the quantum scientists a number of classical problems in machine learning that benefit from noise. We proceed with a brief introduction to standard ways of modelling errors in quantum computing aimed at machine learning practitioners. We conclude by discussing problems where quantum resources allow to perform tasks that would be otherwise hard for a classical learner.§.§ Classical learning can benefit from noise §.§.§ Noisy inputs The first direct link between the addition of noise to the training inputs (x_i)_i=1^n and Tikhonov regularisation was drawn in <cit.>. Here, it is shown that optimising a feed-forward neural network to minimise the squared error on noisy inputs is equivalent (up to the order of the noise variance) to minimising the squared error with Tikhonov regularisation on noiseless inputs.Intuitively, this form of regularisation forces the gradient of the neural network output f(x) with respect to the input x to be small, essentially constraining the learned function to vary slowly with x: neighbouring inputs are encouraged to have similar outputs.An <cit.> also investigated the effects of adding noise to inputs, outputs, weights and weight updates in neural networks and observed that input (and sometimes weight) noise can, in some settings, improve generalisation performance. §.§.§ Noisy parameter updates More recently, in <cit.>, the addition of annealed i.i.d Gaussian noise to the gradients has been empirically shown to help in optimising complex neural network models. Indeed, stochasticity in the optimisation process can also derive from evaluating gradients of the objective function with respect to randomly selected subsets of the training points (as in stochastic gradient descent). This can be intuitively compared to simulated annealing <cit.> since the natural variability in these “partial” gradients can help escape local optima (and saddle points) and the (decreasing) gradient step size can be directly compared to the annealing temperature. The addition of noise to the update of model parameters was also adopted in <cit.>. There, as well as using random subsets of training points to evaluate gradients, at each iteration the parameter update is perturbed with Gaussian noise (with variance equal to the decreasing step size). After the initial stochastic optimisation phase, it can be shown that this method, under specific conditions, will start generating samples from the posterior distribution over model parameters, allowing us to quantify model uncertainty and avoid overfitting at no extra computational cost. §.§.§ Noisy outputs In Gaussian process regression <cit.>, on the other hand, noise in the training outputs (y_i)_i=1^n helps avoid the inversion of an otherwise potentially ill-conditioned kernel covariance matrix K. Assuming additive isotropic Gaussian noise (with variance σ^2), to evaluate model predictions, we only ever need to invert a matrix of the form K + σ^2 I. This can be practical as the kernel matrix is singular or ill-conditioned whenever training inputs are repeated or are very close in the Hilbert space associated with the kernel covariance function. Finally, when training generative adversarial networks (GANs, <cit.>) it has been shown that an overconfident “discriminator” can hinder learning in the “generator”. In GANs in fact, a generative model (the “generator”) is trained by attempting to “deceive” a “discriminator” model into classifying the generated images as coming from the true data distribution. However, especially early on in training, there might be little overlap in the support of the data distribution and the generator. This can result in the discriminator predicting labels with very high confidence and, as well as potentially overfitting, in making the discrimination decision depend very weakly on the generator's parameters.To address this issue, labels (i.e. true, fake) can be “fuzzied”. Specifically, for each training point, the discriminator will assume that all K labels have probability at least ϵ/K of occurring, with the true label having probability (1 - ϵ) + ϵ/K. This corresponds to assuming that with probability ϵ/K labels are sampled at random and, indeed, labels can just be flipped randomly in practice. Effectively, this keeps the model from becoming too confident in its predictions by making it suboptimal to shift all the probability mass on the true label. This technique is called label-smoothing <cit.> and it has been shown to help retain training signal for the generator <cit.>, as well as increasing robustness of classifiers to adversarial examples <cit.>.§.§ A classical/ quantum separation in learning under noise In order to address learning under noise in a quantum setting it is necessary to discuss what type of noise affects quantum computers. The works by Preskill <cit.> and Breuer and Petruccione <cit.> cover the topic extensively. A simple model of quantum errors, usually employed in numerical simulation of noisy quantum devices, makes use of a weighted combination of two kinds of error: bit flips and phase flips. We can justify this simple type of modelling because, in the most common error correcting codes, errors are detected by projecting more complex errors into convex combinations of bit and phase flips. Given a quantum state ψ = α_0 e_0 + α_1 e_1, a bit flip error turns the state into ψ̃ = α_0 e_1 + α_1 e_0. Similarly, a phase flip error changes the relative phase of a quantum state, i.e. the resulting state isψ̃ = α_0 e_0 - α_1 e_1. More complex and realistic models of errors include amplitude damping, leakage to higher levels, and loss . Many authors have studied how noise affects the learnability of a function in the quantum setting. The already mentioned work by Bshouty and Jackson <cit.> showed that DNF formulas can be efficiently learned under the uniform distribution using a quantum example oracle. This contrasts with the classical case (although proved in the statistical query model of learning) where Blum and co-authors showed that DNF are not learnable under noise with respect to the uniform distribution <cit.>.Another result that points to a separation between classical and quantum for a noisy learning problem has been recently proved by Cross, Smith, and Smolin <cit.>. In this case, it is discussed the learnability of parity functions under noise. It is widely believed that learning parity function under noise is not classically efficient <cit.> and the best classical algorithm run in subexponential, but superpolynomial, time. Furthermore, the problem is an average case version of the -hard problem of decoding a linear code <cit.>, which is also known to be hard to approximate <cit.>. Both the classical and quantum problem are easy without noise. In <cit.> was shown that in the quantum PAC model parity functions can be learned efficiently under the uniform distribution (with logarithmic overhead over the noiseless runtime). Their results have been generalised to linear functions and to more complex error models by Grilo and Kerenidis <cit.>.To summarise, in this section we surveyed a number of classical results showing that noise in the inputs, outputs or in the parameters can have positive effects onlearning algorithms. It would be interesting to investigate whether the type of noise encountered in quantum systems has a similar distribution and structure to the one commonly encountered in classical settings. In this case machine learning algorithms would become ideally suited to run on non fault-tolerant quantum hardware. Finally, further research is needed to identify new, noisy, problems that only a learner equipped with quantum resources can solve.§ COMPUTATIONALLY HARD PROBLEMS IN MACHINE LEARNINGAlgorithms whose runtime is upper bounded by a polynomial function of N are said to be efficient. Problems for which there exists an efficient algorithm are easy. Conversely, hard problems are those where no polynomial algorithm is known. An important class of easy problem is called . The class of problems that are efficiently solvable by a quantum computer includes some problems that are not known to be in . The quantum algorithms surveyed in this review speed up efficient classical algorithms. Two types of speedups are obtained: polynomial or exponential. Polynomial speedups, although important from a practical point of view, do not prove that quantum computers are able to turn hard learning problems into easy ones. On the other hand, exponential speedups of algorithms that are already efficient face important challenges. Indeed, as we have seen for the matrix inversion algorithm discussed in Section <ref>, quantum algorithms for the analysis of classical data running in logarithmic time require an equally fast access to the memory. This can be obtained using a QRAM that, however, presents a number of issues (see Section <ref>). In order to achieve an exponential speedup despite the computational costs arising from accessing the memory we are restricted to hard algorithms. This is because, for these algorithms, the polynomial time construction of the quantum state that encodes the dataset does not dominate over the speedup. We discussed an example with such a property: the learnability of DNF formulas (Section <ref>). Classically, the best algorithm for learning DNFs runs in superpolynomial time. With quantum resources we can learn the same problem polynomially. Although these types of learning problems have limited practical applications, they suggest that an exponential separation between classical and quantum models of learning might hold in real world problems.In this section, we present a number of problems in machine learning that are believed to be computationally hard and are receiving considerable interest in the classical community. We do not expect that these problems, some of which are -hard, can be solved efficiently with a quantum computer. Recall that -hard is a class of problems for which there is strong evidence of a separation with <cit.>. Our hope is to spark interest in the search for hard problems in machine learning with the kind of structure (see Section <ref>) that can be exploited by quantum computers. We also decided to include problems that are not hard in the computational complexity sense but whose high degree polynomials runtime make them intractable. For these cases, where slow (i.e. polynomial) memory access times can still be tolerable, even polynomial speedups might be of great practical relevance.§.§ Tensor factorisation As modern datasets grow not only in terms sheer dimension but also in the complexity of the structures required to store such data (e.g. multi-modal data, social networks, recommender systems, and relational data <cit.>), it becomes ever more critical to device methods able to distil concise and interpretable representations of this information. Tensor models offer a powerful way to address these learning problems. For instance, tensors naturally generalize the concept of adjacency matrix for multi relational graphs <cit.>. However, given the intrinsic multi dimensional nature of these objects, tensor based learning problems are typically computationally hard and require large amounts of memory, therefore become quickly impractical in most applications. To this end, finding low rank approximations of tensors (or more generally multi-linear operators), a natural generalisation of the problem of matrix factorisation (see <cit.> and references therein), has recently received significant attention from the fields of machine learning, inverse problems and compressive sensing. However, while for the matrix case the problem is amenable to efficient computations, moving to higher orders becomes significantly challenging. Indeed, in contrast to its matrix counterpart, low rank tensor factorisation, even when relaxed to a nuclear norm regularised optimization problem, has been recently shown to be -hard <cit.>. Approaches have attempted to circumvent these issues by considering further relaxation of the factorisation problem <cit.>, but to this day a standard solution has yet to be proposed.§.§ Submodular problemsRecently, several machine learning problems have been addressed via submodular optimization. Examples of such applications are very diverse, such as document summarisation <cit.>, social networks <cit.>, or clustering <cit.> to name a few. Submodularity characterises a family of discrete optimisation problems, typically entailing cost functions on sets, in which the target functional exhibits a structure akin to that of convexity (or rather concavity) for continuous functions. We refer to <cit.> for an in-depth introduction on the topic. For many submodular problems it is possible to identify a corresponding convex problem via the so-called Lovàsz extension <cit.>. As a consequence, such problems can be solved using convex optimisation methods, leading to efficient learning algorithms. However, for a wide range of these problems, the corresponding computational complexity, albeit polynomial, is of high order (e.g. 𝒪(n^5) with respect to the number n of the parameters, see for instance <cit.>), making them remarkably slow in practice. In this sense, an exponential (or even polynomial) decrease in the number of computations to solve a submodular problem, analogous to the one observed for fast linear algebra using quantum algorithms, could be key to tackle practical applications. §.§ Inference in graphical models Probabilistic models in machine learning can be encoded in graphs. Graphical models of particular use are Bayesian networks <cit.> and Markov random fields <cit.>: directed acyclic and undirected graphs, respectively, where nodes represent random variables and edges denote dependence between variables. Operations like marginalisation and conditioning can be performed by algorithms taking into account the specific connectivity of the given graph (i.e. message passing). While this offers a general framework for inference (i.e. evaluating the distribution of latent variables conditioned on observed ones), it has been shown, by reduction to Boolean satisfiability <cit.>, that exact inference in these networks is -hard and that evaluating the normalising constant Z (or partition function) for the joint distribution is in # (a family of hard counting problems).§ CONCLUSIONS AND OUTLOOKIn this review, we surveyed a number of different quantum methods to tackle learning problems.Despite a number of promising results, the theoretical evidence presented in the current literature does not yet allow us to conclude that quantum techniques can obtain an exponential advantage in a realistic learning setting. Even in the case of quantum algorithms for linear algebra, where rigorous guarantees are already available, issues related to data access and restrictions on the types of problems that can be solved might hinder their performance in practice. In fact, near future advances in quantum hardware development will be important to empirically assess the true potential of these techniques. In this regard, we note how the great majority of the quantum machine learning literature has been developed within the quantum community. We believe that further advances in the field will only come after significant interactions between the two communities. For this reason, we tried to structure the review in a way that presents the different topics in a way familiar to both quantum scientists and machine learning researchers. In order to achieve this goal we put great emphasis on the computational aspects of machine learning. Although this perspective has the obvious advantage of providing an agile way for discussing quantum algorithms (that mostly focus on accelerating the runtime with respect to their classical counterparts), the reader should keep in mind that statistical problems (like determining the generalisation performance of an algorithm) are equally relevant. The approach taken in this review has also left some interesting papers aside (see for example <cit.>). We invite the reader to consult <cit.> for a review that includes these works. In Section <ref> we discussed how the computational cost represents one of the major challenges for the future of machine learning. In particular, polynomial scaling in the number of data points might not be adequate in the age of large scale machine learning. The quantum algorithms presented here allow to reduce the complexity of some, currently used, regularisation methods. We classified the quantum approaches into four main categories: linear algebra, neural networks, sampling, and optimisation. The quantum machine learning algorithms based on linear algebra subroutines are those that promise the greatest computational advantages (i.e. exponential). However, it is not clear whether fundamental limitations related to how quickly these algorithms need to access the memory might compromise their ability to speed up the analysis of classical data. Quantum methods for training neural networks, for sampling, and for optimisation, provide so far mostly quadratic advantages and some of these might be implementable on first generation quantum computers. Unfortunately, the theoretical framework on which they are based is not yet well established (e.g. the quantum Boltzmann machines described in Section <ref>) and only practical experiments will determine their true performance. To summarise, the works surveyed in this review, including the theoretical evidence presented in Section <ref>, suggest the possibility of a quantum speedup for some machine learning problems. However, the extent of these speedups, and consequently the impact of these methods on practical problems, remains an open question. We identified a number of promising directions for the field. First, exploring the tradeoffs between noise, generalisation performance and hardness in a quantum context (Section <ref>). This is particularly interesting for first generation quantum hardware that most likely will not be fault-tolerant.Second, deepening our understanding of how quantum resources can affect sample and time complexity, even for problems that are already known to be efficient.Significant work has already been done but some areas like statistical learning theory are yet to receive a thorough analysis in a quantum context. Third, determining whether a QRAM of the size required to handle large datasets can be constructed on a physical device (Section <ref>). Fourth, understanding whether there exist non polynomial problems in machine learning that can be tackled efficiently using quantum resources (Section <ref>). This direction is arguably the most relevant for finding quantum algorithms capable of demonstrating an uncontroversial speedup in a learning context, and this is indeed the general quest of quantum computation.§.§ Data accessibility This paper has no data. §.§ Competing interests The authors declare no competing interests. §.§ Authors' contributions AR and LW conceived the project. All authors contributed to the literature review. All authors wrote the manuscript.§.§ Acknowledgements We thank Scott Aaronson, David Barber, Marcello Benedetti, Fernando Brandão, Dan Brown, Carlos González-Guillén, Joshua Lockhart, and Alessandro Rudi for helpful comments on the manuscript.§.§ Funding statement AI is supported by the Cambridge-Tuebingen Fellowship and the Qualcomm Innovation Fellowship. AR is supported by an EPSRC DTP Scholarship and by QinetiQ. CC and MP are supported by EPSRC. SS is supported by The Royal Society, EPSRC, Innovate UK, Cambridge Quantum Computing, and the National Natural Science Foundation of China. | http://arxiv.org/abs/1707.08561v3 | {
"authors": [
"Carlo Ciliberto",
"Mark Herbster",
"Alessandro Davide Ialongo",
"Massimiliano Pontil",
"Andrea Rocchetto",
"Simone Severini",
"Leonard Wossnig"
],
"categories": [
"quant-ph",
"cs.LG",
"stat.ML"
],
"primary_category": "quant-ph",
"published": "20170726174825",
"title": "Quantum machine learning: a classical perspective"
} |
firstpage–lastpage Anisotropic EM Segmentation by 3D Affinity Learning and Agglomeration Toufiq Parag^1, Fabian Tschopp^4, William Grisaitis^2, Srinivas C Turaga^2, Xuewen Zhang^5Brian Matejek^1,Lee Kamentsky^1, Jeff W. Lichtman^3, Hanspeter Pfister^1^1School of Engg and Applied Sciences, Harvard University, Cambridge, MA ^2Janelia Research Campus, Ashburn, VA ^3Dept of Molecular and Cellular Biology, Harvard University, Cambridge, MA^4 Institute of Neuroinformatics, University of Zurich and ETH Zurich, Switzerland. ^5 Chester F. Carlson Center for Imaging Science, RIT, Rochester, NY. email: [email protected] 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We analyze particle acceleration in explosive reconnection events in magnetically dominated proton-electron plasmas. Reconnection is driven by large-scale magnetic stresses in interacting current-carrying flux tubes. Our model relies on development of current-driven instabilities on macroscopic scales. These tilt-kink instabilities develop in an initially force-free equilibrium of repelling current channels. Using MHD methods we study a 3D model of repelling and interacting flux tubes in which we simultaneously evolve test particles, guided by electromagnetic fields obtained from MHD. We identify two stages of particle acceleration; Initially particles accelerate in the current channels, after which the flux ropes start tilting and kinking and particles accelerate due to reconnection processes in the plasma. The explosive stage of reconnection produces non-thermal energy distributions with slopes that depend on plasma resistivity and the initial particle velocity. We also discuss the influence of the length of the flux ropes on particle acceleration and energy distributions. This study extends previous 2.5D results to 3D setups, providing all ingredients needed to model realistic scenarios like solar flares, black hole flares and particle acceleration in pulsar wind nebulae: formation of strong resistive electric fields, explosive reconnection and non-thermal particle distributions. By assuming initial energy equipartition between electrons and protons, applying low resistivity in accordance with solar corona conditions and limiting the flux rope length to a fraction of a solar radius we obtain realistic energy distributions for solar flares with non-thermal power law tails and maximum electron energies up to 11 MeV and maximum proton energies up to 1 GeV. instabilities – MHD – acceleration of particles – magnetic reconnection –methods: numerical§ INTRODUCTIONMagnetic reconnection causes a magnetic field to rapidly and violently rearrange its topology. This topological change affects plasma energetics and is one of the processes controlling energy exchange between different plasma system constituents (; ). The main process of interest here is the conversion of energy available in the magnetic field into non-thermal particle distributions in magnetically dominated plasmas (low plasma-β). This phenomenon causes violent energy releases in a wide range of astrophysical events, including various kinds of flares and bursts of high-energy (UV, X-ray and gamma-ray) (). Solar flares are the most prominent and well studied classical examples of reconnection (; ). Observations reveal that 10 % - 50% of magnetic energy is converted into energetic charged particles () and that particles develop a power-law energy distribution containing energy of the same order as the converted magnetic energy (; ). In some observations of solar flares the emission has no distinguishable thermal part and almost all electrons are accelerated to non-thermal energies (; ). Electrons in solar flares can reach energies up to 5 – 50 MeV, while protons gain energies up to several GeV (, ). Reconnection has also been proposed as a mechanism for powerful flares and particle acceleration in more extreme settings like pulsar wind nebulae (; ; ), gamma-ray bursts (), magnetospheres of magnetars (; ; ; ) and in coronae and jets of accreting black holes and active galactic nuclei (; ; ). Flares from astrophysical objects require energy from macroscropic scales to be transferred to the microscopic scales on which particles are accelerated. The change of topology of the magnetic field configuration on large scales is well described by magnetohydrodynamics (MHD) and the dissipation at small scales is described by kinetic (particle) theory. The energetics of the plasma can be split into the part relevant at the fluid level plus non-thermal particle distributions. The MHD approach covers the overall scales and energetics of the system but does not give any information on particle dynamics. Kinetic approaches fully describe the microscopic scale, but are too costly to cover full astrophysical systems. In this work, we treat electrons and ions as test particles embedded in a thermal (MHD) plasma. Particle acceleration associated with reconnection and shocks in magnetically dominated Newtonian plasmas in the solar corona is studied extensively with test particle approaches (, , , , , , , , , ) and even in relativistic plasmas in the context of pulsar wind nebulae (, ). The test particles are guided by MHD fields without giving feedback to these fields. To initiate reconnection in the MHD plasma, we perturb an equilibrium of two adjacent, anti-parallel and repelling current channels (, , ). Translation and rotation of the currents cause a plasma disruption. <cit.> showed that reconnection occurring due to this tilt instability is an efficient source of highly energetic particles in 2.5D settings. In 3D configurations, the kink instability interacts with the tilt instability, redistributing the poloidal magnetic field. Current channels undergoing a tilt or tilt-kink instability typically show a growth phase on Alfvénic timescales in which kinetic energy grows exponentially. This energy is expected to be released and transferred to charged particles via magnetic reconnection. In the stellar corona context, the kink instability is one of the well-explored routes to initiate flares. Accessing a kink instability via anti-parallel, repelling flux ropes is intimately connected to the coalescence instability of two attracting and merging flux ropes. The current filaments eventually attracting or repelling typically form as a result of turbulent plasma processes or instabilities as the tearing mode. Both the tilt and coalescence instability have been studied extensively in 2D configurations (see e.g., ; ; ; ). Here we propose that such anti-parallel currents may also form in stellar coronae where they play a role in magnetic island interactions and can cause particle acceleration. The evolution that is observed shows that repelling currents lead to sudden release of magnetic energy and current dissipation. The repelling islands typically cause localized reconnection, thin current sheet development and strongly curved magnetic fields. On a large scale the current interaction can lead to rapid transfer of significant amounts of energy from the currents to the particles. Our model is representative for the top parts of adjacent flux ropes as seen in extreme ultraviolet observations of the solar corona. If two of these loops develop antiparallel currents, the tilt instability route to reconnection is accessible (). This model was studied in 2D by <cit.> and extended to 3D by <cit.> and <cit.>. Here we investigate the effect of an additional kink instability on particle acceleration as well as the effects of boundary conditions, initial conditions and resistivity models on particle dynamics during the full 3D MHD evolution. We use high-resolution MHD results of <cit.> for the evolution of the repelling current channels. For test particle simulations we make use of the latest addition to the MPI-AMRVAC code () to dynamically evolve test particle populations during MHD evolution. We apply the guiding centre approximation in which particle gyration is neglected and only the particle velocity parallel to the magnetic field is evolved. This approach is valid in typical non-relativistic, magnetically dominated plasmas where the particle energy density is less than the energy density of the underlying MHD fluid. The numerical methods employed are described in detail in Section <ref>. In Section <ref> 2.5D test particle simulations are discussed, in Section <ref> we discuss 3D simulations and the effect of the kink instability on test particle acceleration and energetics. We compare the results of the guiding centre approximation to solutions obtained from the full particle equation of motion in Appendix <ref>.§ NUMERICAL SETUP For the MHD background in which test particles are evolved we use a simulation as described in <cit.>. Two parallel, adjacent, repelling current channels are initiated in a region [-3L,3L] × [-3L,3L] × [-3L,3L] in Cartesian coordinates (x,y,z) with the depth of the current channel in the z-direction orthogonal to the plane. The equilibrium is described by the initial conditions for the flux function ψ_0(x,y) for both 2.5D and 3D setupsψ_0(x,y) = 2/j_0^1 J_0 (j_0^1)J_1(j_0^1 r) cos(θ) r < 1 (r-1/r) cos(θ) r ≥ 1,with J_1 is the Bessel function of the first kind and j_0^1 ≈ 3.831706 is the first root of J_1. The magnetic field is obtained as 𝐁 = ∇ψ_0 ×ẑ + B_z ẑ, resulting in a current distribution J_z = (∇×𝐁)_z = ∇^2 ψ_0. In one half of the unit circle J_z < 0 and in the other half J_z > 0 and initially there are no currents in the region r ≥ 1. An ideal MHD equilibrium is established by postulating a force-free magnetic field with spatially varying, vertical component B_z(x,y) and a uniform plasma pressure p_0 such that the Lorentz force 𝐉×𝐁 = ∇ p = 0, hereB_z(x,y) =(j_0^1)(ψ_0(x,y)) r < 1. 0 r ≥ 1.In 2.5D configurations a uniform resolution of 2400^2 is used and in 3D setups we use an effective resolution of 300^3 with one level of mesh refinement. L = 10 Mm is chosen as a typical unit of length for the astrophysical systems under consideration. The dimensionless density ρ is equal to unity initially and the ratio of specific heats is Γ=5/3. We fix a constant pressure p_0 = 0.01/Γ such that we reach a low plasma-β = 0.04 in an initially force-free equilibrium. The normalization used implies the sound speed outside the current channels as the unit of speed, the radius of the double current channel as the unit of length and the density to fix the unit of mass. Magnetic units where μ_0=1 are employed. MHD fields and particles are evolved for the typical time 10 t_S = 10 L/c_S ≈ 852.6 seconds, with c_S the sound speed outside the current channels. We apply a uniform resistivity η_MHD=10^-4. The low-β and force-free conditions are in accordance with astrophysical systems which are magnetically dominated, such as magnetospheres of black holes and pulsars and the solar corona. The boundary conditions for the MHD evolution in the (x,y)-plane imply a zero gradient. The boundaries in the z-direction are periodic in 3D configurations, whereas in 2.5D the z-direction is invariant. The initially force-free equilibrium is unstable to a tilt instability () and in a 3D configuration also to an additional kink instability bending the field lines with respect to the vertical direction (; ). Once the instabilities develop and the physics become naturally nonlinear, it allows for fast reconnection of the field lines. From solving the set of resistive, compressible MHD equations we obtain the magnetic and electric fields 𝐁 and 𝐄 and the total current density 𝐉 = ∇×𝐁. The MHD data are scaled to CGS units before being used in the test particle calculations. To analyze the energetics and acceleration of electrons and protons in the MHD simulations we follow the orbits of test particles in the MHD flow, similar to the approach of <cit.> and <cit.>. In 3D simulations particles are injected uniformly in the domain. To analyze the behavior of particles in the reconnection zones specifically, several runs are performed with a fraction of 0.99 of the ensemble uniformly distributed in space in a rectangular block, encapsulating the two (displaced) current channels and the areas with the largest current density, x ∈ [-1L,1L], y ∈ [-2L,2L], z ∈ [-3L,3L]. The other fraction of 0.01 of the ensemble is uniformly distributed over the full domain x ∈ [-3L,3L], y ∈ [-3L,3L], z ∈ [-3L,3L], including the surrounding background. In 2.5D simulations, with translational invariance in the z-direction for MHD evolution, the particles are distributed in the (x,y)-plane in accordance with the 3D simulations, at z = 0. In case I3De (see Table <ref>) the current channels are twice as long as in the reference cases, meaning that the particles are distributed over the domain x ∈ [-3L,3L], y ∈ [-3L,3L], z ∈ [-6L,6L] and the MHD resolution is 300 × 300 × 600.Typical parameters for low plasma-β plasma in coronal loops are, for magnetic field magnitude B=0.03 T, temperature T = 10^6 K, number density n = 10^16 m^-3 and plasma-β = 0.0004 (). Particles are injected from a Maxwell-Boltzmann velocity distribution in accordance with solar corona conditionsf(v) = N(2v/v^2_th) exp(-v^2/v^2_th).with thermal speed v_th,p = √((2 k_B T ρ_0/m_p p_0))∼ 10^7 m/s for protons in a fluid with temperature T = 10^6 K, the proton rest mass m_p = 1.6726 · 10^-24 g, dimensionless pressure p_0 and fluid density ρ_0. For electrons we either assume energy equipartition, meaning they have a thermal speed v_th,e = √(m_p/m_e)× v_th,p∼ 10^9 m/s, or we assume that both electrons and protons initially have the typical thermal speed v_th,e = v_th,p = √((2 k_B T ρ_0/m_p p_0)). Both resulting in a thermal Lorentz factor of γ_th≈ 1. The particle gyroradius in the plasma settings we assume, R_L = γ m_0 v_⊥/(Bq) = 10^-3 m for electrons and R_L = 4.4 × 10^-2 m for protons, is small compared to the typical size over which the MHD fields change. The particles have a uniform pitch angle distribution α∈ [-π/2, π/2] with α = arctan(v_⊥/v_), the angle between the velocity vector of a particle and the unit vector parallel to the magnetic field. The particles are advanced according to the Lorentz force resulting from the MHD fields 𝐄 and 𝐁 (e.g. ):d𝐔/dt = q/m_0 c(𝐄 + 𝐔×𝐁/cγ),where 𝐔 = γ𝐯/c is the particles four-velocity, c the speed of light in vacuum and q/m_0 is the charge to mass ratio. We apply the guiding centre approximation, in which the gyration of the particles is neglected, to equation (<ref>) to obtain the relativistic guiding centre equations of motion describing the (change in) guiding centre position 𝐑, parallel relativistic momentum p_ = m_0γ v_ and relativistic magnetic moment μ_r = m_0 γ^2 v^2_⊥/2B in three-space ()d𝐑/dt = (γ v_)/γ𝐛̂+𝐛̂/B(1-E_⊥^2/B^2)×{ -(1-E_⊥^2/B^2)c𝐄 + . cm_0γ/q(v_^2(𝐛̂·∇)𝐛̂+v_(𝐮_𝐄·∇)𝐛̂ + v_(𝐛̂·∇)𝐮_𝐄 + (𝐮_𝐄·∇)𝐮_𝐄) + . μ_r c/γ q∇[B(1-E_⊥^2/B^2)^1/2]+ v_E_/c𝐮_𝐄}, d (m_0 γ v_)/dt =m_0γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂+(𝐮_𝐄·∇)𝐛̂) + qE_ -μ_r/γ𝐛̂·∇[B(1-E^2_⊥/B^2)^1/2], d (m_0 γ^*2 v^*2_⊥/2B^*)/dt = d μ_r^*/dt = 0.Here, 𝐛̂ is the unit vector in the direction of the magnetic field and v_ the component of the particle velocity vector parallel to 𝐛̂. The magnitude of the electric field 𝐄 = -𝐯×𝐁 + η_p𝐉 is split as E = √(E^2_⊥ +E^2_) where the component parallel to the magnetic field, 𝐄_, comes solely from resistive contributions η_p𝐉·𝐛̂ and is therefore also called the resistive electric field. The resistivity η_p is either equal to the resistivity set for the MHD evolution η_p = η_MHD or it is an anomalous resistivity η_p ≠η_MHD that does not affect MHD fields. The drift velocity, perpendicular to 𝐁 is written as 𝐮_𝐄 = c𝐄×𝐛̂/B and v^*_⊥ is the perpendicular velocity of the particle, in the frame of reference moving at 𝐮_𝐄. The magnetic field in that frame is given by B^* = B(1-E^2_⊥/B^2)^1/2 up to first order. The relativistic magnetic moment μ_r^* is an adiabatic invariant and is proportional to the flux through the gyration circle, again in the frame of reference moving at 𝐮_𝐄. The oscillation of the Lorentz factor at the gyrofrequency is averaged out as well, giving γ = γ^*(1-E^2_⊥/B^2)^-1/2. We assume the MHD fields to be slowly varying compared to the particle dynamics, allowing to neglect temporal derivatives in equations (<ref>-<ref>). However, we do treat dynamic MHD evolutions, so we interpolate the MHD variables in time. In the case of the guiding centre approximation, the gyroradius of the particle, R_L = γ m_0 v_⊥/(Bq), is assumed to remain smaller than the typical cell size of the MHD simulation. To confirm validity of the guiding centre approximation results of both equations (<ref>) and (<ref>-<ref>) will be compared. For more information on the guiding centre approach used here and its validity we refer to <cit.> and for its mathematical background to <cit.>.Equations (<ref>) are advanced with a second-order symplectic Boris scheme (e.g. ). Each particle is advanced with an adaptive, individual time step, ensuring that a single gyration is resolved by at least 60 steps, to ensure numerical stability. Equations (<ref>-<ref>) are advanced with a fourth order Runge-Kutta scheme with adaptive time stepping. Here, the particle timestep δ t is determined based on its parallel acceleration a = d v_/dt and velocity v = √((v_)^2 + (v_⊥)^2) as the minimum of δ r / v and v / a, where δ r is the grid step. This grid step is restricted such that a particle cannot cross more than one cell of the MHD grid in one time step. The fields 𝐄 and 𝐁, and for the GCA equations their spatial derivatives, are obtained at the particles position via linear interpolations in space and time between the fluid steps limited by the CFL condition. The particles gyroradius is also calculated at every timestep and compared to the typical cell size to monitor the validity of the guiding centre approximation.In the (x,y)-plane we employ open boundary conditions, in which the particles leaving the physical domain are destroyed. In 2.5D simulations we limit the length of the flux ropes in the z-direction, which is invariant for MHD fields. A particle crossing an artificially set boundary, at z=3 L or z=-3L (consistent with the z-boundaries in 3D simulations), is destroyed. For each destroyed particle a new particle is injected at the opposite z-boundary with a thermal velocity from a Maxwellian distribution. This thermal bath boundary condition limits the length of the flux rope to 6L and counteracts particles accelerating indefinitely in the invariant z-direction. In 3D configurations, we have periodic boundary conditions for MHD fields, where particles leaving a z-boundary are periodically injected at the opposite z-boundary, consistent with MHD. In specific cases (see Table <ref>) we employ a similar boundary condition as in 2.5D configurations where particles leaving a z-boundary are destroyed. For each destroyed particle a new particle is injected at the opposite boundary with a thermal velocity according to a Maxwellian. In this way a large enough ensemble of particles is retained at all times, to achieve accurate statistics. Consequently the length of the flux rope is limited to 6L or equivalently ∼ 0.1 solar radii. This boundary condition realistically mimics the injection of thermal particles in a (curved) flux rope in the corona of a star. To go to more realistic solar corona conditions we set a particle resistivity η_p, that does not affect the MHD evolution. The factor η_p appears in all terms in equations (<ref>-<ref>) as E_ = η_p𝐉·𝐛̂ where the current 𝐉 is interpolated at the particle position from MHD fields. This parameter is either set to be equal to the resistivity used for the MHD evolution η_p = η_MHD = 10^-4, or set smaller than the MHD resistivity as η_p = 10^-5η_MHD = 10^-9. Decreased resistivity avoids artificially large energies due to a large resistive electric field. The magnetic Reynolds number, describing the ratio of advective to diffusive terms in the induction equation is defined byR_m = 𝒰ℒ /η_D.with 𝒰 and ℒ the characteristic velocity and length scale respectively and η_D = η L c_S the dimensional resistivity and c_S the sound speed time used as unit of velocity. In the solar corona it is typically 𝒪(10^8) – 𝒪(10^12) (). Large values of the magnetic Reynolds number mean that resistive effects are restricted to thin regions with large current density. In our simulations the typical length scale is ℒ = 6 L = 6 · 10^9 cm, the total width of the simulation box (the maximum diameter of the flux tubes). The typical velocity is the Alfvén speed 𝒰 = V_A= B/√(ρ)≈ 43 · 10^6 cm/s. With η_p = η_MHD = 10^-4 we find R_m ≈ 2 · 10^5 and for η_p = 10^-9 we find R_m = 2 · 10^10. Simulations with different R_m are compared to see the effect of the resistivity on particle acceleration, without changing the MHD results. In Table <ref> we list all particle simulations including the dimension, the type of particles (indicated by e^- for electrons and p^+ for protons), the number of particles N_tot, the initial spatial distribution (which is always uniform in the domain given) and the equations of motion solved for the particles (guiding centre approximation, indicated with GCA or full equations of motion indicated with Lorentz). We indicate the typical length l the particles can travel in the current channels in the z-direction. Infinite length corresponds to periodic boundary conditions. A finite value corresponds to distance between the two opposite z-boundaries at which a particle is typically destroyed and a thermal particle is injected respectively. We also mention the resistivity used in the particles equations of motion η_p (either 10^-9 or equal to the resistivity applied for the MHD evolution η_MHD = 10^-4), and the magnetic Reynolds number R_m typical for the simulation parameters. In the last two columns we show the particles maximum kinetic energy ℰ_kin,max/(m_0 c^2) = γ_max -1 and the particles maximum energy ℰ_max = γ m_0 c^2 in MeV for each run. § RESULTS IN 2.5D CONFIGURATIONS Flares are strongly transient phenomena and particles accelerate during such an event. Recently, <cit.> combined MHD and test particle methods to investigate proton and electron acceleration in static MHD snapshots of repelling flux tubes in 2.5D. Here particle dynamics are evolved simultaneously with MHD evolution. The 2.5D results presented by <cit.> show very hard energy distributions and an inverted power law spectrum due to indefinite acceleration in the infinitely long current channels. Here we suggest several solutions to obtain more realistic distributions with a power law spectrum and energies in accordance with observations, both in 2.5D setups and 3D setups. The perturbed equilibrium of adjacent and anti-parallel currents develops a tilt instability in which the current channels start to displace and rotate. This instability is indicated by an exponential growth phase of the kinetic energy, that is reached after t ≈ 4 t_S in our 2.5D simulation (see Fig. <ref>).After this phase, the non-linear regime is reached at t ≈ 6 t_S, showing highly chaotic behavior and magnetic energy is converted into kinetic energy. In 3D both phases are delayed until t ≈ 6 t_S and t ≈ 8 t_S respectively, due to the magnetic tension caused by the kinking of the channels (see Fig. <ref>). We evolve particles during the whole evolution shown in Fig. <ref>, however we are mainly interested in particles accelerating due to the tilt instability from t ≈ 6 t_S onwards. Particle acceleration is quantified by means of energy distributions and pitch angle distributions. Electron energy distributions associated with solar flares typically have a high-energy tail that partially can be fitted with a power law function f(ℰ) ∝ (ℰ)^-p with p ≥ 1. A longer time spent in the current channels corresponds to higher energies, and harder energy distributions. In <cit.> it is shown that interacting flux ropes in 2.5D configurations are an efficient mechanism to accelerate particles. However, the spectra found are very hard and the power law slope is even inverted (p < 0) compared to what is expected based on observations, even on very short time scales Δ t ≪ 0.1 t_S. The maximum energies found also exceed electron energies of 5 – 50 MeV associated with solar flares. The main cause mentioned is the 2.5D character of the setup, meaning that the current channels have an infinite length and hence, particles can accelerate indefinitely. Here the length of the flux ropes is limited to 6L by applying a thermal bath at z = ± 3L, in accordance with the periodic boundaries in 3D simulations. In Fig. <ref> we show the kinetic energy (ℰ_kin/(m_0 c^2) = γ - 1) distribution (left-hand panel) and the pitch angle (α = arctan(v_/v_⊥)) distribution (right-hand panel) counted by particle number, for electrons in case A2De. The spectra are coloured by the MHD time t_S, from magenta to red, with magenta corresponding to early times and red to late times. The initial distributions are depicted by a dashed black line. Initially the electrons are distributed in the regions of the (displaced) current channels -1L ≤ x ≤ 1L; -2L ≤ y ≤ 2L from a Maxwellian with thermal speed v_th,e = v_th,p = √(2 k_B T ρ_0/(m_p p_0)), to improve statistics of particles in reconnection regions. We have confirmed that a simulation with an initially uniform electron distribution gives similar results, with the same maximum energy γ_max but a smaller fraction of particles in the high energy tail. The limited flux rope length bounds the kinetic energy to γ - 1 ≲ 10^2, corresponding to ℰ_max≈ 50 MeV, which is three orders of magnitude smaller than in the case with infinitely long flux ropes (). Acceleration in the current channels at early times causes the slope of the spectrum to be inverted, p < 0. Acceleration in the direction parallel to the magnetic field is dominant over particle drifts as can be seen from the pitch angle distribution in the right panel of Fig. <ref> that is strongly peaked around α = 0.For protons (case B2Dp in Table <ref>) the kinetic energy spectra look fairly similar, with two major differences; The maximum kinetic energy is limited by γ - 1 ≲ 10^-2, corresponding to E_max≈ 957 MeV, due to the mass difference between electrons and protons, and less protons have left the domain through the open x-, and y-boundaries. The power law index of the high energy tail is inverted, p < 0. In the pitch angle spectra an asymmetry with respect to α = 0 can be observed, due to electrons and protons accelerating in opposite directions. This observation is visible in all simulations carried out in this work. The tendency for particle to accelerate along the magnetic field lines, and hence obtain a very small pitch angle, is in agreement with the findings of <cit.>. However, curvature acceleration resulting in increasing parallel velocity is neglected in their setup and particle collisions are incorporated. This asymmetry develops directly after t=0, when protons accelerate parallel to the magnetic field. Therefore there are more particles with a pitch angle slightly larger than zero. For electrons this asymmetry is present as well, with more particles with a pitch angle slightly smaller than zero. However, because electrons develop a larger parallel velocity than protons, the α=0 peak is sharper for electrons and the asymmetry around α=0 is less pronounced than for protons. For protons the α = 0 peak is represented by a peak at cos(ζ) = 1 and for electrons at cos(ζ) = -1 if we define cos(ζ) = v_ / v with ζ the angle between the velocity vector and the magnetic field vector. § RESULTS IN 3D CONFIGURATIONS In 3D the perturbation in the velocity field consists of a z-component and dependency. This introduces variations in the z-direction which is invariant in 2.5D setups. The repelling and rotating current channels develop an additional kink instability that causes reconnection. Strong and thin current sheets develop at the boundaries and in between the two repelling islands (see Fig.<ref> for the total current density magnitude in 2.5D and in 3D at t = 9 t_S). In <cit.> it is shown that reconnection in this setup is indicated by a non-zero resistive electric field parallel to the magnetic field. This resistive electric field is plotted in Fig. <ref>, with selected reconnecting magnetic field lines, initially at t = 0 t_S and far into the nonlinear regime at t = 9t_S. Particles mainly accelerate parallel to the resistive electric field and hence parallel to the magnetic field (). In this section we investigate the effect of the kink instability on particle acceleration as well as the influence of the initial particle velocity distribution, the length of the flux ropes and resistivity for both electrons and protons. The effect of the initial spatial distribution and total number of particles on particle statistics are reported in Appendix <ref> – <ref>. The effect of gyration is monitored in specific cases in Appendix <ref>. §.§ Effect of the kink instability on particle distributionsThe effect of the kink instability is best visible for electrons. We show the kinetic energy spectra and the pitch angle spectra obtained from guiding centre simulations, with thermal velocity v_th,e = v_th,p = √(2 k_B T ρ_0/(m_p p_0)) with initially a fraction of 0.99 of the particles in the current channel area and flux ropes with length 6L in Fig. <ref> for 20.000 electrons (case A3De). The main observable difference due to the kink instability is the development of a medium energy tail in the electron energy distribution with 10^-5≤γ -1 ≤ 1 starting at t ≈ 8 t_S in the nonlinear regime. A second noticeable effect is the redistribution of electrons in the thermal distribution γ-1 ≤ 10^-5 after t ≈ 8 t_S. This is attributed to the curvature of the magnetic field due to the kink, expelling particles from the current channels. These particles lose their energy in the ambient medium and either remain there and eventually leave the domain through the open boundaries, or they are caught again in either of the two current channels and re-accelerate. The electron energy distributions develop a high energy peak at γ-1 ≈ 60 (∼ 31 MeV) and the high energy tail has an inverted power law index p < 0. The difference in maximum energy compared to 2.5D results can be explained by the peak current reached in the MHD evolution. The current sheet is narrower in 2.5D due to a higher resolution (), resulting in a larger peak current that accelerates particles to higher energy. In Fig. <ref> we show kinetic energy (left-hand panel) and pitch angle distributions (right-hand panel) for case B3Dp, with 20.000 protons with thermal speed v_th,p = √(2 k_B T ρ_0/(m_p p_0)) in the same setup as electrons in case A3De.The proton distributions are very similar to the 2.5D results. The maximum kinetic energy in the high energy peak for protons is γ - 1 ≲ 2 · 10^-2 or ℰ≲ 957 MeV and the high energy tail has an inverted power law index p < 0, similar to 2.5D case B2Dp. The pitch angle distributions are strongly peaked around α = 0. However, the asymmetry is less pronounced in 3D. This is attributed to randomization of pitch angles due to 3D effects in the nonlinear regime after t ≈ 8 t_S. The particles form two distinct populations. Protons and electrons moving inside the current channels develop a very high parallel velocity and a pitch angle very close to zero (positive for protons and negative for electrons). This is visible in the pitch angle distributions through the asymmetry around α = 0 at early times (before the linear growth phase of the tilt-kink instability at t ≈ 5 t_S). This population of particles is also observed in the kinetic energy distributions at γ_max - 1 ≈ 2 · 10^-2 for protons and γ_max - 1 ≈ 6 · 10^1 for electrons. During the nonlinear phase of the tilt-kink evolution (t ≳ 5 t_S) particles are expelled from the channels due to the kink and reach less high energies (γ_max - 1 ≈ 2 · 10^-3 for protons and γ_max - 1 ≈ 2 · 10^1 for electrons). In the pitch angle distributions this is observed through the more symmetric and less high peak around α = 0. This is in accordance with <cit.> concluding that the width of the peak in the pitch angle distribution depends on the (maximum) particle energy. For higher maximum energy the particle species show a narrower peak. This is explained by taking into account that particles accelerate mostly along the magnetic field and drift velocities are negligible before reconnection occurs. Proton pitch angle distributions are wider than electron pitch angle distributions because electrons reach a larger parallel velocity and therefore the asymmetry at α = 0 is more pronounced for protons.From the peaked pitch angle distributions in the right-hand panels of Figures <ref> and <ref> we conclude that parallel acceleration is dominant. However, which effect causes this is not clear by just comparing the particle drifts from equation (<ref>). To analyze which acceleration mechanism causes this peak, the electrons are split in high energy particles, with γ -1 ≥ 1 (ℰ≳ 0.5 MeV), and low energy particles, with γ - 1 < 1. There are four contributions to the parallel acceleration in the momentum equation (<ref>); The first two m_0γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂) and m_0γ𝐮_𝐄·((𝐮_𝐄·∇)𝐛̂) due to the change of direction of the magnetic field (curvature and polarization effects respectively). The third q E_ due to resistive electric field and the last one -μ_r 𝐛̂·∇[B(1-E^2_⊥/B^2)^1/2]/γ is the mirror deceleration effect. In Fig. <ref> the spatial distribution of high energy electrons (left-hand panel) and the low energy electrons (right-hand panel) is shown, coloured by (γ - 1) representing the particles kinetic energy (top panels) and by the curvature term γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂) in the guiding centre momentum equation (<ref>) (bottom panel). This curvature term is found to be dominant at all time due to the initially curved magnetic field and the kink instability adding further curvature (see also further on in Section <ref> in Fig.<ref>). In the left-hand channel (seen from the top), the particles move upwards, because of the direction of the magnetic field and hence the current density; In the right-hand channel, they move downwards. The particles are assigned a thermal speed every time they cross a z-boundary and therefore all fast particles are on the left of Fig. <ref> and all the slow particles are on the right, with the thermal particles at the foot points. The magnetic curvature is equally divided between left and right channels. However the fast particles in the left-hand panel of Fig <ref> traveled for a longer time in the current channels and obtained a larger acceleration and energy, mainly from the curvature. The curvature is stronger on the outside of the channels than on the inside and the fastest particles (indicated in red in the left-hand top panel) are therefore located at the foot points of the channels, on the outside, where also the current density is largest ().§.§ Individual particle dynamics To analyze how individual particles energize in the magnetic field of two interacting flux ropes we look at electron trajectories in run A3De. It is interesting to look at the energy evolution of a particle that is initially traveling in the current channel but is then expelled from the current channel. In Fig. <ref> we show the trajectory of such a particle from a side-view, coloured by its Lorentz factor. The electron cycles several times through the current channels until t = 9.177 t_S. From then onwards the trajectory is portrayed by thick squares coloured by the time t counted in t_S, until t = 9.192 t_S.This time interval corresponds to the red rectangle in the left-hand and the middle panel in Fig. <ref>, where the evolution of the Lorentz factor is shown. On the right-hand axis of the middle panel of Fig. <ref> the z-position of the particle is depicted with a magenta dashed line to indicate where the particle is on the trajectory in Fig. <ref>. The particle is first injected with a thermal speed at the bottom boundary at z = -3. The particle accelerates and travels upwards until it is mirrored by the magnetic field at t ≈ 9.179 t_S. Then it starts traveling downwards towards z = -3 and decelerates. At t ≈ 9.181 the particle reaches z = -3, it is injected at z = 3 with a thermal speed. It is then mirrored again towards z = 3. At t ≈ 9.181 it leaves at z = 3 and is injected with a thermal speed at z = -3. It accelerates until it reaches a Lorentz factor of γ≈ 2 along the same trajectory upwards through the current channel. At t = 9.1835 it is expelled from the kinking current channel. Outside the current channel the particle moves downwards, following the magnetic field, until it reaches z = -3 at t ≈ 9.187 t_S. It is injected again with a thermal speed at the top boundary at z = 3 and accelerates by following a field line outside the current channel. The particle Lorentz factor increases with time spent in the flux tube and the z-position increases (or decreases depending on the orientation) linearly in the current channels. In the right-hand panel of Fig. <ref> the evolution of the magnitude of the drift terms from the right-hand-side of equation (<ref>) is shown. The parallel velocity is dominant and quickly approaches the speed of light (black dashed line) when the particle accelerates.§.§ Effect of resistivity on particle distributions A resistivity of η_p = 10^-4 results in a magnetic Reynolds number of R_m =2 · 10^5, which is three order of magnitude lower than in the solar corona. To moderate the acceleration parallel to the magnetic field, and therewith the peaked pitch angle spectra and the hard, inverted energy spectra, a resistivity model is proposed that lowers the resistivity to realistic solar corona values. To restrict the contribution of a resistive electric field the resistivity in the GCA equations is set to η_p=10^-9 resulting in a Magnetic Reynolds number R_m =2 · 10^10 in the solar corona regime. The energy spectra for case C3De with η_p = 10^-9 and periodic (infinite) flux ropes are shown in the left-hand panel of Fig. <ref> and the pitch angle spectra in the right-hand panel. A high energy peak develops at t ≈ 5 t_S with maximum Lorentz factor γ -1≲ 10^3, resulting in ℰ_max≈ 500 MeV for electrons and a medium energy part with an inverted slope. The pitch angle is still strongly peaked at α=0 and the energy spectra still show an inverted slope. In case D3Dp 20.000 protons are evolved in similar settings as case C3De for electrons. We find a final kinetic energy distribution (see left-hand panel of Fig. <ref>) with maximum Lorentz factor γ -1 ≤ 2 · 10^-2 (ℰ≲ 957 MeV) and an inverted power law slope. The pitch angle spectrum (see right-hand panel of Fig. <ref>) is peaked around α = 0 but the peak is less dominant than in case B3Dp, where η_p = η_MHD = 10^-4 but the length of the flux ropes is limited to 6L.The high energy particles with γ -1 ≥ 1 (ℰ≳ 0.5 MeV) in case C3De are depicted in Fig. <ref>, coloured by the kinetic energy (γ-1) in the left-hand panel and by the magnitude of the curvature acceleration m_0γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂) in the right-hand panel. Unlike for case A3De (compare to Fig. <ref>), the fast particles are not just located at the foot points, but distributed over the whole area of the current channels, with the fastest particles in the middle (see the left-hand top panel). This area corresponds to the region with the strongest current density (and hence resistive electric field) and magnetic field curvature (see Fig. <ref> for the resistive electric field with magnetic field lines in this setup and <cit.> for more detail on MHD results). The slow particles, with γ - 1 < 1, are residing outside the current channels. Particles mostly accelerate in the regions with strongest curvature, at the outside of the kinked channels. This can also be seen from the distribution of the acceleration terms in the bottom panel of Fig. <ref>. For case C3De the distribution of curvature acceleration at t=9 t_S shows a tail consisting of the fast particles (in magenta with crosses as indicators). Compared to case A3De (Fig. <ref>) there is a clearer separation between slow and fast particles and the contribution of curvature acceleration to the fast particles. In case A3De (Fig. <ref>), all particles are accelerated by the magnetic curvature and the thermal bath prevents a tail to arise in the acceleration distribution. The limited length of the flux ropes and the realistic particle resistivity separately counteract the strong parallel acceleration and the hard spectra found by <cit.>; These two solutions are compared by quantifying the contributions of the four separate terms in the momentum equation (<ref>). We can determine whether the peaked distributions have a physical cause or are due to a too high MHD resistivity set for computational purpose. In Fig. <ref> the distribution of the four terms contributing to the particle momentum in the momentum equation (<ref>) are shown for case C3De and A3De at t=6, before the current channels start kinking and at t=9, in the nonlinear regime, respectively. We see that at t=6 the acceleration mechanisms show a similar distribution, compared between runs with thermal bath and η_p = 10^-4 (case A3De) and the runs with periodic boundary conditions and η_p = 10^-9 (case C3De), except the term due to the resistive, parallel electric field. For this term, q E_/m_0 = q η_p 𝐉·𝐛̂/m_0 we can see the direct effect of decreased resistivity. At t=9 however, in the bottom panel of Fig. <ref>, the distributions of the curvature acceleration contribution γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂) and the acceleration due to polarization γ𝐮_𝐄·((𝐮_𝐄·∇)𝐛̂), are shifted by two orders of magnitude for case C3De with η_p = 10^-9. For the case with thermal bath (A3De) they are not. The mirror effect -μ_r 𝐛̂·∇[B(1-E^2_⊥/B^2)^1/2]/γ is negligible in both cases C3De and A3De. The resistive acceleration qE_/m_0 has not changed with respect to the spectrum before the channels started kinking at t=6 t_S. Conclusively, the resistive electric field is not the main cause for the hard spectra obtained. The periodic boundary conditions and therewith the indefinite acceleration in the z-direction, in the current channels seem to be dominant and this effect is counteracted by thermal bath boundary conditions. However, even with a thermal bath applied, an inverted slope is observed in the energy spectra. §.§ Realistic conditions for solar flaresIn case E3De we combine a limited flux rope length and resistivity for realistic solar corona conditions. In a flux rope of length 6L and a resistivity of η_p = 10^-9 we evolve 20.000 electrons, with the aim to restrict a high energy peak and the accompanying inverted power law index for the high energy tail. The pitch angle distributions (right-hand panel of Fig. <ref>) are still dominated by α = 0, but there are more particles with a nonzero pitch angle, compared to cases A3De and C3De. The high energy peak observed in case A3De and C3De has now disappeared in the kinetic energy distributions (left-hand panel of Fig. <ref>). The slope of the high-energy tail at times t ≳ 7 t_S (coloured green to red), when the channels start kinking, is still inverted. However, at this time there are not many particles left in the thermal part of the distribution. The maximum kinetic energy is bounded by γ - 1 ≲ 5 (ℰ≲ 3 MeV), due to the thermal bath and the length of the channels. How many particles remain thermal is affected by the initial velocity distribution. The particles maximum energy is bounded by the time the particles spend in the flux rope, and hence by the length of the flux rope. §.§ Effect of initial velocity distribution on particle distributionsUp to now it is assumed that all particles, electrons and protons, have the typical MHD velocity v_th,p=√(2 k_B T ρ_0/(m_p p_0)) as thermal speed. Assuming energy equipartition between electrons and protons results in a larger thermal speed for electrons v_th,e= √(m_p/m_e)× v_th,p. In case G3De we explore the effect of a larger initial electron velocity with flux rope length 6L and particle resistivity set as η_p = 10^-9. Electrons are initialized from a Maxwellian with thermal speed v_th,e = √(m_p/m_e)× v_th,p in accordance with energy equipartition and they are uniformly distributed over the spatial domain -3L ≤ x ≤ 3L; -3L ≤ y ≤ 3L; -3L ≤ z ≤ 3L. Few particles accelerate inside the current channels at early times t < 6 t_S in the linear phase (see Fig. <ref>). The maximum electron energy is limited by the length of the current channels and the thermal bath to γ - 1 ≲ 7 or ℰ≲ 4.1 MeV, which is in the range of observed electron energies coming from solar flares. New thermal particles are injected from the thermal bath for every destroyed high energy particle leaving a periodic boundary, maintaining a thermal distribution dominant in number of particles. The high energy tail that develops during the exponential growth phase of the tilt-kink instability after t ≥ 6 t_S has a power law distribution with spectral index p > 1 (indicated by the dotted black line in the left-hand panel of Fig. <ref>). The high energy particles are not dominant in number of particles, nor in energy content due to decreased particle resistivity and the thermal bath. The pitch angle distributions in the right-hand panel of Fig. <ref> are initially (nearly) uniform and the peak due to particles accelerated parallel to the magnetic field at α = 0 develops only after t ≥ 6 t_S. The peak at α = 0 is less dominant than in all other electron cases. In case F3Dp we explore the same configuration for protons, initiated uniformly from a Maxwellian with v_th,p. For protons assuming energy equipartition or a generic fluid velocity as thermal speed results in the same initial energy distribution. The results are similar to electron case G3De with the difference that the maximum energy is limited to γ -1 ≲ 3 · 10^-2 or ℰ≲ 941 MeV (see the left-hand panel of Fig. <ref>) due to the mass difference and consequently the lower thermal speed. Few particles accelerate inside the current channels at early times t < 6 t_S in the linear phase. In the nonlinear phase from t > 6 t_S onwards, a high energy tail develops with a power law distribution with index p > 1 (indicated by the dotted black line in the left-hand panel of Fig. <ref>). Because of the lower maximum energy reached, less protons leave the domain through the open x,y-boundaries, compared to electrons in case G3De and the thermal distribution remains dominant at all times. The pitch angle distribution (see the right-hand panel of Fig. <ref>) remains (nearly) uniform till t = 6 t_S and afterward it is peaked around α = 0 but at least an order of magnitude smaller than in all other proton cases. In Fig. <ref> contributions of the four separate terms in the momentum equation (<ref>) are quantified. Comparing to Fig. <ref> for cases A3De and C3De the effect of a finite flux rope length and low resistivity combined is shown for the four acceleration mechanisms. We see that at t=9 the magnetic curvature is the dominant acceleration mechanism. There is no peak in the distribution. The magnetic curvature acceleration is three orders of magnitude lower than in case C3De (with η_p = 10^-9 and periodic flux ropes) and one order of magnitude lower than in case A3De (with finite flux rope length and η_p = 10^-4). The other three acceleration mechanisms are at least three orders of magnitude smaller than the curvature acceleration. The curvature acceleration term in the GCA momentum equation (<ref>) is proportional to the parallel velocity of the particle. A particle following a (curved) field line accelerates parallel to the field line. Even in the initially straight current channel the field lines are curved due to the initial magnetic field distribution in equation (<ref>). Conclusively, a realistic magnetic Reynolds number and finite flux rope length result in a realistic maximum energy reached, power law index of the high energy tail of the kinetic energy distribution and the shape of the pitch angle distributions. Case G3De for electrons and case F3Dp for protons give the most realistic results for particle acceleration due to interacting flux ropes in the solar corona. In the next section we explore the effect of the finite length of a flux rope for the settings of case G3De. §.§ Effect of the length of the flux ropes on particle distributionsIn case of periodic boundary conditions particles are either expelled from the flux rope or they travel through the flux rope until the simulation ends. When they are expelled from the flux rope they thermalize in the ambient medium until they leave the domain or they are caught into the flux ropes again. The particles causing the high energy peak are in the current channels for a long time and typically cycle through the current channel many times. These fast electrons traveling in a flux rope typically accelerate up to γ_max≈ 1 · 10^3 if the flux rope is infinitely long (i.e. if no thermal bath is applied as in case C3De for electrons). During the simulation an electron with a parallel velocity v_≈ c travels a maximum distance of Δ l ≈ c × 10 t_S ≈ 2.6 · 10^13 cm, or over 4000 cycles through the flux ropes, if it is not expelled at some point. This is equal to Δ l≈ 3.6 · 10^2 solar radii in 10 soundcrossing times t_S. The medium energy tail in case C3De (see Fig. <ref>) consists of particles with an average Lorentz factor of γ - 1 ≈ 10^-2. This is equal to a parallel velocity of v_≈ 0.14 c. Even these particles can travel a maximum distance of Δ l ≈ 0.14c × 10 t_S ≈ 3.4 · 10^12 cm, or 52 solar radii through the flux ropes. Thermal bath boundary conditions limit this to one cycle, or Δ l = 6L ≈ 8.6 · 10^-2 solar radii. To analyze how the length of the flux rope affects the energy distribution, case H3De has settings similar to case G3De, with the total length of the current channels 12L (or 1.7 · 10^-1 solar radii) instead of 6L. All particles are initially in the current channel area. To obtain equal accuracy the resolution is also doubled in the z-direction. The shape of the distributions for case H3De in Fig. <ref> is similar to case G3De (Fig. <ref>). The maximum energy reached by the particles in the flux ropes with length 12L is approximately doubled compared to case G3De (γ_max - 1 ≲ 2 · 10^1 or ℰ≲ 11 MeV). Initially few particles accelerate in the flux tubes and the energy distribution in the left-hand panel remains largely Maxwellian. In the nonlinear phase, after t = 6 t_S a non-thermal power-law tail forms due to the tilt-kink instability and subsequent reconnection. The pitch angle distribution in the right-hand panel remains nearly flat until t = 6 t_S, after which the peak around α = 0 becomes dominant due to particles accelerating along the magnetic field. § CONCLUSIONS We find that reconnection in the low plasma-β regime drives efficient energy conversion from magnetic energy to kinetic energy and that electrons and protons are efficiently accelerated to non-thermal energy distributions. We observe two populations of high energy particles; A high energy peak of particles trapped inside the current channels at early times, where they accelerate efficiently along the magnetic field. Electrons reach maximum energies between ∼10 and ∼500 MeV. The maximum energy depends strongly on initial velocity distribution, plasma resistivity and the length of the flux tubes. A second population consists of electrons accelerating in the reconnection zones at late times in the nonlinear phase. These electrons generate a high energy tail in between the peak and the Maxwellian part of the distribution due to particles accelerating, in quite a narrow range between ∼0.5 MeV and ∼10 MeV. Protons reach maximum energy of approximately 1000 MeV in all cases.We proposed two solutions to limit indefinite particle acceleration due to infinitely long flux ropes as found in <cit.> for 2.5D simulations of low plasma-β environments and extended this setup to full 3D simulations for electrons and protons. One solution is to apply a thermal bath along the length-direction of the flux ropes at the periodic boundaries (case A3De for electrons and B3Dp for protons). This solution effectively limits the length of the flux ropes to realistic scales and destroys fast particles leaving the domain. For every particle ending up in the thermal bath, a new thermal particle is injected at the opposite periodic boundary. This assures to keep a steady total number of particles. The particles maximum kinetic energy grows linearly with the length of the flux ropes and the time spent in the flux rope. In a flux rope of length 6L the maximum particle energy is limited to γ - 1 ≲ 100, or ℰ≲ 50 MeV for electrons and γ - 1 ≲ 2 · 10^-2, or ℰ≲ 960 MeV for protons. The majority of particles reach a non-thermal energy before they reach the end of the flux rope. A high energy tail forms in all cases with limited flux rope length. The shape of the high energy tail and maximum energy depend on the initial conditions and resistivity set for the simulation.The second solution counteracts acceleration in the direction of the resistive electric field, parallel to the magnetic field, as proposed by <cit.>. A particle resistivity is applied that is typically lower than the MHD resistivity and results in realistic magnetic Reynolds number for solar corona conditions (case C3De for electrons and D3Dp for protons). This particle resistivity effectively lowers the electric field that is felt by the particles, but does not affect the MHD evolution and the formation of strong and thin current sheets. It limits the number of particles that accelerate to high energy. Despite having less particles in the high energy peak, the maximum particle energy is not limited due to the infinite length of the (periodic) flux ropes. Electrons accelerate to γ - 1 ≲ 1000 or ℰ≲ 500 MeV and protons to γ - 1 ≲ 2 · 10^-2, or ℰ≲ 960 MeV. A high energy distribution forms in all cases and its shape and maximum energy depends on the initial conditions and the resistivity set.Despite limited maximum particle energy an inverted power law index (p < 0) is found for the non-thermal distribution in the cases with low resistivity or limited flux rope length. This is due to the high energy peak developing from particles accelerating in the current channels at early times. These high energy particles dominate the energy census at late times. To limit the number of particles in the high energy tail we combined limiting the flux rope length and the resistivity to realistic values (cases E3De, G3De and H3De for electrons and F3Dp for protons). This results in setups with realistic magnetic Reynolds number and a flux rope length limited to a fraction of a solar radius. The maximum energy grows with the length of the flux ropes and for electrons ℰ≲ 4 MeV for length 60 Mm and ℰ≲ 11 MeV for length 120 Mm and for protons ℰ≲ 1 GeV. The shape of the slope of the high energy tail formed depends on the initial velocity distribution. Assuming energy equipartition between electrons and protons we find high energy power-law distributions f(ℰ) ∼ (ℰ)^-p withp ≥ 1, as opposed to the inverted spectra found if the flux rope length is not limited and the resistivity is not lowered to realistic solar corona values. The high energy particles are not dominant in number, nor in energy, such that the test particle approximation is valid. In all these cases a part of the particle ensemble and its energy content is in the non-thermal distribution. These findings are in good agreement with the 2D kinetic PIC results of <cit.> for particle acceleration in force-free current sheets in low-β electron-proton plasmas. The non-thermal particle distributions found can explain the efficient electron acceleration in low plasma-β environments such as solar flares.Without applying the two solutions proposed the whole ensemble of particles contains so much energy that kinetic feedback of the particles to the electromagnetic fields cannot be neglected. The assumption that the energy content of the particles is much lower than that of the fluid is invalid. A kinetic description allows particles to lose their energy to the fields through kinetic instabilities and particle-field interaction that is ignored in the test particle approximation. However at solar length scales, considering that reconnection occurs globally in the simulation box, PIC simulations are extremely demanding numerically. In cases with realistic magnetic Reynolds numbers and solar length scales for the flux ropes the number of accelerated particles is small compared to the total number of particles. The fields induced by these particles should not substantially affect the MHD evolution and reconnection. We applied a guiding centre approximation, ignoring the gyration of particles and reducing computing time. Via the guiding centre approximation we demonstrated that magnetic curvature is the leading acceleration mechanism in all cases. The magnetic field inside the current channels is curved initially and the kink instability introduces more curvature, making the curvature acceleration dominant at all times. The curvature acceleration is proportional to the velocity of a particle parallel to the magnetic field, enhancing the effect of the curvature. Particles are mainly accelerated parallel to the magnetic field and hence the resistive electric field, resulting in pitch angle distributions dominated by a peak at α = 0 in all cases. The width and height of the pitch angle peak depends on the maximum parallel particle velocity and hence on particle energy. The α = 0 peak shows an electron/proton asymmetry, previously observed by <cit.>, caused by protons predominantly moving with positive parallel velocity and electrons mostly with negative parallel velocity along the magnetic field.The guiding centre approximation is valid since the gyration radius remains much smaller than typical cell size in all runs. For protons, the gyration might not be negligible due to a larger mass and hence a larger gyroradius. To monitor the validity of our approach, results are compared to a run where proton gyration is not neglected and the full equations of motion (<ref>) are solved in Appendix <ref> (case K3Dp). Applying the GCA has little to no effect on the energy distribution, compared to a case where particle gyration is fully resolved. The GCA is accurate in magnetically dominated, Newtonian plasmas under solar corona conditions. Results are also confirmed for both an ensemble of 200.000 particles (Appendix <ref>, case I3De) and for an initially uniform spatial distribution (Appendix <ref>, case J3De) to show that obtained statistics are accurate. Since kinetic feedback of the particles on the fields is neglected the accumulated current from energetic particles is unimportant and it is not necessary to increase the total number of particles. § ACKNOWLEDGEMENTSThis research was supported by projects GOA/2015-014 (2014-2018 KU Leuven) and the Interuniversity Attraction Poles Programme by the Belgian Science Policy Office (IAP P7/08 CHARM). OP is supported by the ERC synergy grant `BlackHoleCam: Imaging the Event Horizon of Black Holes' (Grant No. 610058). The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government - department EWI. BR likes to thank Lorenzo Sironi, Fabio Bacchini, Norbert Magyar, Jannis Teunissen, Kirit Makwana, Matthieu Leroy and Dimitris Millas for fruitful discussions and comments.mnras§ VALIDATION TESTS §.§ Number of particlesTo assure that the results are statistically accurate, a setup similar to case A3De is evolved with an initial distribution of 200.000 electrons for case I3De. Minor differences are visible in Fig. <ref>, mainly due to smoother distributions. The energy range and global features of both the kinetic energy distributions and the pitch angle distribution are in agreement with results for 20.000 electrons. Evolving more particles has no effect on the physical results, other than improving statistics, since there is no feedback of the particles on the fields, nor any particle-particle interaction. §.§ Spatial distributionBased on the findings of <cit.> for 2.5D configurations, a fraction of 0.99 of the particles is initially distributed in the area of the current channels -1L ≤ x ≤ 1L; -2L ≤ y ≤ 2L; -3L ≤ z ≤ 3L. In this way using 20.000 particles results in accurate statistics, similar to a run with 200.000 particles, and computational resources are mainly used on particles that are likely to accelerate. In 3D configurations a larger area of the box is filled with reconnecting magnetic field due to the kink instability. To monitor whether it is accurate to distribute most particles in the region of the current channels in 3D configurations a setup similar to case E3De with thermal bath and η_p=10^-9 is ran for case J3De, now with an initially uniform spatial distribution. Electrons are randomly and uniformly initialized in the whole box -3L ≤ x ≤ 3L; -3L ≤ y ≤ 3L; -3L ≤ z ≤ 3L. The resulting energy distributions in Fig. <ref> show the same shape and trend and the maximum energy γ-1 ≲ 5 is equal to the maximum energy obtained in case E3De in Fig. <ref>. The main difference is that there are less particles in the high energy tail. However, there are not more particles remaining in the thermal distribution; The particles in he ambient medium, outside the current channels, leave the domain guided by the field lines on the timescales considered. Also the pitch angle is still peaked around α = 0, but the peak is an order of magnitude lower than in case E3De. Distributing particles in the area of the current channels rather than in the ambient medium improves statistics on high-energy particles, but does not alter the physical features of the obtained spectra. This is in accordance with the 2.5D results of <cit.>. §.§ Gyration Protons have a larger gyroradius than electrons due to their mass and therefore the GCA is most likely to break down in proton simulations. To monitor the validity of the GCA, the full particle equations of motion (<ref>) including gyromotion are evolved with a Boris scheme for protons in the same MHD background. In Fig. <ref> we show the kinetic energy distribution for 20000 protons in a setup where particle gyration is resolved. Case D3Dp is chosen for comparison to confirm that the high energy peak found is not a numerical artifact. The energy distribution shows a very similar shape. A peak develops due to particles accelerating in the current channels and the maximum energy reached is limited by the length of the current channels and the thermal bath boundary conditions, such that the maximum energy is similar to case D3Dp, with γ - 1 ≤ 4 · 10^-2 (ℰ≲ 976 MeV). The high energy tail shows an inverted power law index, developing at similar time as in case D3Dp. Besides minor differences in the thermal part of the spectrum, due to the random assignment of a thermal velocity component v_x, v_y and v_z rather than v_ and v_⊥ in the GCA setups, the global aspects are similar to the GCA results in Fig <ref>. For electrons the differences are expected to be even smaller, due to a smaller mass and hence completely negligible gyroradius.§ ERRATUM: RECONNECTION AND PARTICLE ACCELERATION IN INTERACTING FLUX ROPES – II. 3D EFFECTS ON TEST PARTICLES IN MAGNETICALLY DOMINATED PLASMAS The paper "Reconnection and particle acceleration in interacting flux ropes – II. 3D effects on test particles in magnetically dominated plasmas" was published in MNRAS, Volume 471, Issue 3 (2017). After the publication of the article an error was found in a postprocessing script affecting several figures. The numerical data and the results of the calculations are correct and the conclusions drawn are too. Nevertheless, the magnitude of the curvature acceleration term (|γ𝐮_𝐄·(v_(𝐛̂·∇)𝐛̂)|) and the magnitude of the polarization acceleration term (|m_0γ𝐮_𝐄·((𝐮_𝐄·∇)𝐛̂)|), were taken too large by a factor c (speed of light) while manufacturing Figures 7, 12, 13 and 17 in the original manuscript. This affects the scale of the bottom panels in Figure 7 and the right-hand panel in Figure 12 in the original manuscript, where the colour bar ticks should be divided by the speed of light c in CGS units. The conclusions drawn from these figures remain fully valid since only the scale has changed.Removing the erroneous factor c in the curvature acceleration term and the polarization acceleration term in the postprocessing script also affects Figures 13 and 17 where the number distributions of the curvature acceleration and the polarization acceleration (magenta and red curves in the original Figures respectively) are shifted to the right by a factor c. The conclusion that curvature acceleration is dominant at all times is therefore incorrect and should be replaced by the conclusion that the curvature acceleration is the second most dominant mechanism of particle acceleration in cases A3De and C3De (Fig. <ref>) and that it grows strongly in the nonlinear regime after t ≈ 6t_S in case C3De with a lowered resistivity. In case G3De (Fig. <ref>) the parallel acceleration is limited by both a lowered resistivity and a limited length of the flux rope, resulting in a less dominant curvature acceleration and a more dominant mirror mechanism (green curve). The resistive electric field acceleration is dominant as was already remarked in the conclusions of the original manuscript, with the remark that this term depends linearly on the resistivity that is arbitrarily set. Therefore the approach to lower the particle resistivity to a value realistic for the solar corona is still justified by the correction made here. The general conclusions in the final section of the original manuscript are unaffected by this correction. | http://arxiv.org/abs/1707.08920v2 | {
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"title": "Reconnection and particle acceleration in interacting flux ropes -- II. 3D effects on test particles in magnetically dominated plasmas"
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=1 ./graphs1/ =10000 propProposition 𝔐 Φ øω I ϵ θ ∂ łλ ØΩ ŁΛ Σ σ 1 2 α β̱ δ̣ γ √(λ)ß𝗌 𝗊 | http://arxiv.org/abs/1707.08584v3 | {
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[ [ December 30, 2023 ===================== We present novel retrospective change point detection approach based on optimal transport and geometric discrepancy.The method does not require any parametric assumptions about distributions separatedby change points. It can be used both for single and multiple change point detection and estimation, while the number of change points iseither known or unknown. This result is achieved by construction of a certain sliding window statisticfrom which change points can be derived with elementary convex geometry ina specific Hilbert space. The work is illustrated with computational examples, both artificially constructed and based on actual data.Introduction Change point problem was firstly mentioned by Stewhart <cit.> as a problem of industrial quality control in the year 1928.This problem was solved by Girshick and Rubin <cit.> in 1953.The optimal solutions for parametriconline formulation of the problem were provided by Shiryaev and Pollak <cit.>,<cit.>. Asymptotically optimal posteriori change point detection method was developed by Borovkov <cit.>.This method requires knowledge of parametric likelihood functions, however itcan be translated to nonparametric case with empiricallikelihood <cit.>. The fundamental results in nonparametric study of change-point problem are due Brodsky and Darkhovsky <cit.>, and Horvath et al. <cit.>.These methods are generally tailoredfor one-dimensional data.Nonparametric change point inference with multidimensional data isstill an open problem.New methods are still being developed usually with applications of ideas from other scopes to change point data.For example, <cit.>, where divergence from cluster network analysis is used in order to estimate both the number and the locations of change points or <cit.>, where multivariate version of Wilcoxon rank statistic is computed. This work is focused on nonparametric change point detection in multivariate data. This problem has large scope of applications includingfinance <cit.>,genomics <cit.> and signal processing <cit.>.The problem of optimal transportation were introduced by Monge <cit.> in the year 1781. The modern statement of the problem is due to Kantorovich as it was stated in the seminal paper in 1941 <cit.>. We recommend <cit.> and <cit.> as modern references to the subject Vector rank and quantile functions were introduced by Serfiling <cit.> as part Depth-Outlyingness-Quantile-Rank (DOQR) paradigm. Monge-kantorovich vector ranks were introduced by Chernozhukov et al. <cit.>.Study of discrepancy were started by H. Weyl <cit.> in 1916with number theory as intended area of application,more computational aspects of discrepancywere pioneered by Niederreiter <cit.> (quasi- and pseudo- random number generation) and Hlawka <cit.> (numeric integration).We say that discrepancy is geometric when we talk about continuous distributions with fixed well-identified support, this is the opposition to the case of combinatorial discrepancy which study nonuniformity of finite distributions. The main contribution to the theory of quadratic generalized geometric discrepancy is by Hickernel <cit.>. This is a work of a highly synthetic nature.This Synthetic nature is coming from the combinationof vector ranks and geometric discrepancy,optimal transportand change point problems.All this concepts are rarely to never seen together in scientific publications,however in this paper they are intertwined in a singular computational methodology.But why these results come to existence only now?The answer to this questions in our case is rooted into the fact that computing discrepancy used to be computationally challenging task for non-uniform probability distributions.However, in the recent publication <cit.> by Chernozhukov,Carlie, Galichon and Halin the Monge-Kantorovich vector rank functions were introduced.Vector rank function can be understood in the context of this work as homeomorphic extension of classical one-dimensional cumulative distribution functions to multidimensional spaces.The idea of using low-discrepancy sequences as a tool for the estimation of the vector rank functions was already present in the <cit.>.As the name suggests discrepancy is the natural measure of consistency of the sequence to be low-discrepancy sequence in the same way as the order statistic is the natural measure of the sequence to be ordered.Thus, asclassical one-dimensional ranks and order statistic is used in the construction of the classical Kolmogorov-Smirnov and Crámer-Von Mizes tests of goodness-of-fit,the vector rank function and discrepancy can be used in order to construct natural extension of this tests in the multidimensional context.Thus, the main objective of the work is the exploration of vector rank discrepancy asmethodology for nonparametric high-dimensional statistic. We select change point problem as the test subject for this inquiry.The strict measure theoretic approach mentioned above is strengthened in this work as we speak about probability measuresinstead of probability distributions and densities whenever is possible which contradicts current trends in the statistical science.This can be justified by convenience of understandingmeasures as points in certain multidimensional spaces bearing intrinsic geometric structure.On the other hand, we view all mathematical constructions in this work as theoretical prototypes of certain concrete algorithms ad data structures which can beimplemented in order to solve practical tasks.Our approach to the problem is highly inspired by <cit.> and we use convex geometric intuition whenever possible.acknowledgement Underlying research was conducted during author's work for master's degree in"Mathematical methods of Optimization and Stochastics" program in NRU Higher School of Economics, Moscow, Russia.Author wants to thank head of the program Vladimir Spokoiny, his scientific advisor Andrei Sobolevski for suggesting the research topic and support, Alexey Naumov and Maxim Panov for the discussion in IITP RAS;Yuri Yanovich,Alexandra Suvorikova , Elena Chernousova and Alexey Kroshnin for fruitful discussion at the IUM seminar. NotationThis is notation for the use in the sequel.^d is a d-dimensional euclidean space with d < ∞ . I = [0,1] is a unit interval and I^d is a unit hypercube in ^d.In line with Kolmogorov's axiomatics triple (Ø,,) is a probability space, and every observation as a random variable is assumed to bea Borel measurable map from Ø to ^d. For brevity Ω denotes the whole probability measure structure (Ø, , ).Every random variable ξ : Ø→^d defines the probability measure P = ξ_# that isreferred to as the probability law of ξ, where ξ_# denotes a pushforward of the measureby ξ . That is, for every Borel set B ⊂^d it holdsthat P(B) = (ξ^-1(B)). In this case the notation ξ∼ P is used. In case ξ is a random process over domainnotation ξ∼ Pmeans that P defines the hierarchicalprobability distribution system of ξ as a whole.In this case for all t,s ∈ the measure P_t is the distribution of a single observation ξ_t and P_t,sis the joint distribution of ξ_t and ξ_s i.e. ξ_t ∼ P_t, (ξ_t, ξ_s) ∼ P_t,s.ξ P means that ξ is independently identically distributed sample(i.i.d) with probability law P.A set of all probability laws with finite first and second moments is denoted by ^d. A subset of ^dof probability measures absolutely continuous with respect to Lebesgue measure λ is denoted by ().is the uniform distribution over I^d. It is obvious that ∈^d.In the sequel I^n is assumed to be implicitly equipped with 𝒰^n for each n ∈, which makes it into a probability space,andI^0 = {∅} is equipped with the counting measure.If ξ_n is ansequence of random elements in some metric space with distance metric ρ, when it is said that ξ converges to x in probability if for each ϵ∈_++ it holds lim_n →∞( ρ(ξ_n, x) >ϵ) = 0, which is denoted asξ_n →_ x. In caseρ(ξ_n,x) is not a measurable function for all n the convergence in outer probability may be introduced and denoted by ξ→_∙ x. Having A^ϵ_n = {ω∈Ω : ρ(ξ_n(ω),x) > ϵ} the convergence in outer probability is equivalent to lim_n →∞inf{(B) | B ∈ : A_n^ϵ⊂ B } = 0 for all ϵ > 0. If ξ is a discrete time ergodic process with convergence in probability, 1/n∑^n_i = 1 f(ξ_i) →_∫_^d fdP̅for every bounded Lipschitz function f, we say that P̅ is the weak limit distribution ofξ.Problem Formulation In this section a brief review for common variations of change point problem is provided.Ordered Setis associated with time. It is possible to discuss change point problem with continuous time <cit.>. However, this work deals only with case of discrete time. It is also possible to consider a change point detection for random fields<cit.>.Two main types of change point problem are offline and online detection. In offline detectionis assumed to be a finite set of size || = T.Without loss of generality, it is assumed that = {1, …, T }. Initial observations X are treated as a time series indexed by .Firstly,we discuss the case of a single change point.The change point of X is an unknown moment of time ∈ such that (X_t)_ t = 1 ^∼ P^1 and (X_t )^T_t =+ 1∼ P^2. There are two possibilities. Firstly, both subsamples may be i.i.d with(X_t)^_t = 1P̅^1 and (X_t)^T_t =+ 1P̅^2.It is also possible to consider unknowns P̅^1 andP̅^2 to be ergodic weak limit distributions of(X_t)^_t = 1 and (X_t)^T_t =+ 1respectively. Then offline single change point detection problem is a hypothesis test for H_0 : P̅_1 = P̅_2 versus H_1 : P̅_1 ≠P̅_2 for every possible estimateofhaving 1 < < T.Retrieving valid value ofis a related problem which will be referred as a change point estimation.In case of multiple change pointsexistence of up to N change moments_1< … < _n< …_N ∈ is assumed. In simpler formulation of problem value of N is assumed to be known, while in more complicated one N is also an object of estimation. As the previous variationthe problem is a hypothesis test about distribution of samples(X_t)_t = _n - 1 + 1^_n∼P̅_n where _0 = 1 and _N + 1 = T for simplicity. The problem splits into multiple hypothesis tests for H^n_0 : P̅_n = P̅_n + 1 versus H^n_1 : P̅_n ≠P̅_n + 1 for all possible estimates ofin case of known N. For unknown N the problem is structured around testing against H_0 :P̅_1 ≠P̅_2≠…≠P̅_N̂ for all estimates ( N̂, ) of (N,) where possible values of N̂ areconstricted to some meaningful finite set. For online change point problem≅ as an ordered set, and X can be thought as infinite times series unfolding inreal time. In this case the goal is to identify change point as soon as possible which means using minimal amount of observations X_t with t ≥. Branding every moment of time as a change point will achieve zero delay. However, this detection procedure is unacceptable as it achieves maximal false alarm rate.This means that change-point detection algorithm needs to be tuned in for minimal delay with fixed false alarm rate.In a more theoretical frameworka single change point _0 can be considered, which implies that infinite set { X_t : t < 0 } of observations with prior distribution can be used for detection.However, in more practical situation the stream of chanege ponts : → is considered, with only finite sample { X_t : _n - 1 < t ≤_n } available for detection of _n.Generally, methods of change point detection are focused on detection of change in such properties of statistical distributions as mean, variance and spectral density. The method presented in this work is focused on change in ergodicweak limit distribution of data.Which means that change in mean and variance must be detectable, although,change in spectral density is not.Moreover, changes restricted to such properties of distributionas median or higher moments also must be traceable. Another restriction of proposed solution is a demand for all measuresP̅_n to belong to ^d.Construction of proposed change point detection starts withoffline single detection problem.We restrict our attention to models with i.i.d distributed observations along one side of change point.In order to give meaningful characteristic to point estimates of change point this problem can be reformulated as certain probabilistic game against Nature. Twosample spaces (Ø^1, ^1, ℙ^1) and (Ø^2, ^2, ℙ^2) with corresponding families of random variables ξ^1 and ξ^2 taking values in ^d are assumed to exist. We assume that both families are intrinsically i.i.d which means that ξ^1P_1 and ξ^2P_2 and the same holds for ξ^2. This means that both sample probability spaces can be decomposed as Ω^1 = ∏^∞_t = 1Ω^1_t andΩ^2 = ∏^∞_t = 1Ω^2_t such that for each ω∈Ø^1the random variable ξ_t^1(ω)depends only onω^t.The common sample space is constructed by the ruleΩ_t = Ω^1_t ⊔Ω^2_t= ({1}×Ω^1_t ) ∪( { 2 }×Ω^2_t ), Ω = ∏^∞_t = 1Ω_t with σ-algebras generated by a set{{1 }× A : A ∈_1}∪{{2 }× B : B ∈_2} . Note, that the only possible probability distribution on the set {1,2 } isa Bernoulli distribution Bern(r) with a parameter r ∈ [0,1]. So probability measure on Ω can be defined by ℙ(S) = rℙ_1(ι^-1_1 S) + (1 -r)ℙ_2(ι^-1_2 S), where ι_j is a natural injection defined by ω↦ (j,ω).Nature defines value of r and distribution lawsP_1 and P_2. Random variables (b_t,X_t) are constructed byletting (b_t,X_t)(j, ω) = ( j, ξ^j_t(ω) ). Then, nature generates sample of T observations of (b,X), sorts it by b, then erases values of b which retains observable sample of X. Observer either claims that there were no change point which leads to a victory in caseP_1 = P_2 or equivalently r is too close to either 0 or 1. Otherwise, the observer gives point-estimate r̂ of r which can be converted to a classical change-point estimate = r̂ T. This construction also can be understood as a Bayesian assumption ∼B(T,r), where B(T,r) stands for binomial distribution.However, the change point problem is a single element sample problem, which means that only minimal properties of the parameter can be inferred from the data. This does not change the problem except for treatingas an estimate of the mean. Obviously,theobserver chooses strategy which maximizes chances of victory. This can be interpreted in terms of loses L, which are equal to L(0) = min{1 - r, r }= L(1) in case observer claims no change point and L(r̂) =| r̂ - r| otherwise. LetL(r̂) = min1 - r̂, r̂in case the observer gives estimate when no real change point exists.If one preserves r while increasing T the empirical distribution of X will converge to a measure μ_r = r P̅_1 + (1 - r) P̅_2, the convex combination of measures. It possible to think of μ_r as contained inside one-dimensional interval [P̅_1, P̅_2] embedded into the infinite-dimensional convex space of probability measures. In order to construct such measures from random variables we will use random mixing map I^s_Z : Y ↦ b_sY + (1 - b_s)Z in the sequel. Here b_s ∼Bern(s) and s ∈ [0,1].In order to propose the solution strategy forthe change point problem weintroduce vector rank functionR : ^d →^d such that R_#μ_r =, a known reference distribution of simple structure. For our applicationis a uniform distribution over a unit hypercube I^d. It was shown in <cit.> and <cit.> that R̂, an estimate of R , can be recovered from data without any parametric assumptions.If data is i.i.d distributed with laws P_1 and P_2 on each side of change point, whenfigure 1 willexhibit relations between probability distribution laws discussed so far. The diagram shows that application of vector rank function can be thought as a probabilist's "change of basis". The pragmatic value for change point problem is in elimination of unknown distribution μ_r from the model.It is obvious that the diagram depicted at fig. 1 commute. As the pushforward acts as a linear map of signed measures defined over two different measurable spaces. If α and β are measures, x, y ∈, B is a measurable set and f is a measurable function, then f_#(xα + yβ)(B) = x α( f^-1(B))+ y β(f^-1(B)) = x f_#α (B)+ y f_#β(B). Therefore, by linearity of the pushforward = R_#μ_r = R_# ( r P_1 + (1 -r) P_2) = r R_# P_1 + (1 - r)R_# P_2 . Thus, the diagram is correct.The real result of change-point estimation will depend on measurable difference between P_1 and P_2. We introduce notion of Kullback-Leibler divergence in order to quantify this difference. Kullback-Leibler divergence between measures P_1 and P_2 admitting densities p_1 and andp_2 respectively isKL(P_1 , P_2) = ∫_Ø_1logp_1/p_2 dP_1 Notethat while P_1,P_2∈^d they admit densities. Now we formulate a sufficient condition on R that ensures that our transformations preserve divergence between distributions.If R is invertible almost everywhere P_1, when KL(R_#P_1,R_#P_2) = KL( P_1,P_2)Let X be a random variable such thatX ∼ P_1. Then, R(X) ∼ R_# P_1 and by invertibilitya.e. of Rdistribution R_# P_1 has density p_1 ∘ R^-1 a.e. ; similar fact is also true for P_2 . Then, KL( R_# P_1, R_# P_2 ) = log p_1(R^-1RX) / p_2 ( R^-1RX )= log p_1 (X) / p_2( X) =KL( P_1,P_2) Formulation of the method In this chapter our approach to construction of the vector rank function's estimates R̂_n is presented. The main goal of This approach isthe avoidance of approximation of the entire function R.Note, that even the estimation that the aim of this task is actually not a precise estimation of values R(X_i) for each observation X_i but a construction ofan estimate that preserves the relative geometry of a sample X along its change points.The work <cit.>presents continuous, semidiscrete,and discrete methods of estimation of R.The Implementation of change point detection discussed in this section is based solely on discrete approach. The reason behind this choice is computational simplicity.Vector rank function in general can be understood as (P, mathbbU)-almost surely homeomorphismsbetween supports of sample probability distribution P and reference probability law of choice 𝕌 (R is defined almost everywhere P and continuous with respect to subset topology, and admits similar continuous inverse existing almost everywhere 𝕌) ,such thatR_# P = 𝕌 and both depth median and symmetries are preserved.This symmetriesand the concept of depth median need to be additionally defined,which goes beyond of the scope of this paper. In our application uniform measure over unit hypercube I^d is selected as the reference U. In case d = 1 the cumulutive distribution function (cdf) of P is also the 'vector' rank function of the probability distribution of P. The vector rank function is not unique in general. We use Monge-Kantarovich vector rank developed in <cit.>, which is defined as optimal transport betweenP and UR = _R : R_#P = ∫_^d d^2(R(x),x)dP(x).Vector rank role of the optimal transport map R can be intuitively justified by the equivalence of the above optimization problem to the maximization of the integral∫_^d⟨ R(x) - m() , x - m(P)⟩ dP(x),where m() and m(P) stands for depth medians of distributionsand P respectively.Thus, the optimal transport preserves geometric properties of distribution P which canbe expressed by inner products of points relative to its center of symmetry. This is what is understood as the relative geometry of the data,and what we try to preserve duringvector rank estimation. In this work we are focused on the discrete estimation of vector rank function. Let u be an equidistributed sequence of points in I^d. That is, for every Lipschitz continuous functionf defined on I^d it holdslim_n →∞1/n∑^n_i = 1 f(u_i) = ∫_I^d f(x)dx.Then, for T observations define Y_i = R̂_T(X_i) = u_σ^*(i), where σ^* = _σ∈ S^T∑^T_i = 1X_i - u_σ(i)_2^2.In case convergence in (<ref>) is understood as convergence a.s or even as convergence in probability the sequence u can be taken as an i.i.d sequence of random variablesor as an ergodic process with a uniform weak limit distribution.However, thisimplementation is more suited fordeterministic form of u.Discrepancy can be understood as a natural measure of slackness of condition (<ref>) for a fixed value of T.Classical (one-sided) geometric discrepancy of the sequence (u_i)^n_i = 1 is described by the formulaD(u) = max_x ∈ I^d| | { u_i | 1 ≤ i ≤ n}∩ [0,x] |/T- ∏^d_j = 1 x^j|.Note that (<ref>) admits representation D(u) = φ_u _∞ for a certain function φ_u.This idea were used in <cit.> to introduce generalized quadratic discrepancy with supremum-norm replaced by acertain quadratic norm in a certain Sobolev Space. The result can be expressedas(D_2^κ(u) )^2 = = ∬η(x,y)dx dy - 2/n∑^n_i = 1∫η(x,u_i)+1/n^2∑^n_i,j = 1η(u_j,u_i),where η is the reproducing Hilbert kernel of the Sobolev space η(x,y) = ∏^d_i = 1( M + β^2( κ(x^i) + κ(y^i) + 1/2B_2( x^i - y^i1) + B_1(x^i)B_1(y^i) )with β∈ standing fora scale parameter and κ standing for a functional parameter with square-integrable derivativewith ∫^1_0 κ = 0 and the value M defined byM = 1 - β^2 ∫^1_0 (κ')^2,and B_i is the ith Bernoulli polynomial.Note, that in cased = 1 statistic (<ref>) turns into Kolmogorov-Smirnov statistic and (<ref>) turns into Cr/`amer-Von Mizes statistic for uniform distribution test.The function κ is a functional parameter defining exact form of the discrepancy.In this text we use star discrepancy produced byκ^* = 1/6 - 1/x^2and the centred discrepancy produced by selectingκ^c = -1/2 B_2( x - 1/2 1) The case of u being a low-discrepancy sequence is a particularly well suited for our application.low-discrepancy sequence is a deterministic sequence u in I^d designed with a goal of minimizing value the D^κ_p(u_i)^n_i = 1 for each natural number n.Definition above is rather informal. However, it is postulated firmly that for any low-discrepancy sequence u property (<ref>) holds and lim_n →∞ D^κ_p(u_i)^n_i = 1 = 0 for any choice of p and κ.Thus,the rate of convergence of D^κ_p(u_i)^n_i = 1 can be understood as a measure of efficiency of a low-discrepancy sequence u_i.In our application we use Sobol sequence with grey code implementation. Sobol sequence were introduced by Sobol <cit.> and the grey code implementation is due <cit.>.Detailed investigation of the nature of this sequence goes beyond the scope of this work. However, any other low-discrepancy sequence can be used.For a Sobol sequence value D^κ_2(u_i)^n_i = 1converges to zero as O(n^-1 + ϵ),where ϵ depends on d implicitly. While convergence rate of discrepancyfor a random uniform sequenceis O(1/√(n)), for d ≪ 200 valueof ϵ < 1/2,which makes low-discrepancy sequence rather efficient. However, scrambling and effective dimension techniques can increase rate of convergence <cit.>.It can be established that the sequences u and X have no repeating values almost surely,As our data is assumed to come from atomless distributions.This means that the problem of finding Permutation σ^* in (<ref>) is the optimal assignment problem.The Optimal assignment problemis the special case the linear programming,which can be solved in O(n^3) time by the Hungarian algorithm <cit.><cit.>.Amortizations based on applications of parallel programming can improve computation complexity to O(n^2 log n).Approximate algorithms can be used for faster computations, for example <cit.>.Note, that even if sample X has an i.i.d distribution, then theresulting transport Y is notindependent itself.However, the covariance of the elements is converging to zero as T goes to infinity.Let n ≠ m be two indices less or equalto T and i,j be coordinate indices. As (X_k)^T_k = 1 is assumed to be i.i.d, it follows that Y^i_n has a discrete uniform distribution over { u^i_k }^T_k = 1.Then, (Y^i_n,Y^j_m) =Y^i_n Y^j_m -Y^i_nY^j_m= ∑^T_k = 1∑^T_ l ≠ ku^i_k u^j_l/T(T - 1) - ∑^T_k = 1∑^T_l = 1 u_k^i u_l^j /T^2 = = ∑^T_k = 1∑^T_ l ≠ ku^i_k u^j_l/T^2(T - 1) - ∑^T_k = 1 u^i_k u^j_k/T^2,so by bounding from above and below with equidistribution property of u in the limit case(Y^i_n,Y^j_m) ≤∑^T_k = 1∑^T_l = 1u^i_k u^i_l/T^2(T - 1)lim_T →∞ UU/T - 1=lim_T →∞ 1/4(T - 1)= 0, (Y^i_n,Y^j_m) ≥- ∑^T_k = 1 u^i_k u^j_k/T^2≥- ∑^T_k = 1 u^i_k /T^2lim_T →∞ -U /T =lim_T →∞ -1/2T= 0; where U is a uniform random variable on [0,1]. Thus, the value (Y^i_n,Y^j_m) converges to zero. As variance of Y^i_n converges to the variance of a standard uniform distribution on [0,1]we will treat sample Y as uncorrelated in asymptotic context under assumption of i.i.d. distribution of X. It can be easily seen that that quadratic discrepancy admits a degenerateV-statistic representationwith the kernel : ( D^κ_2(y) )^2 =1/n^2∑^n_i,j = 1(y_i,y_j), assuming sample y of n elements. Properties of V-statistic produces theasymptotic resultn ( D^κ_2(y) )^2 V = ∑^∞_i = 1λ_i Z_i^2 ∼(λ), whereZ (0,1) and λ are non-zero eigenvalues of the integral operator 𝒜 defined by the relation 𝒜(f)(x) = ∫^1_0 f(y)(x,y) dy It can be shown that 𝒜 is in fact positive-semidefiniteandtrace-class, which means that all λ_i > 0 and that ∑^∞_n = 1λ_i < ∞ Eigenvalues of 𝒜 can be approximated by afinite collection of N numbers λ̂ with Nyström method.As it was shown in <cit.> the twofold approximation of cdf of V_|N = ∑^N_i = 1λ_i^N Z_i^2 is possible for fixed natural numbers N, K and parameter α∈ (0,1]:(V_|N < x) ≈ ≈1/2 - ∑^K_k = 0sin(1/2∑^N_1 = 1arctan(2(k + 12) αλ^N_i) - (k +1/2) α x) /( k + 1/2) ∏^N_i = 1√(1 +(2(k + 1/2) αλ^N_i )^2 ).This formula can be used for computing quantilesand critical values of (λ) with simple one-dimensional zero-seeking algorithm. Case of A Single Change Point In this chapter an approach for detecting a single change point is presented. We impose a model assumptions that for a fixed τ change pointeither satisfy τ << T - τ or it does not exist . This value τ can be selected in a such way that 1 ≪τ≪ Tand be used as a sliding window bandwidth defininga 'control chart'-like object which we refer to as diphoragram. empirical sliding diphoragram for change point data (X_t)^T_i = 1 is defined byΔ_t^T,τ = ( D^κ_2(R̂_T(X_i))^t + τ_i = t)^2 =( D^κ_2(Y_i)^t + τ_i = t)^2and idealsliding diphoragram byΔ_t^T,τ = ( D^κ_2( R_μ_r(X_i))^t + τ_i = t)^2 for t in range from 1 to T'_τ = T - τ.If κ is a continuous function, thentt as T →∞.With this condition discrepancy is a continuous function of data. So, convergence of vector ranks proved in <cit.> implies convergence of diphoragrams.Note, that with kernel representationcharts admit a difference representationt + 1 -t =1/τ^2(∑^t+1 + τ_i = t + 1(Y_i,Y_t + τ + 1)-∑^t+ τ_i = t (Y_i,Y_t) ).From the computational standpoint this means that computation of the whole time-seriesandtakes only quadratic time O(dT^2) in number in observation.Moreover, if crisp optimal transport with low-discrepancy sequence u is used in estimation of vector rank function, then A_Y = A_u, so all values can be precomputed. Forthe application to the change point problemconsider two increasing sequences of integersT_n and τ_n such that T_n/τ_n =a > 1 for all n ∈ are constructed. Then as Y_i are assumed to be independent t and t + τ_n are also independent random variables. mean sliding discrepancy for set sample (Y_i)_i = 1^T_n is computed by= τ_n^2/T_n∑^a - 1_j = 01 + jτ_n Note, that the mean sliding discrepancy is not the sameas the discrepancy of the whole sample. By independence, in case H_0 holds, as n →∞ ∑^a_j = 1∑^∞_i = 1λ_i/a Z_i,j^2 ∼( ( λ_i/a)^a_j = 1)^∞_i = 1,where Z (0,1). Otherwise, there will be a sliding discrepancies t sampled form the non-uniform data which goes to ∞ in probability as n →∞.So, the whole sum ∞. In casethe single change point exists the statisticis expected to attain the minimal value at such moment of time t^* when theempirical distributionof (Y_t)^t^* + τ_n_t = t^*approaches the empirical distribution of the whole sample, as the empirical distribution of the whole sample is converging to the . Lett^*_n = _1 ≤ t ≤ T_n - τ_n t.It is expected thatthe ratio of numbers of elements from both sides of change pointin the subsample used in the computation of t^*_nand in the whole sample are equal. This can be represented in the algebraic relation_n - t^*_n/τ_n= _n/T_n = r̂_n,where θ̃_n is the produced estimate of the change point. This provides the expression for the estimateθ̃_n =t^*_n/1 - a^-1.We accept H_0 for a fixed γ-level in casep_n = ( V_|N≤) < 1 - γ,where the cdf is numerically estimated as in (<ref>) for some fixed parameters N and K. Otherwise reject H_0 and state that there was a change most probable at _n. In order to reason about properties of the estimate _n. Let ℳ(I^d) denotespace of finite signed measures other the hypercube I^d.Space of nonuniformities 𝒟 isdefined as a quotient of the real vector spaces𝒟 = ℳ(I^d)/endowed with a Hilbert space structure by inner product defined for [ν], [μ] ∈𝒟 by ⟨ [ν], [μ] ⟩ =∫∫(x,y) dν(x)dμ(y) Note, that the relation (<ref>) is well defined as for all a,b ∈⟨[ν + a ], [μ + b] ⟩=∫∫(x,y) dν(x)dμ(y) +b∫∫(x,y) dν(x)d y + + a ∫∫(x,y) d xdμ(y) +ab ∫∫(x,y) d xd y = ∫∫(x,y) dν(x)dμ(y),as ∫(x,y)d x = 0 for any value of y andis symmetric and is indeed an inner product asis a positive-definite kernel.As the lineintersects simplex of probability measures 𝒫(I^d) only in one point ( itself) the natural projection μ↦ [μ] is injective on 𝒫(I^d),so we denote a nonuniformity arising from each μ∈𝒫(I^d) just as μ.If every subsample (Y_i)_i = t^t + τ is associated with an empirical measures μ̂_t = ∑_i = t^t + τ_n δ_Y_i, thent = μ̂_t^2_𝒟 = ⟨μ̂_t,μ̂_t ⟩. By construction of the vector rank function r R_# P_1 + (1 - r) R_# P_2 _𝒟 = 0, hence r R_# P_1 + (1 - r) R_# P_2 = 0 in 𝒟.This produces resultR_# P_2 = -r/1 -r R_# P_1 ⟹ R_# P_2_𝒟^2= r^2/(1 - r)^2 R_# P_1 ^2_𝒟⟹ ⟹ r =R_# P_2 _𝒟/ R_# P_1 _𝒟 +R_# P_2 _𝒟Considering that r = 1/2 it follows that [R_# P_1]=- [R_# P_2], so magnitude of the discrepancy will have the same distribution for sample with equal proportions of elements with distributionsof R_# P_1 and R_# P_2. If t^* = ( 1 - a^-1 ), then it can be shown that estimateis unbiased: _n = ∑^T_n - τ_n_t = 1 t (t = _t' t')1 - a^-1_n = t^* +B^+_n-B^-_n /1 - a^-1_n = t^*/1 - a^-1_n = = T_n/2, whereB^-_n = ∑^⌊ r(T_n - τ_n) ⌋_t = 1 t ( t^*- t = _t't'),B^+_n = ∑^⌊ (1 - r)(T_n - τ_n) ⌋_t = 1t (t^*+ t = _t' t').However, then r ≠ 1/2 the estimate r̂ is projected tobe biased towards 1/2 as the valueB^+ will only increase and thevalue B^- will only decreaseas r decreases. Otherwise, increaseof rincreases the value B^-too, anddecreases the value of B^+.In order to proof consistency of estimator r̂_n we introduce a family of sequences for each s ∈ ((1 -a^-1)^-1, 1)Δ̃^s_n =t'having t' = _1 ≤ t ≤ T_n - τ_n| s - t/T_n(1 - a^-1)|Then by the weak convergence of vector ranks for every s the sequence converges to a value:Δ̃^s_n Δ^s = 1_ s < r - a^-1/1 - a^-1(s)R_# P_1 + 1_ s > r/1 - a^-1(s)R_# P_2 + +1_r - a^-1/1 - a^-1≤ s ≤r /1 - a^-1(s) (r - s(1 - a^-1)/a^-1 R_# P_1 +a^-1 - r + s(1 - a^-1)/a^-1 R_# P_2 ) _𝒟 .Otherwise Δ̃^r_n0.We can assert bystructural uniformity of Δ̃_n thatΔ̃Δ in Skorohod's topology.Thus, r̂_n = _s Δ_n^s _sΔ^s= r providing convergence in probability. This suggests that the bias of r̂_n can be bounded by some sequence β with convergence β_n0:|B^+_n - B^-_n|/T_n≤β_nWithout loss of generality assume that r < 1/2. Then, we separate positive bias into mixing and non-mixing parts:|B^+_n - B^-_n|/T_n <B^+_n/T_n < < ∑_t = 1^⌊(1-r)τ_n ⌋ t(t' < Δ̃^r_n )/T_n+∑_t = ⌊(1-r)τ_n ⌋ + 1^T_n - τ_n - t^* t(t' < Δ̃^r_n ) /T_n.For non-mixing part apply Markov inequality for each fixed value of Δ̃^r_n and some value ζ > 1(t' < Δ̃^r_n ) = ( (t')^-ζ >(Δ̃^r_n )^- ζ) ≤( Δ̃^r_n )^ζ( (t')^-ζ).It is possible to use inverse as Δ̃_n > 0 for each n. With this inequalitythe non-mixing part can be bounded∑_t = ⌊(1-r)τ_n ⌋ + 1^T_n - τ_n - t^* t(t' < Δ̃^r_n ) /T_n≤ T_n (( Δ̃^r_n )^ζ) ( ( Δ̃^1_n )^-ζ) ≤ ≤ C_r T_n (( Δ̃^r_n )^ζ).As Δ̃^r_n → 0 with the rate O(τ_n^-1 + ε) it ispossible to specify a constant ζ in such a way thatT_n(( Δ̃^r_n )^ζ ) ↓ 0 and the expectation( ( Δ̃^1_n )^-ζ ) approaches some constant value by weak convergence and therefore can be bounded.Thus, the non-mixing bias approaches zero as n goes to infinity. Note, that all moments of discrepancy exists as it is bounded on a positive interval for each n. This method can be extended in order to show that the estimate is asymptotically unbiasedunder an assumption of change point slackness,Sequence of diphoragrams t has slackness property iff for some constant γ∈ (0,a^-1)andfor all n large enough there aretime points t^**_n > t^*_n such that :t^**_n - t^*_n/T_n≥γ, ∑_t = 1^t^**_n t( t = _t't') - B^-_n ≤ 0.Then, it is possible to construct a similar bound| B^+_n - B^-_n|/T_n≤ T_n (( Δ̃^r_n )^λ( ( Δ̃^γ + r_n )^-λ| Δ̃^r_n) ),which converges to zero at infinity. alternative methodsOne of the negative properties of the method described in the previous chapter is the requirement of specification of bandwidth τ, which prevents change point detection in the proximity ofthe limit points 1and T.However, as space of nonuniformities𝒟is a metric space, it is possible to determine change point by maximizing distance between two empirical measuresdist() = μ̂^+_ - μ̂^-_^2_𝒟, where μ̂^-_= 1/∑^_t = 1δ_Y_t,μ̂^+_= 1/T - ∑^T_t = + 1δ_Y_t .Then, by definition of the norm the distance is computed as dist() = 1/()^2∑^_n,m = 1𝒦(Y_n,Y_m) - -2/(T - )∑^_n = 1∑^T_m = + 1𝒦(Y_n,Y_m) +1/( T - )^2∑^ T _n,m = + 1𝒦(Y_n,Y_m),leading to a change point estimate = _dist(). As empirical measures of (<ref>) will converge to some points of interval [R_# P_1, R_# P_2] as T_n goes to infinity, the estimate _n converges to true valuein probability.Change points detection methods of this forms were explored in the work <cit.>. Thus, we will not explore it in further depth. The important properties of this statistic is that it still can be computed in O(T^2) time and that dist() is a U-statistic. Geometric idea of (<ref>) suggests that the value ς() =/T - μ̂^+__𝒟/μ̂^-__𝒟 + μ̂^+__𝒟 approaches zero as data size grows to infinityiff r̂ approaches the true ratior. Thus, if change point exists, then change point can be estimated as = _ |ς()|.Or alternatively compute r̂^m by applying iteration r̂^m = μ̂^+_r^m -1_𝒟/μ̂^-_r^m-1_𝒟 + μ̂^+_r^m - 1_𝒟, where μ^+_r^m - 1 and μ^-_r^m - 1 are empirical measures corresponding to the estimates obtained at previous iterations. The initial value r̂^0 can be selected to be equal to 1/2.It is interesting, that true location of change point can be estimated by the value of the error ς(). Assume, that >, then the value μ^-__𝒟 should not distinct dramatically from the value | μ^-__𝒟. However,it holds that | μ^+__𝒟 < | μ^+__𝒟 with high probability as μ^+_ is closer to the uniform distribution than μ^+_. Thus, 0 ≈ς() > ς() with high probability.Otherwise, then < thenthe statistic ς() grows similarly to the function f(x) = Cg(x) -x/c + x, where g is monotonically decreases, as grows to T.Thus 0 ≈ς() < ς() withhight probability. This suggests, that it must be possible to establish an iterative procedure for finding optimal estimate . Note, that in casechange point does not exist the values of |ς()| will be lowfor all possible values of . So, in case change point exists it is possible to establish an algorithm: Multiple Change PointsIn this chapter the situation of K possible consecutivechange points = (^1,…,^K) in the data is considered. Now, r denotes a list of positive values r =(r^1, …, r^K) = ( ^1/T, ^2 - ^1/T,…,^K - ∑^K-1_k = 1^k/T), which can be understood as the first K coefficients in the convex combinationof K + 1 probability distributions P_1, …, P_K + 1. That is, defineμ_r = ∑^K_k = 1 r^k P_k+ ( 1 -∑^K_k = 1 r^k)P_K + 1.Furthermore, estimatesand r̂ are treated as lists of corresponding structure. By definition of vector rank function R_#μ_r =.However, in order to apply methods similar to ones developed in the previous chapter we need one more property of the model.K probability measures P_1, … P_K are said to be convexly independent if for all lists of K coefficient λ∈^K_+, such that ∑^K_k =1λ_k = 1, for each i equalityP_i = ∑^K_k = 1λ_k P_k,holds only if λ_j = δ_i,j for each j, where δ_i,j is the Kronecker delta.Assume that the true value of K ∈𝔎. Then, by construction K^SMA≤ K with probability α. It can be postulated, that 0 of 𝒟lies in the convex hull of R_# P_i. That is 0 ∈conv{R_# P_k}^K + 1_k = 1. It suggests that there are projections of 0 to the edges of the convex polytope π_k0 ∈ [R_# P_k, R_# P_k + 1] which minimizes π_k 0 _𝒟. Hence, if τ is taken to be small enough then local minimas of Δ^T,τ will happen in the proximity of the change points with high probability. The problem with this method is that proportions of points from different sides ofa change point at the local minimise are given by the relationπ_k 0 = R_# P_k + 1^2_𝒟 - ⟨ R_# P_k,R_# P_k + 1⟩_𝒟/d^2_𝒟(R_# P_k,R_# P_k + 1)R_# P_k + + R_# P_k^2_𝒟 - ⟨ R_# P_k,R_# P_k + 1⟩_𝒟/d^2_𝒟(R_# P_k,R_# P_k + 1)R_# P_k + 1,which can not be recovered from the diphoragram. For this reason we propose an iterative procedure. Letμ̂_^k,^k + 1^n denote empirical measures of points from the interval bounded by change point estimates [^k]^n and [^k + 1]^n . * start with 𝒯_1 = 𝒯.* for each k ≤ K select t^*_k = _t ∈𝒯_kΔ̃^T, τ_t and update T_k + 1 = { t ∈𝒯_k : | t - t^*_k | > τ}.* make initial change point estimation with blind adjustment [^k]_1= t^*_k + τ/2.* readjust change points for a fixed number of iterations Nwith nth plus one readjustment being[^k]_n + 1 =t^*_k + [λ̂^k_1 ]_n τ, where [λ̂^k_1 ]_n =μ^n__k , _k + 1^2_𝒟 - ⟨μ^n__k -1 , _k ,μ^n__k , _k + 1⟩_𝒟/d^2_𝒟(μ^n__k - 1 , _k,μ^n__k , _k + 1) with surrogate change points being [ ^0]_n-1 = 0 and[ ^K + 1]_n - 1 = T .In order to approach problem of model misspecification we apply smallest accepted model (SMA) methodology.For a collection of K we acquire minimizing time points t^*. Then, test for change points in the intervals bounded by _i and _i + 1 with surrogate values as above by computing a p-value approximations.Thus, the model can be estimated while new local minimizers are being recovered and the process can be stopped as minimal accepted value of K̂ has been achieved. Computational Results In order to conduct computational experiments themethods discussed in the previous chapter were implemented in Python programming language with use of numpy and scipy libraries.We use package in order togenerate Sobol sequences. Simulations with Zero Change PointsFor simulations with zero change points we are interested in measuring statistical significance or confidence of the test statistic T which can be understood as significance(T) = ( TrejectsH_0 | H_0is true), confidence(T) = (TacceptsH_0 | H_0is true).In simulations with zero change points H_0 obviously is true. So, for a run of n simulations we estimate confidence as conf(T,n) = #{simulations with no change points detected}/n.We run simulations without change points and vary certain fixed parameter while measuring confidence and inverse p-value for each value of parameter. The only nontrivial results were obtained for change in data dimension d.Additional experiments were conducted with growing variance and changes in covariance structure of the observations, however no dependencies were identified. This could be due to stabilizing effect of vector rank functions. Simulations with One Change Point While running a simulations with one change point we are naturally interested in the estimation of statistical power of the test T which can be understood aspower(T) = ( TrejectsH_0 | H_0is false),and can be estimated for n simulations as pow(T,n) =#{simulations with change points detected}/n.We investigate dependence of pow(T,n) on differences between distributions from opposite sides of the change point.The figures shows that star discrepancy outperforms symmetric discrepancyin detecting change both in expectation and in variance. However empirical results in <cit.> indicates that in some situations symmetric discrepancy may turn out tob be a better tool.In the situation of existing change point not only power of the test is of interest but also a precision of change point estimations.As it was shown in previous chapter bias of a change point estimate increases as true ratio r ofdistributionsin the sample diverges from the value of 1/2.We provide a computational illustrations: Simulations with Multiple Change PointsFor case of multiple change point we provide only example with diphoragrams: Examples with financial data: In order to provide examples, which are not artificially constructed, financial data similar to one in <cit.>.This data was acquired from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.htmlData libraryofKeneth R. French. It contains mean monthly returns from five portfolios each composed of one offive industrial sectors in the USA, which are: * Finance, * Manufacturing, * Retail, wholesaleand some services, * Utilities, * Other.This provides data dimension of d = 5 and total number ofobservations of T = 1091 as they were recorded monthly from July of the year 1925 to the may of the year 2017. In the original paper <cit.> the parametric Bayesian inference was used too estimate nine and seven change points, and the total number of change point was only guessed and not inferred.Our results differ significantly from this original results, however different time range was used.It can be projected that all above change points can be attributed to the important events in the economic history. For example, change point at July 1954 can be related to the end of recession of 1953,which itself can be explained by the change of interrelationsof the industrial sectors leading to the change in the structure of the observed distribution.Hence, we can describe performance of the SMA-procedure as satisfactory. Conclusion and DiscussionAs result of the work a collection of methods of change point detection methods was developed. All this methods are based on interaction of vector ranks and geometric discrepancy which is a novel result. Certain basic consistency result were proved for the method in its basic form designed for detecting a single change point. However, they also can be applied for detecting and estimating multiple change points. Computational results shows applicability of the method both for simple artificial change point problems an problems concerning actual data from applications.It is strictly indicated by resulting experience that the method is much more potent in situations then the distributions are not concentric.Empiricalevidences of our theoretical findings were also observed.These theoretical results include expression of relation on different sides of change point through inner product in the Hilbert space 𝒟.Another Theoretical result concerns dependence of estimate's bias on ratio in which change point separates sample. Another positive results is the discovery of 𝒟, which can be used for establishing alternative iterativeprocedures defined by relations in this Hilbert space. Furthermore, this Hilbert space structure on empirical measures can be used for proving more theoretical results in the future.Better proofs which cover ergodic processes and provide exact rates of convergence a still need to be worked out for the methods.Furthermore concentration results for change point estimates might render this algorithms interesting for practical application.However, they are absent in the current work. Finally, online version of method can be implemented. The only requirement for this algorithm is fast online computation of the optimal assignment problem.It can be projected that such algorithm can bederived from the Sinkhorn distance regularised algorithm which were designed by Cuturi <cit.>. | http://arxiv.org/abs/1707.08658v1 | {
"authors": [
"Nikita Pronko"
],
"categories": [
"stat.ME",
"62H30 (Primary), 62G05 (Secondary)"
],
"primary_category": "stat.ME",
"published": "20170726223938",
"title": "Change Point Detection with Optimal Transport and Geometric Discrepancy"
} |
firstpage–lastpage Binary Hermitian Forms and Optimal Embeddings Michael Zhao December 30, 2023 =============================================The migration of planets on nearly circular, non-inclined orbits in protoplanetary discs is entirely described by the disc's torque. This torque is a complex function of the disc parameters, and essentially amounts to the sum of two components: the Lindblad torque and the corotation torque. Known torque formulae do not reproduce accurately the torque actually experienced in numerical simulations by low- and intermediate-mass planets in radiative discs. One of the main reasons for this inaccuracy is that these formulae have been worked out in two-dimensional analyses. Here we revisit the torque formula and update many of its dimensionless coefficients by means of tailored, three-dimensional numerical simulations. In particular, we derive the dependence of the Lindblad torque on the temperature gradient, the dependence of the corotation torque on the radial entropy gradient (and work out a suitable expression of this gradient in a three-dimensional disc). We also work out the dependence of the corotation torque on the radial temperature gradient, overlooked so far. Corotation torques are known to scale very steeply with the width of the horseshoe region. We extend the expression of this width to the domain of intermediate mass planets, so that our updated torque formula remains valid for planets up to typically several tens of Earth masses, provided these relatively massive planets do not significantly deplete their coorbital region. Our torque expression can be applied to low- and intermediate-mass planets in optically thick protoplanetary discs, as well as protomoons embedded in circumplanetary discs.Planetary systems: formation – Planetary systems: protoplanetary discs – Accretion,accretion discs – Methods: numerical – Hydrodynamics – Planet-disc interactions § INTRODUCTION An important ingredient for studies of planetary population synthesis are the so-called migration maps, which are two-dimensional functions Γ(M_p,r) that provide the torque exerted by the disc on a planet on a nearly circular orbit, as a function of its mass M_p and of its distance r to the central star.A migration map is intrinsic to a given disc model. Over most of the (M_p,r) domain, the torque is generally a negative quantity, corresponding to an orbital decay of the planet. Nevertheless, there may be some regions where the torque is positive (named islands of outward migration). These regions broadly occur where the disc's temperature drops faster than r^-1, and span a mass range from a few Earth masses to potentially several tens of Earth masses. The outcomes of models of planet population synthesis depend sensitively on the outline of these islands <cit.>. Migration maps can be established in one of two ways: they can either be obtained by three dimensional (3D) simulations, including all the physical ingredients needed for a correct description of the disc <cit.>, or they can be obtained using torque formulae. The first approach is naturally not suitable to the exploration of a large number of disc models, owing to its considerable computational cost, and one must resort to torque formulae <cit.>. Numerical simulations of low-mass planets embedded in radiative discs allows to assess the validity of torque formulae. <cit.> have compared the torque experienced by a 20 M_⊕ planet in a radiative disc to the predictions of <cit.> and <cit.>. The former displays a broad agreement with the outcome of the simulations. The latter, however, was found to be at odds with the simulations outcome, a discrepancy identified as too saturated corotation torques, and later resolved when estimates of the width of the horseshoe region in 3D discs became available <cit.>. Despite their broad agreement with simulation outcomes, the torque formulae of <cit.> and <cit.> lack the accuracy required by models of planetary population synthesis. One of the main reasons is that these analyses are based on results of two-dimensional analysis. Even though they capture most of the mechanisms that contribute to the torque, the dimensionless coefficient that they feature in many places may be significantly off.The purpose of this paper is to update the torque formula of <cit.> by means of 3D numerical simulations, successively tailored to determine the value of each of the different dimensionless coefficients that we want to update. It is organised as follows. In section <ref> we introduce and discuss the different torque components that we want to update and we describe the numerical code and setups that we use for this purpose. In section <ref> we present our different results, and provide an updated torque formula in section <ref>. For the convenience of the reader mainly interested in the formula, this section is self-contained.We compare the results of our updated torque formula to published numerical simulations in section <ref>, and draw our conclusions in section <ref>.§ NOTATION We consider a planet of mass M_p on a circular orbit of radius r_p around a central star of mass M_⋆, with angular speed Ω_p, embedded in a disc of surface density Σ(r) which follows the power law:Σ(r)=Σ_0(r/r_p)^-α.The disc's midplane temperature T(r) obeys the law:T(r)=T_0(r/r_p)^-β.We denote with ν the kinematic viscosity of the disc, with χ its thermal diffusivity and with κ its opacity.The disc's pressure scale length is H, and is given by:H=c_s/Ω_p,where c_s is the disc's isothermal sound speed:c_s=√(%s/%s) RTμ,R being the constant of ideal gases and μ the mean molecular weight. We will also make use of the disc's aspect ratioh=H/r,and of the planet-to-star mass ratioq=M_p/M_⋆. § FOCUS OF OUR TORQUE UPDATE Both the Lindblad and corotation torques can be normalised to:Γ_0=Σ_0Ω_p^2r_p^4(q/h)^2=Σ_0Ω_p^4r_p^6q^2c_s^-2.We draw hereafter a list of the different updates that we perform on the normalised torques, which are dimensionless quantities.§.§ Lindblad torque The normalised Lindblad torque depends on α and β, which are respectively the slopes of surface density and temperature <cit.>. Its dependence on α has already been worked out by <cit.> by means of semi-analytical calculations in the linear regime for planets in globally isothermal discs, and we do not revise here this (weak) dependence. On the contrary, its dependence on the temperature slope has been worked out by <cit.> by a linear analysis in two-dimensional discs with a softened potential.Here we reevaluate the dependence of the Lindblad torque on the gradient of midplane temperature. We undertake this analysis in section <ref>. We note that the Lindblad torque can exhibit a dependence not only on α and β, but also on αβ <cit.>. This dependence is very weak, however, and we neglect it. More generally, we seek, and restrict ourselves to, linear dependencies of the different torque components on the radial gradients of physical quantities at the planet's location.The Lindblad torque scales with the inverse squared of the sound speed (see Eq. <ref>). It therefore depends on the thermal diffusivity of the disc: when the latter is large enough that the disturbances triggered by the planet at Lindblad resonances behave isothermally, it scales with the inverse squared of the isothermal sound speed. On the other hand, when the thermal diffusivity is small and the disturbances behave adiabatically, the Lindblad torque scales with the inverse squared of the adiabatic sound speed, and is therefore a factor of γ smaller, where γ is the disc's adiabatic index. This dependence has been considered by <cit.> and <cit.>. We do not revise it here, and we will adopt for the updated expression of the Lindblad torque (see section <ref>) the dependence given by <cit.>. §.§ Corotation torque The corotation torque is the sum of three main components: one which scales with the radial gradient of vortensity, one which scales with the radial gradient of entropy, and one which scales with the radial gradient of temperature. In addition, there is a contribution to this torque arising from the vortensity viscously created at the contact discontinuities which appear on the downstream separatrices in the presence of an entropy gradient. These four contributions are represented in Fig. <ref>. Each of the first three components is a blend of a linear component and of a horseshoe drag. Each horseshoe drag is itself the product of the unsaturated horseshoe drag by a saturation function which is generally a number between 0 and 1 (corresponding respectively to a totally saturated and to an unsaturated horseshoe drag).Each torque component can in principle have its own saturation function. We neither update, in this work, the saturation functions nor the blending coefficients. <cit.> gave an expression of the former, obtained from first principles by a two-dimensional analysis of a simplified model of the horseshoe flow. The horseshoe flow has since been shown to be essentially two-dimensional even in 3D discs <cit.>. On another hand, the flow arising from turbulence in three dimensions and that arising from a kinematic viscosity in a laminar 3D disc are markedly different <cit.>, so a 3D study of the torque saturation as a function of kinematic viscosity would probably be pushing the viscous model of protoplanetary discs beyond its domain of validity.The blending coefficients were determined through a fit of numerical experiments by <cit.> and by <cit.>. <cit.> use the weights ε and 1-ε respectively for the horseshoe drag and for the linear corotation torque (where ε depends on the torque component). <cit.> use the weights 1-K and G, so that their sum is not necessarily 100 %. This allows them to reproduce the fact that the horseshoe drag may be larger than its unsaturated value. In the analysis of <cit.>, this fact is incorporated at a different place (namely the fact that the saturation functions can have values slightly larger than unity). Here we stick to the blending coefficients of <cit.>.The vortensity component of the corotation torque does not need any update. <cit.> have shown that the 3D horseshoe drag in a barotropic disc has the same expression in a two dimensional disc, and that the horseshoe region has a vertical separatrix, hence a constant width on the whole vertical extent of the disc <cit.>. Naturally, an updated width of the horseshoe region is used to evaluate this torque (more detail on this is given in section <ref>), but no amendment has to be done to the horseshoe drag expression. For the sake of simplicity, we consider here that the gradient of vortensity[More specifically, the quantity dlog(Σ/B)/dlog r, where B, the second Oort's constant, is half the flow's vertical vorticity.] is equal to 3/2-α. This is true in any part of the disc where the surface density and angular frequency are power laws of the distance to the star.The temperature component of the corotation torque has been overlooked in previous torque formulae. Here we obtain the value of this torque as a by-product of the study of the Lindblad torque of section <ref> and present its properties in section <ref>. This torque component arises from the production of vortensity by the radial temperature gradient, and appears in locally isothermal calculations. The vortensity thus produced is concentrated near the (downstream) separatrices. Although it is not singular as the vortensity arising from non-barotropic effects, which can formally be represented by a delta-function at the separatrix <cit.>, it has nearly same spatial distribution and it is reasonable to expect it to saturate in a similar manner. We therefore assume that this torque has same dependence on the viscosity as the entropy torque, and focus here exclusively on the magnitude of the unsaturated horseshoe drag and that of the linear torque.Finally, we reevaluate the dependence of the entropy component on the entropy gradient, a much needed determination since this component plays a preponderant role in the appearance of outward migration islands in migration maps. We also reevaluate how to determine the entropy gradient in 3D disc. Whereas its expression is straightforward in 2D discs, one may contemplate different ways of evaluating it in a 3D disc (such as the gradient in the midplane, or of the vertically averaged quantity, etc.). Here we simply adopt the expression that yields to a one-to-one relationship (with the least possible dispersion) of the torque excess in a non-barotropic simulation with respect to a barotropic simulation. This analysis is presented in section <ref>. §.§ Additional analysis In addition to the aforementioned reevaluations of key dependencies of the torque, we have performed two side studies in order to improve the accuracy and predictive power of the torque formula, which we present below.§.§.§ Width of horseshoe region The different components of the horseshoe drag scale with the width of the horseshoe region to the fourth power. It is therefore crucial to get an accurate estimate of this width. Besides, as soon as the ratio q/h^3 is larger than ∼ 0.1, the law that gives the width for small mass planets <cit.> is no longer valid and the horseshoe region is actually larger than what is predicted by the low-mass law, resulting in a boost of the corotation torque <cit.>. The threshold of 0.1 is particularly stringent for typical protoplanetary discs, and translates into 4 M_⊕ only for a disc with an aspect ratio of h=0.05 (which shows, incidentally, that the comparisons mentioned in section <ref> were performed much beyond the domain of validity of the torque formulae for low-mass planets). In order to relax such a stringent condition on the planetary mass, we study here the transition between the low-mass regime, for which the width scales as (q/h)^1/2, and the high-mass regime, for which, as in the restricted three body problem, the width scales as q^1/3. This study is presented in section <ref>.§.§.§ Thermal diffusivity Since the saturation of the corotation torque depends on the value of the disc's thermal diffusivity, it is important to determine this value as accurately as possible. We have studied the radial spread of an initially localised excess of temperature in order to assess the accuracy of standard estimates which relate the disc's thermal diffusivity to the disc's temperature, density and opacity. This study is presented in section <ref>.§.§ Numerical detailsWe conduct all our numerical experiments with the public hydrocode FARGO3D[<http://fargo.in2p3.fr>] <cit.> with orbital advection enabled <cit.>. Here we use a spherical mesh that covers the full azimuthal range [-π,π]. We denote N_ϕ the number of cells in azimuth, N_r the number of cells in radius and N_θ the number of cells in colatitude. We always simulate a half-disc in colatitude (since the planet is always coplanar with the disc), and use reflecting boundary conditions at the midplane. All our calculations involving a planet are performed in the corotating frame.According to our needs, we either use a locally isothermal equation of state, or we solve an energy equation. For the first case we use the closure relationship between the pressure p and density ρ:p( r,t)=c_s^2(r)ρ( r,t),where c_s(r) is the isothermal sound speed, which depends on the distance to the central star, and which is constant in time. In such case there is no need to solve an energy equation. In the second case, we solve the time evolution of the internal equation, and in the absence of some form of thermal diffusion, the flow behaves adiabatically. In this case we assume the gas to be ideal, so that the pressure is given by:p( r, t)=(γ-1)e( r,t),where e is the internal energy density.Finally, in order to avoid the divergence of the planetary potential Φ_p in the vicinity of the planet, we soften it over a length scale ϵ:Φ_p(r)=-GM_p/√(|r-r_p|^2+ϵ^2).In three-dimensional calculations, ϵ has to be chosen small compared to the pressure scale length, and larger or comparable to the resolution. In all our numerical experiments we have ϵ=0.1H.§ RESULTS We present hereafter the results of the numerical experiments conducted to address the points listed in section <ref>. §.§ Dependence of the Lindblad torque on the temperature gradient For this experiment, we use a locally isothermal setup, and use the fact that at larger time the corotation torque saturates, so that the total torque essentially amounts to the Lindblad torque.In a locally isothermal setup, the Lindblad torque has the form:Γ_L/Γ_0=-(2.34-0.099α+kβ),where as discussed in section <ref> the first two terms of the right hand side come from the work of <cit.>, and where k is the coefficient that we want to determine. In principle, it suffices to perform two runs which have different values of β (all other parameters being the same) to infer k.Our choice of parameters results from a trade-off between different effects. We want the planetary mass to be sufficiently large so that the horseshoe region is well resolved and can be saturated in a reasonable amount of time. On the other hand, we want the planetary mass to be sufficiently small so that the planet does not significantly perturb the disc by carving a gap.There is another difficulty inherent to this experiment: when the disc is not globally isothermal (β 0), it is prone to a baroclinic instability <cit.> which results in considerable noise in the torque curves, precluding any converged measurement. However interesting this instability may be, it has for us a parasitic character, and we aim at getting rid of it, while preserving an almost complete saturation of the corotation torque. We prevent the appearance of the instability by using some amount of viscosity, and determine by dichotomy the minimal amount of viscosity required to quench the instability. Also, the strength of the instability scales with |β|, so one has to choose values of β sufficiently large to allow an accurate measurement of k, but also sufficiently small to avoid the development of the baroclinic instability, once the viscosity has been chosen.With these considerations in mind, we have chosen for this numerical experiment the following values: q=2.4· 10^-5 (which corresponds to 8 M_⊕ if the central star has a solar mass) and ν=7· 10^-8r_p^2Ω_p. The disc's aspect ratio is h=0.05. Although the kinematic viscosity is quite small, we do not expect the planet to carve a significant gap in the disc over the time scale required to achieve the saturation of the corotation torque. The Hill radius of the planet [r_p(q/3)^1/3] is much smaller than the pressure length scale, and the expression 3h/[4(q/3)^1/3]+50ν/(r_p^2Ω_p q) is a factor of 2 above the critical value for gap opening <cit.>. Even under these conditions a gap may eventually be opened <cit.>, but this will occur on time scales much longer than those of our numerical experiments. As we shall see below, we do find evidence for secular effects in our torque measurements, which affect our results at the percent level.Our mesh extends from π/2-3h to π/2 in colatitude, over N_θ=40 cells, and from 0.6r_p to 1.4r_p in radius, over N_r=200 cells. In addition, we have N_ϕ=900 cells. We use wave-killing boundary conditions in radius <cit.>, in order to avoid reflection of the wake at the radial boundaries. We therefore expect our results to be nearly insensitive to the exact location of the mesh radial boundaries, as these are located at a large number of pressure scale lengths from the orbit. We perform two simulations, with β=1/2 and β=-1/2 respectively. Each one is run over 1000 orbital periods of the planet. We choose α=3/2 so that there is no vortensity gradient, and the only corotation torque present before saturation is the temperature component of the corotation torque.This choice of parameters implies that the horseshoe region is resolved over 11 zones, which is sufficient to allow for a nearly complete saturation of the torque <cit.>. We have q/h^3=0.19, so that the flow is weakly non-linear. The horseshoe libration time is ∼ 60 orbits, so that our runs duration allows for a full saturation of the corotation torque. The ratio of the viscous time scale across the horseshoe region to the libration time scale is ∼ 7, which implies that the corotation should reach a degree of saturation of ∼ 90 %. We show the torque as a function of time for the two runs in Fig. <ref>. The corotation torque saturates, leaving essentially the Lindblad torque. We see a systematic, slow increase of the torque in both cases, which we attribute to the slow opening of a very shallow dip (we find that the rate of this increase decays as the viscosity increases).The value of k can be directly obtained by subtracting the two normalised torques. We show their difference in Fig. <ref>. From this figure we infer a value of k≈ 1.4, comparable to, and marginally smaller than the value of 1.7 found by <cit.> for 2D discs. §.§ Dependence of the corotation torque on the temperature gradient We can further exploit the runs presented at the previous section to work out the dependence on β of Γ _T^UHD and Γ _T^lin (see Fig. <ref>). Over the first half libration time, the torque is the sum of the Lindblad torque and the unsaturated horseshoe drag. Since there is no vortensity gradient in these runs, the horseshoe drag is entirely attributable to the temperature gradient. As we now have an expression of the Lindblad torque that takes the temperature gradient into account, we can subtract this torque from the torque measured to get an estimate of the corotation torque. We can also slightly improve upon the results of the previous section by taking into account the residual value of the corotation torque at larger time, as it does not fully saturate.We write the torque over the first half libration time as:Γ^(U)_β=Γ_L+Γ _T^UHD,where the (U) superscript stands for “unsaturated”, and where the β subscript conveys that in this numerical experiment the only parameter that is varied is β. In Eq. (<ref>), we do not have a vortensity component for the reasons exposed above, and we do not have an entropy component either because the setup is isothermal. The planetary mass is sufficiently large, and the viscosity sufficiently small, for the torque to be the (unsaturated) horseshoe drag <cit.>.Writing the unsaturated corotation torque asΓ _T^UHD =k'βΓ_0,where k' is the dimensionless coefficient that we want to determine, we can write:Γ^(U)_β/Γ_0=K-kβ+k'β, where K=-2.34+0.099α <cit.>. We can also write the torque value at larger time asΓ^∞_β/Γ_0=K-kβ+ϵ k'β,where ϵ≪ 1 represents the amount of saturation of the corotation torque. From these relations we infer:k-ϵ k' = Γ^∞_-1/2-Γ^∞_1/2/Γ_0≡ΔΓ^∞/Γ_0k-k' = Γ^(U)_-1/2-Γ^(U)_1/2/Γ_0≡ΔΓ^(U)/Γ_0,hencek = ΔΓ^∞-ϵΔΓ^(U)/Γ_0(1-ϵ)k' = ΔΓ^∞-ΔΓ^(U)/Γ_0(1-ϵ).We recover the fact that k=ΔΓ^∞/Γ_0 when ϵ=0, that is to say when the corotation torque is fully saturated at larger time. Fig. <ref> shows the torque behaviour at early time, from which we infer ΔΓ^(U)=0.3Γ_0. When ϵ=0, this yieldsk'=1.1. Evaluating k and k' when ϵ=0.1 (a reasonable amount of residual corotation torque, as discussed in section <ref>), we get k=1.5, and k'=1.2. This shows that the coefficients that we have obtained for the temperature dependence of the normalised Lindblad and corotation torques are accurate to within ∼ 0.1. We conclude this section by determining in a similar manner the linear corotation torque. We simply have to amend the parameters so that the corotation torque remains linear. This is achieved by adopting a much smaller planet mass and a large viscosity <cit.>. The results of these new runs are displayed in Fig. <ref>, in which one can see that indeed the torques remain nearly constant after a dynamical timescale and do not show the variation toward a different value on O(10) orbits typical of the horseshoe drag <cit.>. These results imply that the temperature component of the linear corotation torque scales as:Γ_T^lin=1.0βΓ_0,where again the numerical coefficient is determined to within ∼ 0.1.At this stage, considering that we use the saturation function and the weights provided by <cit.>, we have completed our determination of the terms entering in Γ _T^CR (see Fig. <ref>, third frame). §.§ Torque dependence on the radial entropy gradient We now perform a 3D version of the study presented by <cit.> and update the terms entering in Γ_S^CR (see Fig. <ref>, second frame). We take an adiabatic index γ=1.4, unless stated otherwise. We consider a large number of disc models (here N=80) for which each value of the surface density and temperature slopes are chosen randomly (namely, α is given by a random variable uniformly distributed over [-1.5,1.5], while β is uniformly distributed over [-2,2]). For each disc model, two runs are performed: one with a locally isothermal equation of state, and one with an adiabatic flow, which provide respectively the torque Γ_iso and Γ_adi. The parameters and the duration of the runs are such that the corotation torque is in the regime of unsaturated horseshoe drag. We then seek an appropriate linear combination ξ of α and β such that the torque excess, defined as ΔΓ=γΓ_adi-Γ_iso, be a one-to-one map of ξ. We illustrate this process in Fig. <ref>. We find that an appropriate expression for ξ is the following:ξ=β-0.4α-0.64,and that in this case we have:ΔΓ=4.6ξΓ_0. By construction, our procedure implies that there is a linear relationship between ξ and the torque excess, and therefore the calculation of the adiabatic torque is straightforward once one knows the isothermal torque. In a 2D situation, the excess has been found to scale with the radial entropy gradient <cit.>. Here, its expression resembles the gradient of entropy evaluated in the midplane[Here this gradient is normalised so that the coefficient of β is one.], which is β-2(γ-1)/(3-γ)α-3(γ-1)/(3-γ)=β-0.5α-0.75, but differs slightly from it.By extension, we call ξ the radial entropy gradient, but it must be kept in mind that this is an abuse of language, as this quantity has not been obtained by considerations about the entropy, but by a fit of numerical simulations. As mentioned in section <ref>, we assume that this corotation torque component saturates as the entropy-related torque in 2D discs.We have not investigated in a systematic manner the dependency of the different coefficients of Eq. (<ref>) on the adiabatic index. Nonetheless, for the reader interested in the torque in a different kind of discs, we mentioned that we have conducted a similar study for the case γ=5/3, and found in that case that ξ=β-0.58α-0.85 provides the one-to-one linear relationship ΔΓ=4.1ξΓ_0.At this stage we see that the horseshoe drag in the adiabatic case is the sum of a term that scales with ξ (combination of α and β) and of the isothermal horseshoe drag, itself sum of a term that scales with the vortensity gradient (3/2-α in power law discs) and a term that scales with the temperature gradient (β) as found in section <ref>. We have three components and only two slopes (α and β). It may seem desirable at first glance to simplify the expression for the net corotation torque into a dependence on α and one on β. This, as discussed by <cit.>, would obfuscate the physical meaning of each term. Besides, each component involves a different distribution of vortensity perturbation within the horseshoe region, and thus in principle saturates in a different manner, so the simplification could only be valid for the regime of unsaturated horseshoe drag, not for the general regime. We shall therefore keep the three components in the general expression of the corotation torque.We have performed a numerical experiment similar to the one described above, with a lower mass (q=10^-6) and a higher viscosity (ν=2· 10^-5r_p^2Ω_p), so that the corotation torque remains linear <cit.>. We find in that case:ΔΓ^lin=0.8ξΓ_0. §.§ Width of the horseshoe region As said in section <ref>, the different components of the horseshoe drag scale with the width of the horseshoe region to the fourth power. It is therefore important to have an accurate value of this width, so we study its scaling as a function of the planet mass in the regimes of low and intermediate masses, corresponding respectively to q≪ h^3 and q≲ h^3 (we shall provide slightly more quantitative definitions at the end of this section). For this purpose we use globally isothermal simulations, and perform a streamline analysis in the midplane, exploiting the fact that the horseshoe region has a width which is independent of the altitude in this case <cit.>. Our runs have a setup similar to the setup outlined in section <ref>, except for the resolution (N_ϕ,N_r,N_θ)=(1320,440,40). In particular, we still have a null vortensity gradient. <cit.> have found that the horseshoe region is asymmetric when the vortensity gradient is finite. Even though we average the width of the horseshoe region in front and at the rear of the planet in order to minimise the effect of an asymmetry, we prefer not to introduce a source of asymmetry of the horseshoe region[The result of <cit.> was obtained in two-dimensional discs. No similar result has been obtained in three-dimensional discs, so we assume that adopting a null vortensity gradient keeps the degree of asymmetry to an acceptable level.]. The planet mass is introduced progressively over two orbital periods, and the width of the horseshoe region is measured 20 orbits after the introduction of the planet. We perform a systematic study for three values of the aspect ratio: h=0.03, h=0.04 and h=0.05. For each of these values, we perform 30 calculations with a planetary mass varying in a geometric sequence from 0.05h^3 to 4h^3. The width of the horseshoe region is determined automatically by dichotomy, averaging the results in ϕ=1 rad and ϕ=-1 rad. We find that more accurate results are obtained if the disc is inviscid when q<0.4h^3, and viscous for larger masses. For lower mass indeed, a viscous drift of the disc can distort severely the horseshoe region <cit.>, whereas for high masses, the planet can trigger vortices on the edges of its horseshoe region if the disc is inviscid, which precludes an accurate determination of the width of the horseshoe region.For low masses, we have the scaling <cit.>:x_s=1.05r_p√(%s/%s)qh,whereas for large masses we recover the scaling found in 2D simulation with softened potential <cit.>:x_s≈ 2.5r_p(q/3)^1/3.Using the width normalised to the pressure length scale X_s=x_s/(hr_p) and denoting with Q the planetary mass normalised to the thermal mass M_th=c_s^3/(GΩ_p):Q=M_p/M_th=q/h^3,we can rewrite Eqs. (<ref>) and (<ref>) as:X_s=1.05Q^1/2Q≪ 1andX_s=1.7Q^1/3Q≫ 1.The lack of an explicit reference to h in these expressions suggests that the normalised width of the horseshoe region has a universal dependence on the mass expressed in thermal masses. Fig. <ref> confirms this expectation. We find that a function that gives the asymptotic behaviours of Eq. (<ref>) and (<ref>) and that matches satisfactorily the width behaviour for intermediate masses is a blend of the asymptotic values at low- and large-mass with weights respectively ϵ and 1-ϵ, where ϵ=1/(1+2Q^2). This yields:X_s=1.05Q^1/2+3.4Q^7/3/1+2Q^2,orx_s=1.05(q/h)^1/2+3.4q^7/3/h^6/1+2q^2/h^6r_p. From now on we will use Eq. (<ref>) to evaluate the width of the horseshoe region.As a side result, this leads us to a reasonable definition of an intermediate mass protoplanet as a protoplanet that has a horseshoe width intermediate between the widths given by the low-mass and high-mass scaling (Eqs. (<ref>) and (<ref>)). Examination of Fig. <ref> shows that this broadly corresponds to 0.2h^3≲ q≲ 2h^3. We assume that the corotation torque tends toward its linear value when dissipation (either viscous or thermal) is large, regardless of the planetary mass. This result has been shown by <cit.> for low-mass planets, but it may be questionable for intermediate mass objects. Although we have not explored this issue in a systematic manner, we have performed simulations with a planet of half the thermal mass (Q=0.5), and found that it is subjected, at very large viscosity, to a vortensity corotation torque comparable to the linear estimate. This result was obtained considering globally isothermal discs with and without vortensity gradients[The unsaturated total torque in isothermal simulations when Q∼ 0.5 can be off by several Γ_0 from its expected value, whereas we found with subsidiary calculations of isentropic discs that the total torque is correctly reproduced. These different behaviours have also been observed by <cit.> and are linked to small scale features of the flow. Since the realistic discs in which we apply our torque formula are not isothermal, we believe that this discrepancy is unimportant, but it should be kept in mind when dealing with intermediate mass planets embedded in isothermal discs.].We note, however, that a substantial amount of dissipation is required for a fall back of the corotation torque of an intermediate mass object to its linear estimate. The decay toward the linear regime is obtained for <cit.>:hr_pν/Ω_px_s^3≫ 0.1,and a similar condition holds, obtained by writing χ in stead of ν, for the decay of the entropy torque to its linear value. We introduce the disc's alpha parameter α_ss <cit.>, and write ν=α_ssr_p^2Ω_ph^2. Using for x_s the conservative value given by the low-mass estimate, we recast the condition of Eq. (<ref>) as:α_ss≫ 0.1Q^3/2.For Q=0.2, which corresponds to the mass at which departure from the small mass regime becomes noticeable, this translates into α_ss≫ 10^-2. This implies that under most circumstances the corotation torque exerted on intermediate mass objects should essentially be the horseshoe drag.§.§ Thermal diffusion coefficient Previous estimates of the thermal diffusion coefficient have been obtained by using values at the midplane of the disc <cit.>. We have undertaken simulations to check whether heat diffusion can indeed be described by a diffusion coefficient evaluated at the disc's midplane.Following <cit.>, we write the thermal diffusivity, in the absence of scattering, as (their Eq. B.2): χ=D/c_vρ=λ4acT^3/ρ^2c_vκ,where D is the heat diffusion coefficient, λ, the flux-limiter, is 1/3 in the optically thick regions where the planet is located, a is the radiation constant and c_v is the specific heat at constant volume. We can recast Eq. (<ref>) as:χ=16(γ-1)σ T^3/3ρ^2( R/μ)κ,where σ is Stefan's constant. We note that this expression has oftentimes been transformed into an expression involving the disc's thickness H and orbital frequency Ω at the denominator. These expressions, in general, overestimate by a factor γ the thermal diffusivity of Eq. (<ref>). The equation that describes the hydrostatic equilibrium of the disc does not involve the adiabatic index, and it yields the relationship H=c_s/Ω, rather than H=c_s^adi/Ω=√(γ)c_s/Ω, independently of the fact that an energy equation is used to describe the flow. However, we suggest to simply use Eq. (<ref>) rather than the transformed expression, as the latter, which features all the variables of Eq. (<ref>) as well as new ones, does not bring any simplification.Here, we specifically study the radial diffusion of heat, as it is mainly this effect which determines (together with viscous diffusion) the degree of saturation of the horseshoe drag.Namely, we study the radial spread of a radially localised temperature excess in order to determine experimentally the thermal diffusivity of the disc and compare it to the theoretical estimate.In this section only we use a module of radiative transfer in the flux limited diffusion approximation <cit.> with the flux limiter of <cit.>, and solve the equation of radiative energy in addition to that of internal energy (two-temperature approach) as in <cit.>.We use a meridional mesh of size (N_ϕ,N_r,N_θ)=(1,800,43) spanning a radial range of 0.75 au to 3 au, and a range in colatitude [π/2-0.1,π/2]. The surface density at 1 au is Σ_0=1700 gcm^-2, the Rosseland opacity is fixed to 1.8 cm^2g^-1, and the kinematic viscosity is ν=4.46· 10^14 cm^2s^-1.The slope of surface density is α=0.3. The temperature profile of the disc at equilibrium depends on a balance between viscous heating and radiative losses through the disc's photospheres. We firstly run our initial setup, which has an arbitrary aspect ratio, in order to relax the disc toward hydrostatic and thermal equilibria. Once these are reached, we increase by 1 % the temperature of a three-zone wide ring at r=1.5 au (over the whole colatitude range), and restart the simulation (hot ring run). We also perform a “neutral” restart in which the temperature has not been altered. The radial profile of the temperature excess obtained by subtracting the temperature profile of the neutral run from the hot ring run (at the same date), quickly adopts a nearly Gaussian profile:δ T(r,t)∝1/σ(t)exp[-(r-r_0)^2/σ^2(t)].In linear diffusion theory, σ^2(t)=4χ t. We use this relationship to infer the value of χ. Fig. <ref> shows the time behaviour of σ^2, which is nearly linear in time, as expected. From the slope of the linear regression fit, we can estimate that the value of χat r=r_0 is:χ≈ 1.03· 10^15 cm^2 s^-1.An estimate based on the temperature excess averaged in colatitude, rather than measured at the midplane, yields nearly exactly the same result. At r=1.5 au, we have in the neutral run a midplane temperature T=509 K, a density ρ=4.33· 10^-10 gcm^-3, and R/μ=3.615· 10^7 erg K^-1 g^-1. Eqs. (<ref>) yields a value 27 % larger than the value found experimentally.The disc of this numerical experiment is in radiative equilibrium and its only source of heating is viscous friction. Its temperature therefore decays from the midplane to low values near the boundaries in colatitude. We have repeated this numerical experiment with a nearly isothermal vertical profile, obtained by lowering the viscosity by a factor of ten and by imposing a temperature at the upper boundary T_b such that T_b^4=0.9T^4_midplane. The vertical heat flux is therefore lowered by an order of magnitude with respect to the previous disc, the temperature is nearly constant with respect to colatitude, and the resulting disc has same midplane temperature as the previous one. We measured a thermal diffusivity nearly equal to the one of the previous disc, which shows that this quantity is rather insensitive to the vertical profile of temperature, and determined by the midplane temperature.§ NEW TORQUE FORMULA We hereafter summarise our results and provide an updated torque formula for different cases: the general case, the linear regime for a radiative disc, and the linear regime for an isothermal disc. §.§ General case Usually torque expressions are given in terms of the reference torque Γ_0 of Eq. (<ref>). For the horseshoe drag, this expression arises naturally when using the low mass value of the horseshoe width (Eq. <ref>), and an horseshoe drag expression, which involves the product Σ_0Ω_p^2x_s^4. Here, however, we use a more complex law (Eq. <ref>) for the horseshoe width, so that casting our results in terms of Γ_0, for the different components of the corotation torque, would lead to cumbersome expressions. We therefore rather cast the different components of the horseshoe drag in terms of x_s.The total torque is a function of the slopes of surface density and temperature α and β, of the disc's opacity, viscosity, aspect ratio and surface density (respectively κ, ν, h and Σ_0) and of the planet orbital frequency, orbital radius and mass to star mass ratio (respectively Ω_p, r_p and q). It requires the prior evaluation of the thermal diffusivity using Eq. (<ref>), and that of the half width of the horseshoe region x_s, which is given, when the flow is adiabatic or behaves nearly adiabatically over the horseshoe U-turn timescale, by a modification of Eq. (<ref>) that uses the adiabatic sound speed, rather than the isothermal one:x_s=1.05(q/h')^1/2+3.4q^7/3/h'^6/1+2q^2/h'^6r_p,where h'=h√(γ).The total torque is the sum of the Lindblad torque Γ_L and corotation torque Γ_C:Γ_tot=Γ_L+Γ_C.The Lindblad torque is given by:Γ_L=-(2.34-0.1α+1.5β)Γ_0f(χ/χ_c),where the first two coefficients in the factor of the right hand side (2.34 and -0.1) are given by <cit.>, and where an approximate value of the third coefficient (the factor of β) has been obtained in section <ref> and subsequently slightly improved in section <ref>. The function f(x) and critical diffusivity χ_c are given by <cit.> and have respectively the expressions:f(x)=(x/2)^1/2+1/γ/(x/2)^1/2+1,and:χ_c=r_p^2h^2Ω_p.The function f(χ/χ_c) can be regarded as the inverse of an effective adiabatic index γ_eff <cit.>.We now turn to the evaluation of the four terms that compose the corotation torque (see Fig. <ref>). The vortensity component Γ_V^CR of the corotation torque is given by: Γ_V^CR=ε_bΓ_V^HD+(1-ε_b)Γ_V^lin.In this expression, the blending coefficient ε_b, as advertised in section <ref>, has the expression worked out by <cit.>:ε_b=(1+30hz_ν)^-1,where z_ν is given by <cit.>:z_ν=r_pν/Ω_px_s^3.The (vortensity components of the) horseshoe drag Γ^HD_V and linear corotation torque Γ_V^lin are respectively:Γ_V^HD= F_VΓ_V^UHD,andΓ_V^lin=(0.976-0.640α)Σ_0Ω_p^2r_p^4(h')^-2=(0.976-0.640α)Γ_0/γ,where the numerical coefficients of the right hand side of Eq. (<ref>) can be obtained from the data given by <cit.>.In Eq. (<ref>) the two factors of the right hand side are respectively the saturation function of the vortensity component of the horseshoe drag, and the unsaturated horseshoe drag. The former has the expression <cit.>:F_V=8π/3z_ν F(z_ν),where F(x) is defined as:F(x)= {[ 1-x^1/2; 4/(27x) ].and the latter has the expression:Γ^UHD_V=3/4(3/2-α)Σ_0Ω_p^2x_s^4.This last expression, initially established for two dimensional discs <cit.>, has been generalised to three dimensional discs <cit.>. Eqs. (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) correspond to the five cells of the first frame of Fig. <ref>, from left to right and bottom to top.Next, we give the expression of the entropy component of the corotation torque, corresponding to the second frame of Fig. <ref>. It reads[Note that in <cit.>, the thermal diffusivity was denoted by κ, whereas here it is denoted by χ, while κ is the opacity. As a consequence, ε_κ (z_κ) has been changed into ε_χ (z_χ).]: Γ_S^CR=ε_νε_χΓ_S^HD+(1-ε_νε_χ)Γ_S^lin.As for the vortensity torque, we keep for the blending coefficients the expressions given by <cit.>:ε_ν=[1+(6hz_ν)^2]^-1andε_χ=(1+15hz_χ)^-1,where z_χ is defined in a similar fashion as z_ν <cit.>:z_χ=r_pχ/Ω_px_s^3.The linear component of the torque, as seen in section <ref>, is:Γ_S^lin=0.8ξΓ_0/γ,where ξ is given by:ξ=β-0.4α-0.64.The horseshoe drag Γ_S^HD of Eq. (<ref>) is simply the product of the unsaturated horseshoe drag by the saturation function of the entropy corotation torque:Γ_S^HD= F_SΓ_S^UHD.In two dimensions, the expression of the unsaturated horseshoe drag can be written KξΣ_0Ω_p^2x_s^4, where K is a numerical constant <cit.>. We assume this dependency to hold in three dimensions, and seek the value of K using the results of section <ref>. In the numerical exploration that we performed, we had, using Eq. (<ref>), x_s=1.086r_p(q/h')^1/2, from which we infer K=3.3. The unsaturated torque expression is therefore:Γ_S^UHD=3.3ξΣ_0Ω_p^2x_s^4.For the saturation function we keep the dependence given by <cit.>:F_S=1.2×1.4z_χ^1/2×1.8z_ν^1/2,where x=min(1,x). Eqs. (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) correspond to the five cells of the second frame of Fig. <ref>, from left to right and bottom to top.The third frame of Fig. <ref> corresponds to the temperature component of the corotation torque, that we now evaluate:Γ_T^CR=ε_νΓ_T^HD+(1- ε_ν)Γ_T^lin.Since the vortensity distribution associated to this torque component is roughly similar to that of the entropy torque (it is concentrated near the separatrices), we assume that it behaves like the entropy torque as a function of viscosity. However, we discard the dependence on thermal diffusivity. Indeed, in the limit of a large diffusivity, the disc behaves isothermally and the temperature torque subsists (as we saw in section <ref>), on the contrary of the entropy torque. The linear component that features in Eq. (<ref>) reads (see section <ref>):Γ_T^lin=1.0βΓ_0/γ,while the horseshoe drag reads:Γ_T^HD= F_TΓ_T^UHD.From the above discussion we adopt:F_T=1.2×1.8z_ν^1/2.For the unsaturated horseshoe drag, we assume the form K'βΣ_0Ω_p^2x_s^4, i.e. similar to that of the entropy, but with a different numerical constant and a scaling on the temperature gradient. The toy model presented by <cit.> suggests it is reasonable, at least in the low mass limit, as it displays a scaling in x_s^2Δ P, where Δ P is the perturbation of pressure at the stagnation point. This quantity, in the low mass limit, scales as Σ_0Ω_p^2x_s^2 if the stagnation point is typically at a pressure scale length away from the planet. We seek the value of K' using the results of section <ref>. In that section, we had, using Eq. (<ref>) (rather than Eq. <ref> since the disc was isothermal), x_s=1.13r_p(q/h)^1/2. Equating the form given above with that of Eq. (<ref>), we infer K'=k'/1.13^4=0.73 and thus:Γ_T^UHD=0.73βΣ_0Ω_p^2x_s^4.Eqs. (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) correspond to the five cells of the third frame of Fig. <ref>, from left to right and bottom to top. In a fourth and last stage, we evaluate the additional component, arising from the viscous creation of vortensity at the abrupt density jumps that appear at the contact discontinuities at the separatrices of the horseshoe region. For brevity and definiteness we hereafter call this term the viscous coupling term (VCT). <cit.> give an expression of this term (the second part of the bracket of their Eq. 129). This term corresponds to an asymptotic value (at larger time) and therefore already embeds its own saturation function. It naturally scales with the entropy gradient, as it is this gradient which primarily determines the magnitude of the density jumps at the separatrices. As emphasised in section <ref>, the quantity ξ was determined from a best fit of numerical data, and named entropy gradient on the grounds of the very close resemblance of the effect studied in that section with the well-studied two-dimensional process that triggers the appearance of an additional torque component that scales with the radial entropy gradient <cit.>. We therefore suggest that the three-dimensional version of the VCT scales with ξ.Our choice of normalisation of ξ is that the coefficient of β is unity, whereas it is -1/γ with the normalisation of <cit.>. As this torque behaves as a bulk term (it implies a smooth distribution of vortensity from corotation to the separatrices), and since it has no known linear equivalent, we follow <cit.> and assume it to decay like the vortensity component of the horseshoe drag, proportionally to ε_b:Γ_VCT^CR=4πξ/γΣ_0Ω_p^2x_s^4ϵ_bz_ν[z_ν F(z_ν)-z_χ F(z_χ)/z_ν-z_χ]The full corotation torque is finally given by:Γ_C=Γ_V^CR+Γ_S^CR+Γ_T^CR+Γ_VCT^CR.Eqs. (<ref>) to (<ref>), together with Eq. (<ref>), provide an expression for the total tidal torque exerted on a low to intermediate mass planet in circular orbit in an optically thick disc in hydrostatic and radiative equilibrium. §.§ Linear regime in the general case We give hereafter the simpler expression of the torque in the linear regime. It reduces to:Γ_tot^lin=Γ_L+Γ_V^lin+Γ_S^lin+Γ_T^lin,which simplifies as:Γ_tot^lin=Γ_L+(0.46-0.96α+1.8β)Γ_0/γ,where Γ_L is given by Eq. (<ref>). §.§ Linear regime in an isothermal disc We can further simplify this expression in the case of an isothermal disc (in which case we discard the entropy component of the corotation torque). This yields:Γ_tot^lin,iso=-(1.36+0.54α+0.5β)Γ_0.This result is in good agreement with the result obtained by <cit.> in three-dimensional simulations of locally isothermal discs.§ DISCUSSION We compare here the torque value provided by our formula of section <ref> to published studies of the torque value in radiative discs, either as a function of the planetary mass or of the orbital radius. Fig. <ref> shows the comparison to results recently published by <cit.>. They show a broad agreement between the torque measured in their simulations and our torque formula. They also show that the boost of the horseshoe width, described by Eq. (<ref>), is an essential ingredient of the torque formula above 10-15 M_⊕ (i.e. for intermediate mass planets), and that torque formulae based on a low-mass expression of the width of the horseshoe region hardly yield a migration reversal.It should be kept in mind that the torque measured in numerical simulations is subjected to some inaccuracy, as can be seen from the results for 20 M_⊕ (in the first two plots), for which two torques values are displayed. This inaccuracy arises from a variety of numerical effects, such as resolution (in particular the number of zones over which the radial width of the horseshoe region is resolved), or the recipe for the torque calculation (in particular whether it includes all zones or whether it excludes those located near the planet), etc.Yet, some points or parts of the torque curves are outliers with respect to our formula: * At low planetary mass (on the left plot), the torque measured is somehow below the value predicted by our formula. This behaviour was already discussed by <cit.>, and attributed to the “cold-finger” effect <cit.>, which is observed on low-mass planets when there is thermal diffusion in the disc. We note that <cit.> have obtained their Eq. (156) — here Eq. (<ref>) — using a fit of numerical simulations with different values of the thermal diffusivity, and may have unwittingly included a two-dimensional version of the “cold-finger” effect. It is therefore unclear to which extent the discrepancy found at low planetary mass is due to this effect. We also note that for the lowest planetary mass considered here (5 M_⊕), <cit.> found that the cold finger effect nearly vanishes. This issue requires further work, and will be presented elsewhere.* The plateau value for the mass range 15-35 M_⊕ on the middle plot, or the elbow of the left plot (at 38 M_⊕ according to our formula, and at ∼ 50 M_⊕ in the results of <cit.>) are also discrepant features, which are unlikely to be explained by the errors and systematic effects on the torque measurement. Beside the “cold-finger” effect, which might account for the mismatch at low mass, there is a number of simplifications in our torque formula which can explain the residual discrepancies between our torque formula and the results of numerical simulations.We neglect the feed back of the planet's torque on the disc density profile. At intermediate masses, this feed back tends to create a dip around the orbit (precursor of the gap that would be carved at larger masses), which has an effect on the different components of the torque. In particular the effect can be different on the outer Lindblad torque and on the inner Lindblad torque, resulting in a non-trivial effect on the net Lindblad torque. Also, the slight decrease of density can alter the thermal diffusivity at the planetary orbit, by virtue of Eq. (<ref>). On the other hand, as we saw in section <ref>, this equation is accurate only to within ∼ 25 %, and both the saturation functions and the blending coefficients between the linear torque and the horseshoe drag depend sensitively on the value of the thermal diffusivity.Also, we use for these functions the analytic dependence worked out in two dimensions by <cit.>, which can constitute another reason for the discrepancy. Lastly, we have used for all components of the corotation torque the actual adiabatic index of the gas γ, instead of an effective adiabatic index as done for the Lindblad torque <cit.>. We comment that we have performed runs with planets of mass 1 M_⊕ to 20 M_⊕ embedded in radiative discs with different opacities (namely 1.8 cm^2.g^-1 and 5 cm^2.g^-1). We have found that the width of the horseshoe region at low mass corresponds indeed to x_s=1.05√(q/h)/γ^1/4 rather than x_s=1.05√(q/h)/γ^1/4_eff. At larger masses, it displays a growth above this value, as expected for the transition from the low-mass to the high-mass branch of the horseshoe width. Nonetheless, owing to thermal diffusion, the actual law might be more complex than the one of Eq. (<ref>) and this can have an impact on the torque value, which is a very sensitive function of x_s.We finally compare the torque predicted by our formula to that measured by <cit.> in Fig. <ref>. For radii below 2.5 a_Jup, there is a satisfactory agreement between the formula and the numerical simulations. Beyond that radius, however, the predicted torque plummets to low values, whereas the torque measured in the simulations plateaus at an intermediate, negative value. We note that this happens when the disc's optical depth drops below value of a few (at r=a_Jup it is τ_eff∼ 4). Note that the actual optical depth of the disc is somehow lower than the one displayed, as the opacity law adopted by the authors was: κ/(1 cm^2 g^-1)=2.0· 10^-4[T/(1 K)]^2 (Bitsch, private communication). The heat source of the disc being viscous friction, it is hotter at the midplane, and therefore its opacity is largest at the midplane. This example illustrates the fact that Eq. (<ref>), and therefore our torque formula, are valid when the disc is optically thick. Fig. <ref> also shows that torque formulae based on the low-mass scaling of the horseshoe width hardly predict any migration reversal.We comment that the dependence of the corotation torque on viscosity and thermal diffusivity is entirely contained in the saturation functions and blending coefficients. Since those are not reconsidered in the present work, our torque has same dependence on viscosity and thermal diffusivity as the dependence found by <cit.>. Namely, Figs. 13, 17 and 18 of that work show the dependence of the vortensity torque in the viscosity, and the dependence of the entropy torque on the viscosity and thermal diffusivity. The temperature term of the corotation torque was not considered in that work, but it exhibits same dependence on viscosity as the entropy torque.§ CONCLUSION We provide an updated formula for the torque experienced by a low mass (q < 0.2h^3) or intermediate mass planet (0.2h^3≲ q≲ 2h^3) in circular and non-inclined orbit in an optically thick disc. Our torque formula agrees reasonably well with the torque measured in numerical simulations, as shown by comparison with recently published results. One key ingredient of the formula is an accurate expression of the width of the horseshoe region as a function of the planet-to-star mass ratio and of the disc's aspect ratio.Using in our torque formula the low-mass scaling x_s=1.05r_p√(q/h) clearly yields wrong, largely underestimated results for intermediate mass planets. We have used tailored, three-dimensional explorations of the parameter space to update several numerical coefficients that appear in the torque formula. Our work does not consider the effect found at low planetary mass by <cit.> in discs with thermal diffusion, and called by these authors the “cold-finger” effect. A further step toward accurate torque formulae could be an analytic expression of this torque component, as well as a relaxation of the different simplifying assumptions mentioned in section <ref>. § ACKNOWLEDGEMENTSThe authors thank Elena Lega and Bertram Bitsch for an extremely careful reading of a first draft of this manuscript, and for constructive feedback. Alejandra Jiménez acknowledges financial support from a UNAM DGAPA fellowship. The simulations presented in this work were run on a GPU cluster acquired with CONACyT's grant 178377. 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"authors": [
"María Alejandra Jiménez",
"Frédéric S. Masset"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170727182414",
"title": "Improved torque formula for low and intermediate mass planetary migration"
} |
Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA [email protected] Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia Interferometric phase measurement is widely used to precisely determine quantities such as length, speed, and material properties <cit.>. Without quantum correlations, the best phase sensitivity Δφ achievable using n photons is the shot noise limit (SNL), Δφ = 1/√(n). Quantum-enhanced metrology promises better sensitivity, but despite theoretical proposals stretching back decades <cit.>, no measurement using photonic (i.e. definite photon number) quantum states has truly surpassed the SNL. Rather, all such demonstrations — by discounting photon loss, detector inefficiency, or other imperfections — have considered only a subset of the photons used. Here, we use an ultra-high efficiency photon source and detectors to perform unconditional entanglement-enhanced photonic interferometry. Sampling a birefringent phase shift, we demonstrate precision beyond the SNL without artificially correcting our results for loss and imperfections. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances. Unconditional violation of the shot noise limit in photonic quantum metrology Geoff J. Pryde =============================================================================It has been known for several decades that probing with various optical quantum states can achieve phase super-sensitivity, i.e measurement of the phase with an uncertainty below the SNL: <cit.>. It has been shown theoretically that multi-photon entangled states, such as NOON states, may achieve super-sensitivity and can, in principle, saturate the Heisenberg limit, the ultimate bound on sensitivity <cit.>. For this reason, they are of great interest for maximising the information that can be collected per photon, which is useful for investigating sensitive samples <cit.>. NOON states are superpositions of N photons across two arms of an interferometer, each of which is a single optical mode: |Ψ_NOON⟩=1/√(2)(|N⟩|0⟩+|0⟩|N⟩). We use the term photonic to refer to states like this, because they possess definite photon number, and these photons are counted in detection. By contrast, we exclude from term “photonic” schemes using states of indefinite photon number and continuous wave-like measurement, such as squeezed states and homodyne detection.Such techniques have genuinely beaten the SNL, e.g. refs <cit.>, but work over narrow bandwidths and cannot directly achieve the theoretical maximal sensitivity per resource. The key feature of NOON and similar photonic states <cit.> is that they produce interference fringes that oscillate faster any classical interference pattern, a feature called phase super-resolution <cit.>. Super-resolution interference experiments have been reported using two-<cit.>, three-<cit.>, four- <cit.>, six- <cit.> and eight- <cit.> photon states.Super-resolution, however, is not enough by itself to surpass the SNL <cit.>: a high interference fringe visibility, and high transmission and detection efficiency are also required—they must exceed the threshold at which these imperfections cancel the quantum advantage. For imperfect NOON state interferometry, a handy estimate of this thresholdwas introduced in Ref. <cit.>: a genuine quantum advantage requires the interference visibility v and the combined single-photon transmission and detection efficiency η to satisfyη^N v^2 N > 1. (For precise evaluation of the potential for super-sensitivity, the Fisher information can be used to analyse experimental schemes and data <cit.>, as described below.) Here we performed the first phase sensing experiment with N=2 photon NOON states that unconditionally demonstrates phase uncertainty below the SNL. This result was enabled by construction of a spontaneous parametric downconversion (SPDC) source <cit.>, that was optimised for high photon transmission and required no spectral filtering in order to achieve high quantum interference visibility v. This allowed us to fully exploit the benefits of high-efficiency detector configuration <cit.>, yielding an ultra-high heralding efficiency <cit.>—equivalent to the single-photon efficiency η. We emphasise that high-efficiency (95%) detection at 1550 nm wavelength has been available since 2008 <cit.>, but photonic metrology beyond the SNL has not yet been achieved. This is because the capability to reach the required quantum interference and efficiency simultaneously requires a source capable of producing photons of exceptional spatial and spectral purity without filtering. Unlike previous experiments, our measurement apparatus does not require post-selection to achieve phase uncertainty below that achievablein anideal, lossless classical interferometer. For our experimental apparatus, we expected v≈0.98 and symmetrical interferometer arm efficiencies η≈0.8 (which includes the detector efficiency), resulting in η^N v^2 N ≈ 1.23. Thus, we anticipated a violation of the SNL, which we tested in two experiments described below.For fair comparison with the SNL, an accurate accounting of resources is required. In the archetypal NOON-state phase sensing protocol, preparation and use of an N-photon NOON state constitutes a trial. In the ideal case, each trial leads to a detection event at the output of the interferometer. Since each trial gives only a little information about the phase, a number of such trials may be performed. In our work, two-photon NOON states were generated probabilistically at random times by the SPDC source. Each detection event (i.e. any combination of detector registrations) represented a recorded trial. We counted k such detection events to complete the protocol. However, due to imperfect transmission and detection efficiency η, some NOON states did not lead to detections. Furthermore, due to higher-order SPDC events (the occasional simultaneous emission of 4,6,… photons), the resources equivalent to multiple (2,3,…) trials were overlapped in time and could not be distinguished by our non-photon-number-resolving detectors. Therefore, the actual number of trials (i.e. the number of photon pairs passing the phase shift), k̃, was larger than the number of recorded trials. Because the ideal classical scheme is assumed to be lossless and to use all resources passing the phase shift, it must be attributed an effective number of resources n=Nk̃ = 2k̃. This makes the SNL harder to beat. For the loss and downconversion parameters of our experiment, the worst-case estimate, based on the lowest possible value for the overall experimental efficiency, we determined k̃/k=1.048125.Our experimental scheme is shown in Figure 1. We used collinear type-II parametric down-conversion, producing degenerate 1550 nm photon pairs <cit.>. Careful design and implementation of the source's output mode structure allowed us to achieve high fiber-coupling efficiency and state-of-the-art superconducting nanowire single photon detectors (SNSPDs) <cit.> provided high detection efficiency.The down-conversion process generates two photons in the |1⟩_H|1⟩_V polarisation state (H≡ horizontal; V≡ vertical) in the same spatial mode, which can be written as the NOON-polarisation state |Ψ⟩=1/√(2)(|2⟩_L|0⟩_R+|0⟩_L|2⟩_R). These right- (R) and left-circular (L) polarisation modes constituted the two arms of the interferometer. A half-wave plate (HWP) set at an angle φ/4 relative to its optic axis was used to implement the birefringent phase shift φ between the arms. A common misconception about two-photon NOON states generated from SPDC is that the same phase sensitivity can be usefully achieved by using a pump photon (at half the wavelength) instead of the two-photon entangled state. However, this is clearly not correct for sensing in any material with dispersion.After the phase shift, the modes were interfered on a polarising beam-splitter (PBS) and the output counting statistics were detected with SNSPDs and analysed with coincidence or time-tag logic. The output signal consisted of three possible types of detection outcomes: C_11, a coincidence detection between both output modes; C_20, a detection occurring only in the transmitted output mode; and C_02, a detection occurring only in the reflected output mode. The numbers of each type of detection in a time period τ were, respectively, c_11(φ), c_20(φ) and c_02(φ).In order to test and calibrate our setup we first measured interference fringes. Detection events (≈250000 per phase value) were collected for a fixed amount of time for various φ∈ [0,2π). We observed an interference visibility of(98.9±0.02)%, calculated from fitting to the c_11(φ) detection fringe. The transmissions of the reflected and transmitted outputs of the interferometer were measured to be η_r=(79.41±0.09)% and η_t=(80.26±0.09)%, calculated from c_11(0)/(c_11(0)+c_20(0))and c_11(0)/(c_11(0)+c_02(0)) ratios, respectively (a slight variation of transmission was observed when HWP was rotated, see Methods for details). Calculated transmissions include all the loss in the setup and the non-unit detection efficiency of SNSPDs. Probability fringesp_11(φ), p_20(φ) andp_02(φ) were then obtained by fitting detection signals, c_i(φ), i ∈{11,20,02}, which were appropriately normalised for each phase value. We used the Fisher information per recorded trial,ℱ=∑_i(∂lnp_i/∂φ)^2p_i where i∈{11,20,02}, to quantify the phase sensitivity of our phase measurement setup <cit.>. Our results (Fig. 2b) show a clear violation, for a range of phase values φ, of the adjusted SNL bound that takes into account the information in unrecorded trials: ℱ_SNL=Nk̃/k=2.09625.For the second experiment, we performed phase sensing for individual settings of the phase shifter within the range where we expected to beat the SNL. At each setting, time-tag hardware was used to acquire detection events corresponding to k=10000 trials. From the distribution of C_i events (i ∈{11,20,02}), corresponding P_i probabilities were obtained by normalisation.This set of three probabilities corresponds to a single phase estimate value φ^ est. In practice, finding this estimate required to minimise the squared difference between the measured probabilities and their corresponding calibration curves, p_i(φ) (Fig. 2a).The phase search range was restricted to φ^ est∈[0,π/2].This process was repeated for s=14520 samples (for each angle of the HWP), determining φ^ est_j for each sample j. The mean and standard deviation of the mean (standard error of the mean) {φ^ est_j} are shown in Fig. 3. This measurement procedure was repeated for a range of phase values around the region of interest. When compared to the standard deviation of the mean (=1/√(n^ tot))that is achievable with n^ tot=ns=Nk̃s=304375500 classical resources (adjusted for loss and higher order terms, as before) per data point, our results show a clear advantage of our quantum approach.In conclusion, we demonstrated unconditional violation of the SNL in photonic phase sensing. We recover the estimate of the phase shift applied to the mode, and its corresponding standard deviation of the mean, directly from our measurement data without additional adjustments.Moreover, all the parameters necessary for the calculation of the SNL of our measurement apparatus, such as circuit loss, pair and multi-pair generation probability, were calculated directly from our measurement data. Our results are not only of fundamental interest, but are also directly applicable to a phase measurement scenario where low photon flux is required, such as measurement of light-sensitive materials <cit.>. We note that the N=2 NOON state is also the N=2 Holland-Burnett state; thus, we anticipate that our technique can be extended, using loss-tolerant approaches <cit.> and number-resolving detectors <cit.>, to perform sub-SNL sensing with significantly higher photon numbers in the near future. It can also be applied to multipass metrology protocols <cit.>.§ METHODS §.§ Photon source and characterisationHigh heralding efficiency (Klyshko efficiency <cit.>) and high interference visibility were essential to the success of this demonstration. These features were achieved by using a spontaneous parametric down-conversion (SPDC) source that operated at the group velocity matching conditions to generate frequency uncorrelated photon pairs, removing the need for spectral filtering that is typically required in conventional SPDC sources. The heralded single photon source was pumped by a mode-locked Ti:Sapphire laser with 81 MHz repetition rate, 775 nm wavelength and 6 nm FWHM bandwidth. SPDC from a nonlinear periodically poled KTP crystal (pp-KTP, poling period 46.20 μ m) phase matched for type-II collinear operation produced degenerate photon pairs centred at 1550 nm wavelength and with ≈ 15 nm FWHM bandwidth. We used a pump waist size of 170 μ m and a signal and idler collection waist size of 50 μ m <cit.>. Together with the high efficiency SNSPDs <cit.>, this allowed us to achieve symmetric heralding efficiencies of (82± 2)%.An additional 1 mm KTP crystal, with optic axis rotated by 90^∘ with respect to the optic axis of pp-KTP, was placed after the downconverter to compensate for the temporal walk-off (due to birefringence) in the 2 mm pp-KTP crystal. Non-classical interference visibilities between the signal and idler photons were measured to be (98.9±0.2)%, with no background subtraction. §.§ DetectionThere is one superconducting nanowire single photon detector (SNSPD) coupled to each of the interferometer outputs. We record three types of events: C_11, a coincidence detection between both output modes; C_20, a detection occurring only in the transmitted output mode; and C_02, a detection occurring only in the reflected output mode. Since the SNSPDs cannot resolve photon number, the latter two events may arise from: the case where both photons in the two-photon NOON state went to the same detector; a case where one photon was lost and therefore only one photon is recorded; or the case of a dark count. We do not artificially remove signals from these latter two cases. We note that the rate of dark counts is small compared to the rate of real events.§.§ Resource countingThe total transmission (including detection efficiency) is easily determined when the phase shift is set to zero. However, we have also observed that the circuit transmission can slightly vary with the rotation of the HWP that implements the phase shift. In order to verify transmission at every value of φ deviates only slightly from this, we build a theoretical model of the interferometer, which includes an imperfect interference on the PBS and loss in two of the output modes. Then, at each rotation angle of the HWP we find transmission of both output modes (η_r(φ) and η_t(φ)) by finding the closest theoretical match to the measured data with least squares minimisation. We confirm that the transmission variation stays below ≈1%. Moreover, the standard deviation in total number of detection events per point in Fig. 2, which were acquired over a fixed amount of time, was measured to be ≈0.2%, confirming that the variation of our setup parameters was very small.In order to estimate the true number of resources we use, we calculated the probability of having at least one photon transmitted through the system. For the three possible outcomes, this is given byη_11(φ)=1-(1-η_t(φ))(1-η_r(φ)), η_20(φ)=1-(1-η_t(φ))(1-η_t(φ)), η_02(φ)=1-(1-η_r(φ))(1-η_r(φ)).The lowest efficiency η_min=min_φ,j(η_j(φ)), for j∈{11,20,02} and φ∈[0,2π), observed over entire range of φ out of the three outcomes was η_min≈95.56%. We chose the lowest efficiency η_min(φ)=min_j(η_j(φ)), calculated at each angle of the HWP (i.e. at each φ) in order to calculate the number of trials accordingly. This results in an overestimate of the number of photons used in our experiment—and therefore used to calculate the effective SNL—setting a higher SNL bound than was actually the case, and making it harder to violate the SNL. The uncertainty in the estimated transmission and the SNL bound was calculated by error propagation of the uncertainty in c_i through the theoretical model of the interferometer. We have observed very strong dependence of the estimated parameters on c_11(φ) near the φ=π/2+kπ, k∈ℤ, resulting in an increased parameter uncertainty in those regions.Another important source of additional resources that are not directly detected in the protocol is the possible emission of multiple photon pairs in the SPDC process. 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Lett. volume 116, pages 143601 (year 2016)NoStopAcknowledgements This work is supported by Australian Research Council grant DP140100648. The authors thank Joseph Ho for help with SNSPDs. Author contributionsG.J.P. conceived the idea and supervised the project. S.S. and M.M.W. constructed and carried out the experiment with help from H.M.C. L.K.S., V.B.V. and S.W.N.developed the high-efficiency SNSPDs. All authors discussed the results and contributed to the manuscript.Additional informationThe data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Correspondence and requests for materials should be addressed to G.J.P. Competing financial interestsThe authors declare no competing financial interests. | http://arxiv.org/abs/1707.08977v2 | {
"authors": [
"Sergei Slussarenko",
"Morgan M. Weston",
"Helen M. Chrzanowski",
"Lynden K. Shalm",
"Varun B. Verma",
"Sae Woo Nam",
"Geoff J. Pryde"
],
"categories": [
"quant-ph",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170727180135",
"title": "Unconditional violation of the shot noise limit in photonic quantum metrology"
} |
[email protected] 1Department of Physics, Graduate School of Science and Engineering, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan2Research Center for Space and Cosmic Evolution, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan3Academia Sinica Institute of Astronomy and Astrophysics,PO Box 23-141, Taipei 10617, Taiwan4Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan5Dipartimento di Fisica e Astronomia, Università di Firenze, Via G.Sansone 1, I-50019 Sesto Fiorentino, Italy 6INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5,I-50125 Firenze, ItalyLow-metallicity active galactic nuclei (AGNs) are interesting to studythe early phase of the AGN evolution.However most AGNs are chemically matured and accordinglylow-metallicity AGNs are extremely rare.One approach to search for low-metallicity AGNs systematically is utilizingthe so-called BPT diagram that consists of the [O iii]λ5007/Hβλ4861 and[N ii]λ6584/Hαλ6563 flux ratios.Specifically, photoionization models predict that low-metallicity AGNs show a high [O iii]λ5007/Hβλ4861 ratio and a relatively low [N ii]λ6584/Hαλ6563 ratio, that corresponds to the location between the sequence of star-forming galaxies and that of usual AGNs on the BPT diagram (hereafter “the BPT valley”).However, other populations of galaxies such as star-forming galaxies and AGNs with a highelectron density or a high ionization parameter could be also located in the BPT valley, not only low-metallicity AGNs. In this paper, we examine whether most of emission-line galaxies atthe BPT valley are low-metallicity AGNs or not. We select 70 BPT-valley objectsfrom 212,866 emission line galaxies obtained by the Sloan Digital Sky Survey.Among the 70 BPT-valley objects, 43 objects show firm evidence of the AGN activity; i.e., the He iiλ4686 emissionand/or weak but significant broad Hα emission. Our analysis shows that those 43 BPT-valley AGNs are not characterized by a very high gas density nor ionization parameter, inferring thatat least 43 among 70 BPT-valley objects (i.e., >60%) are low-metallicity AGNs.This suggests that the BPT diagram is an efficient tool to search for low-metallicity AGNs.§ INTRODUCTION The active galactic nucleus (AGN) is one of the most luminous class of objectsin the Universe, whose huge radiative energy is released through the massaccretion onto the supermassive black hole (SMBH). The mass of SMBHs(M_ BH) is tightly correlated with the mass or the stellar velocitydispersion of their host galaxies <cit.>,implying that SMBHs andgalaxies have evolved with closely interacting in each other (the so-calledco-evolution of SMBHs and galaxies). However, the physics behind theco-evolution is still unclear. To understand the total picture of theco-evolution, examining the scaling relations for AGNs in the early phase of theco-evolution is an interesting approach since different theoretical models predictdifferent redshift dependences of scaling relations <cit.>. One simple strategy to explorethe early phase of the co-evolution is measuring the scaling relations at highredshifts, where the typical age of AGNs is much younger than low-redshiftAGNs. Many attempts have been made for measuring the scaling relations forhigh-redshift AGNs <cit.>, and a higher M_ BH with respect to the massor velocity dispersion of host galaxies has been sometimes reported<cit.>. On the other hand, there are some reportsclaiming that such a possible evolution in the scaling relation is a result ofobservational bias through the sample selection <cit.>. Measuring the properties of AGN host galaxies at highredshift is generally very challenging, that prevents us from assessing the scaling relations at high redshifts.Another possible approach to study the early phase of the co-evolution isfocusing on young AGNs at low redshifts, where detailed observations are mucheasier than high redshifts. In this context, low-metallicity (i.e., chemically young)AGNs in the low-redshift Universe are particularly interesting. However, thetypical metallicity of AGNs inferred for broad-line regions (BLRs) and narrow-line regions (NLRs) is high (Z ≳ 2 Z_⊙; e.g.,) and low-metallicity AGNs are very rare <cit.>. <cit.> proposed a method to search for AGNs with a low-metallicity NLR, that utilizesan optical emission-line diagnostic diagram which consists of the flux ratios of [N ii]λ6584/Hαλ6563 and [O iii]λ5007/Hβλ4861. This diagnostic diagram was originally investigated for classifying emission-line galaxies into star-forming galaxies and Seyfert 2 galaxies (BPT diagram, ). <cit.> established the “maximum”starburst line in the BPT diagram by combining stellar population synthesismodels and photoionization models. On the other hand, <cit.> derived empirical classification criteria for star-forminggalaxies while <cit.> derived empiricalclassification criteria for low-ionization nuclear emission-line regions (LINERs; ), using emission-line data taken from the databaseof Sloan Digital Sky Survey (SDSS; ).<cit.> pointed out that AGNs with a low-metallicityNLRs (i.e., characterized by the solar or sub-solar metallicity) should have a flux ratio of[O iii]λ5007/Hβλ4861 as high as usual AGNs (∼ 10^0.5-10^1) but have an intermediate flux ratio of [N ii]λ6584/Hαλ6563 between usual AGNs and low-mass (i.e., low-metallicity) star-forming galaxies (∼ 10^-1-10^-0.5).This is because the nitrogen relative abundance is in proportion to the metallicitydue to its nature as a secondary element <cit.>. Inthe BPT diagram, there are only few objects located at the region characterizedby a high flux ratio of [O iii]λ5007/Hβλ4861 andan intermediate flux ratio of [N ii]λ6584/Hαλ6563 (hereafter “BPT valley”; see Figure <ref>).<cit.> specifically focused on AGNs with a low-mass host galaxy (i.e., M_ host < 10^10 M_⊙), and then they selectedlow-metallicity AGNs using another diagnostic diagram that consists of [N ii]λ6584/[O ii]λ3727 and [O iii]λ5007/[O ii]λ3727 flux ratios. However, it is not clear whether low-metallicity AGNs should be always found in a sample of AGNs with a low-mass host galaxy. Also, the method adopted by <cit.> requires a wide wavelength coverage (λ_ rest∼ 3700-6600 Å),that is not convenient for future applications to expand the search of low-metallicity AGNs toward the high-redshift Universe. Therefore, we focus on BPT-valley selection (requiring a moderately narrowwavelength coverage; λ_ rest∼ 4800-6600 Å) to selectlow-metallicity AGNs without any host-mass cut. However, there is a potentially serious problem in the BPT-valley selection foridentifying low-metallicity AGNs.That is, not only low-metallicity AGNs are located in the BPT valley.As <cit.> showed, star-forming galaxies witha very hard radiation field or high-density H ii regions are expectedto be seen in the BPT valley (see also, e.g., ).Also, star-forming galaxies with a high ionization parameter<cit.>, a high nitrogen-to-oxygenabundance ratio (N/O; e.g., ),or shocks <cit.> are also expected to be seen in the BPT valley.Not only star-forming galaxies, AGNs with a high electron density or high ionization parameter(i.e., not characterized by a low metallicity) could be also seen in the BPT valley<cit.>.Therefore, it is not completely clear whether the BPT-valley objects are really low-metallicity AGNsand whether the BPT diagram is a useful tool to search for low-metallicity AGNs.This problem prevents us from selecting chemically-young AGNs observationally.In this paper, we investigate the optical spectra of BPT-valley objects forexamining whether most of emission-line galaxies at the BPT valley are reallylow-metallicity AGNs or not. Through this examination, it will be tested whetherthe optical BPT diagram is an efficient and appropriate method to search forlow-metallicity AGNs. In Section 2, we present our selection procedure of theBPT-valley sample. In Section 3, we show how we identify BPT-valley AGNs toavoid contaminating star-forming galaxies at the BPT valley. In Section 4, weinvestigate gas properties of the selected BPT-valley AGNs such as electrondensity and ionization parameter, for examining whether the BPT-valley AGNs are characterized by a low metallicity or not.In Section 5, we disccus physical properties of the BPT-valley AGNs. Section 6 describes the summary of this work. § SAMPLE In order to select the BPT-valley objects, we use Max-Planck-Institute forAstrophysics (MPA)-Johns Hopkins University (JHU) SDSS Data Release 7() galaxy catalog[http://www.mpa-garching.mpg.de/SDSS/].The MPA-JHU DR7 catalog of spectral measurements contains various spectral properties such as emission-line fluxes and their errors, based on the analysis for 927,552 objects without showing dominant broad Balmer lines(i.e., star-forming galaxies, composite galaxies, LINERs, and type-2 Seyfert galaxies)in the SDSS DR7. Our sample selection is based on the following procedure (the flow chart of oursample selection process is shown in Figure <ref>).First, we select the initial sample according to the following criteria.We require the reliable redshift measurement (i.e., z_ warning = 0) and also z> 0.02.This redshift limit is required to cover [O ii]λ3727. This results in 906,761 galaxies. Then we require a signal-to-noise ratio (S/N) > 3 for some key emission lines, i.e.,Hβλ4861, [O iii]λ5007,[O i]λ6300, Hαλ6563,[N ii]λ6584 and [S ii]λλ6717, 31 (212,866 galaxies).Next, we classify these 212,866 galaxies and extract the BPT-valley sampleaccording to the following steps.* Using theempirical line, log([OIII]/ Hβ) > 0.61/ log ([NII]/Hα) -0.05+1.3, for removing usual star-forming galaxies (56,217 galaxies).* Using themaximum starburst line, log([OIII]/ Hβ) > 0.61/ log ([NII]/Hα) -0.47+1.19, for removing so-called composite galaxies (22,865 galaxies).* Using theempirical criterion, log([OIII]/ Hβ) > 1.36log([OI]/ Hα) + 1.4, for obtaining Seyfert sample by removing LINERs (14,253 galaxies).* Adopting the following criterion, log([NII]/ Hα) < -0.5, for finally selecting the BPT-valley sample (71 galaxies).Note that 1 object in the 71 BPT-valley objects was observed twice andduplicated in the final sample, i.e., the final BPT-valley sample consists of 70 objects.The BPT-valley criterion (Equation <ref>) is determined empirically,by taking account of the frequency distribution of the [N ii]λ6584/Hαλ6563flux ratio of Seyfert galaxies.Figure <ref> shows the [N ii]λ6584/Hαλ6563 frequency distribution ofSeyfert galaxies,where the average and standard deviation of the logarithmic [N ii]λ6584/Hαλ6563flux ratio are -0.058 and 0.145, respectively. Accordingly, the 3 σ bounding from the average value is -0.493,and therefore we adopt the threshold to categorize BPT-valley objects as describedby Equation <ref>. Figure <ref> shows the finally selected 70 BPT-valley objectsin the BPT diagram that consists of [N ii]λ6584/Hαλ6563 versus[O iii]λ5007/Hβλ4861. Table 1 shows the basic properties of the selected BPT-valley objects.ccrrrrrrrrcThe BPT-valley sample 0ptID SDSS Name Plate MJD Fiber z Hβλ4861 [O iii]λ5007 [O i]λ6300 Hαλ6563 [N ii]λ6584(1) (2) (3) (4) (5)(6) (7) (8) (9) (10)(11)1.......... SDSS J102310.97-002810.8 0272 51941 0238 0.11274 253.972279.29 125.19 881.10 235.202.......... SDSS J111006.26-010116.5 0278 51900 0096 0.10949 382.921837.07 129.44 1484.61438.763.......... SDSS J230321.73+011056.4 0380 51792 0565 0.18136 235.381692.71 12.29816.35 247.334.......... SDSS J024825.26-002541.4 0409 51871 0150 0.02467 10.35 41.33 4.71 47.6514.92 5.......... SDSS J073506.37+393300.8 0432 51884 0316 0.03479 29.46 163.578.55 109.51 21.85 6.......... SDSS J023310.78-074813.4 0455 51909 0388 0.03097 97.71 707.2012.83433.29 104.887.......... SDSS J092907.78+002637.3 0475 51965 0205 0.11732 259.862602.18 56.121173.36274.708.......... SDSS J090613.76+561015.1 0483 51902 0016 0.04668 188.361144.93 73.96621.46 189.469.......... SDSS J144328.78+044022.0 0587 52026 0374 0.11411 36.11 300.133.37 137.53 32.98 10......... SDSS J114825.71+643545.0 0598 52316 0189 0.04169 349.081347.75 60.171302.50392.2411......... SDSS J213439.57-071641.9 0641 52199 0487 0.06377 215.602079.35 78.83800.93 182.9412......... SDSS J001050.35-010257.4 0686 52519 0020 0.11299 223.891669.57 109.74 940.89 275.9013......... SDSS J124738.52+621243.1 0781 52373 0076 0.12112 171.82915.8033.03697.15 162.9114......... SDSS J131659.37+035319.9 0851 52376 0219 0.04541 127.051399.96 80.91698.40 136.7315......... SDSS J115908.55+525823.1 0881 52368 0623 0.06644 392.772812.16 52.131218.45379.5316......... SDSS J104632.21+543559.7 0906 52368 0169 0.14475 83.71 938.6419.52375.92 101.4617......... SDSS J104600.36+061632.0 1000 52643 0035 0.18447 146.67637.7431.28586.86 177.7218......... SDSS J101945.65+520608.6 1008 52707 0378 0.06492 29.60 115.176.75 92.2928.00 19......... SDSS J205111.11+000913.2 1023 52818 0393 0.06644 7.3744.11 4.92 37.119.6820......... SDSS J214930.43+010509.4 1031 53172 0369 0.11399 28.14 210.353.43 109.29 31.13 21......... SDSS J081653.27+285423.1 1206 52670 0530 0.05740 246.451610.19 131.99 1145.34203.6022......... SDSS J095319.42+422912.2 1217 52672 0349 0.22343 207.671835.79 83.29698.87 115.0923......... SDSS J110504.94+101623.5 1221 52751 0359 0.02076 104.02466.3929.14481.20 151.6224......... SDSS J114440.53+102429.3 1226 52734 0435 0.12688 51.67 304.7339.93241.38 67.83 25......... SDSS J124110.10+104143.7 1233 52734 0611 0.15613 214.491004.84 29.37703.65 188.7126......... SDSS J092620.42+352250.3 1274 52995 0147 0.24729 118.98868.1881.24513.89 75.74 27......... SDSS J131756.07+491531.3 1282 52759 0390 0.09231 215.302050.51 105.88 711.65 176.3028......... SDSS J090107.41+085459.2 1300 52973 0335 0.08380 172.611116.69 153.30 822.23 210.3729......... SDSS J120134.05+581421.1 1313 52790 0527 0.04636 28.83 136.205.78 87.4320.49 30......... SDSS J152723.47+334919.1 1354 52814 0044 0.09116 44.63 173.4624.18254.04 79.05 31......... SDSS J112314.89+431208.7 1365 53062 0119 0.08005 56.52 226.8733.20183.08 52.19 32......... SDSS J152328.09+313655.6 1387 53118 0210 0.06850 382.471707.84 67.321300.98322.8933......... SDSS J120900.89+422830.9 1448 53120 0075 0.02364 100.07614.2424.50320.70 61.56 34......... SDSS J121839.40+470627.6 1451 53117 0190 0.09389 478.355046.81 101.68 1861.68380.4535......... SDSS J005231.29-011525.2 1496 52883 0089 0.13485 251.992400.52 55.68826.34 254.4436......... SDSS J011341.11+010608.5 1499 53001 0522 0.28090 191.992118.06 38.27779.72 162.7937......... SDSS J001901.52+003931.8 1542 53734 0375 0.09669 58.06 296.6915.99242.69 58.75 38......... SDSS J034019.39+002530.6 1632 52996 0467 0.35296 40.58 375.4010.78167.65 46.21 39......... SDSS J032224.64+401119.8 1666 52991 0048 0.02608 121.081388.24 50.70428.76 130.7740......... SDSS J135855.82+493414.1 1670 53438 0061 0.11592 56.67 385.5614.74189.76 50.29 41......... SDSS J160452.78+344540.4 1682 53173 0201 0.05493 87.33 437.2037.34364.99 111.9442......... SDSS J132011.71+125940.9 1698 53146 0327 0.11398 25.43 174.934.92 92.9924.68 43......... SDSS J143523.42+100704.1 1711 53535 0306 0.03122 128.18530.6115.31473.30 123.1644......... SDSS J072637.94+394557.8 1733 53047 0326 0.11141 505.823357.26 23.231744.03120.4245......... SDSS J095914.76+125916.4 1744 53055 0385 0.03432 1298.59 8418.94 361.09 4396.861088.11 46......... SDSS J113714.22+145917.2 1755 53386 0463 0.03484 74.19 364.4725.99276.40 77.15 47......... SDSS J120847.79+135906.7 1764 53467 0013 0.29030 136.33659.3128.11554.94 137.9248......... SDSS J135429.05+132757.2 1777 53857 0076 0.06332 312.803422.56 111.50 1005.37306.9449......... SDSS J130431.99+061616.7 1794 54504 0046 0.06283 184.841202.46 41.38702.29 217.2450......... SDSS J134316.52+101440.1 1804 53886 0433 0.08132 186.67960.51170.90 635.68 198.2151......... SDSS J160032.89+052608.8 1822 53172 0012 0.11653 281.021630.88 69.221204.67280.0952......... SDSS J081212.84+541539.8 1871 53384 0060 0.04417 93.10 809.7334.28326.93 94.82 53......... SDSS J084038.99+245101.6 1931 53358 0396 0.04334 137.37770.1139.75579.80 151.5054......... SDSS J122451.88+360535.4 2003 53442 0112 0.15094 25.95 148.2211.11126.62 35.77 55......... SDSS J134237.37+273251.3 2017 53474 0127 0.04947 12.67 97.69 10.2760.0217.32 56......... SDSS J140952.03+244334.6 2128 53800 0358 0.05215 45.42 220.7612.29198.30 41.19 57......... SDSS J142535.21+314027.1 2129 54252 0618 0.03324 91.61 362.1142.31323.51 95.49 58......... SDSS J145505.97+211121.1 2148 54526 0122 0.06751 82.30 441.8012.31437.94 126.6159......... SDSS J083200.51+191205.8 2275 53709 0472 0.03753 549.646069.28 42.4615422.57 419.5860......... SDSS J103731.01+280626.9 2356 53786 0468 0.04263 54.57 447.1024.55216.95 65.48 61......... SDSS J104403.52+282628.3 2356 53786 0618 0.16286 225.461047.00 17.43794.20 193.3062......... SDSS J104724.40+204433.5 2478 54097 0541 0.26515 102.51751.7037.17391.58 66.16 63......... SDSS J160635.22+142201.9 2524 54568 0498 0.03245 162.66621.8341.72517.08 160.4464......... SDSS J171901.28+643830.8 2561 54597 0345 0.08954 152.42709.9721.69586.95 174.3265......... SDSS J084658.44+111457.5 2574 54084 0382 0.06296 130.82638.1541.04557.91 161.2966......... SDSS J095745.49+152350.6 2584 54153 0442 0.05183 96.42 702.5524.58514.46 117.3767......... SDSS J133014.91+242153.9 2665 54232 0388 0.07151 50.49 244.8415.74284.46 76.98 68......... SDSS J135007.07+164227.2 2742 54233 0551 0.13043 99.01 903.0538.82495.52 137.0069......... SDSS J153941.67+171421.9 2795 54563 0509 0.04583 157.66758.8514.76500.33 119.4070......... SDSS J143730.46+620649.4 2947 54533 0227 0.21862 66.97290.5422.39219.25 65.39 Col. (1): Identification number assigned in this paper.Col. (2): Object name.Col. (3)–(5): Plate-MJD-Fiber ID in the SDSS observation for analyzed spectra.Col. (6): Redshift measured by the SDSS pipeline. Col. (7)–(11): Emission-line fluxes in units of 10^-17 erg s^-1 cm^-2.§ SELECTION OF SECURE-AGN SAMPLE As described in Section 1, the BPT-valley sample potentially includes star-forming galaxies withspecial gas properties, not only AGNs.Thus we first select objects showing secure evidence of the AGN from the BPT-valley sample.Specifically, we regard objects showing at least one of the following two featuresin their SDSS spectra as secure AGNs; (1) a broad Hαλ6563 emission,and (2) a He iiλ 4686 emission line.Details of the selection procedure of secure AGNs are given below.§.§ Broad Hαλ6563 emission line The velocity profile of recombination lines is a powerful tool to examinethe presence of AGNs, since star-forming galaxies never show a velocitywidth wider than ∼1000 km s^-1 in full-width at half maximum (FWHM).Generally the optical spectra of type-1 AGNs show broad permitted lines whosevelocity width is ≳ 2000 km s^-1 emitted from BLRs.The origin of recombination lines with FWHM ∼ 1000 - 2000 km s^-1 isnot very clear, since such lines may arise at BLRs in so-called narrow-lineSeyfert 1 galaxies (NLS1s; e.g., Osterbrock & Pogge 1985) orat NLRs in type-2 AGNs with a relatively large velocity width(such as NGC 1068 and NGC 1275; see, e.g., ).However, in either case, the detection of recombination lines withFWHM > 1000 km s^-1 strongly suggests the presence of AGNs.Therefore we search for the broad Hαλ6563 componentin the optical spectrum of the BPT-valley objects. Here we do not search forthe broad component of the Hβλ4861 emission,since it is intrinsically fainter than that of the Hαλ6563 emissionand it is sometimes affected significantly by the Fe ii multiplet emission <cit.>.We use an IRAF routine specfit <cit.> to find the broadHαλ6563 component. Specifically, we fit the SDSS optical spectrum of the BPT-valley objectsin the range of λ_ rest = 6200-6800 Å with andwithout the broad Hαλ6563 component, and examinewhether the addition of the broad component improves the spectral fit significantly.The details of the fitting procedure are as follows.First, we fit the optical spectrum with a linear continuum component andsingle-Gaussian emission-line components for [O i]λ6300,[O i]λ6363, [N ii]λ6548, Hαλ6563,[N ii]λ6584, [S ii]λ6717, and [S ii]λ6731(hereafter “nobroad fitting”).Here we assume that the velocity width of all emission lines is the same,and the relative separation of the emission lines is fixed to be the same asthat of their laboratory wavelengths.The flux ratios of [O i]λ6300 to [O i]λ6363 and[N ii]λ6584 to [N ii]λ6548 are fixed to be 3.00 and 2.96respectively <cit.>, and the flux ratios among the remainingemission lines are kept to be free.Then, we add a broad component for the Hαλ6563 emission tothe nobroad fit, where the flux, wavelength center and width of this additionalcomponent are kept to be free (hereafter “broad fitting”).Here we recognize that the additional broad component significantly improvesthe fit by the following criterion:χ̃^2_ nobroad-χ̃^2_ broad/χ̃^2_ nobroad>0.4,where χ̃^2_ nobroad and χ̃^2_ broad are the reducedchi-square of the nobroad fitting and broad fitting, respectively.Note that the threshold, 0.4, is determined empirically, so that the resultbecomes consistent with the visual inspection.As a result, 13 BPT-valley objects with a broad component are identified from the 70 BPT-valley objects. Figures <ref> and <ref> show the SDSS spectrum with the best-fit resultfor the BPT-valley objects with a broad Hαλ6563 component.Figure <ref> shows an example of objects (ID = 48) whose fitting result does not satisfy thecriterion defined by Equation <ref> (for this case, the improvement of the fit is slightly less thanthe threshold, 0.32). Note that we regard object ID = 8 as an object with a broad Hα component,though the FWHM of the broad Hα component is less than1000km s^-1 (Figure <ref>).This is because this object shows [Fe vii]λ6087 and [Fe x]λ6374lines, that are seen only when the AGN presents. Note that such high-ionization forbiddenemission lines are preferentially seen in type-1 AGNs<cit.>.Note that the [Fe vii]λ6087 line is seen in 8 objectswhile [Fe x]λ6374 line is seen in 3 objects(including ID = 8, note that 2 objects in addition to ID = 8 show both [Fe vii]λ6087and [Fe x]λ6374).The spectral properties of the BPT-valley objects with a broad Hαλ6583component are summarized in Table 2. Only 1 BPT-valley object (ID = 25) shows the broad Hβ component among the 13 BPT-valley objects showing a broad Hα component (see Figure <ref>). §.§ He iiλ4686 emission lineThe presence of a He iiλ4686 emission line indicates the existence ofthe hard ionizing radiation since the ionization potential forHe^+ is 54.4 eV.This hard radiation is naturally produced by AGNs.Therefore, the He iiλ4686 emission line is a good indicator of AGNs.We examine whether the SDSS optical spectrum of the BPT-valley objects showthe He iiλ4686 line by the visual inspection,since the He iiλ4686 information is not given in the MPA-JHU database.As a result, 38 BPT-valley objects with the He ii emission line are identified fromthe 70 BPT-valley objects.Some of the SDSS spectra of BPT-valley objects with the He ii detection are shownin Figures <ref>, while those without the He ii detection are shownin Figure <ref>.§.§ Classification result of the BPT-valley sampleThe results of the classification of the BPT-valley objects are summarized in Table 3. Among the 70 BPT-valley objects, 8 objects show both broad Hα component and He ii emission line, that are now confirmed to be AGNs. There are 5 objects showing the broad Hα component but without He ii emission line, that are also regarded as AGNs. The non-detection of the He ii line is likely due to insufficient signal-to-noise ratio, since the He ii line is very weak. In addition, 30 objects show the He ii line but without broad Hα component, that are thought to be typical type-2 AGNs. Here we should mention that the stellar absorption lines (mainly Hα) are not considered in our fitting procedure. Though the stellar Hα absorption line could impact the narrow component of the Hα emission, the absorption effect is negligible for examining the presence of the broad Hα component. This is because the equivalent width of the detected broad Hα component is higher than 20 Å (the median value of EW_rest(Hα)_b is 44.74 Å, Table 2) while the typical equivalent width of the stellar Hα absorption is ∼ 2-3 Å in nearby galaxies <cit.>.Note that the detected He ii line is not caused by Wolf-Rayet stars, because the typical velocity width of the detected He ii line is not broad (≲ 1000km s^-1). Therefore, at least 43 among the BPT-valley objects are regarded as AGNs. There may be some additional AGNs in the remaining 27 objects, possibly owing to insufficient S/N to detect any AGN indicators in their spectra. Instead, some of those 27 objects could be non-AGNs, i.e., star-forming galaxies with a relatively high N/O ratio or fast shocks. We do not discuss further about those 27 objects since the main interests of this work are on the BPT-valley AGN sample. Accordingly, we conclude that at least 43 objects of the BPT-valley sample (or ∼ 60%, but probably more) are confirmed to be AGNs. As described in Section 3.1, at least one of the [Fe vii]λ6087 and [Fe x]λ6374 lines are seen in 9 BPT-valley objects. Interestingly, a large fraction of objects showing both the broad Hα component and He ii emission show such high-ionization iron lines (5 among 8 objects). On the other hand, objects showing neither the broad Hα component nor He ii emission never shows those high-ionization iron lines. Then, a few objects in the remaining two classes show high-ionization iron lines (4 among 35 objects). This may infer that our classification is well tracing the presence of the AGN, but the absence of high-ionization iron lines could be simply due to a low S/N ratio of the spectra. Figure <ref> shows how various populations of galaxies classified in this work are populated in the BPT diagram. There are no significant segregation except for two BPT-valley objects whose [N ii]λ6584/Hαλ6563 flux ratio is very low, < 0.1. Both of these two galaxies show no broad Hα component nor He ii line, which is consistent with the idea that these two objects are not low-metallicity AGNs but somewhat extreme low-metallicity galaxies, characterized probably by a very high ionization parameter and/or very hard ionization radiation. lrrrrr 0pt 6Broad-line AGNs in the BPT valleyID f( Hα)_n f( Hα)_b FWHM_Hα FWHM_[SII] EW_ rest(Hα)_b(1) (2) (3) (4) (5) (6)6cbroad Hα and He ii3......... 991.96 1834.96 7162.24 245.05 88.108......... 414.20 309.70 875.811 247.31 27.0212........ 524.87 830.66 1682.70 324.79 57.2513........ 748.78 728.55 2033.36 223.68 72.8325......... 607.56 930.70 1974.28 248.66 159.8834........ 2072.41 797.10 2083.37 377.39 33.0665........ 573.27 761.34 4830.02 263.52 44.6166........ 448.38 768.76 2430.79 221.86 47.036cbroad Hα and noHe ii17........ 719.07 392.29 3396.57270.94 44.7421........ 1014.89 1145.11 2440.19 337.97 44.1347........ 515.66 726.01 3569.90 273.99 246.7758........ 390.35 801.51 4347.42 307.51 28.8367........ 276.51 335.10 2372.32 276.41 25.14 1Classified as an object with a broad Hα component through the FWHMof the additional Hα component is less than 1000 km s^-1 (see the main text).Col. (1): Identification number assigned in this paper.Col. (2): Flux of the nallow component of Hα in units of 10^-17 erg s^-1 cm^-2. Col. (3): Flux of the broad component of Hα in units of 10^-17 erg s^-1 cm^-2. Col. (4): FWHM of the broad component of Hα in units of km s^-1. Col. (5): FWHM of the [S ii]λ6717 (i.e., narrow component) in units of km s^-1. Col. (6): Rest-frame equivalent width of the broad component of Hα in units of Å. lcc 3 0pc Classification result of the BPT-valley samplebroad nobroad He II8 30 noHe II 5 27 § SELECTION OF LOW-METALLICITY AGNS The 43 BPT-valley objects confirmed to be AGNs are not necessarily low-metallicity AGNs,because AGNs with a very high electron density or very high ionization parameter arealso expected to be populated in the BPT valley as mentioned in Section 1.More specifically, the [N ii]λ6584 emission in AGNs with a density higher thanthe critical density of the [N ii]λ6584 transition (∼8.7 × 10^4 cm^-3)is significantly suppressed due to the collisional de-excitation effect.On the other hand, a very high ionization parameter results in a higher relative ionic abundance ofN^2+ (i.e., a lower relative ionic abundance of N^+), that results in a weaker[N ii]λ6584 emission. Therefore, in this section, we examine whether the 43 BPT-valley AGNs are characterizedby a very high electron density or very high ionization parameter or not, and test whetherthe AGNs in the BPT-valley are characterized by low-metallicity gas or not. §.§ Electron densityThe emission-line flux ratios of [S ii]λ6717/λ6731 and[O ii]λ3729/λ3726 are famous good indicators of the electron density<cit.>. In this work, we use the [S ii]λ6717/λ6731 line ratio to estimate electron density,because the wavelength separation of the [O ii] doublet is too small to be well resolved withthe SDSS spectral resolution. We use an IRAF routine temden for deriving the electron density from the[S ii]λ6717/λ6731 ratio, by assuming the electron temperature of 10,000 K.Here we derive the electron density whose [S ii]λ6717/λ6731 ratio iswithin the range of 0.5–1.4.No BPT-valley objects show the [S ii]λ6717/λ6731 ratio lower than 0.5(i.e., the high-density limit) while 11 among the 70 BPT-valley objects show the flux ratiohigher than 1.4 (i.e., the low-density limit).Among 14,252 Seyfert sample, only 12 objects show the [S ii]λ6716/λ6731ratio lower than 0.5 while 2,880 objects show the flux ratio higher than 1.4.Figure <ref> shows the frequency distribution of the inferred gas density for objects whose[S ii]λ6717/λ6731 ratio is within the range of 0.5–1.4; i.e.,41 BPT-valley AGNs(showing a broad Hα component and/or He ii emission), 59 BPT-valley objects(including objects without any AGN signatures), and 11,360 Seyfert galaxies.Here we show the histograms for both BPT-valley AGNs and BPT-valley objects, because some ofBPT-valley objects without any AGN signatures could be also AGNs (see Section 3.3).The median density of the BPT-valley AGNs, BPT-valley objects, and Seyfert galaxies are210 cm^-3, 210 cm^-3, and 270 cm^-3, respectively.In order to investigate whether the frequency distribution of the gas density is statistically differentamong the samples, we apply the Kolmogorov-Smirnov (K-S) statistical test with a null hypothesis thatthe frequency distribution of the gas density of two classes of objects comes fromthe same underlying population.The derived K-S probability for the BPT-valley AGNs and Seyferts is 0.207, while that forthe BPT-valley objects and Seyferts is 0.146.These results strongly suggest that the BPT-valley sample is not characterized by the higher gas densitywith respect to the Seyfert sample. §.§ Ionization parameterThe ionization palameter is the ratio of the number density of hydrogen-ionizing photonsto that of Hydrogen atoms.In order to investigate the ionization parameter, the [O iii]λ5007/[O ii]λ3727flux ratio is a useful indicator because this ratio does not suffer significantly from chemical properties ofthe gas in both AGNs and star-forming galaxies<cit.>. Note that this flux ratio is sensitive also to the gas density if the density is higher thanthe critical density of [O ii] (∼10^3.5 cm^-3),but the typical density of NLRs inferred from the [S ii] doublet ratio is much lower thanthat as described in Section 4.1. Though the dust reddening is not corrected to study the BPT diagram due to the small wavelengthseparation of emission-line pairs used for the BPT diagram (Section 2), we should correct for the reddeningeffect to investigate the [O iii]λ5007/[O ii]λ3727 flux ratio.For this correction, we assume R_V = A_V/E(B-V) = 3.1 and the intrinsic flux ratioof Hαλ6584/Hβλ4861 = 3.1, and adopt the reddening curveof <cit.>.Figure <ref> shows the histogram of the [O iii]λ5007/ [O ii]λ3727 lineratio of the BPT-valley AGNs, BPT-valley objects, and Seyferts, with S/N([O ii]λ3727) > 3. Here it should be noted that the BPT-valley objects showlog([O iii]λ5007/Hβλ4861) > 0.5 by definition,while the Seyfert galaxies could have much lower [O iii]λ5007/Hβλ4861 flux ratiosdown to ∼ -0.2.This may introduce a selection effect in the sense that strong [O iii] emitters could be selectivelyincluded in the BPT-valley sample.Therefore, for reducing this selection effect, only objects withlog([O iii]λ5007/Hβλ4861) > 0.5 are examined for assessingthe ionization parameter.After adopting this additional criterion, the numbers of the BPT-valley AGNs, BPT-valley objects, andSeyferts examined in Figure <ref> are 42, 69, and 8,500, respectively.This figure shows that the BPT-valley samples seem to show systematically higher[O iii]λ5007/[O ii]λ3727 flux ratios than the Seyfert sample.The median values of the logarithmic [O iii]λ5007/[O ii]λ3727 flux ratios ofthe BPT-valley AGNs,BPT-valley objects, and Seyferts are 0.67, 0.65, and 0.46, respectively.In order to investigate whether or not the distributions of the[O iii]λ5007/[O ii]λ3727 line ratio arestatistically different between BPT-valley sample and Seyfert sample, we apply the K-S statistical test.The K-S probability that the underlying distribution of these two distributions is the same is 3.925× 10^-6 for the BPT-valley AGNs and Seyferts,while 1.803 × 10^-5 for the BPT-valley objects and Seyferts.These results suggest that the BPT-valley samples have statistically higher[O iii]λ5007/[O ii]λ3727 flux ratios, i.e., the ionization parameter,than the Seyfert sample.Note that it is well known that low-metallicity galaxies are generally characterized by a relativelyhigh ionization parameter, at least for star-forming galaxies <cit.>.It may be interesting that the BPT-valley objects show a clear edge at the lower side of the[O iii]λ5007/[O ii]λ3727 distribution in Figure <ref>. However, probably this feature is not statistically significant, because the number of BPT-valley objectsis not enough to discuss the tail of the frequency distribution of the[O iii]λ5007/[O ii]λ3727 flux ratio.In the next subsection, we will examine whether or not this difference in the ionization parameter can beresponsible for the lower [N ii]λ6584/Hαλ6563 ratio observed in theBPT-valley samples with respect to the Seyfert sample.§.§ Model calculationsAs shown in Section 4.2, the ionization parameter of the BPT valley sample is higher than that ofthe Seyfert sample.Since it is interesting to examine either the BPT-valley AGNs are characterized by a low metallicity ora high ionization parameter, we perform photoionization model calculations.We perform photoionization model calculations for simulating the NLR of AGNs,using the code CLOUDY version 13.03 <cit.>.Here the main parameters for CLOUDY calculations are as follows<cit.>:* The hydrogen density of the cloud (n_ H). * The ionization parameter (U). * The chemical composition of the gas. * The shape of the input SED.We calculate photoionization models covering the following ranges of parameters: 10^1cm^-3≤ n_ H≤ 10^6cm^-3 and 10^-4≤ U ≤ 10^-1.We set the gas-phase elemental abundance ratios to be the solar ones.The adopted solar abundances relative to hydrogen are taken from <cit.>with extensions by <cit.>. The adopted metallicity (i.e., the solar one) is not typical for usual Seyfert galaxies(whose NLR metallicity is generally higher than the solar metallicity), possibly nor BPT-valley AGNs(that could have sub-solar metallicity).However, as described below, it is useful to fix the metallicity to examine whether the ionizationparameter alone can account for the difference in the emission-line flux ratios betweenBPT-valley objects and Seyferts.For the input SED, we adopt the following one:f_ν=ν^α_UVexp( -hν/kT_ BB) exp( -kT_ IR/hν)+ aν^α_Xas a typical spectrum of AGNs (see ).kT_ IR is the infrared cutoff of the big-blue bump, and we adoptkT_ IR=0.01 ryd <cit.>.α_ UV is the slope of the low-energy side of the big-blue bump. We adopt α_ UV = 0.5, which is typical for AGNs<cit.>.α_ ox is the UV–to–X-ray spectral slope, which determines the parameter a in equation (6). We adopt α_ ox=-1.35, which is the average value of nearby Seyfert 1 galaxies <cit.>. α_ x is the X-ray slope, and we adopt α_ x=-0.85 (see ). T_ BB is the characterizing the shape of the big-blue bump, and we adopt 490,000 K(see ). The calculations end at the depth where the temperature falls to 3,000 K,below which gas does not contribute significantly to the flux of optical emission lines.Figure <ref> shows the results of the photoionization model calculations, overlaidon the BPT diagram.Though the density effect is not significant in the range of10^1 cm^-3 < n_ H < 10^5 cm^-3,we can see the effect of the collisional de-excitation at n_ H > 10^4 cm^-3.However, this figure suggests that the difference in the [N ii]λ6584/Hαλ6583flux ratio is more easily explained by the difference in the ionization parameter rather than by the differencein the gas density.More specifically, a higher ionization parameter by 0.5–1 dex in the BPT-valley objects with respect tothe Seyfert sample is required to explain the lower [N ii]λ6584/Hαλ6583flux ratio of the BPT-valley objects. For examining whether the BPT-valley objects have a higher ionization parameter than the Seyfert sample,we investigate another diagnostic diagram that consists of the emission-line flux ratios of[O iii]λ5007/[O ii]λ3727 and [O i]λ6300/[O iii]λ5007(Figure <ref>).This diagram is useful to examine the effect of ionization parameter without suffering fromthe metallicity effect, because only oxygen lines are used and thus less sensitive to the metallicity.Figure <ref> shows that the BPT-valley sample and Seyfert sample have a similar gas density,that is consistent with our analysis presented in Section 4.1.More interestingly, Figure <ref> shows that the BPT-valley sample shows a systematicallyhigher ionization parameter than the Seyfert sample, but the inferred difference in the ionizationparameters is only less than 0.5 dex.This strongly suggests that the lower [N ii]λ6584/Hαλ6563 flux ratioin the BPT-valley sample with respect to the Seyfert sample is not explained by the ionization parameter(nor the gas density, as described in Section 4.1).Therefore we conclude that the BPT-valley AGNs are characterized by a systematlcally lower metallicitythan the Seyfert sample, as originally proposed by <cit.>. § DISCCUSIONSAs mentioned Section 1, low-metallicity AGNs are interesting to study the early phase ofthe AGN evolution. However low-metallicity AGNs are very rare, so that little has been reported on physical property of low-metallicity AGNs.In this section, we present some basic properties of BPT-valley objects which are expectedto be low-metallicity AGNs. §.§ Stellar massNaively it is expected that the stellar mass of low-metallicity AGNs is expected to berelatively low, as suggested by the mass-metallicity relation seen in star-forming galaxies<cit.>. Accordingly <cit.> introduced a mass criterion(i.e., M_* < 10^10M_⊙) to select low-metallicity AGNs.However, it is not clarified whether low-metallicity AGNs should be always found ina sample of AGNs with a low-mass host galaxy.Therefore, in this paper, we select low-metallicity AGNs without stellar-mass cut andinvestigate the mass distribution of host galaxies of low-metallicity AGNs.Here the stellar mass has been measured and given in the MPA-JHU DR7 catalog<cit.>. Among the 43 BPT-valley AGNs and 70 BPT-valley objects, the host mass isavailable for 39 and 64 objects, respectively.Figure <ref> shows the histogram of the stellar mass of the 39 BPT-valley AGNs,64 BPT-valley objects and 13,662 Seyferts. The median of the stellar mass of the BPT-valley AGNs, BPT-valley objectsand Seyferts are 10^10.15M_⊙, 10^10.07M_⊙ and10^10.77M_⊙, respectively.This result clearly shows that the stellar mass of the BPT-valley AGNs is systematically lowerthan that of Seyferts.However, interestingly, a substantial fraction of the BPT-valley AGN (23 among 39 objects)are actually hosted by galaxies with M_* > 10^10M_⊙,suggesting that low-metallicity AGNs are not necessarily hosted by low-mass galaxies.Note that such low-metallicity AGNs with a relatively massive host galaxy cannot be selectedby the criteria of <cit.> due to the mass criterion ofM_* < 10^10M_⊙.Such low-metallicity AGNs hosted by a relatively massive host galaxy may be realized bytaking into account of the inflow of low-metallicity gas from the surrounding environment <cit.>.§.§ Electron temperatureConsidering the effect of the metal cooling,low-metallicity AGNs are expected to be characterized by the higher electron temperature. Hence we investigate the [O iii]λλ(4959+5007)/[O iii]λ4363 lineratio which is very sensitive to the gas temperature. Here it should be mentioned that, [O iii]λλ(4959+5007)/[O iii]λ4363 lineratio also depends on the electron density <cit.>.Therefore we investigate the[O iii]λλ(4959+5007)/[O iii]λ4363 and[S ii]λ6717/[S ii]λ6731 line ratios simultaneously in Figure <ref>.Here this figure shows the emission-line flux ratios of BPT-valley objects and Seyferts but only forobjects with a significant detection of the [O iii]λ4363 line (S/N > 3). As described in Section 4.2, only objects with log ([O iii]λ5007/Hβ) > 0.5 are used(that results in 9,043 Seyferts and 70 BPT-valley objects). Note that [O iii]λλ(4959+5007)/[O iii]λ4363 line ratio is correctedfor the reddening effect in the same way as Section 4.2. The median values of log ([S ii]λ6717/[S ii]λ6731) of the BPT-valley AGNs,BPT-valley objects and Seyferts with a [O iii]λ4363 detection are 0.088, 0.088 and 0.055,respectively. Therefore the electron density of the BPT-valley sample is slightly higher than that of Seyfertsas already mentioned in Section 4.1.The median of log ([O iii]λλ(4959+5007)/[O iii]λ4363) ofthe BPT-valley AGN, BPT-valley objects and Seyferts are 1.77, 1.77 and 1.79, respectively. This result suggests that the electron temperature of the BPT-valley objects is not significantly higherthan that of Seyferts.However, the fraction of objects showing a significant (S/N > 3) [O iii]λ4363 emissionis very different between the Seyferts and BPT-valley objects.More specifically, 44 among the 70 BPT-valley objects (∼ 63 %) show the [O ii]λ4363emission while only 1,516 among 9,043 Seyferts (∼ 17 %) show the [O iii]λ4363 line.This difference infers that generally the gas temperature of the NLR in BPT-velley objects tends to beso high that the [O iii] λ4363 line is detected in most cases,while the typical gas temperature of the NLR in Seyferts may be lower than that in BPT-valley objectsand only highly biased objects with a relatively high temperature in the Seyfert sample show the[O iii]λ4363 line.This result is consistent to our expectation that the BPT-valley objects is actually characterized by arelatively high gas temperature, due to the low gas metallicity. § CONCLUSIONS In this paper, we focus on low-metallicity AGNs (Z_ NLR ≲ 1 Z_⊙)which are very rare but important since they are in the early phase of the galaxy-SMBH co-evolution. Specifically, in this work it is examined whether the BPT-valley selection is an effective and reliableway to identify low-metallicity AGNs, as proposed by <cit.>.The main results are as follows:* We select 70 BPT valley sample which expected low metallicity AGN from14,253 Seyfert galaxies of MPA-JHU SDSS DR7 galaxy catalog. * Out of 70 BPT-valley objects, 43 objects show clear evidence of the AGN based on the detection of the broad Hα component and/or He iiλ4686 emission.* The typical gas density of the BPT-valley sample (∼210 cm^-3) is not higher than that ofthe Seyfert sample (∼270 cm^-3), suggesting that the lower[N ii]λ6584/Hαλ6563 ratio in the BPT-valley AGNs with respect tothe Seyfert sample is not caused by the collisional de-excitation effect. * The higher [O iii]λ5007/[O ii]λ3727 ratio in the BPT-valley sample(∼4.5) with respect to that in the Seyfert sample (∼2.9) suggests a typically higherionization parameter of the BPT-valley sample; however, photoionization models suggest thatthe inferred difference in the ionization parameter between the BPT-valley sample andSeyfert sample is not enough to explain the observed lower[N ii]λ6584/Hαλ6563 ratio of the BPT-valley sample.* The BPT-valley selection for identifying low-metallicity AGNs is thus confirmed to be a useful method;in our analysis, more than 60% of the BPT-valley sample are low-metallicity AGNs(Z_ NLR ≲ 1 Z_⊙).We would like to thank the anonymous referee for her/his careful reading this paper and useful suggestions, and also Masaru Kajisawa and Kazuyuki Ogura for their useful comments. TN is financially supported by JSPS grants Nos. 25707010, 16H01101, and 16H03958.KM is also supported by JSPS grant No. 14J01811.Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. | http://arxiv.org/abs/1707.08731v1 | {
"authors": [
"Kota Kawasaki",
"Tohru Nagao",
"Yoshiki Toba",
"Koki Terao",
"Kenta Matsuoka"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727073356",
"title": "Active Galactic Nuclei with a Low-Metallicity Narrow-Line Region"
} |
Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Max-Planck-Institut fr Physik komplexer Systeme, Nthnitzer Strasse 38, 01187 Dresden, GermanyNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23,10691 Stockholm,SwedenOptical conductivity (OC) can serve as a measure of correlation effects in a wide range of condensed matter systems. We here show that the long-range tail of the Coulomb interaction yields a universal correction to the OC in a three-dimensional Weyl semimetal σ(Ω)=σ_0(Ω)[ 1+1/N+1], where of σ_0(Ω)=Ne^2_0 Ω/(12 h v) is the OC in the non-interacting system, with v as the actual (renormalized) Fermi velocity of Weyl quasiparticles at frequency Ω, and e_0 is the electron charge in vacuum. Such universal enhancement of OC, which depends only on the number of Weyl nodes near the Fermi level (N), is a remarkable consequence of an intriguing conspiracy among the quantum-critical nature of an interacting Weyl liquid, marginal irrelevance of the long-range Coulomb interaction and the violation of hyperscaling in three dimensions, and can directly be measured in recently discovered Weyl as well as Dirac materials. By contrast, a local density-density interaction produces a non-universal correction to the OC, stemming from the non-renormalizable nature of the corresponding interacting field theory. Optical conductivity of an interacting Weyl liquid in the collisionless regime Vladimir Juričić December 30, 2023 ==============================================================================§ INTRODUCTION Optical conductivity (OC) stands as an indispensable experimental probe of electromagnetic response in a wide range of materials, including high-T_c cuprate superconductors <cit.>, heavy fermion compounds <cit.>, Fe-based superconductors <cit.>, graphene <cit.> and three-dimensional Weyl and Dirac systems <cit.>. This is so because charge dynamics has a direct impact on the OC, which then thus provides a rather comprehensive picture of electronic band structure, low-energy quasiparticle dynamics and nature of correlations in these systems. In topological semimetals, which have recently attracted ample attention <cit.>, the imprint of electronic interactions on the OC may be important because undoped Weyl and Dirac semimetals at zero temperature (T=0) are inherently quantum critical states living in three dimensions [see Fig. <ref>], where hyperscaling is violated <cit.>. Concomitantly, the thermodynamic potentials carry anomalous logarithmic corrections <cit.>. In addition, the long-range tail of the Coulomb interaction in these critical systems is marginally irrelevant, leading to a logarithmically slow vanishing of the fine structure constant due to the screening of the Coulomb charge and a simultaneous, also logarithmically slow growth of the Fermi velocity.As we show, the Coulomb interaction causes a universal (independent of frequency and the fine structure constant) enhancement of the OC in an interacting Weyl liquid, arising from a subtle interplay between its marginal irrelevance and the violation of hyperscaling in three dimensions. The OC (σ) at frequency Ω is given byσ(Ω) = σ_0 (Ω) [ 1+ 1/N+1],after we account for the leading order correction due to the Coulomb interaction, and σ_0(Ω)∼Ω/v is the OC in the noninteracting Weyl semimetal, featuring N Weyl nodes in the Brillouin zone, with v as the renormalized (experimentally measured) Fermi velocity of the Weyl quasiparticles at frequency Ω in the interacting system. This is the central result of our work. Although Coulomb interaction enhances the OC, for sufficiently large number of Weyl nodes (N ≫ 1), the interaction driven correction to the OC scales as ∼ 1/N, which then vanishes as N →∞. Such peculiar scaling stems from the dynamic screening of the electronic charge by massless Weyl fermions in the medium. Hence, the scaling in Eq. (<ref>) can be viewed as the leading term of a systematic and controlled 1/N-expansion of the OC in an interacting Weyl semimetal. Since the long-range Coulomb interaction is expected to be always marginally irrelevant [see, for example, Ref. <cit.> for such conclusion in two dimensions], we are compelled to believe that interaction mediated enhancement of OC possibly remains valid beyond the leading order in 1/N and thus should be observable in recently discovered Weyl and Dirac materials <cit.>. Recent experiment on ZrTe_5 <cit.>, a predicted Dirac semimetal <cit.>, found a large enhancement of the OC. By contrast, a weak short range interaction is an irrelevant perturbation at the Weyl or Dirac quantum critical point (QCP), see Fig. <ref>, and provides only a non-universal correction to the OC which rapidly vanishes as Ω→ 0 [see Eq. (<ref>)]. In the next section we introduce the low-energy theory of an interacting Weyl liquid, and discuss the scaling of OC in non-interacting system. In this section, we also briefly review the renormalization group flow of Coulomb interaction. Sec. <ref> is devoted to the discussion on the scaling of OC and its correction due to Coulomb interaction to the leading order. We summarize our findings and present discussion on related systems (such as Dirac semimetals) in Sec. <ref>. Technical details of our analysis are displayed in the Appendices <ref>-<ref>.§ INTERACTING WEYL FERMIONS AND OPTICAL CONDUCTIVITY Weyl semimetal can be envisioned as the simplest example of a QCP, separating an electron- and a hole-doped chiral Fermi liquids, that supports linearly dispersing sharp low-energy quasiparticles, with dispersion E_ k=v | k|, up to a high energy cutoff E_Λ [see Fig. <ref>]. The Weyl QCP is therefore characterized by the dynamical exponent z=1, which determines the relative scaling between energy and momentum. The corresponding Euclidean action isS_0= ∫ dτ d r ψ^†(τ,r)[∂_τ± (-i) v σ·∇ + μ] ψ(τ,r),with τ as the imaginary time and ± denoting the two chiralities of the Weyl cones which on a lattice always appear in pairs <cit.>. Here, μ is the chemical potential, measured from the apex of the conical dispersion, σs are standard Pauli matrices acting on the two-component spinors ψ(τ,r) representing (pseudo-)spin. The chemical potential with positive scaling dimension [μ]=z=1 is the relevant perturbation at Weyl QCP point that controls a quantum phase transition (QPT), characterized by the correlation length exponent ν=1, from a hole- to an electron-doped chiral Fermi liquid. Together, these two exponents (ν and z) define the universality class of this QPT, as well as determine the crossover boundaries at frequency Ω^∗∼ v|n|^1/3 or temperature T^∗∼(ħ v/k_B)|n|^1/3, among various phases in terms of the carrier density, n; see Fig. <ref> <cit.>. The signature of Weyl fermions in transport and thermodynamic quantities can therefore be observed for Ω>Ω^∗ and T> T^∗. Specifically, we here focus on the OC of such critical Weyl liquid in the collisionless regime (Ω≫ T), with T=0 from outset.The scaling form of the OC (σ) can be inferred from the gauge invariance which dictates that[σ]=d-2 exactly <cit.>, or σ(Ω)=σ_Q ℓ^-1 in units of quantum conductance σ_Q=e_0^2/h. Here, ℓ is a characteristic length scale inside the Weyl critical fan [shaded region in Fig. <ref>] at finite frequency, and thus ℓ∼ v/Ω. The OC of a noninteracting Weyl liquid is then given by σ_0(Ω)= σ_Q c_0 Ω/v, with c_0=N/12 as a universal number <cit.> and the system behaves as a power-law insulator, since σ_0(Ω→ 0) → 0 [see Appendix <ref>].In the presence of generic density-density interaction, captured by the imaginary-time actionS_ int=∫ dτ d r d r' ρ(τ, r)V( r- r')ρ(τ, r'),where ρ(τ, r)=ψ^†(τ, r)ψ(τ, r) is the electronic density, the correction to the OC depends crucially on its range. For the long-range Coulomb interaction V( r- r')=e^2/| r- r'|, and thus [e^2]=z-1, implying that the dimensionless coupling is the fine structure constant α=e^2/v. Furthermore, in the reciprocal space the Coulomb interaction V( k)∼ e^2/k^2 is an analytic function of the momentum and therefore charge is dynamically screened by Weyl fermions, as opposed to the situation in two dimensions <cit.>, which together with the logarithmically slow increase of the Fermi velocity makes the fine structure constant marginally irrelevant in a Weyl fluid. These key features, even though believed to be true in general, can qualitatively be appreciated from the leading order flow equations for v and α respectively given by <cit.> [see Appendix <ref>]dv/dl= α v/3π, d α/dl=-N+1/3 πα^2,where l=log(E_Λ/Ω) is the logarithm of the renormalization group length scale. On the other hand, for a contact interaction V( r- r')=g_0δ( r- r'), the scaling dimension of the coupling is [g_0]=z-D <cit.>, which makes it irrelevant close to the Weyl QCP in D=3. Thus, the dimensionless short range coupling g=g_0Ω^2/v^3, yielding dg/dl=-2 g to the leading order. Consequently, while its long-range tail provides the leading correction to physical observables in the noninteracting system, such as the OC as we demonstrate here, the short-range pieces of the Coulomb interaction give rise to only subleading corrections.§ SCALING AND CORRECTION TO OPTICAL CONDUCTIVITY General scaling arguments suggest that OC in an interacting Weyl liquid assumes the following form in terms of the renormalized couplings and Fermi velocityσ(Ω,α,g)=σ_0(Ω)F(α,g),where F(g,α) is a universal scaling function, with F(0,0)=1. We then recover the OC in the noninteracting system. Since in three spatial dimensions hyperscaling hypothesis is violated, the above scaling function receives logarithmic corrections (besides the usual power-law ones), which to the order n in the perturbation theory have the formF_n(g,α)=∑_m=0^n (α^n C_n,m+g^n G_n,m)log^m(E_Λ/Ω),with C_n,m and G_n,m as the real coefficients, and F(g,α)=∑_n≥0F_n(g,α). We here determine these coefficients perturbatively to the leading order in the coupling constants (α and g), i.e. for n=1.To find the OC in the presence of interactions, we first compute the correction to the current-current correlation function Π_μν(iΩ, q), with Ω as Matsubara frequency, q as momentum, which is the Fourier transform of Π_μν(τ, r)=⟨ j_μ(τ, r) j_ν(0,0) ⟩, and μ,ν=0,1,2,3. Here, “four"-current j_μ(τ, r)=[ρ(τ, r), j(τ, r)], with the spatial components j(τ, r)=v ψ^†(τ, r)σψ(τ, r). The charge conservation -i∂_τρ+∇· j=0 then implies-(iΩ)^2Π_00(iΩ, q)+q_l q_m Π_lm(iΩ, q)=0,with l,m=1,2,3, which constraints physically relevant regularizations of the theory. We here employ the dimensional regularization scheme in spatial dimensions D=3-ϵ, as it manifestly preserves the U(1) symmetry of the theory <cit.> and obtainΠ_00(iΩ, q)=q^2Π̃(iΩ),Π_lm(iΩ,0)=-δ_lmΩ^2Π̃(iΩ),withΠ̃(iΩ)=N/12π^2v[ e^2/6π v(1/ϵ^2+b/ϵ) + g_0Ω^2/24π^2v^3(1/ϵ^2+a/ϵ) ],where a=[5-3γ_E+3log(4π)]/3≈ 3.62069, b=-[1+2γ_E-2log(4π)]/2≈ 1.454, and γ_E≈0.577 is the Euler-Mascheroni constant [see Appendix <ref>, <ref>, <ref>]. The terms proportional to 1/ϵ∼log(E_Λ /Ω) capture the logarithmic divergent pieces of the current-current correlator. This result is consistent with the charge conservation condition, displayed in Eq. (<ref>). Subsequently, we use the Kubo formulaσ_lm(Ω)=2πσ_Qlim_δ→0Π_lm(iΩ→Ω+iδ)/Ωto find interaction driven leading order correction to the OC, as given by Eq. (<ref>), with [see Appendix <ref>, <ref>, <ref>]C_1,1=1/3π, C_1,0=-b/6π,G_1,1=-1/12π^2, G_1,0=a/24π^2.We note that computation of the current-current correlator using the hard cut-off method violates the charge conservation condition <cit.>, and may lead to unreliable values of the above coefficients <cit.>.Finally, we recall that the couplings entering the scaling function [see Eq. (<ref>)] are the renormalized ones and thus scale dependent. From the leading order renormalization group flow equations [see Appendix (<ref>)], we obtainα(Ω) ≈3 π/(N+1) log(E_Λ/Ω),g(Ω)=ĝ_0 (Ω/E_Λ)^2,where ĝ_0 is the dimensionless bare short-range coupling. Then together with Eqs. (<ref>), (<ref>), (<ref>), above running couplings in turn yield the leading correction to the OC in a Weyl liquid, givingσ(Ω) =σ_0(Ω)[1+1/N+1-b/2 (N+1) log(E_Λ/Ω)..-ĝ_0Ω^2/12π^2 E^2_Λ{log(E_Λ/Ω)-a/2}],which simplifies to Eq. (<ref>) for Ω≪ E_Λ.Therefore, the long-range tail of the Coulomb interaction yields a universal (independent of frequency and the strength of the fine structure constant) correction to the OC of the noninteracting Weyl fluid, which is a remarkable consequence of an intriguing conspiracy among the quantum-critical nature of a Weyl semimetal [see Fig. <ref>], marginal irrelevance of the long-range Coulomb interaction [see Eq. (<ref>)] and the violation of hyperscaling in three dimensions [see Eqs. (<ref>), (<ref>) and (<ref>)]. In particular, the logarithmically slow decrease of the fine structure constant at low energy precisely cancels the perturbatively obtained logarithmic correction to the OC, ultimately producing a finite result. This outcome is staunchly suggestive of the renormalizability of the field theory describing an interacting Weyl liquid in the presence of only long-range Coulomb interaction. Note that such correction is operative in the entire quantum-critical regime of the Weyl fluid, shown in Fig. <ref> [the shaded regime], making our prediction relevant for real Weyl materials where chemical potential often (if not always) is placed away from the band touching diabolic points. For an analogous problem in two dimensions, the fine structure constant is also marginally irrelevant, but hyperscaling holds, giving rise to a positive, but logarithmically slowly vanishing correction to the OC <cit.>. On the other hand, the short-range piece of the Coulomb interaction in a Weyl fluid produces only a power-law correction of the conductivity, which ultimately vanishes in the Ω→ 0 limit. Moreover, the explicit dependence of this correction on the ultraviolet cutoff (E_Λ) cannot be eliminated through a redefinition of the bare coupling constant, reflecting non-renormalizability of the field theory of (quasi-)relativistic fermions coupled via short-range interaction in three dimensions <cit.>. In other words, the lattice details, such as the bare strength of the coupling and the ultraviolet cutoff (E_Λ), enter this contribution to the conductivity, which makes the correction to the OC due to local interaction a nonuniversal quantity. Otherwise, the short-range piece causes a reduction in the OC, which is qualitatively consistent with the fact that it can trigger a QPT from a Weyl semimetal to a translational-symmetry breakingaxionic insulator, when sufficiently strong <cit.>. The Coulomb correction to OC scales as ∼ 1/N for N ≫ 1 and therefore vanishes in the limit when the Brillouin zone accommodates a large number of Weyl points (N →∞), since in this limit the Coulomb interaction suffers complete dynamic screening by massless Weyl fermions. Furthermore, using the general scaling argument we can speculate that the n^ th order correction goes as ∼ 1/N^n, with the coefficients that remain to be determined for n>1. Nevertheless, existence of a plethora of Weyl compounds with diverse flavor number (N), such as N=24 and N=6 respectively in all-in all-out and spin-ice ordered Weyl phase in 227 pyrochlore iridates <cit.>, N=24 in inversion asymmetric Weyl semimetal (such as TaAs, NbAs, etc.) <cit.>, and topological Dirac semimetal (that at low energies can be considered as two superimposed Weyl semimetals) such as Cd_2As_3 and Na_3Bi with N=4 <cit.>, endows an unprecedented opportunity to extract the scaling of OC as a function of N and test the validity of our prediction [Eq. (<ref>)], with prior notion of the Fermi velocity (v) available from the APRES measurements, for example.§ DISCUSSIONS For completeness, we also report the correction to the dielectric constant in Weyl liquid, given by ε(Ω)=1+2N e_0^2/3 h v[1+1/N+1]log( E_Λ/Ω),due to the long-range Coulomb interaction, which can directly be obtained from Eq. (<ref>) by applying the Kramers-Kronig relation [see Appendix <ref>]. Notice that a logarithmic enhancement of ε(Ω), observed in a recent experiment <cit.>, is a clear manifestation of the violation of the hyperscaling hypothesis in three dimensions. Our conclusions regarding the scaling of OC and dielectric constant are also directly applicable for interacting massless Dirac fermions that can be represented as a band gap (Δ) tuned QCP, separating two topologically distinct insulating phases in three dimensions, see Fig. <ref>. Since Dirac semimetal supports linearly dispersing quasiparticles and the scaling dimension of the band gap in this system is [Δ]=1, the Dirac QCP is also characterized by z=1 and ν=1.Consequently, in the presence of Coulomb interaction the correction to the OC is given by Eq. (<ref>) or (<ref>), and that for the dielectric constant takes the form of Eq. (<ref>), but with a modification N → 2 N, and N now counts the number of four-component Dirac fermions. Therefore, N=1 for Bi_2Se_3, Bi_1-xSb_x, Hg_1-xCd_xTe, ZrTe_5, N=4 for Pb_1-xSn_xTe <cit.>, andN=3 for SmB_6, YbB_6 <cit.>. Notice that even on the insulating sides of the phase diagram (|Δ| ≠ 0) and/or when the chemical potential lies in the valence or conduction band, the proposed scaling of OC remains valid as long as Ω>Ω^∗∼ |Δ| or v|n|^1/3, see Fig. <ref>. Therefore, existence of ample Dirac materials with different number of Dirac nodes (N) and residing in the close proximity to the topological QPT, which can also be tuned externally by applying hydrostatic pressure or changing the chemical composition, constitutes an ideal platform to test the validity of proposed scaling of the OC and dielectric constant. To summarize, we here explicitly demonstrate that an intriguing confluence among the quantum critical nature of an interacting Weyl or Dirac liquid, hyperscaling violation in three dimensions and marginal nature of the long-range Coulomb interaction endows these systems with a universalinteraction mediated enhancement of the OC. The scaling of this correction with the flavor number can directly be probed in a large number of discovered and predicted Dirac and Weyl materials <cit.> as well as in numerical simulations <cit.>, making our results relevant to recent and ongoing experiments <cit.>. Finally, our findings may also motivate future investigations of the interaction effects on magneto-transport and on hydrodynamic transport in interacting Weyl/Dirac liquids. B. R. was supported by the Welch Foundation Grant No. C-1809 and by NSF CAREER Grant no. DMR-1552327 of Matthew S. Foster (Rice University). § LEADING ORDER RENORMALIZATION GROUP FLOW EQUATIONS FOR FERMI VELOCITY AND FINE-STRUCTURE CONSTANT The Euclidean action for the Weyl quasiparticles interacting with the long range Coulomb interaction has the formS= ∫ dτ d rψ^†(τ, r)[∂_τ-i a_0-i v σ·∇]ψ(τ, r) + 1/2∫ d r a_0( r)|∇|^2/2π e^2 a_0( r),where τ is the imaginary time, r is the spatial coordinate, ψ(τ, r) is a two-component Weyl spinor, v is the Fermi velocity of the Weyl quasiparticles, and σs are the Pauli matrices. The partition function is 𝒵=∫𝒟Φe^-S[Φ], where Φ denotes all the fields in the action. Auxiliary gauge field a_0 is chosen so that after integrating it out, the Coulomb interaction has the usual 1/k^2 form in three spatial dimensionsS_c= ∫ d^3 kd^3 k' ρ( k) 2π e^2/| k- k'|^2 ρ( k') ≡ ∫ d kd k' ρ( k) V_C( k- k') ρ( k'),with ρ( k) as the density operator in momentum space, and V_C( k)=2π e^2/ k^2.The propagators for the Weyl fermion and the gauge field in terms of a Matsubara frequency iω and a momentum k, respectively, readG_f(iω, k)=iω+vσ· k/ω^2+v^2k^2, G_a_0( k)=2π e^2/ k^2. We now perform a renormalization group analysis using the dimensional regularization in D=3-ϵ dimensions and a minimal subtraction scheme to find the leading order flow equation for the Fermi velocity and the Coulomb charge. The one-loop β function for the Fermi velocity follows from the leading order correction to the self-energy for Weyl fermions due to the long-range tail of the Coulomb interaction which reads asΣ( iω, 𝐪) = i^2 ∫d^D 𝐤/(2 π)^D∫^∞_-∞dω/2 π V_C( k)G_f(i(ω+Ω), k+ q).An explicit calculation then yieldsΣ( iω, 𝐪) =-∫d^D 𝐤/(2 π)^D∫^∞_-∞dω/2 π2 π e^2/|𝐤|^2 i (ω+Ω) + v σ· (𝐤+𝐪) /(ω+Ω)^2+ v^2 (𝐤+𝐪)^2= -1/2 v∫d^D 𝐤/(2 π)^D2 π e^2/|𝐤|^2 σ· (𝐤+𝐪)/|𝐤+𝐪|=-π e^2/2 ∫^1_0 dx∫d^D 𝐤/(2 π)^D x^-1/2 (1-x) σ·𝐪/[ k^2 + x(1-x) q^2 ]^3/2= -σ·𝐪/|𝐪|^3-D π e^2 Γ( 3-D/2)/2(4π)^D/2Γ(3/2) ∫^1_0 dx (1-x)^D-1/2 x^D-4/2= -σ·𝐪/|𝐪|^3-Dπ e^2 Γ( 3-D/2)/2(4π)^D/2Γ(3/2)Γ(D/2-1)Γ(D+1/2)/Γ(D-1/2)=-( e^2/3π)[ σ·𝐪/|𝐪|^ϵ]1/ϵ,for D=3-ϵ. On the other hand, the one-loop polarization bubble, leading to charge renormalization, reads asΠ( iω, 𝐪)=- i^2Tr∫d^D 𝐤/(2 π)^D∫^∞_-∞dω/2 π[ σ_0 G_f(iω, k) × σ_0G_f(i(ω+Ω), k+ q) ].To find the renormalization factor that ultimately gives the beta function for the charge, it is sufficient to keep the terms in the polarization up to q^2 order, leading toΠ( iω, 𝐪)=- i^2 Tr∫d^D 𝐤/(2 π)^D∫^∞_-∞dω/2 π [ σ_0i ω + v σ·𝐤/ω^2+ v^2 𝐤^2× σ_0 i (ω+Ω) + v σ· (𝐤+𝐪) /(ω+Ω)^2+ v^2 (𝐤+𝐪)^2] =- Π_00(iω, q)=-N q^2/12 π^2 v[ q^-ϵ/ϵ+ 𝒪(1)],where the density-density correlator Π_00(iω,q) to the q^2 order is given by Eq. (<ref>).From Eq. (<ref>), after re-exponentianting it, introducing the wavefunction renormalization Z_ψ and the renormalization factor for the velocity Z_v via Z_v v=v_0, with v_0 as the bare velocity, we findZ_Ψ(-iω+Z_v v σ· q)+e^2 q^-ϵ/3π v ϵv σ· q=-iω +v σ· q,which then yieldsZ_ψ=1, Z_v=1-e^2/3π v ϵq^-ϵ.Using renormalization condiction Z_v v=v_0, we then readily obtain the infrared β-function for the velocity to the leading orderβ_v≡-dv/dlog q=e^2/3π=v α/3 π,where α=e^2/v is the fine structure constant. The flow equation for charge is obtained from Eq. (<ref>) after re-exponentiating it, and recalling that the form of the action for the gauge field is given by Eq. (<ref>). The renormalization condition for the charge then reads1/2π e_0^2+N q^-ϵ/6π^2 vϵ=1/2π e^2,with e_0 denoting the bare charge, from which we findrenormalization constant for the charge Z_e^2e^2=e_0^2 to be of the formZ_e^2=1+N e^2/3π vϵq^-ϵ,which yields the leading order beta function for the chargeβ_e^2=-d e^2/d log k=-N e^4/3π v.Finally, using Eqs. (<ref>) and (<ref>), we obtain the flow equation for the fine structure constant α=e^2/vβ_α=-1/3π(N+1)α^2≡ -Aα^2,which therefore yieldsthe coefficient A=(N+1)/3π that we use in the main text.On the other hand, the imaginary time action in the presence of only local density-density interaction reads asS_SR=g_0 ∫ dτd r [ ψ^†(τ, r) ψ(τ, r) ]^2,where g_0 denotes the strength of contact interaction. In this notation g_0>0 represents repulsive interaction. The scaling dimension of any short-range interaction is [g_0]=z-D, implying [g_0]=-2 (and thus an irrelevant perturbation) in a three-dimensional (D=3) Weyl liquid (z=1), yielding the following leading order beta function (infrared) for the dimensionless coupling constant, defined as g=g_0 Ω^2/v^3 for example,β_g=-2 g + 𝒪(g^2).§ OPTICAL CONDUCTIVITY IN NONINTERACTING SYSTEM In two subsequent sections, we will first provide the detailed derivation of the components polarization tensor corresponding to (a) current-current correlator at zero momentum and (b) density-density correlator at finite but small momenta, which enter the Kubo formula for the optical conducitivity. Subsequently, we present a proof that the polarization function in a noninteracting system computed using dimensional regularization about D=3 spatial dimensions is consistent with the charge conservation. Finally, we show how to obtain the expression for OC from the expression of the current-current correlator at zero momentum and finite frequency after the analytic continuation. §.§ Current-current correlator at q=0 The current-current correlation function for noninteracting Weyl fermions at a finite Matsubara frequency and a momentumreadsΠ_lm(iΩ, q) =- ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πTr [(v σ_l) G_f(i(ω+Ω), k+ q) ( v σ_m ) G_f(iω, k)],since the current operator is of the formj=ψ^†v σψ. To find optical conductivity from the current-current correlator we compute it only at the zero momentum, which yieldsΠ_lm(iΩ,0 )= - ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πTr [ ( v σ_l )i (ω+Ω) + v σ·𝐤/(ω+Ω)^2 + v^2 𝐤^2( v σ_m )i ω + v σ·𝐤/ω^2 + v^2 𝐤^2 ] =-2 N δ_l,mv^2∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 π [ -ω(ω+Ω) + ( 2/D-1 ) v^2 𝐤^2/[ (ω+Ω)^2 + v^2 𝐤^2 ][ ω^2 + v^2 𝐤^2]] =-4 N/v^D-2δ_l,m ( 1/D-1 )2 π^D/2/Γ( D/2) (2 π)^D ∫^∞_0 dk k^D/4 k^2+Ω^2= -N Ω^2/v^D-2δ_l,m ( 1/D-1 )2^1-Dπ^1+ D/2/Γ( D/2) (2 π)^D ( π D/2) = -N Ω^2/12 π^2 vδ_l,m[ 1/ϵΩ^-ϵ+ a/2],for D=3-ϵ. Here, D denotes the spatial dimensionality of the system and N is the number of Weyl points. In the above expression a=[5-3 γ_E + 3 log(4π) ]/3 ≈ 3.62069, where γ_E ≈ 0.577 is the Euler-Mascheroni constant.§.§ Density-density correlator at Ω=0 The density-density correlation function for noninteracting Weyl fermions at a finite Matsubara frequency and a momentumreadsΠ_00(iΩ, q) =- ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πTr [σ_0G_f(i(ω+Ω), k+ q) σ_0 G_f(iω, k)]. Explicit calculation for thedensity-density correlator (with external frequency and momentum) yieldsΠ_00(iΩ, 𝐪)=- ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 π 𝐓𝐫 [ σ_0 i (ω+Ω) + v σ·( 𝐤 +𝐪) /(ω+Ω)^2 + v^2 ( 𝐤 + 𝐪)^2σ_0i ω + v σ·𝐤/ω^2 + v^2 𝐤^2 ] =-2N ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 π -ω(ω+Ω)+ v^2 𝐤·( 𝐤+ 𝐪)/[ (ω+Ω)^2 + v^2 ( 𝐤 + 𝐪)^2 ][ ω^2 + v^2 𝐤^2 ]= N/v^D∫d^D 𝐤/(2π)^D |𝐤|+ |𝐤+𝐪|/Ω^2 + ( |𝐤|+ |𝐤+𝐪| )^2 [1- 𝐤·( 𝐤 + 𝐪) /|𝐤|+ |𝐤+𝐪|] =N/v^D∫d^D 𝐤/(2π)^Dq^2 k^2 - ( 𝐤·𝐪)^2/k^3 ( Ω^2 + 4k^2 ) + 𝒪( q^4 ).While arriving at the final expression we perform Taylor series expansion in powers of q and kept the terms only to the order q^2. In the intermediate step we have rescaled the momentum as v 𝐤→𝐤 and v 𝐪→𝐪. Now performing the integral over 𝐤 (and restoring the factor of v^2 in front of q^2) we arrive at the final expression for the density-density correlatorΠ_00(iΩ, 𝐪) = N/v^D ( v^2 q^2 ) ( 1/D-1 ) 2^1-Dπ^1+ D/2/Γ( D/2) (2 π)^D ( π D/2)= N q^2/12 π^2 v[ 1/ϵq^-ϵ+ a/2],where D=3-ϵ.We notice here that charge conservation∂_τρ(τ, r)+∇· j(τ, r)=0 implies thatthe polarization tensor satisfies -iΩΠ_0μ(iΩ, q)+q_lΠ_lμ(iΩ, q)=0,and Ω^2Π_00(iΩ, q)+q_l q_m Π_lm(iΩ, q)=0, where index μ includes also the imaginary time, and summation over repeated indices is assumed. Therefore, divergent parts of the current-current and the density-density correlation functions in Eqs. (<ref>) and (<ref>) are consistent with charge conservation. However, we will now compute the polarization function Π(iΩ,q) for a noninteracting Weyl at a finite frequency and momentum with D=3-ϵ regularization, and showthat the entire function at arbitrary momentum and frequency is consistent with charge conservation.§.§ Polarization tensor for noninteracting system We start with the expressionfor the components of the polarization tensor that includes Eqs. (<ref>) and (<ref>) as special casesΠ_μν(iΩ, q) =- ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πTr [σ_μG_f(i(ω+Ω), k+ q) σ_ν G_f(iω, k)] ≡ - Tr(P_μ(iΩ, q)σ_ν),where σ_μ=(σ_0,v σ). Following the steps outlined in Appendix A of Ref. <cit.>, we then computeP_μ(iΩ, q)=∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πTr[G_f(iω, k) σ_μ G_f(i(ω+Ω), k+ q)].After introducing using the Feynman parameters, we obtainP_μ(iΩ, q)=1/v^D∫_0^1 dx ∫dω/2π ∫d^D k/(2π)^D [iω+σ· k]σ_μ[iω+iΩ+σ·( k+ q)]/[(ω+xΩ)^2+( k+x q)^2+Δ]^2,with v q→ q and Δ=x(1-x)(Ω^2+q^2). After integrating over ω, shifting the momentum k+x q→ k, and using the subsequent rotational symmetry of the integrand,we arrive at the following expressionP_μ(iΩ, q)=1/4v^D∫_0^1dx ∫d^D k/(2π)^D[-σ_μ/√(k^2+Δ)+ (σ_l σ_μσ_l)k^2/D(k^2+Δ^2)^3/2-x(1-x)(iΩ+σ· q)σ_μ (iΩ+σ· q)/(k^2+Δ^2)^3/2].After integrating over the momentum, we obtainP_μ(iΩ, q)=F(D)/v^D (Ω^2+q^2)^3-D/2[(σ_lσ_μσ_l-σ_μ)(Ω^2+q^2)+(D-1)(iΩ+σ· q)σ_μ (iΩ+σ· q)],whereF(D)=Γ[1-D/2] Γ^2[D+1/2]/4(4π)^D/2 Γ[1/2] Γ[D+1].From here, using that σ_l σ_l=D, we findP_0(iΩ, q)=2F(D)(D-1)/(Ω^2+q^2)^3-D/2 [q^2+iΩσ· q],which then yields for the components of the polarizationΠ_00(iΩ, q)=-4(D-1)F(D)/v^D-2(Ω^2+q^2)^3-D/2q^2,andΠ_0m(iΩ, q)=-4(D-1)F(D)/v^D-2(Ω^2+q^2)^3-D/2iΩ q_m,where we restored q→ v q. To obtain P_m(iΩ, q), we have to recall that since we consider D=3-ϵ, all three Pauli matrices should be used in the scalar products such as σ· q (unlike in D=2) which then yields(iΩ+σ· q)σ_m (iΩ+σ· q)=(-Ω^2+q^2)σ_m+2iΩ q_m.Ultimately, we obtainP_m(iΩ, q)=2(D-1)F(D)/v^D-1(Ω^2+q^2)^3-D/2(iΩ q_m-σ_m Ω^2),which then yieldsΠ_lm(iΩ, q)=4(D-1)F(D)/v^D-2(Ω^2+q^2)^3-D/2Ω^2δ_lm,andΠ_m0(iΩ, q)=-4(D-1)F(D)/v^D-2(Ω^2+q^2)^3-D/2iΩ q_m. It is now easy to check that the obtained polarization tensor is consistent with charge conservation, i.e. that it obeys Eqs. (<ref>) and (<ref>), and is therefore manifestly gauge invariant. Furthermore, taking D=3-ϵ, we obtain the results for the q=0 current-current and the Ω=0 density-density correlators in Eqs. (<ref>) and (<ref>). §.§ Analytic continuation and optical conductivity We now proceed with the computation of the optical conductivity from the obtained polarization tensor. In order to extract the OC from the polarization tensor, we first identify the ultraviolet divergence appearing as 1/ϵ as1/ϵ≡ -1/2 log[1+4 v^2 Λ^2/Ω^2],where Λ is the ultraviolet momentum cutoff up to which the energy dispersion of Weyl fermions scales linearly with momentum. Now we perform the analytic continuation to real frequency according to i Ω→Ω +i δ and subscribe to the definition of the real part of the optical conductivityσ_jj = [ e^2_0/h× 2 π]lim_δ→ 0[ Π_lm( i Ω→Ω +i δ) ]/Ω = N e^2_0/12 h v Ω≡σ_0 (Ω).In the final expression we denote the OC in a noninteracting Weyl liquid as σ_0, and e_0 is the external test charge.§ OPTICAL CONDUCTIVITY DUE TO LOCAL DENSITY-DENSITY INTERACTION Now we systematically incorporate the correction to OC in a Weyl semimetal due to electron-electron interactions. For the sake of simplicity and to establish the methodology, we first focus on the short-range component of the density-density Coulomb interaction. It should be noted that in any lattice system the long range tail of the Coulomb interaction is always accompanied by its short-range component. Therefore, inmaterial-based and numerical experiments one needs to account for both components of the density-density interaction. §.§ Interaction correction to current-current correlator The correction to the current-current correlator due to the short-range component of density-density interaction (characterized by strength g_0) is given by δΠ_lm (i Ω,0)=δΠ^SE_lm(i Ω,0) + δΠ^V_lm(i Ω,0). Respectively, δΠ^SE_lm(i Ω,0) and δΠ^V_lm(i Ω,0) accounts for self-energy and vertex diagrams. The contribution from the self-energy diagram reads asδΠ^SE_lm (i Ω,0)=(-1)^2 2 g_0 v^2 ∫d^D 𝐤/(2π)^D ∫d^D 𝐩/(2π)^D ∫^∞_-∞dω/2 π ∫^∞_-∞dω^'/2 π 𝐓𝐫[ i ω + v σ·𝐤/ω^2+v^2 𝐤^2 σ_li (ω+Ω) + v σ·𝐤/(ω+Ω)^2+v^2 𝐤^2 i ω^' + v σ·𝐩/(ω^')^2+v^2 𝐩^2 ×i (ω+Ω) + v σ·𝐤/(ω+Ω)^2+v^2 𝐤^2σ_m ]≡0.On the other hand, the contribution from the vertex diagram goes asδΠ^V_lm (i Ω,0)=(-1)^2 g_0 v^2 ∫d^D 𝐤/(2π)^D ∫d^D 𝐩/(2π)^D ∫^∞_-∞dω/2 π ∫^∞_-∞dω^'/2 π 𝐓𝐫[i ω + v σ·𝐤/ω^2+v^2 𝐤^2 σ_li (ω+Ω) + v σ·𝐤/(ω+Ω)^2+v^2 𝐤^2 i ω^' + v σ·𝐩/(ω^')^2+v^2 𝐩^2×σ_mi (ω^' -Ω) + v σ·𝐩/(ω^'-Ω)^2+v^2 𝐩^2] = g_0 v^2𝐓𝐫[ I_l( iΩ)I_m( -iΩ) ],where [4]I_l( iΩ)= ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 π i ω + v σ·𝐤/ω^2+v^2 𝐤^2 σ_li (ω+Ω) + v σ·𝐤/(ω+Ω)^2+v^2 𝐤^2 = σ_l/v^D 2^2 π^D/2 (1-D)/D Γ( D/2)(2π)^D ∫^∞_0 dk k^D/4 k^2 + Ω^2= σ_l Ω^2/v^3 [(1/D-1 )2^-1-D π^1+D/2/Γ( D/2) (2 π)^D( π D/2)].The factor of (-1)^2 in the correction arises because of the fermion loop, which gives a factor -1, and the Taylor expansion to the first order in the coupling, which also gives such a factor. Therefore, the net contribution from the vertex diagram for D=3-ϵ reads asδΠ^V_lm (i Ω,0) = δ_l,m2NΩ^4/v^4 [(1/D-1 )2^-D π^1+D/2/Γ( D/2) (2 π)^D( π D/2)]^2 = δ_l,m [ 2N g_0 Ω^4/576 π^4 v^4] [ 1/ϵ^2 + a/ϵ].§.§ Interaction correction to density-density correlator We now present the computation of the correction to the polarization tensor due to short-range interaction from density-density correlator. The contribution arising from the self-energy diagram isδΠ^SE_00 (i Ω, 𝐪)=(-1)^2 2 g_0 ∫d^D 𝐤/(2π)^D ∫d^D 𝐩/(2π)^D ∫^∞_-∞dω/2 π ∫^∞_-∞dω^'/2 π 𝐓𝐫[ i ω + v σ·𝐤/ω^2+v^2 𝐤^2 σ_0i (ω+Ω) + v σ·( 𝐤 +𝐪)/(ω+Ω)^2+v^2 ( 𝐤+ 𝐪)^2 ×i ω^' + v σ·𝐩/(ω^')^2+v^2 𝐩^2 i (ω+Ω) + v σ·( 𝐩 + 𝐪)/(ω+Ω)^2+v^2 ( 𝐩 +𝐪)^2σ_0 ]≡0.On the other hand, contribution from the vertex diagram reads asδΠ^V_00 (i Ω, 𝐪)=(-1)^2 g_0 𝐓𝐫{∫d^D 𝐤/(2π)^D∫^∞_-∞dω/2 π [i ( ω -Ω) + v σ· (𝐤-𝐪)/(ω-Ω)^2+v^2 (𝐤-𝐪)^2 σ_0i ω + v σ·𝐤/ω^2+v^2 𝐤^2] × ∫d^D 𝐩/(2π)^D ∫^∞_-∞dω^'/2 π [ i ω^' + v σ·𝐩/(ω^')^2+v^2 𝐩^2 σ_0i ( ω^' -Ω) + v σ· (𝐩-𝐪)/(ω^'-Ω)^2+v^2 (𝐩-𝐪)^2] }.Note that for short-range interaction the contribution from the vertex diagram breaks into two pieces. We now show some essential steps of the analysis, which will also be useful while we compute the same diagram, but in the presence of the long-range tail of the Coulomb interaction. At this stage we first rescale the momentum according to v 𝐤→𝐤, v 𝐩→𝐩, v 𝐪→𝐪. Note thatI_1(𝐩, 𝐪, Ω)=∫^∞_-∞ d ω^'/2 π i ω^' +σ·𝐩/(ω^')^2+ 𝐩^2 i ( ω^' -Ω) + σ· (𝐩-𝐪)/(ω^'-Ω)^2+ (𝐤-𝐪)^2 = 1/2 [ Ω^2 + (p+|𝐩-𝐪| )^2 ]×[ - ( p+|𝐩-𝐪| ) [ 1 - 𝐩· (𝐩-𝐪)/p |𝐩-𝐪|] + i σ·{Ω( 𝐩-𝐪/|𝐩-𝐪| - 𝐩/p) - ( 𝐩×𝐪)p+|𝐩-𝐪| /p |𝐩-𝐪|}] ≡ a + σ·𝐛Similarly, after completing the integral over Matsubara frequency ω we can writeI_2(𝐩, 𝐪, Ω)= ∫^∞_-∞dω/2 π i ( ω -Ω) + σ· (𝐤-𝐪)/(ω-Ω)^2+ (𝐤-𝐪)^2 i ω + σ·𝐤/ω^2+ 𝐤^2≡ c + σ·𝐝,witha= -p+|𝐩-𝐪|/2 [ Ω^2 + (p+|𝐩-𝐪| )^2 ][ 1 - 𝐩· (𝐩-𝐪)/p |𝐩-𝐪|], b=i Ω( 𝐩-𝐪/|𝐩-𝐪| - 𝐩/p) - i ( 𝐩×𝐪)( 1/|𝐩-𝐪| + 1/p)/2 [ Ω^2 + (p+|𝐩-𝐪| )^2 ], c=-k+|𝐤+𝐪|/2 [ Ω^2 + (k+|𝐤+𝐪| )^2 ][ 1 - 𝐤· (𝐤+𝐪)/k |𝐤+𝐪|], d= i Ω( 𝐤+𝐪/|𝐤+𝐪| - 𝐤/k) + i ( 𝐤×𝐪)( 1/|𝐤+𝐪| + 1/k) /2 [ Ω^2 + (k+|𝐤+𝐪| )^2 ].In terms of above parameters δΠ^V_00 (i Ω, 𝐪) can be written asδΠ^V_00 (i Ω, 𝐪) =g_0/v^6∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D 𝐓𝐫[ ( a + σ·𝐛) ( c + σ·𝐝) ]=- 2N g_0/v^6∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D[ ac + 𝐛·𝐝].Since we are interested in contribution proportional to q^2, next we expand all these quantities to the order q^2, yieldinga= 4 (𝐩·𝐪)^2+p^2 q^2/p^3 (Ω^2 +4 p^2 ),b=iΩ[ 𝐩 (𝐩·𝐪)- p^2 𝐪 ]-2 p^2 (𝐩×𝐪)/p^3 (Ω^2 +4 p^2 ), c= 4 (𝐤·𝐪)^2+k^2 q^2/k^3 (Ω^2 +4 k^2 ),d=iΩ[ 𝐤 (𝐤·𝐪)- k^2 𝐪 ]+2 k^2 (𝐤×𝐪)/k^3 (Ω^2 +4 k^2 ).Notice that the product ac ∼𝒪(q^4) and therefore does not contribute to the conductivity. We can compactly write𝐛·𝐝=-Ω^2 [ 𝐩( 𝐩·𝐪) -p^2 𝐪] ·[ 𝐤( 𝐤·𝐪) -k^2 𝐪]/ 4 k^3 p^3 (Ω^2 +4 k^2 ) (Ω^2 +4 p^2 ) - ( 𝐩×𝐪) ·( 𝐤×𝐪) /k p (Ω^2 +4 k^2 ) (Ω^2 +4 p^2 ) - Ω𝐪· (𝐤×𝐩)/2 k^3 p^3[ p^2 (𝐤·𝐪 ) + k^2 (𝐩·𝐪 ) ]/(Ω^2 +4 k^2 ) (Ω^2 +4 p^2 ).As the last term is odd under the exchange of 𝐤 and 𝐩, it does not contribute after the momentum integral. Also the two pieces in the second term in the last expression are individually odd functions of 𝐤 and 𝐩, and therefore both vanish. After these simplifications the net contribution from the vertex diagram goes asδΠ^V_00 (i Ω, 𝐪)=-N g_0 Ω^2/2 v^6 [ ∫d^D 𝐤/(2π)^D 𝐤 (𝐤·𝐪) -k^2 𝐪/k^3 (Ω^2 +4 k^2 )] ·[ ∫d^D 𝐩/(2π)^D 𝐩 (𝐩·𝐪) -p^2 𝐪/p^3 (Ω^2 +4 k^2 )] =-2 N g_0 Ω^2 (v^2 q^2)/ v^6[(1/D-1 )2^-D π^1+D/2/Γ( D/2) (2 π)^D( π D/2)]^2 = -[ 2N g_0 Ω^2 q^2/576 π^4 v^4] [ 1/ϵ^2 + a/ϵ]. Notice that the results for the interaction correction to the current-current and the density-density correlators in Eqs. (<ref>) and (<ref>), respectively, are consistent with the charge conservation, Eq. (<ref>).§.§ Optical conductivity From expression for the polarization tensor, now identifying the ultraviolet divergence following Eq. (<ref>) and following the definition of the real part of optical conductivity [see Eq. (<ref>)], we find the correction of the OC due to the short-range component of the density-density interaction to beδσ_jj(Ω)=σ_0 (Ω)[ g_0 Ω^2/24 π^2 v^3]{ a- 2 log( E_Λ/Ω) },where E_Λ=2 Λ v and the quantity inside the straight bracket “[]" is the dimensionless short-range coupling constant in D=3.§ OPTICAL CONDUCTIVITY DUE TO LONG-RANGE TAIL OF THE COULOMB INTERACTION Finally, we turn our focus to the computation of the correction to the OC due to the long-range tail of the Coulomb interaction. This is the most challenging part of the analysis. Nonetheless, we have already developed a vast amount of technical aspect of this problem to facilitate the discussion in this section. Once again we will compute the polarization tensor from (a) current-current correlator and then (b) density-density correlator. And finally from either of these two expressions we will compute the correction to the OC.§.§ Current-current correlator The correction to the current correlator has the self-energy and the vertex parts, which we compute separately.Contribution to the current-current correlator, for which we consider only one of the diagonal components due to isotropy, at zero momentum arising from the self-energy diagram is of the formδΠ^SE_xx(iΩ,0)= -i^2 (2N)∫dω/2π∫dω'/2π∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D Tr[G_f(iω, k)vσ_xG_f(iω+iΩ, k) ×vσ_x G_f(iω, k) G_f(iω', p)V_C( k- p)].Here, a fermion loop gives a factor of -1, while the Coulomb vertex which appears twice to the leading order produces the factor i^2. Theintegrals over the Matsubara frequencies are performed from -∞ to ∞, i.e. ∫ dω≡∫_-∞^∞ dω. Explicit form of this contribution then readsδΠ^SE_xx = -2N/v^2𝐓𝐫∫d^D 𝐤/(2π)^D∫dω/2 πi ω +σ·𝐤/ω^2+ 𝐤^2σ_x i (ω+Ω) +σ·𝐤/(ω+Ω)^2+ 𝐤^2σ_x i ω +σ·𝐤/ω^2+ 𝐤^2[ i^2∫d^D 𝐩/(2π)^D∫dω^'/2 πi ω^' +σ·𝐩/(ω^')^2+ 𝐩^22 π e^2/|𝐤-𝐩|^2] =-2N/v^2𝐓𝐫{∫d^D 𝐤/(2π)^D∫^∞_-∞dω/2 πi ω +σ·𝐤/ω^2+ 𝐤^2σ_x i (ω+Ω) +σ·𝐤/(ω+Ω)^2+ 𝐤^2σ_x i ω +σ·𝐤/ω^2+ 𝐤^2Σ( 𝐤) },where Σ(𝐤) is the self-energy correction due to long-range interaction, given by Eq. (<ref>), which for convenience we write againΣ(𝐤)=-π e^2/2 [ Γ( 3-D/2) Γ( D/2-1) Γ( D+1/2)/(4 π)^D/2Γ(3/2) Γ(D-1/2)]σ·𝐤/|𝐤|^3-D≡ -E(D)σ·𝐤/|𝐤|^3-D.After performing the trace algebra we arrive at the following compact expression for the self-energy diagramδΠ^SE_xx=4 N E(D) ∫d^D 𝐤/(2π)^D∫dω/2 π [ (ω^2-k^2)(k^2-2k^2_x) -2ω(ω+Ω)k^2 ]/k^3-D[ ω^2+k^2 ]^2 [(ω+Ω)^2+ k^2 ].Performing the integral over the Matsubara frequency ω we obtainδΠ^SE_xx =-2N π e^2/v^2[ Γ( 3-D/2) Γ( D/2-1) Γ( D+1/2)/(4 π)^D/2Γ(3/2) Γ(D-1/2)] ( 1-1/D)∫d^D 𝐤/(2π)^Dk^D-2(4k^2-Ω^2 )/(Ω^2+4k^2 )^2= 2 N π e^2 Ω^2/v^2[ Γ( 3-D/2) Γ( D/2-1) Γ( D+1/2)/(4 π)^D/2Γ(3/2) Γ(D-1/2)2 π^D/2/4 Γ( D/2) (2π)^D( 1-1/D) ] ∫^∞_0 dkk^2D-5( Ω^2+12 k^2)/(Ω^2+4k^2 )^2= 2 N π e^2 Ω^2/v^2[ Γ( 3-D/2) Γ( D/2-1) Γ( D+1/2)/(4 π)^D/2Γ(3/2) Γ(D-1/2)2 π^D/2/4 Γ( D/2) (2π)^D( 1-1/D) ] 2^3-2Dπ (2D-3)cosec(π D) = N e^2 Ω^2/72 π^3 v^2 [ 3/ϵ^2 + 1/ϵ[ 7-3 γ_E + 3 log(4 π)]],for D=3-ϵ. We point out that while arriving to the second line, we subtract the Ω=0 piece of the bubble. Now we turn our attention to the vertex diagram. The expression for the polarization bubble arising from this diagram reads asδΠ^V_xx (i Ω,0)=-i^2 N ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D∫dω/2 π∫dω^'/2 π Tr[G_f(iω, k) vσ_x G_f(iω+iΩ, k) G_f(iω+iΩ, p) ×vσ_xG_f(iω, p)V_C( k- p)].Again, a fermion loop gives a factor of -1, while the Coulomb vertex which appears twice to the leading order produces the factor i^2. We now explicitly write this expression asδΠ^V_xx (i Ω,0) = -i^2N/v^2∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D∫dω/2 π∫dω^'/2 π Tr{i ω +σ·𝐤/ω^2+ 𝐤^2σ_x i (ω+Ω) +σ·𝐤/(ω+Ω)^2+ 𝐤^2 i (ω+Ω) +σ·𝐩/(ω+Ω)^2+ 𝐩^2× σ_x i ω^' +σ·𝐩/( ω^')^2+ 𝐤^2} 2 π e^2/|𝐤-𝐩|^2.After performing the trace algebra and completing the frequency integrals using the residue technique we findδΠ^V_xx (i Ω,0) = 4 π N e^2/v^2∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D Ω^2 (p_x k_x -𝐤·𝐩) + 4 ( k_x p_x 𝐤·𝐩 + k^2 p^2 -p^2_x k^2-k^2_x p^2 )/k p ( Ω^2 +4k^2 ) ( Ω^2 +4p^2 ) |𝐤-𝐩|^2.Now we subtract the zero frequency piece of δΠ^V_xx (i Ω,0), given byδΠ^V_xx (0,0)= 4 π N e^2/v^2∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D4 ( k_x p_x 𝐤·𝐩 + k^2 p^2 -p^2_x k^2-k^2_x p^2) /16 k^3 p^3 |𝐤-𝐩|^2,to arrive at the following compact expression for the vertex diagram [4]δΠ^V_xx (i Ω,0) = -4 π N e^2 Ω^2/v^2∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D4 k^2 p^2 ( 𝐤·𝐩 -p_x k_x ) + [ Ω^2 + 4( k^2 +p^2 )] ( k_x p_x 𝐤·𝐩 + k^2 p^2 -p^2_x k^2-k^2_x p^2)/4 k^3 p^3 ( Ω^2+4p^2 ) ( Ω^2+4k^2 ) |𝐤-𝐩|^2=-1/v^2[ δΠ^V,1_xx,1 (i Ω,0) + δΠ^V,2_xx,1 (i Ω,0) + δΠ^V,3_xx,1 (i Ω,0) + δΠ^V_xx,2 (i Ω,0) ].Various pieces appearing in the above expression read as [4]δΠ^V,1_xx,1 (i Ω,0)= NΩ^2/4∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2 k_x p_x 𝐤·𝐩 + k^2 p^2 -p^2_x k^2-k^2_x p^2 /k^3 p^3 [ k^2 + (Ω/2)^2 ],δΠ^V,2_xx,1 (i Ω,0)=-NΩ^4/32∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^21/k p [ k^2 + (Ω/2)^2 ] [ p^2 + (Ω/2)^2 ] , δΠ^V,3_xx,1 (i Ω,0)=-NΩ^4/32∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2k_x p_x 𝐤·𝐩 - 2 p^2 k^2_x/k^3 p^3 [ k^2 + (Ω/2)^2 ] [ p^2 + (Ω/2)^2 ],δΠ^V_xx,2 (i Ω,0)= NΩ^2/8∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2𝐤·𝐩 -p_x k_x/k p [ k^2 + (Ω/2)^2 ] [ p^2 + (Ω/2)^2 ].Now we present some details of the computation of each term.Note that following identity involving Feynman parameters is extremely useful to compute these terms1/A^αB^βC^γ = Γ(α+β+γ)/Γ( α) Γ( β) Γ( γ)∫^1_0 dx ∫^1-x_0 dy (1-x-y)^α-1 x^β-1 y^γ-1/[ (1-x-y)A + x B + y C ]^α+β+γ.Now the term δΠ^V,1_xx,1 (i Ω,0) can be written asδΠ^V,1_xx,1 (i Ω,0) = NΩ^2/4 (2π e^2) Γ( 7/2)/Γ( 3/2)∫^1_0 dx ∫^1-x_0 dy ∫d^D 𝐩/(2π)^Dp^2-p^2_x/p^3∫d^D 𝐤/(2π)^D[ ( 1-1/D) k^2 + x^2 p^2 ] (1-x-y)^1/2/[ k^2 + x(1-x) p^2 + y (Ω/2)^2 ]^7/2= NΩ^2/4 (2π e^2) (1-1/D) Γ( 5-D/2)/Γ( 3/2) (4π)^D/22 π^D/2/Γ(D/2) (2 π)^D∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dp p^D-2√(1-x-y)/[ x(1-x)p^2+y(Ω/2)^2]^5-D/2[ D-1/2+ 5-D/2 x^2 p^2/x(1-x)p^2+y(Ω/2)^2] =NΩ^2 e^2 π^1+D/2Γ(3-D) Γ( 1+D/2) (1-1/D)/Γ( 3/2) Γ(D/2) (2 π)^D (4π)^D/2∫^1_0 dx x^1-D/2/(1-x)^D+1/2∫^1-x_0 dy y^D-3√(1-x-y)=NΩ^2 e^2 π^1+D/2Γ(3-D) Γ( 1+D/2) (1-1/D) Γ(3-D/2) Γ(D/2-1 )/Γ( 3/2 ) Γ(D/2 ) (2 π)^D (4π)^D/2Γ( 1/2 )Γ(D-2)=Ne^2 Ω^2/72 π^3[ 2/ϵ^2 + 6-2γ_E+2 log(4π)/ϵ].Next we compute δΠ^V,2_xx,1 (i Ω,0) given byδΠ^V,2_xx,1 (i Ω,0) = -2 π e^2 NΩ^4/32Γ( 5/2)/Γ( 1/2)∫^1_0 dx ∫^1-x_0 dy/√(1-x-y)∫d^D 𝐤/(2π)^D1/k [ k^2 + (Ω/2)^2 ]∫d^D 𝐩/(2π)^D1/[ p^2+x(1-x)k^2 +y(Ω/2)^2 ]^5/2= -2 π e^2 N Ω^4/322 π^D/2Γ(5-D/2)/Γ(D/2) (2 π)^D Γ(1/2) (4π)^D/2∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dkk^D-2[ 1-x-y ]^-1/2/[ k^2 + (Ω/2)^2 ] [ x(1-x) k^2 + y(Ω/2)^2 ]=Ne^2Ω^2D-42^4-2Dπ^1+D/2/(2π)^D (4π)^D/2Γ(D/2)∫^1_0 dx ∫^1-x_0 dy/√(1-x-y)[ π/sin(π D)Γ( 5-D/2) [ x(1-x)]^D-5/2- Γ(3-D) Γ( D-1/2) y^D-3[ x(1-x)]^1-D/2_2F_1 [ 1,D-1/2,D-2,y/x(1-x)] ]= - N e^2 Ω^27ζ( 3)/128 π^2.Here, _2F_1(a,b,c,z) is the ordinary hypergeometric function. Next we compute δΠ^V,3_xx,1 (i Ω,0) which can be expressed as [4]δΠ^V,3_xx,1 (i Ω,0) =- π e^2 N Ω^4/16Γ( 7/2)/Γ( 3/2)∫^1_0 dx ∫^1-x_0 dy ∫d^D 𝐩/(2π)^D[ 1-x-y]^-1/2/p^3 [ p^2 + (Ω/2)^2 ]∫d^D 𝐤/(2π)^D1/D[k^2p^2_x -2 p^2 k^2 ] -x^2 p^2 p^2_x/[ k^2+ x(1-x)p^2+y (Ω/2)^2 ]^7/2=- π e^2 N Ω^4/162 π^D/2Γ( 5-D/2)/(2π)^D (4π)^D/2Γ(D/2)∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dpp^D-2[ 1-x-y]^-1/2[ x(1-x) ]^D-5/2/[ p^2+ (Ω/2)^2 ] [ p^2+ y/x(1-x)(Ω/2)^2 ]^5-D/2× [ 1/2( 1/D-2) -x^2/D5-D/2p^2/x(1-x) p^2 + y (Ω/2)^2] ≡δΠ^V,3,1_xx,1 (i Ω,0)+δΠ^V,3,2_xx,1 (i Ω,0).After the integral over the radial momentum variable p, one of the two entries in the final expression for δΠ^V,3_xx,1 (i Ω,0) reads as [4]δΠ^V,3,1_xx,1 (i Ω,0) = N e^2/32Ω^2D-4 2^8-2Dπ^1+D/2( 1/D-2) /Γ(3/2) (4π)^D/2 (2 π)^D Γ(D/2)∫^1_0 dx ∫^1-x_0 dy √(1-x-y)[ [x(1-x)-y ]^D-5/2 πΓ( 5-D/2) /sin (π D)- Γ(3-D) Γ( D-1/2) y^D-3[ x(1-x) ]^1-D/2_2F_1 [1,D-1/2,D-2,y/x(1-x)]] =N e^2 Ω^2 5/192 π^3,while the second term goes asδΠ^V,3,1_xx,1 (i Ω,0) = N/32Ω^2D-4 2^9-2Dπ^D/2Γ( 5-D/2) ( 5-D/2D)/Γ(3/2) (4π)^D/2 (2π)^D Γ( D/2) ∫^1_0 dx x^2 ∫^1-x_0 dy (1-x-y)^1/2[- [x(1-x)-y ]^D-7/2× π/sin (π D) + y^D-3[ x(1-x)]^-D+1/2Γ(3-D)Γ( D+1/2)/Γ( 7-D/2)_2F_1 [1,D+1/2,D-2, y/x(1-x)] ]= Ne^2 Ω^2 [7 ζ(3)-6 /384 π^3].Therefore, the net contribution from δΠ^V,3_xx,1 (i Ω,0)δΠ^V,3_xx,1 (i Ω,0)=Ne^2 Ω^2/π^3[ 5/192-6/384 + 7 ζ(3)/384] =N e^2 Ω^2/384 π^3[ 4+ 7 ζ(3) ]is finite. Finally, the last term in the expression of δΠ^V_xx (i Ω,0) is given byδΠ^V_xx,2 (i Ω,0) = π e^2 N Ω^2/4Γ( 5/2)/Γ( 1/2)∫^1_0 dx ∫^1-x_0 dy ∫d^D 𝐤/(2π)^D[1-x-y ]^-1/2/k [ k^2+(Ω/2)^2 ] ∫d^D 𝐩/(2π)^D x [k^2-k^2_x ]/[ p^2 + x(1-x)k^2+ y (Ω/2)^2 ]^5/2= π e^2 N Ω^2/4 2 π^D/2( 1-1/D)Γ( 5-D/2)/ (2π)^D (4π)^D/2Γ( 1/2) Γ( D/2)∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dk k^d x [1-x-y ]^-1/2[ x(1-x)]^D-5/2/[ k^2+ (Ω/2)^2 ] [ k^2+ y/x(1-x)(Ω/2)^2 ]^5-D/2= N e^2 Ω^2/8(D-1) 2^7-2Dπ^1+D/2/D (2π)^D (4π)^D/2Γ( 1/2) Γ( D/2) ∫^1_0 dx ∫^1-x_0 dy x [ 1-x-y]^-1/2{[x(1-x)-y ]^D-5/2πΓ( 5-D/2)/sin(π D)+y^D-2[x(1-x) ]^-D+1/2Γ(D-2) Γ( D+1/2) _2F_1 [ 1, D+1/2, D-1, y/x(1-x)]} = N e^2 Ω^2/72 π^3[ 3/2 ϵ]. Therefore the net divergent contribution arising from the vertex diagram is given byδΠ^V_xx (i Ω,0)= -N e^2 Ω^2/72 π^3 v^2[ 2/ϵ^2 + 15-4 γ_E + 4 log(4π)/2 ϵ].After accounting for the divergent piece coming from the self-energy diagram we obtain the net leading order correction to the polarization bubble due to the long-range tail of the Coulomb interaction to beδΠ_xx (i Ω,0)= N e^2 Ω^2/72 π^3 v^2 [ 1/ϵ^2 -1/2 ϵ[1+2 γ_E -2log(4π) ] ].§.§ Density-density correlator Contribution to the density-density correlator from the self-energy diagram reads asδΠ^SE_00(iΩ,𝐪) = -i^2 (2N)∫dω/2π∫dω'/2π∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D Tr[G_f(iω, k) σ_0G_f(iω+iΩ, k+ q) × σ_0 G_f(iω, k) G_f(iω', p)V_C( k- p)],or explicitlyδΠ^SE_00(iΩ,𝐪) = - 2 N/v^4Tr{∫d^D 𝐤/(2π)^D ∫dω/2 π i ω +σ·𝐤/ω^2+ 𝐤^2 [i^2 ∫d^D 𝐩/(2π)^D ∫dω^'/2 π2 π e^2/|𝐤-𝐩|^2 i ω^' +σ·𝐩/ω^'^2+ 𝐩^2]i ω +σ·𝐤/ω^2+ 𝐤^2× i (ω+Ω) + σ·( 𝐤 + 𝐪)/(ω+Ω)^2+ ( 𝐤 +𝐪)^2} =-2N/v^4Tr{∫d^D 𝐤/(2π)^D ∫dω/2 π i ω +σ·𝐤/ω^2+ 𝐤^2 Σ(𝐤) i ω +σ·𝐤/ω^2+ 𝐤^2 i (ω+Ω) + σ·( 𝐤 + 𝐪)/(ω+Ω)^2+ ( 𝐤 +𝐪)^2},with Σ( k) given by Eq. (<ref>). After performing the trace algebra we arrive at the following compact expression for the self-energy contributionδΠ^SE_00(iΩ,𝐪) = -2 E(D)∫d^D 𝐤/(2π)^D ∫dω/2 πk^D-3[ -2 ω(ω+Ω) k^2 + (k^2-ω^2)𝐤· (𝐤+𝐪) ] /(ω^2 + k^2)^2[ (ω+Ω)^2 + (𝐤+𝐪)^2].After expanding the above expression in powers of q and retaining the terms only to the order q^2, we find δΠ^SE_00(iΩ,𝐪)=δΠ^SE,1_00(iΩ,𝐪)+δΠ^SE,2_00(iΩ,𝐪), where [4]δΠ^SE,1_00 (iΩ,𝐪) =4 E(D)q^2/D ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 πk^D-1 (k^2-ω^2)/(ω^2 + 𝐤^2)^2[ (ω+Ω)^2 + 𝐤^2]^2= E(D)q^2/D 2 π^D/2/Γ( D/2) (2 π)^D ∫^∞_0 dk32 k^4-12 k^2 Ω^2 -Ω^4/k^5-2D(4 k^2+Ω^2 )=E(D) q^22^4-2Dπ^1+D/2/DΓ( D/2) (2 π)^D[ (5-7D+2 D^2 ) Ω^2D-6cosec (π D) ]=-N e^2 q^2/36 π^3 v^2[ 1/ϵ^2 + 1-γ_E + log(4π)/ϵ],for D=3-ϵ and after taking q → v q. The remaining contribution from the self-energy diagram goes as [4]δΠ^SE,2_00(iΩ,𝐪) =-2 E(D)q^2/D ∫d^D 𝐤/(2π)^D ∫^∞_-∞dω/2 π k^D-1[ -2 ω(ω+Ω) + k^2-ω^2 ]/(ω^2 + 𝐤^2)^2[ (ω+Ω)^2 + 𝐤^2]^2 [4 (𝐤·𝐪)^2/(ω+Ω)^2 + k^2 -q^2] =2 q^2 E(D) 2 π^D/2/Γ( D/2) (2 π)^D∫^∞_0 dk16 (D-5) k^4 + 24 k^2 Ω^2 -(D-3)Ω^4/4 D k^5-2D[ Ω^2 + 4 k^2]^3=-q^2 Ω^2D-6 E(D)2^3-2Dπ^1+ D/2/DΓ( D/2) (2 π)^D (D-1)(2D-5) cosec (π D) = N e^2 q^2/72 π^3 v^2[ 1/ϵ^2 + 1-γ_E + log(4π)/ϵ].Therefore, net self-energy correction reads asδΠ^SE_00(iΩ,𝐪) =-N e^2 q^2/72 π^3 v^2 [1/ϵ^2 +1-γ_E+log(4π)/ϵ].Next we turn our attention to the vertex diagram. The contribution from the vertex diagram in the presence of external frequency and momentum reads asδΠ^V_00 (i Ω,𝐪) = -i^2 N∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D∫dω/2 π∫dω^'/2 π Tr[G_f(iω, k) σ_0 G_f(iω+iΩ, k) G_f(iω+iΩ, p) × σ_0G_f(iω, p)V_C( k- p)].We now write this term asδΠ^V_00(iΩ,𝐪)= N/v^4Tr{∫d^D 𝐤/(2π)^D ∫d^D 𝐩/(2π)^D 2 π e^2/|𝐤-𝐩|^2 [ ∫^∞_-∞dω/2 π i ( ω -Ω) + σ· (𝐤-𝐪)/(ω-Ω)^2+ (𝐤-𝐪)^2i ω +σ·𝐤/ω^2+ 𝐤^2] × [ ∫^∞_-∞dω^'/2 π i ω^' + σ·𝐩/(ω^')^2+ 𝐩^2i ( ω^' -Ω) +σ· (𝐩-𝐪)/(ω^'-Ω)^2+ (𝐤-𝐪)^2]}.The frequency integrals in the above expression can readily be performed following Eqs. (<ref>) and (<ref>). Upon expanding four parameters a,b,c,d defined through Eq. (<ref>), up to the quadratic order in q^2 as shown in Eq. (<ref>), we arrive at the following compact expression for the vertex functionδΠ^V_00(iΩ,𝐪) = N/2 v^4∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2 -Ω^2 [ 𝐩( 𝐩·𝐪) -p^2 𝐪] ·[ 𝐤( 𝐤·𝐪) -k^2 𝐪] + 4 k^2 p^2 ( 𝐩×𝐪) ·( 𝐤×𝐪)/ k^3 p^3 [ Ω^2 + 4 k^2 ] [ Ω^2 + 4 p^2 ].The momentum integrals can be performed most efficiently by separating the above expression into two pieces yielding δΠ^V_00(iΩ,𝐪)=[ δΠ^V,1_00(iΩ,𝐪)+δΠ^V,2_00(iΩ,𝐪) ]/v^4. Now we present evaluation of each term. The two terms are of the formδΠ^V,1_00(iΩ,𝐪)=N ∫d^D 𝐤/(2π)^D[ 𝐤( 𝐤·𝐪) -k^2 𝐪]/k^3[ Ω^2 + 4 k^2 ]·𝐈_1 (𝐪, 𝐤,Ω),δΠ^V,2_00(iΩ,𝐪)=2N ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D 2 π e^2/|𝐤-𝐩|^2 ( 𝐩×𝐪) ·( 𝐤×𝐪)/k p [ Ω^2 + 4 k^2 ] [ Ω^2 + 4 p^2 ].The quantity 𝐈_1 (𝐪, 𝐤,Ω) appearing in the expression of δΠ^V,1_00(iΩ,𝐪) reads as [4]𝐈_1 (𝐪, 𝐤,Ω) = -Ω^2/8∫d^D 𝐩/(2π)^D 2 π e^2/|𝐤-𝐩|^2 [ 𝐩( 𝐩·𝐪) -p^2 𝐪]/p^3 [ Ω^2 + 4 p^2 ]=-2 π e^2 Ω^2/8Γ( 7/2)/Γ( 3/2) ∫^1_0 dx ∫^1-x_0 dy √(1-x-y) ∫d^D 𝐩/(2π)^D 𝐩(𝐩·𝐪) + x^2 𝐤(𝐤·𝐪) -p^2 𝐪 -x^2 k^2 𝐪/[ p^2 + x(1-x) k^2 + y ( Ω/2 )^2 ]^7/2=-2 π e^2 Ω^2/8 Γ( 3/2) (4π)^D/2∫^1_0 dx ∫^1-x_0 dy √(1-x-y)[ (1-D) Γ( 5-D/2) 𝐪/2 [ x(1-x) k^2 + y ( Ω/2 )^2]^5-D/2 + x^2 [ 𝐤( 𝐤·𝐪) -k^2 𝐪] Γ( 7-D/2)/[ x(1-x) k^2 + y ( Ω/2 )^2]^7-D/2].After some algebraic simplifications we arrive at the following expression forδΠ^V,1_00(iΩ,𝐪)= N q^2 Ω^2 V(D) ∫^1_0 dx ∫^1-x_0 dy √(1-x-y)/[ x(1-x) ]^7-D/2∫^∞_0 dkA(x, D) k^D + B(y,D) Ω^2k^D-2/[ k^2+ ( Ω/2)^2 ] [ k^2+ y/x(1-x)( Ω/2)^2 ]^7-D/2,where x and y are Feynman parameters andV(D)= π e^2 (1- 1/D)π^1+D/2/16 Γ( D/2) (2 π)^D (4π)^D/2, A(x,D)=x(1-x)D-1/2Γ(5-D/2) -2 x^2 Γ(7-D/2)/2, B(y,D) = (D-1) Γ(5-D/2) /8.After some lengthy algebra it can be shown that δΠ^V,1_00(iΩ,𝐪) = 𝒪(ϵ) and thus does not contribute to the conductivity. On the other hand, after algebraic manipulation the second term in the expression of δΠ^V_00(iΩ,𝐪) becomesδΠ^V,2_00(iΩ,𝐪)=N π e^2 Γ( 5/2)/Γ( 1/2)∫^1_0 dx∫^1-x_0 x dy/√(1-x-y)∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D ( 𝐤×𝐪)^2/[ p^2 + x(1-x) k^2 + y (Ω/2)^2 ]^5/2=N e^2 q^2 ( 1-1/D)Γ( 5-D/2) π^1+ D/2/2 Γ( 1/2) Γ( D/2) (2π)^D (4 π)^D/2∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dkx [ x(1-x) ]^D-5/2[ 1-x-y]^-1/2/[ k^2+ ( Ω/2)^2 ] [ k^2+ y/x(1-x)( Ω/2)^2 ]^5-D/2k^D.After a lengthy calculation and at the end setting D=3-ϵ and q → v q we arrive at the final expression for the vertex correction to the density-density correlator coming from the Coulomb interactionδΠ^V_00(iΩ,𝐪)= N e^2 q^2/72 π^3 v^2 [3/2×1/ϵ] + 𝒪 (1).Therefore, after accounting for the contribution from self-energy diagram we obtain the net correction to the polarization tensor due to the long-range tail of the Coulomb interaction from density-density correlation to beδΠ_00 (i Ω, 𝐪) = δΠ^SE_00 (i Ω, 𝐪) + δΠ^V_00 (i Ω, 𝐪) =- N e^2 q^2/72 π^3 v^2 [ 1/ϵ^2 - 1/2 ϵ[ 1+ 2 γ_E -2 log(4π) ] ] ≡ -N e^2 q^2/72 π^3 v^2 [ 1/ϵ^2 + b/ϵ],where b=-[ 1+ 2 γ_E -2 log(4π) ]/2 ≈ 1.454.We point our here that the results for the Coulomb correction to the current-current correlator in Eq. (<ref>) and to the density-density correlator in Eq. (<ref>) are consistent with the charge conservation, Eq. (<ref>), and thereforedimensional regularization employed manifestly preserves gauge invariance of the theory. §.§ Correction to optical conductivity Upon obtaining the same expression for the polarization bubble from both current-current and density-density correlators we can proceed with the computation of the correction to the OC due to the long-range piece of the Coulomb interaction. Following the steps highlighted in Sec. <ref>, we findσ_jj (Ω)= σ_0 (Ω)[ 1- α/6π{ b-2 log( E_Λ/Ω) }],where α=e^2/(ħ v) is the fine structure constant in the Weyl medium. Here E_Λ=2 v Λ, where Λ is the ultraviolet momentum cut-off and therefore E_Λ is the band width of a Weyl semimetal (WSM) in linearized approximation. If we also account for the correction to the OC due to the short-range piece of the density-density interaction the OC of an interacting Weyl liquid (to the leading order in coupling constants) is given byσ_jj (Ω)= σ_0 (Ω)[ 1- α/6π{ b-2 log( E_Λ/Ω) } + ( g_0 Ω^2/24 π^2 v^3){ a- 2 log( E_Λ/Ω) }],with a ≈ 3.62069 and b ≈ 1.454. Recall σ_0 (Ω) is the conductivity of a N-flavored non-interacting Weyl semimetal, given by σ_0(Ω)=N e^2_0/(12 h v), where N is the number of Weyl points in the Brillouin zone. We realize that coupling constants appearing in the above expression for the optical conductivity are renormalized ones. Therefore, two dimensionless couplings, namely α and g=g_0 Ω^2/v^3 in the above expression needs to be substituted by their scale dependent strengths, which we can readily obtained from the renormalization group flow equations of these two couplings appearing respectively in Eq. (<ref>) and Eq. (<ref>). After expressing β_x as dx/dl, where x=α, ĝ_0 and l is the logarithm of the renormalization scale, and therefore l=log( E_Λ/Ω), we obtain the running couplingsα(Ω)= α_0/1+ α_0 N+1/3 πlog( E_Λ/Ω) ≈1/N+1/3 πlog( E_Λ/Ω),g(Ω)=ĝ_0( Ω/E_Λ)^2,where the quantities with subscript “0" denote their bare strength. Upon substituting these running couplings back in Eq. (<ref>), we findσ_jj (Ω)= σ_0 (Ω)[1+1/N+1-b/2 (N+1) log(E_Λ/Ω)-ĝ_0Ω^2/12π^2 E^2_Λ{log(E_Λ/Ω)-a/2}],which matches with Eq. (13) of the main text, and ultimately simplifies to Eq. (1) from the main text for Ω≪ E_Λ. § ALTERNATIVE COMPUTATION OF CURRENT-CURRENT CORRELATOR WITH LONG-RANGE COULOMB INTERACTION We here present an alternative route to compute the current-current correlator in the presence of long-range Coulomb interaction. In the previous calculation we have chosen l=m=x at the fermion-current vertex and performed the analysis. Alternatively, we can sum over this contribution for all spatial components of the current operator, and then devide the final expression by D (spatial dimensionality) to obtain the contribution of each diagram/term to longitudinal conductivity.If we sum over the contribution from the self-energy diagram from all spatial components of current operators, its contribution to conductivity goes as (after performing the trace algebra)δΠ^SE_lm(iΩ,0)= 4 N E(D)/D δ_lm ∫d^D 𝐤/(2π)^D∫^∞_-∞dω/2 π [ -2D k^2 ω(ω+Ω) -(2-D)( ω^2 k^2 - k^4) ]/k^3-D[ ω^2+k^2 ]^2 [(ω+Ω)^2+ k^2 ]= N e^2 Ω^2/72 π^3 v^2 [ 3/ϵ^2 + 1/ϵ[ 7-3 γ_E + 3 log(4 π)]] δ_lm,which is identical to the result obtained in Eq. (<ref>). Since the frequency integral produces a lengthy expression, we here do not wish to present all the intermediate steps, which, however, follow those in Eq. (<ref>). We can proceed with the same strategy for the vertex diagram by computing the terms in Eqs. (<ref>)-(<ref>). In this framework δΠ^V,2_lm,1 (i Ω,0) remains unchanged, while three other components appearing in δΠ^V_lm,1 (i Ω,0) read as [4]δΠ^V,1_lm,1 (i Ω,0)= NΩ^2/4 Dδ_lm∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2 ( 𝐤·𝐩)^2 +(D-2) k^2 p^2 /k^3 p^3 [ k^2 + (Ω/2)^2 ],δΠ^V,3_lm,1 (i Ω,0)=-NΩ^4/32 Dδ_lm∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2( 𝐤·𝐩)^2 - 2 p^2 k^2/k^3 p^3 [ k^2 + (Ω/2)^2 ] [ p^2 + (Ω/2)^2 ],δΠ^V_lm,2 (i Ω,0)= NΩ^2/8 D (D-1) δ_lm∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D2 π e^2/|𝐤-𝐩|^2𝐤·𝐩/k p [ k^2 + (Ω/2)^2 ] [ p^2 + (Ω/2)^2 ].Now we present some key steps of the computation of each term and display the final result.After some algebraic simplification δΠ^V,1_lm,1 (i Ω,0) reads asδΠ^V,1_lm,1 (i Ω,0) = N Ω^2/4D [ 1/4I_1 + 1/4 I_2 + (D-2/3) I_3 ] δ_lm.The integral I_1 in the above expression goes asI_1= ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^Dk/ p^3 |𝐤-𝐩 |^2 [ k^2+(Ω/2)^2 ] =2 π^D/2Γ( 5-D/2) Γ( D-3/2) Γ( D/2-1)/(4π)^D/2 (2π)^DΓ( 3/2) Γ( D/2) Γ( D-5/2) ∫^∞_0 dk k^2D-5/k^2+(Ω/2)^2= 2 π^D/2Γ( 5-D/2) Γ( D-3/2) Γ( D/2-1)/(4π)^D/2 (2π)^DΓ( 3/2) Γ( D/2) Γ( D-5/2)[ Ω^2D-62^5-2D π cosec(π D) ].The quantity in the straight bracket comes from the integral over radial momentum variable k. The following integral identity will be extremely useful to compute I_2 and I_3∫d^D 𝐩/(2π)^D 1/|𝐤-𝐩 |^2 ( |𝐩|^2 )^a = Γ( 1+a-D/2) Γ(D/2-1 ) Γ(D/2-a )/Γ(a) Γ(D-1-a) (k^2 )^-1-a-D/2.With the help of the above integral identity, the second term is given byI_2 = ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^Dp/ k^3 |𝐤-𝐩 |^2 [ k^2+(Ω/2)^2 ] =2 π^D/2Γ( 1-D/2) Γ( D+1/2) Γ( D/2-1)/(4 π)^D/2 (2 π)^D Γ( D/2) Γ( -1/2) Γ( D-1/2) ∫^∞_0 dkk^2D-5/k^2+(Ω/2)^2= 2 π^D/2Γ( 1-D/2) Γ( D+1/2) Γ( D/2-1)/(4 π)^D/2 (2 π)^D Γ( D/2) Γ( -1/2) Γ( D-1/2)[ Ω^2D-62^5-2D π cosec(π D) ].The last term in the expression of δΠ^V,1_lm,1 (i Ω,0) assumes the formI_3= ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D1/ k p |𝐤-𝐩 |^2 [ k^2+(Ω/2)^2 ] = 2 π^D/2Γ( 3-D/2) Γ( D-1/2) Γ( D/2-1)/(4 π)^D/2 (2 π)^D Γ( D/2) Γ( 1/2) Γ( D-3/2)∫^∞_0 dkk^2D-5/k^2+(Ω/2)^2= 2 π^D/2Γ( 3-D/2) Γ( D-1/2) Γ( D/2-1)/(4 π)^D/2 (2 π)^D Γ( D/2) Γ( 1/2) Γ( D-3/2)[ Ω^2D-62^5-2D π cosec(π D) ].Now combining the contributions from I_1, I_2 and I_3, we obtainδΠ^V,1_lm,1 (i Ω,0)= Ne^2 Ω^2/72 π^3 [ 2/ϵ^2 + 6-2γ_E +2 log(4π)/ϵ] δ_lm,for D=3-ϵ, which is in agreement with the result inEq. (<ref>).Now we proceed with the computation of δΠ^V,3_lm,1 (i Ω,0). After some simple algebraic manipulation we findδΠ^V,3_lm,1 (i Ω,0)= - 2π e^2NΩ^4/32D[K_1-2 K_2 ] δ_lm,where K_1=1/2( J_1 + J_2 + J_3 ), withJ_1=-∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D1/k^3 p^3 [ k^2+(Ω/2)^2 ] [ p^2+(Ω/2)^2 ] = Ω^2D-8(16 π)^2-D ^2 ( π D/2)/[ Γ( D/2) ]^2.The second term in K_1 reads as [4]J_2= ∫d^D 𝐤/(2π)^Dk/[ k^2+(Ω/2)^2 ]∫d^D 𝐩/(2π)^D1/p^3 |𝐤-𝐩|^2 [ p^2+(Ω/2)^2 ] = Γ( 7-D/2)/(4π)^D/2Γ( 3/2)2 π^D/2/(2π)^D Γ( D/2)∫^1_0 dx ∫^1-x_0 dy ∫^∞_0 dk k^D√(1-x-y) [ x(1-x)]^D-7/2/[ k^2+(Ω/2)^2 ] [ k^2 + y/x(1-x) (Ω/2)^2 ]^D-7/2.Further analysis of J_2 produces extremely lengthy expression and we perform the analysis in mathematica. The last term in K_1 goes asJ_3= ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D1/k p |𝐤-𝐩|^2 [ k^2+(Ω/2)^2 ] [ p^2+(Ω/2)^2 ] = [ N e^2 Ω^27 ζ(3)/128 π^3] ×32/2 π N Ω^2 e^2,and K_2= J_3. Upon collecting all these contribution we arrive at the final expression for δΠ^V,3_lm,1 (i Ω,0), given byδΠ^V,3_lm,1 (i Ω,0)=-2π e^2NΩ^4/32D[ 1/2(J_1+J_2 )-3/2 J_3 ]δ_lm≈N e^2 Ω^2/π^3[ -0.000531776/96π^4 + 7 ζ(3)/256]δ_lm≈ N e^2 Ω^2/π^3(0.0323292 ) δ_lm = N e^2 Ω^2/384 π^3 [ 4+ 7 ζ(3) ] δ_lm,in agreement with the result in Eq. (<ref>).Finally we come to the computation of δΠ^V_lm,2 (i Ω,0), which after some algebraic simplification can be expressed asδΠ^V_lm,2 (i Ω,0)= -π e^2 N Ω^2/8( 1-1/D)[ ( L_1 )^2 -2 L_2 + 2 L_3] δ_lm,whereL_1=∫d^D 𝐤/(2π)^D 1/k [ k^2 + (Ω/2)^2 ] = 2 π^D/2/Γ( D/2) (2 π)^D[ - π2^2-D Ω^D-3 sec( π D/2) ].The second entry in the expression of δΠ^V_lm,2 (i Ω,0) goes asL_2= ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D 1/k p |𝐤-𝐩|^2 [ p^2 + (Ω/2)^2 ]= Γ( 5-D/2) /(4π)^D/2Γ( 1/2) 2 π^D/2/(2 π)^DΓ( D/2) ∫^1_0 dx [ x(1-x)]^D-5/2∫^1-x_0 dy∫^∞_0 dk k^D-2 [1-x-y ]^-1/2/[ k^2 + y/x(1-x)( Ω/2)^2 ]^5-D/2= 2 π^D/2 Γ(3-D) Γ( D-1/2)/(2 π)^D (4π)^D/2Γ( D/2) Γ( 1/2)2^5-2D(Ω^2)^D-3∫^1_0 dx [x(1-x) ]^1-D/2∫ ^1-x_0 dy y^D-3 [ 1-x-y ]^-1/2= 2 π^D/2 Γ(3-D) Γ( D-1/2)/(2 π)^D (4π)^D/2Γ( D/2) Γ( 1/2)2^5-2D(Ω^2)^D-3 Γ(D-2) Γ( 3-D/2) Γ( D/2-1)/Γ( D-3/2).The last entry in the expression of δΠ^V_lm,2 (i Ω,0) is given byL_3= ( Ω/2)^2 ∫d^D 𝐤/(2π)^D∫d^D 𝐩/(2π)^D1/k p |𝐤-𝐩|^2 [ p^2 + (Ω/2)^2 ] [ k^2 + (Ω/2)^2 ]= ( Ω/2)^2 Γ( 5-D/2) /(4π)^D/2Γ( 1/2) 2 π^D/2/(2 π)^DΓ( D/2) ∫^1_0 dx∫^1-x_0 dy ∫^∞_0 dk k^D-2 [ x(1-x)]^D-5/2[1-x-y ]^-1/2/[ k^2 + (Ω/2)^2 ] [ k^2 + y/x(1-x)( Ω/2)^2 ]^5-D/2= 1/32 π^4∫^1_0 dx ∫^1-x_0 dy[1-x-y ]^-1/2/x(1-x) log[ y/x(1-x)]/y/x(1-x)-1 =7 ζ(3)/32 π^4.Now collecting the contributions from L_1, L_2 and L_3 we obtain the divergent piece of δΠ^V_lm,2 (i Ω,0) to beδΠ^V_lm,2 (i Ω,0) =N e^2 Ω^2/72 π^3 [ 3/2 ϵ] δ_lm,in agreement with the result in Eq. (<ref>). Thus in this alternative approach to compute the current-current correlator we obtain identical results for each and every contribution toboth self-energy and vertex diagram. § KRAMERS-KRONIG RELATIONS AND DIELECTRIC CONSTANT Finally, we present the computation of the imaginary part of the optical conductivity [(σ)], which is tied with the real part of the dielectric constant [ε(Ω)] according toε(Ω)=1-4π/Ω (σ),from its real component [(σ)] by applying the Kramers-Kronig relation(σ) = -2 π/Ω 𝒫 ∫^E_Λ_0 dΩ^' (σ)/Ω^'^2 -Ω^2.In the above expression 𝒫 denotes the principle value of the integral. Since we are interested in the regime Ω≪ E_Λ so that signature of Weyl fermions are prominent, we take the simplified expression for the real part of the optical conductivity after accounting for the leading correction due to Coulomb interaction, given by [Eq. 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"Vladimir Juricic"
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"published": "20170726175958",
"title": "Optical conductivity of an interacting Weyl liquid in the collisionless regime"
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figuresfigures | http://arxiv.org/abs/1707.08804v1 | {
"authors": [
"Irénée Frérot",
"Tommaso Roscilde"
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"categories": [
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"primary_category": "quant-ph",
"published": "20170727100004",
"title": "Quantum critical metrology"
} |
firstpage–lastpage Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ========================================================================================================================================The episodic dynamics of the magnetic eruption of a spinning black hole (BH) accretion disks and its associated intense shapeup of their jets is studied via three-dimensional general-relativistic magnetohydrodynamics (GRMHD). The embedded magnetic fields in the disk get amplified by the magnetorotational instability (MRI) so large as to cause an eruption of magnetic field (recconection)and large chunks of matter episodically accrete toward the roots of the jets upon such an event. We also find that the eruption events produce intensiveAlfvén pulses, which propagate through the jets.After the eruption, the disk backs to the weakly magnetic states. Such disk activitiescauseshort time variabilities in mass accretion rate at the event horizon as well as electromagnetic luminosity inside the jet. Since the dimensionless strength parameter a_0=eE/m_e ω c of these Alfvén wave pulses is extremely high for a substantial fraction of Eddington accretion rate accretion flow onto a supermassive black hole, the Alfvén shocks turn into ultrarelativistic (a_0≫ 1) bow wake acceleration, manifestinginto the ultra-high energy cosmic rays and electrons which finally emit gamma-rays. Since our GRMHD model has universality in its spatial and temporal scales, it is applicable to a wide range of astrophysical objects ranging from those of AGN(which is the primary target of this research), to micro-quasars. Properties such as time variabilities of blazar gamma-ray flaresand spectrum observed by Fermi Gamma-ray Observatory are well explained by linear acceleration of electrons by the bow wake.(magnetohydrodynamics) MHD –accretion discs – (galaxies:) quasars: supermassive black holes — galaxies: jets§ INTRODUCTIONActive galactic Nuclei (AGNs) are high energy astronomical objects, so that they emit non-thermal radiation in any frequency ranges of radiation, in other words, radio, infrared, visible lights, ultraviolet, X-rays, and gamma-rays <cit.>. These central engines are believed to be accreting supermassive black holes, with relativistic jet whose bulk Lorentz factor is ∼ 10 <cit.>. Their jets show strong time variabilities in the timescales from days to years <cit.>. In the extreme cases, blazars show bursts of hours <cit.>. These radiations are believed to be emitted by a bunch of electrons with strongly relativistic motions.The system of AGN jets is also believed as a cosmic ray accelerator <cit.>. Although it has a potential to accelerate highest energy to the energy of ∼ 10^20 eV, it is still not well understood what is physical mechanism for the particle acceleration and where the acceleration site is. Many studieshave been done for diffusive shock acceleration model based on conventional Fermi acceleration mechanism <cit.>. In the Fermi acceleration, charged particles interact with magnetized clouds, and are vented random directions by their magnetic field. Since the head-on collisions, which the particles gain the energy, are more frequent than rear-end collisions, which they loose their energies, the particles statistically gain the energy step by step and eventually obtain very high energy to be cosmic ray particles. However, this Fermi acceleration mechanism is difficult to explain highest energy particles ∼ 10^20 eV,because of 1) the large number of scatterings necessary to reach highest energies, 2) energy losses through the synchrotron emission at the bending associated with scatterings, and 3) difficulty in the escape of particles which are initially magnetically confined in the acceleration domain (e.g., <cit.>).On the other hand, <cit.> proposed that particles can be accelerated by the wakefield induced by an intense laser pulse, see review by <cit.>. This long lasting energy elevated state of wakefield may be regarded a Higgs state <cit.> of plasma. In particular, ponderomotive force which is proportional to gradient of E^2 works to accelerate the charged particles, effectively, where E is electric field of the electromagnetic wave. The acceleration towards relativistic regime by the ponderomotive force is confirmed by recent experiments by ultra intense lasers for electrons <cit.>. Positron acceleration driven by wakefields by electrons is also reported by <cit.>.<cit.> appliedthis mechanism to magnetowave induced plasma wakefield acceleration for ultra high energy cosmic rays. Recently, <cit.> applied this wakefield acceleration theory to the relativistic jets launched from an accreting black hole. In such astrophysical context going far beyond the laboratory scales of wakefields (see, for example, <cit.>), the relativistic factors that characterize the dynamics a_0=eE/ m_e ω c becomes far greater than unity. This has not been achieved in the laboratory yet, though simulations began to peek into it. In this regime <cit.> the ponderomotive acceleration has advantages over the Fermi mechanism.In close scrutiny, the wakefields are composed with two parts the frontal bow part and the following stern (wake) part. Here we may call simply the bow wake and stern wake. In the work of high a_0 simulation, it was shown in <cit.> that the bow wake(it is driven directly by the ponderomotive force) is dominant over the stern wake. The advantages of this bow wake acceleration over the Fermi mechanism are: * The ponderomotive field provides an extremely high accelerating field (including the wakefield).* It does not require particle bending, which would cause strong synchrotron radiation losses in extreme energies.* The accelerating fields and particles move in the collinear direction at the same velocity, the speed of light, so that the acceleration has a built-in coherence called “relativistic coherence” <cit.>; in contrast, the Fermi acceleration mechanism, based on multiple scatterings, is intrinsically incoherent and stochastic.* No escape problem <cit.> exists. Particles can escape from the acceleration region since the accelerating fields naturally decay out. They found that protons can be accelerated even above ZeV ∼ 10^22 eV in the bow wake of a burst of Alfvén waves emitted by an accretion disk around a black hole with the mass of 10^8 M_⊙. <cit.> used three major assumptions based on the standard α-disk model <cit.>.* Assumption A: the magnetic field energy ℰ_B included in an Alfvén wave burst is assumed as:ℰ_B=(B_D^2/4π)π(10R_s)^2Z_D=1.6× 10^48(ṁ/0.1)(m/10^8)^2erg,where B_D is the magnetic field stored in the inner most regions of the accretion disk, R_ s=2R_ g is the Schwarzschild radius of the black hole, Z_ D is the thickness of the disk, R_ g is gravitational radius,ṁ is the accretion rate normalized by the Eddington luminosity, and m is the mass of the black hole in the unit of solar mass. * Assumption B: they assumed that the angular frequency ω_ A of the Alfvén wave corresponds to that excited by magnetorotational instability (MRI <cit.>), which takes place in a magnetized accretion disk, in other words:ω_ A=2πc_ A_D/λ_ A∼ 2.6×10^-5(m/10^8)^-1 Hz,where λ_ A is the wavelength of the Alfvén wave, and c_ A is speed of Alfvén wave. <cit.> showed that the Alfvén shock gives rise to electromagnetic wave pulse with ω=ω_ A along the propagation in the jets through the mode conversion, as the density and magnetic fields in the jets decrease during the jet propagation. * Assumption C: the recurrence rate ν_ A of the Alfvén burst is evaluated as:ν_ A=ηc_ A_D/Z_ D Hz,where η is the episode-dependent parameter of the order of unity. They found that the non-dimensional strength parametera_0=e E / m_e ω c is as high as 10^10 for the case of ṁ=0.1 and m=10^8, where e is electric charge, E is the intensity of the electric field, m_e is mass of electron, and c is speed of light. The ponderomotive force of this extremely relativistic waves co-linearly accelerate to the jet particle up to the maximum energy:W_ max=2.9× 10^22q(Γ/20) (ṁ/0.1)^4/3(m/10^8)^2/3 eV,where q is the charge of the particle and Γ is the bulk Lorentz factor of the jet. Recent one-dimensional particle in cell (PIC) simulation shows maximum energy gain via a ponderomotive force in the bow wakeand the maximum energy is almost proportional to a_0^2 <cit.>. Based on the above estimation, <cit.> concluded that the accreting supermassive black hole is the ZeV (10^22 eV) linear accelerator. GRMHD simulations of accretion flows onto the black hole have been done since early works by <cit.>. Some improvements in the numerical method for solving GRMHD equations made it possible to follow the long-term dynamics of magnetized accretion flows <cit.>. 2D axis-symmetric and full 3D simulations have been done to study properties of accretion disk, Blandford-Znajek efficiency, jet and so on.<cit.> have studied outward going electromagnetic power through the event horizon, i.e,Blandford-Znajek process by 2D axis-symmetric simulations. <cit.> studied long term magnetized jet propagation launched from the disk and black hole system. The properties of the accretion flows, and outflows are studied by a series of papers <cit.> Long-term 3D GRMHD simulations have been done by <cit.> <cit.> pointed out that the initial magnetic field topology strongly affects the results, especially outflows, see also <cit.>. Recently <cit.> have tried to comparethe results by GRMHD simulations with observational results. Major motivation of the present paper is to verify the assumptions of <cit.> (Equations (<ref>) and (<ref>)) by the 3D GRMHD simulations of accretion disk around a supermassive black hole. This paper is organized as follows. We describe our physical models and numerical details in 2.The results are shown in <ref>. The application to the ultra high energy cosmic ray acceleration, blazars, and gravitational waves is discussed in <ref> and 5. § GENERAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATION METHOD§.§ Basic Equations We numerically solve general relativistic magnetohydrodynamic equations, assuming a fixed metric around a black hole. The unit in which GM_ BH and c are unity is adopted, where G is gravitational constant, M_ BH is mass of the central black hole. The scales of length and time are R_ g=R_ s/2=GM_ BH c^-2 and GM_ BH c^-3, respectively. The mass and energy is scale free. The achieved mass accretion rate at the event horizon is used to scale of the mass and energy, for example.The metric around a rotating black hole whose dimensionless spin parameter is a can be described by Boyer-Lindquist (BL) coordinate or Kerr-Schild (KS) coordinate as follows.The line element in BL coordinate isds_ BL^2=g_ttdt^2+g_rrdr^2+g_θθdθ^2 +g_ϕϕdϕ^2+2g_tϕdtdϕ,where g_tt=-(1-2r/Σ), g_rr=Σ/Δ, g_θθ=Σ, g_ϕϕ=Asin^2θ/Σ, g_tϕ=-2rasin^2θ/Σ, Σ=(r^2+a^2cos^2 θ), Δ=r^2-2r+a^2, and A=(r^2+a^2)^2-a^2Δsin^2θ. We follow standard notation used in <cit.>, i.e, metric tensor for Minkowski space is diag(-1,1,1,1).The line element in Kerr-Schild coordinate isds_ KS^2=g_ttdt^2+g_rrdr^2+g_θθdθ^2 +g_ϕϕdϕ^2+2g_trdtdr +2g_tϕdtdϕ+2g_rϕdrdϕ,where g_tt=-(1-2r/Σ), g_rr=1+2r/Σ, g_θθ=Σ, g_ϕϕ=Asin^2θ/Σ, g_tr=2r/Σ, g_tϕ=-2rasin^2θ/Σ, and g_rϕ=-asin^2θ(1+2r/Σ).We also use so-called modified Kerr-Schild (mKS) coordinate (x_0,x_1,x_2,x_3) so that the numerical grids are fine near the event horizon and the equator. The transformation between Kerr-Schild coordinate and modified Kerr-Schild coordinate is described as t=x_0, r=expx_1, θ=π x_2 +1 2(1-h)sin(2π x_2), and ϕ=x_3, where h is a parameter which controls how the grids are concentrated around the equator. We have done three cases of resolution in the polar and azimuthal angle grid. Constant grid for polar angle, i.e., h=1 are used for two lower resolution cases. The grid numbers areN_1=124, N_2=124, and N_3=60,andN_1=124, N_2=252, and N_3=28 which are uniformly spaced for both cases. In another case we set h=0.2 so that the polar grids concentrates around the equator. The grid numbers are N_1=124, N_2=252, and N_3=60 which are uniformly spaced. Since the resolution of polar grid with h=0.2 is about 10 and 5 times better than that with h=1 at the equator for the cases with N_2=124 and N_2=252, respectively, we can capture shorter wavelengthand faster growing mode of MRI for poloidal direction by the highest resolution case. The highest resolution is comparable to those used in recent 3D GRMHD simulations<cit.>. Computational domain covers from inside the event horizon to r=30000 R_ g, [0.01π, 0.99π] in polar angle, and [0, 2π] in azimuthal angle. Contravariant vectors in Boyer-Lindquist coordinate and Kerr-Schild coordinate are related with u^t_KS=u^t_BL+(2r/Δ)u^r_BL, u^r_KS=u^r_BL, u^θ_KS=u^θ_BL, and u^ϕ_KS=(a/Δ)u^r_BL+u^ϕ_BL. Contravariant vectors in Kerr-Schild coordinateand modified Kerr-Schild coordinate are related with u^t_KS=u^0_mKS, u^r_KS=ru^1_mKS, u^θ_KS=(π(1-(1-h)cos(2π x_2)))u^2_mKS, and u^ϕ_KS=u^3_mKS.Mass and energy-momentum conservation laws are,1√(-g)∂_μ (√(-g)ρ u^μ)=0, ∂_μ (√(-g)T^μ _ν)=√(-g)T^κ_λΓ^λ _νκ,where T^μν is energy momentum tensor, ρ is rest mass density, u^μ is fluid 4-velocity, g is determinant of metric tensor, i.e., g_BL=g_KS=-Σ^2sin^2θ, and g_mKS=-π^2r^2(1-(1-h)cos2π x_2)^2Σ^2sin^2θ, and Γ ^k _ij is Christoffel symbol which is defined as Γ ^k _ij =(1/2) g^kl (∂ g_jl/∂ x^i +∂ g_li/∂ x^j -∂ g_ij/∂ x^l). Energy momentum tensor which includes matter and electromagnetic parts is defined as followsT^μν=T_ MA^μν+T_ EM^μν,T_ MA^μν=ρ h u^μ u^ν + p_ th g^μν,T_ EM^μν=F^μγF^ν_γ -1 4g^μνF^αβF_αβ,where h(≡ 1+U/ρ+p_ th/ρ) is specific enthalpy, U is thermal energy density, p_ th is thermal pressure, and F^μν is the Faraday tensor and a factor of √(4π) is absorbed into the definition of F^μν. The dual of the Faraday tensor is^*F^μν=1 2e^μναβF_αβ,where e^μναβ=√(-g)ϵ^μναβ and ϵ^μναβ is the completely antisymmetric Levi-Civita symbol (ϵ^0123=-ϵ_0123=-1). The magnetic field observed by normal observer isℬ^μ=-^*F^μνn_ν=α ^*F^μ t,where n_μ=(-α, 0,0,0) is the normal observer's four-velocity and α≡√(-1/g^tt) is the lapse. Note the time component of ℬ is zero, since ℬ^t=α ^*F^tt=0. Here we introduce another magnetic field which is used in <cit.> as B^μ=^*F^μ t=ℬ^μα.The time component of B^μ is also zero. We also introduce four magnetic field b^μwhich is measured by an observer at rest in the fluid,b^μ=-^*F^μνu_ν.B^i and b^μ are related withb^t≡ B^μ u_μ,b^i≡ (B^i+u^i b^t)/u^t.b^μ u_μ=0 is satisfied. By using this magnetic four vector electromagnetic component of energy momentum tensor and the dual of Faraday tensor can be written asT_ EM^μν=b^2 u^μ u^ν + p_ b g^μν-b^μ b^ν,^*F^μν=b^μu^ν-b^νu^μ = ℬ^μu^ν-ℬ^νu^μα u^t= B^μu^ν-B^νu^μ u^t,where the magnetic pressure is p_ b=b^μ b_μ/2 =b^2/2. The Maxwell equations are written as^*F^μν_;ν=0.By using Eqs. (<ref>) these equations give∂_i (√(-g)B^i)=0, ∂_t (√(-g)B^i) +∂_j (√(-g) (b^i u^j-b^j u^i))=0.These are no-monopole constraint equation and time evolution of spacial magnetic field equations, i.e, the induction equations, respectively.In order to close the equations, ideal gas equation of state p_ th=(γ-1)U is adopted, where γ is the specific heat ratio which is assumed to be constant (γ=4/3) [The calculation with γ=5/3 shows some minor differences compared to that with γ=4/3as discussed by <cit.>. Similar time variavlities in the inflows and outflows discussed below are observed by adopting two different specific heat ratio in the equation of state.]. We ignore self-gravity of the gas around the black hole and any radiative processes, assuming radiatively inefficient accretion flow (RIAF) in the disk <cit.>, although the effects of radiation have been discussed by <cit.>.We numerically solve these equations by GRMHD code developed by one of authors <cit.>. Magnetohydrodynamic equations are solved by using shock capturing method (HLL method), applying 2nd order interpolation to reconstruct of physical quantities at the cell surfaces and 2nd order time integration by using TVD (total variation diminishing) Runge-Kutta method, see also <cit.>. The boundary conditions are zero gradient for x_1 and periodic one for x_3. §.§ Initial ConditionWe adopt the Fishbone-Moncrief solution as an initial condition for hydrodynamic quantities as adopted for recent GRMHD simulations <cit.>. This solution describes the hydro-static torus solution around a rotating black hole. The gravitational force by the central black hole, the centrifugal force and the pressure gradient force balance each other. There are some free parameters to give a solution. We assume the disk inner edge at the equator is at r=6.0 R_ g anda constant specific angular momentum (l^*≡ |u^tu_ϕ|=4.45). The 4-velocity is firstly given at Boyer Lindquist coordinate then transformed to Kerr-Schild one and to modified Kerr-Schild one. The dimensionless spin parameter is assumed to be a=0.9. The radii of event horizon and the innermost stable circular orbits (ISCO) at the equator are at r_ H(a=0.9)∼ 1.4 R_ g and r_ ISCO(a=0.9)∼ 2.32 R_ g, respectively. The initial disk profile is on the plane including polar axis shown in Fig. <ref> which shows mass density contour. The disk is geometrically thick and is different from standard accretion disk <cit.> in which the geometrically thin disk is assumed. As we have discussed in Sec. <ref> magnetic fields play an important role not only for the dynamics of the accretion flows but also for the dynamics of the outflows. We impose initially weak magnetic field inside the disk as a seed. This weak magnetic field violates the initial static situation and is expected to be amplified by winding and MRI.Here we introduce the four-vector potential A of the electromagnetic field. The Faraday tensor is defined by this vector potential as follows.F_μν=∂_μ A_ν-∂_ν A_μ. In this study we assume initially closed poloidal magnetic field, i.e., the toroidal component of the initial vector potential is,A_ϕ∝max ( ρ/ρ_max-0.2, 0 ),where ρ_max is maximum mass density in the initial torus. Other spacial components are zero, i.e., A_r=A_θ=0. The minimal plasma β which is the ratio of the thermal pressure to the magnetic pressure is 100 in the disk. Since the vector potential has only toroidal components, the poloidal magnetic field is imposed. To violate axis-symmetry maximally 5% amplitude random perturbation is imposed in the thermal pressurei.e., thermal pressure is p_ th= p_0 (0.95+0.1C), where p_0 is equilibriumthermal pressure derived by Fishbone-Moncrief solution andC is random numberin the range of 0≤ C≤ 1. This perturbation violates axis-symmetry of the system and triggers generation of non axisymmetric mode. § EPISODIC ERUPTION OF DISKS AND JETSAt first we discuss the results based on the highest resolution calculation. The resolution effect, i.e., comparison with the results by lower resolution calculations is briefly discussed later. The main properties, such as the amplification and the dissipation of the magnetic field inside the disk, Alfvén wave emission from the disk, and these time variavilities, which will be discussed below are common for all calculations by different resolutions, although characteristic timescales are different each other. §.§ B-field amplification and mass accretionThe magnetic field in the disk is amplified by winding effect and MRI, as follows. The initially imposed poloidal magnetic field is stretched in the toroidal direction, generating toroidal components, since there is a differential rotation inside the accretion disk. Although initially imposed magnetic field is weak, i.e., the minimum plasma β is 100 inside the disk,the strength of the magnetic field quickly increases by the winding effect <cit.> and MRI. Soon stratified filaments which is parallel to the equatorial planeappear around the equator in the magnetic pressure contour. Fig. <ref>(a) shows the volume averaged strength of the magnetic field ⟨(B^iB_i)^1/2⟩at the equator i.e., averaged (B^iB_i)^1/2over r_ ISCO≤ r ≤ 10R_ g, θ=π/2, and 0 ≤ϕ≤ 2π. Since the magnetic field is stretched by the differentialrotation in the disk, toroidal component dominates over the poloidal component after t=100 GM_BHc^-3, though both components show strong time variability. Fig. <ref>(b) shows themass accretion rate at the event horizon Ṁ(r_ H, t), where the mass accretion rate at a radius is defined asṀ(r_ H, t)=-∫√(-g_ KS)ρ(r_ H,t) u^r(r_ H,t) dA_ KS=-∫√(-g_ mKS)ρ u^1 dA_ mKS,where dA is area element, for example, dA_ mKS=Δ x^2 Δ x^3, and the sign is chosen so that the case of mass inflowis positive mass accretion. The mass accretion rate Ṁ(r_ H, t) also shows strong time variability, and synchronized withmagnetic field strength near the ISCO.Bottom two panels in Fig. <ref> show the contours of 1/β at the equator at t=7550 GM_ BHc^-3 and t=7640 GM_ BHc^-3. Between these two figures the state of the disk near the disk inner edge (r ∼ 6 R_ g) transitioned from high β state (β^-1∼ 10^-2) to low β state (β^-1∼ 1). Since we sometimes observe that the plasma β in the disk is more than 100, we use “low” β state for the disk with a plasma β of order of unity at which the disk is still gas pressure supported. It should be noted that “low β state” is defined as a magnetically supported disk with β≲ 1, for example, <cit.>. The characteristics of two states presented in <cit.> are listed in the top of Fig. <ref>. Middle two panels in Fig. <ref> show the toroidal magnetic field lines of two disks from MHD simulations by <cit.>. At the low β state (right panels in Fig. <ref>) the toroidal magnetic field is stretched to the limit and magnetic field energy is stored. Since some field lines are almost anti-parallel and very close to each other, the reconnection will happen eventually. After magnetic field energy is dissipated via reconnection, the system becomes high β state (left panels in Fig. <ref>). Bar structures near the disk inner edge can be seen (bottom panels in Fig. <ref>). The non-axis-symmetric mode is excited, as shown in global hydrodynamic and magnetohydrodynamic simulations of accretion disks, for example <cit.>. Figure <ref> shows the contours of 1/β (at the equator),mass density (y-z plane), and magnetic pressure (x-z plane) at two different times as shown in Fig. <ref>. Along the polar axis low density and highly magnetized region appears. It corresponds to the Poynting flux dominated jet. Thus baryonless and highly magnetized jet is formed along the polar axis. In this region the Alfvén speed is almost speed of light ∼ c.A disk wind blows between the jet and the accretion disk. Filamentaly structures which are excited by MRI can be seen in the magnetic pressure contours. The thickness of filaments is ∼ 0.1 R_ g, as shown in Fig. <ref>.Both the averaged strength of the magnetic field near the ISCO at the equator and the mass accretion rate at the event horizon show synchronous time variability. This is because that the mass accretion rate at the event horizon is strongly affected by the activities of magnetic field amplification near the disk inner edge. The magnetic field amplification via MRI enhances the specific angular momentum transfer inside the accretion disk, resulting in the increase of the accretion rate at the event horizon. This means that the amplification of magnetic fields acts as a viscosity which is introduced as α-viscosity in <cit.>. Typical increase timescales are 20-60 GM_ BH c^-3. While the magnetic field is amplified, the mass accretion rate at the event horizon rapidly increases. As we will show later, the outflow properties are also show intense time variability which is strongly related with particle acceleration via wakefield acceleration.The mass accretion rate repeatedly shows sharp rises followed by gradual falling down. The rising timescale for the quick increase of mass accretion rate corresponds to the value of the growth timescale of MRIτ_1 ∼ f_ MRIΩ_ MRI^-1,where f_ MRI is order of unity, Ω_ MRI is growth rate of MRI, and Ω_ MRI=3Ω_ K/4 for the fastest growing mode and Ω_ K(r)=r^-3/2 is Newtonian Keplerian angular velocity. The timescale for the fastest growing MRI mode isτ_1 ∼ 4.7 f_ MRI(r r_ ISCO)^3/2[ GM_ BH c^-3].This timescale at around r= 6-8 R_ g is almost as same as the timescale of increase of the strength of magnetic field in the disk. By the analysis of MRI for Newtonian MHD, the angular frequency of the mode is k_z c_ A_z=√(15/16)Ω_ K∼Ω_ Kfor the fastest growing mode for z (parallel to polar axis) direction in Keplerian accretion disk, where k is wave number and c_ A_z is Alfvén speed of z component. The volume averaged Alfvén speed ⟨c_ A_z⟩∼⟨√(b^θ b_θ/(ρ h +b^2))⟩ near the ISCO (r_ ISCO<r<10 R_ g) at the equator is typically 3× 10^-3 c. Thus the wavelength of this mode isλ=2π/k_z∼2π⟨c_ A_z⟩ /Ω_ K∼ 0.022 (r/r_ISCO)^1.5 R_ g with a grid size near the ISCO and at the equator rΔθ=0.0057(r/r_ ISCO)R_ g for higher resolution case. Our simulation shows that the typical rising timescale in poloidal magnetic field amplification ∼ 30GM_ BH c^-3. Corresponding wavelength of MRI is estimated ∼ 0.14R_ g at r=8 R_ g. This structure is well resolved by more than 8 gridsand is consistent with thickness of the filamentary structure near the equator shown in magnetic pressure panel in Fig. <ref>.Since the episodic period of this quick increase of strength of poloidal magnetic field, i.e., large peak to peak, is τ_2 ∼ 100-400 GM_ BH c^-3 which is about 2-6 times longer than the Keplerian orbital period at r=6 R_ g near the ISCO (∼ 22 (r/r_ ISCO) GM_ BH c^-3). This timescale for the recurrenceis roughly consistent with the analysis by local shearing box <cit.> in which about 10 times orbital period at the radius of magnetic field amplfication is observed as repeat timescale.Along the polar axis funnel nozzles appear (Fig. <ref>). The outward going electromagnetic luminosity, opening angle of this jet are also time variable like as the mass accretion rate. The radial velocity just above the black hole and becomes positive at typically 10≲ r ≲ 20 R_ g, i.e., stagnation surfaces.Figure <ref>(b) shows radial electromagnetic luminosity calculated by the area integration only around polar region (0≤θ≤ 20^∘) at the radius r=15R_g. Electromagnetic luminosity shows similar short time variability with the magnetic field amplification near the disk inner edge. We can see typical rising timescale of the flares is as same as that for rising timescale of magnetic field in the disk, i.e., τ̅_1 ∼ 30 GM_BH c^-3 and the typical cycle of flares is also as same as repeat cycle of magnetic field amplification τ̅_2 ∼ 100 GM_BH c^-3. We have observed some active phases in electromagnetic luminosity in the jet. In these active phase, the averaged radial electromagnetic flux increases and becomes about a few tens percent of the averaged disk Alfvén flux at the equator at around t=1300, 3000, 4000, and 8200GM_BHc^-3, as shown in the Fig. <ref> (a). The disk Alfvén flux at the equator is evaluated as an average of z component of Alfvén energy flux at the equator⟨ E_ EM/dV⟩ times half ofAlfvén speed ⟨c_ A_z⟩/2 inside the disk (r_ ISCO<r<10 R_ g). A large fraction of emitted Alfvén waves goes into the jet, when the level of the electromagnetic luminosity in the jet becomes a few tens percent of Alfvén flux in the disk. This is almost consistent with the assumption A as we discuss in the next section.Figure <ref> shows time evolution and the vertical structure of the averaged toroidal magnetic field (⟨(B^ϕB_ϕ)^1/2⟩) as shown in <cit.>, i.e, butterfly diagram. The average is takenat 0≤ϕ≤ 2π and 3R_ g≤ R≤ 3.2R_ g, where R is distance from the polar axis. Although initially no magnetic field is imposed at R=3R_ g, soon the magnetic field is transported to this site with accreting gas due to MRI growth and angular momentum exchange. After that the magnetic fields quasi-periodically goes up to the north and goes down to the south from above and below the equator due to the magnetic buoyancy, i.e., Parker instability <cit.>, as shown by <cit.>. Each episode corresponds to the one cycleof the disk state transitions. The magnetic fields sometimes changes its sign, although it happens much less than that observed in <cit.> who did high resolution simulations in local hearing box. Around the equator magnetic fields rise up with a speed ∼ 10^-3 c which corresponds to theaveraged Alfvén speed of z component (c_ A_z) there. Strong magnetic fields sometimes goes up or goes downfrom the equator to outside of the disk. The appearance of the flare in the Poynting luminosity (Fig. <ref> (b) and (c)) in the jet corresponds to this strong magnetic field escape from the disk. The magnetic field lines in the jets are connected not with the disks but with themiddle and high latitude of the central black hole. The outward going electric magnetic luminosity from themiddle and high latitude of the central black hole is not so high as compared with that in the jet.TheAlfvén waves emitted from the disks do not directly goes into the jets as assumed in <cit.>. As shown above the time variavilities of the Poynting flux in the jet is as same as that of the magnetic fields strength in the disk and shows strong correlation. The amplification of Poynting fluxabove the black hole and the disks occurs by the Alfvén fluxes from the disk. Another possibility is that the blobs falling onto the black hallinteract with magnetic fields which are connected with those in the jets. §.§ resolution effectIn this subsection we discuss the resolution effect. We have performed calculations by three different grid types in polar and azimuthal angle as described in sec. <ref>. The highest resolution case for which the results are shown above is by non-uniform grids in polar angle for which the grids concentrates around the equator, resulting in about 10 or 5 times better in the polar angle around the equator than that for other two cases in which constant polar angle grids are adopted with around a half or same number of polar grids. In all cases we have observed properties discussed above, such as, time variable magnetic field amplification in the disk, disk state transition between low and high plasma β states, time variable mass accretion onto black holes, and Poynting flux dominated jet with some flares.The timescale of the fastest growing mode(30GM_BHc^-3) is observed in the amplification of magnetic fields in the disk for highest resolution case, whereas longer timescales (typically 50GM_BHc^-3 for 2nd highest resolution caseand 80GM_BHc^-3 or longer time scale for the lowest resolution case) are observed. The thinnest and multiple filamentsin magnetic pressure contour around the equator are observed for the highest resolution case. These results mean the longer wavelength mode of MRI is observed in the lower resolution cases. The recurrence timescale for higher resolution case is also faster than that by low resolution cases.§ PARTICLE ACCELERATION As shown in Fig. <ref> (b) flares in electromagnetic power in the jet are observed, where the Alfvén speed is almost speed of light because of the low mass density. Large amplitude Alfvén waves become electromagnetic waves by mode conversion of strongly relativistic waves <cit.>. The interaction of the electromagnetic waves and the plasma can result in the acceleration of the charged particles by the ponderomotive force i.e., wakefield acceleration <cit.>. The key for the efficient wakefield acceleration is the Lorentz invariant dimensionless strength parameter of the wave <cit.>,a_0=eEm_e ω c.The velocity of the oscillation motion of the charged particles via electric field becomes speed of light, when a_0 ∼ 1. If the strength parameter a_0 highly exceeds unity, the ponderomotive force works to accelerate the charged particles to relativistic regime to wave propagating direction.§.§ Comparison with <cit.>In order to evaluate a_0, <cit.> used three assumptions A, B and C. Based on the results of numerical simulation, we intend to confirm three assumptions. First, assumption A tells us that the Alfvén flux in the jet is equal to that in the disk. As shown in Fig. <ref>(a), electromagnetic flux in the jet becomes a few tens percent of Alfvén flux in the accretion disk at the some active phases of the electric magnetic luminosity in the jet. Thus most Alfvén waves emitted from the disk via Alfvén burst goes to the jet as assumed in <cit.> at this epoch, in other words, assumption A. in which all Alfvén waves are assumed to go into the jet.Second, <cit.> assumed that magnetic field amplification occurs at R=10R_s=20R_g for the standard disk model <cit.>. In <cit.> the timescale of the magnetic field amplification (τ_1)and frequency of the Alfvén wave are determined by theMRI growth rate (Eq. (<ref>)). Although they evaluated it around 10 R_ s, the magnetic field amplification occurs at any radius in the disk. Since magnetic field amplification which affects to the mass accretion rate and the time variavilities in the jet mainly occurs inside compared with the assumption by <cit.>, the timescales are shorter than those of them due to faster rotation period. If we apply R=6.4 R_g=3.2 R_s instead of R=20 R_g=10 R_s for <cit.> model, the timescales are close to our numerical resultsas shown in table <ref>. The reason why the magnetic field amplification at smaller radiusmay be due to the high spinning of the black hole, i.e., a=0.9 for whichboth the event horizon and the radius of ISCO is smaller than those for non-spinning black hole case (r_ ISCO(a=0)= 6 R_ g). This timescale is well consistent to the rising timescales of blazar flares observed for 3C454.3 which will be discussed in next subsection. In other words, assumption B is OK in qualitatively, but Eq. (<ref>) must to be ω_A∼ 1.0×10^-4(m/10^8)^-1 Hz (see also table <ref>).Finally, <cit.> estimated the repeat timescale as the crossing time of the Alfvén wave in the disk, i.e, Z_ D/ c_ A_z (assumption C) for the standard disk model <cit.>. When we apply the radius R=6.4 R_g=3.2 R_s instead of R=20 R_g=10 R_s in Eq. (<ref>), we obtain Z_ D/ c_ A_z=354 GM_ BH c^-3, ignoring factor η which is order of unity. This value is close to our typical τ̅_2=100 GM_ BH c^-3. The case of R=20 R_g=10 R_sis also listed in the table <ref>.We can reevaluate the strength parameter a_0 asa_0=eE m_e ω c=1.4× 10^11(M_ BH 10^8 M_⊙)^1/2(Ṁ_ avc^20.1 L_ Ed)^1/2.Here electric field is estimated as E=(⟨c_ A_ D⟩/ c)^1/2⟨ B_ D⟩. The angular frequency of the pulsed electromagnetic wave originating from the Alfvén shock (see <cit.>) ω_ D=2πc_ A_ D/λ_ A_ D, where λ_ A_ D is assumed to be 0.14 R_ g. We use the values at t=7900GM_ BH c^-3 andthe time averaged mass accretion rate7750 GM_ BH c^-3≤ t ≤ 8300 GM_ BH c^-3) at the event horizon Ṁ_ av=55.8 is used as a normalization to 10% Eddington luminosity for M_ BH=10^8 solar masses. The estimated strength parameter highly exceeds unity as discussed in <cit.>. This suggests efficient particle acceleration via wakefield acceleration can occur in the jet. Since both electrons and protons are acceleratedat the large amplitude electromagnetic flares via ponderomotive forces and these particle move with the waves, high energy non-thermal electrons are concentrated at these waves.If we apply the radius for the magnetic field amplification at R=6.4 R_g=3.2 R_s instead of R=20 R_g=10 R_s for Eq. (<ref>), the estimated values such asthe angular frequency of Alfvén wave in the jet (ω_J), recurrence rate of the burst (1/ν _ A), acceleration time D_3/c, maximum energy of accelerated particle W_ max, total accretion power L_ tot, Alfvén luminosity L_ A in the jet, gamma-ray luminosity L_γ, and UHECR luminosity L_ UHECR in <cit.> are revised. Table <ref> is revised version of Table 1 in <cit.>. Figure <ref> is the revised version of Fig. 4 in <cit.> which shows the relation of the maximum energy of accelerated particle W_ max as a function of mass of central black hole and accretion power L_ tot=Ṁ c^2. Since we do not consider any radiative processes, i.e., RIAF (L_ tot≤ 10% Eddington luminosity), the gray shadowed region shows the objects for the UHECR (W_ max≥ 10^20 eV) accelerators. §.§ gamma-raysBlazar is a subclass of AGNs for which the jets are close to our line of sight. A blazar jet is very bright due to relativistic beaming effect, including high energy gamma-ray bands. They are observed in multi-wavelength from radio to TeV gamma rays. Short time variabilities and polarization are observed, including recent AGILE and Fermi observations for high energy gamma ray bands <cit.>. These non-thermal emissions are usually explained by the internal shock model <cit.>, i.e., two shell collisions <cit.> for which a rapid shell catches up a slow shell, forming relativistic shocks. At the shocks particle accelerations, generating non-thermal particles, are expected by Fermi acceleration mechanism. Finally non-thermal emission is produced via synchrotron emission and inverse Compton emission. Other gamma ray emission from accretion disks and related processes have been suggested by <cit.>.Our model can naturally explain properties ofobserved active gamma-ray flares, i.e., spectrum and timescales of flares. For electrons energy loss via synchrotron radiation causes a cutoff around PeV regime <cit.>, although heavier particles, such as protons and heavier nuclei are accelerated up to ultra high energy cosmic ray regime (∼ 10^20 eV) and beyond. The accelerated electrons emit radiation from radio to high energy gamma-rays via synchrotron radiation and inverse Compton emission mechanism. The distribution of accelerated non-thermal particles becomes power law with a power law index ∼ -2 <cit.>, which is consistent with the observed blazar spectrum with the power law index close to -2. The photon index becomes close to -2, when the light curve of gamma-rays becomes active phase <cit.>. This anti-correlation betweenthe gamma-ray light curves and the photon index also supports our results.From our numerical simulations the rising timescale of electromagnetic flares in the jet is as same as rising timescale of magnetic field amplification in the disk, i.e., typically τ̅_1 ∼ 30 GM_ BH c^-3 and timescales of peak to peak in the flares are as same as timescale of repeat cycle of magnetic field amplification in the disk, i.e., typically τ̅_2 ∼ 100 GM_ BH c^-3.As for comparison with observed gamma-ray flares of blazars the rising timescale of flares and timescales of the cycle of the flares are normalized by the (1+z)GM_ BH c^-3, where z is cosmological redshift of the object. The redshifts for two objects are z_ 3C454.3=0.86 <cit.> and z_ AO0235+164=0.94. From the observations of line widths of Hβ in the broad line region (BLR), the mass of central black hole in 3C454.3 is estimated from 5× 10^8M_⊙ <cit.> to 4× 10^9M_⊙ <cit.>. In this paper we adopt 5× 10^8 M_⊙ as a mass of central black hole in 3C454.3, since the estimation of the BLR was done by using C_ IV line with less contamination by the non-thermal continuum in <cit.>. The mass of central black hole in AO0235+164 is also derived toM_ BH(AO0235+164)=5.85 × 10^8 M_⊙ <cit.> from the line width of Hβ in the BLR.We adopted 7 days as repeat time and 3 days as rise time for 3C454.3, though various timescales with different times are observedfrom 3C454.3. Among them, the seven days flare observed by <cit.> is the most energetic. The estimated apparent isotropic gamma-rayenergy in the seven days flare is4 times or more higher than those of flares reported in <cit.>. Sub-energetic and shorter timescale flares reported in <cit.> can be explained the result of the magnetic eruption via reconnection at the smaller region in the accretion flows.For AO0235+164 we compare the flare reported in <cit.>, since the flare is the most energetic one in the apparent isotropic energy compared with other flares of AO0235+164, for example <cit.>. The rising timescale is three weeks and repeat timescale is four weeks. Table <ref> summarize of the comparison between our results, theoretical model by <cit.> and observations. Both timescales for3C454.3are good agreement with our results. For AO0235+164 the timescales are longer than those for our results which suggests the magnetic field amplification may occur outward where the timescale of MRI growth becomes longer. § DISCUSSIONS AND SUMMARY We have performed 3D GRMHD simulation of accretion flows around a spinning black hole (a=0.9) in order to study the AGN jets from the system of supermassive black hole and surrounding accretion disk as an ultra high energy cosmic ray accelerator via wake field acceleration mechanism.We start our simulation from a hydrostatic disk, i.e., Fishbone-Moncrief solution with a weak magnetic field and random perturbation in thermal pressure which violate the hydrostatic state and axis-symmetry. We follow the time evolution of the system until 8300 GM_ BH c^-3. Initially imposed magnetic field is well amplified, due to differential rotation of the disk. Non axis-symmetric mode, i.e., bar mode near the disk edge grows up. For highest resolution calculation case, the typical timescale of the magnetic field growth near the disk edge is τ̅_1 = 30 GM_ BH c^-3 which corresponds to inverse of the growth rate of the MRI for the almost fastest growing mode. For lower resolution calculation case, the time scale of the magnetic field growth near the disk edge becomes longer one. And the thickness of the filamental structure in the magnetic pressure around the equator increases by the lower resolution calculations. Amplified magnetic field once drops then grows up again. The typical repeat timescale is τ̅_2 = 100 GM_ BH c^-3 which corresponds tothe analysis by high resolution local shearing box simulations. The transition between low β state and high β state repeats. This short time variability for the growth of poloidal magnetic field neat the disk edge also can be seen in the mass accretion rate at the event horizon which means the mass accretion seen in our numerical simulation triggered by angular momentum transferby the growth of the magnetic fields.We have two types outflows as shown in Fig. <ref>. One is the low density, magnetized, and collimated outflow, i.e, jets. The other is disk winds which are dense gas flow and between the disk surface and jets. In the jet short time variabilities of the electromagnetic luminosity are observed. The timescales are similar with those seen in the mass accretion rate at the event horizon and and poloidal Alfvén energy flux in the disk near the ISCO, i.e., typical rising timescales of flares and typical repeat cycleare as same as the rising timescales of the magnetic field amplification and repeat cycle of the magnetic field amplification, τ̅_1 =30 GM_BHc^-3 andτ̅_2 =100 GM_BHc^-3, respectively. Thus short pulsed relativistic Alfvén waves are emitted from the accretion disk, when a part of stored magnetic field energy is released. Since the strength parameter of these waves extremely highas ∼ 10^11 for the 10^8 solar masses central black hole and 10% Eddington accretion rate accretion flows, the wakefield acceleration proposed by <cit.> can be applied in the jet after mode conversion from Alfvén waves to electromagnetic waves. There are some advantages against Fermi acceleration mechanism <cit.>. When we apply this mechanism to the cosmic ray acceleration, the highest energy of cosmic ray reaches 10^22 eV which is enough high to explain the ultra high energy cosmic rays. We observe magnetic field amplification occurs inner radius compared with the assumption by <cit.>, i.e., R=20 R_g. If we apply the model by <cit.> assuming that magnetic field amplification occurs much inside the disk. The two timescales are consistent with our numeral results.Since both protons and electrons are accelerated via ponderomotive force in the jet. High energy gamma-ray emission are observed if we see the jet almost on-axis, i.e., blazars. The observed gamma-ray flare timescales such as rising timescales of flares and repeat cycle of flaresfor 3C454.3 by Fermi Gamma-ray Observatory <cit.> are well explained by our bow wake acceleration model. Telescope Array experiment reportedthat there is hotspot for the cosmic ray with the energy higher than 57EeV <cit.>. This observation supports that AGN jet is the origin of cosmic ray.Lastly, the consequences from our present work include the following implication on the gravitational observation.Since non-axis-symmetric mode grows in the disk, mass accretion onto the black hole causes the emission of gravitational waves <cit.>. We estimate the levels of the signal of these gravitational waves, assuming the black hole and accretion disk system. The dimensionless amplitude of the gravitational wave at the coalescing phase can be estimated as <cit.>:h_ coal=5.45× 10^-21( ϵ_ GW 0.01 ) ( 4Gpc R ) (μ√(2)× 10^3 M_⊙),whereϵ_ GW is efficiency and we here assume as 1%, μ is reduced mass of the unit of solar mass for the black hole and the blob ∼ mass of the blob. The mass of the blob is estimated by Ṁτ̅_1. Figure <ref> (a) shows time evolution of estimated amplitude of the gravitational waves from our mass accretion rate for the blazar 3C454.3. The mass accretion history from t=6500 GM_ BH c^-3 to t=8300 GM_ BH c^-3 is used for this plot.Figure <ref> (b) shows the estimated amplitude of the gravitational wave for some objects, such as gamma-ray active blazars AO0235+164 (=0.94) and 3C454.3 (z=0.86), nearby AGN jet M87, and famous stellar black hole Cygnus X-1. We assume1% Eddington mass accretion rate and typical frequency of the gravitational wave signal is τ̅_1^-1/(1+z), where z is redshift of the object. Approximated sensitivity curves of the KAGRA <cit.> proposed space gravitational wave detectors such as eLISA <cit.>, preDECIGO <cit.>, DECIGO <cit.> and BBO <cit.> are also presented. The signal level is so far small compared to the limit of the presently operating or proposed gravitational antennas.Our model cal be applied tomagnetic accretion flows onto the central objects, arising from other events such as black hole or neutron star collisions. For example, recently the gravitational waves from neutron star merger have been detected by LIGO and Virgo gravitational wave detectors <cit.>. Short gamma-ray burst followed this event just 1.7 s later of the two neutron stars merger <cit.>. If the jets which emit gamma-rays are powered by magnetic accreting flows onto the merged object, strong and relativistic pulses of Alfvén waves would be emitted like our analysis of accretion disks and then charged particles in the jet are accelerated by the electromagnetic waves as <cit.> discussed. We see the present acceleration mechanism and its signature of gamma-ray bursts in an ubiquitous range of phenomena. § ACKNOWLEDGEMENTSThe authors wish to acknowledge the anonymous referee for his/her detailed and helpful comments to the manuscript. We thank Shinkai H. for introducing some references for the fitting formula on the sensitivities of gravitational wave detectors. We appreciate useful comments by K. Abazajian and B. Barish. This work was carried out on Hokusai-greatwave system at RIKEN and XC30 at CFCA at NAOJ. 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"authors": [
"Akira Mizuta",
"Toshikazu Ebisuzaki",
"Toshiki Tajima",
"Shigehiro Nagataki"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170727095026",
"title": "Production of intense episodic Alfvén pulses: GRMHD simulation of black hole accretion disks"
} |
Approximation of SDEs with smooth coeffients with derivatives of linear growth ]A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives Müller-Gronbach and Yaroslavtseva] Thomas Müller-Gronbach and Larisa YaroslavtsevaFakultät für Informatik und MathematikUniversität PassauInnstrasse 33 94032 PassauGermany [email protected], [email protected], it has been shown in [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of autonomousstochastic differential equations (SDE) on the time interval [0,T] with infinitelydifferentiable and bounded coefficientssuch that no strong approximation methodbased on evaluation of the driving Brownian motion at finitely many fixed times in [0,T], e.g. on an equidistant grid,can converge in absolute mean to the solution at the final time with a polynomial rate in terms of the number of Brownian motion values that are used. In the literature on strong approximation of SDEs, polynomial error rate results are typicallyachieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition. This assumption is violated for thepathologicalSDEs fromthe above mentioned negative result. However, in the presentarticle weconstruct an SDE with smooth coefficients that have first order derivatives of at most linear growth suchthatthe solution at the final time can not be approximated with a polynomial rate, whatever method based on observations of the driving Brownian motion at finitely many fixed times is used.Most interestingly, it turns out that using a method that adjusts the number of evaluations of the driving Brownian motion to its actual path, the latter SDE can be approximatedwith rate 1 in terms of the average number of evaluations that are used. To the best of our knowledge, this is only the second example in the literature of an SDE for which there exist adaptive methods that perform superiorto non-adaptive ones with respect to the convergence rate.[ [ December 30, 2023 =====================§ INTRODUCTION Let d,m∈, T∈(0, ∞), consider a d-dimensional system of autonomous stochastic differential equations (SDE)dX(t) = μ(X(t)) dt + σ(X(t)) dW(t),t∈ [0,T],X(0) = x_0with a deterministic initial value x_0∈^d, a drift coefficient μ^d→^d, a diffusion coefficient σ^d→^d× m and an m-dimensional driving Brownian motion W, and assume that (<ref>) has a unique strong solution (X(t))_t∈[0, T].A fundamental problem in the numerical analysis of SDEs is to characterize whenthe solution at the final time X(T) can be approximatedwith a polynomialerror rate based onfinitely many evaluations of the driving Brownian motion Win terms of explicit regularity conditions on the coefficients μ and σ. It is well-known that if the coefficients μ and σ are globally Lipschitz continuous then the classical Euler-Maruyama scheme achieves the rate of convergence 1/2, see <cit.>. Moreover, the recent literature on numerical approximation of SDEs contains a number of results on approximation schemes that are specifically designed for SDEs with non-Lipschitz coefficients and achieve polynomial convergence ratesunder weaker conditions on μ and σ,see e.g. <cit.> for SDEs with globally monotone coefficients and e.g. <cit.>for SDEs with possibly non-monotone coefficients.On the other hand, it has recently been shown in <cit.>that for any sequence(a_n)_n∈⊂ (0,∞), which may converge to zero arbitrarily slowly,there exists an SDE (<ref>) with d=4 and m=1 and with infinitelydifferentiable and bounded coefficients μ and σ such that no sequence of approximations X_n(T) of X(T), where X_n(T) is based on n evaluations of the driving Brownian motion W at fixed time points in [0,T], can converge to X(T) in absolute meanfaster than the given sequence (a_n)_n∈.More formally, for this SDE one has for every n∈,inf_ s_1,…,s_n∈ [0,T]u^n→^4measurable[ | X( T ) - u( W( s_1 ), …, W( s_n ) ) |] ≥a_n.In <cit.> it has been proventhat the negative result (<ref>) can even be achievedwith m=1 and d=2 in place of d=4.In particular, (<ref>) implies that there exists an SDE (<ref>) with infinitelydifferentiable and bounded coefficients μ and σ such that its solution at the final time can not be approximated with a polynomial mean error ratebased onevaluations of the driving Brownian motion W at finitely many fixed time points in [0,T], i.e., for every α>0,lim_n→∞( n^α·inf_ s_1,…,s_n∈ [0,T]u^nm→^dmeasurable[|X(T)-u(W(s_1), …, W(s_n))|]) =∞. We add that the latter statement for the special case when the approximation u(W(s_1), …, W(s_n)) is given by the Euler-Maruyama scheme withtime step 1/n has first been shown in <cit.>.The proof of the negative result (<ref>) in <cit.> is constructive. Each of the respective SDEsis given by X(0)=0 anddX_1(t) = dt, dX_2(t)= f(X_1(t)) dW(t), dX_3(t)= g(X_1(t)) dW(t),dX_4(t) = h(X_1(t)) ·cos(X_2(t)·ψ(X_3(t))) dtfor t∈ [0,T], where f,g,h→ are infinitelydifferentiable, bounded, nonzero and satisfy {f ≠ 0}⊂ (-∞, τ_1],{g ≠ 0}⊂ [τ_1, τ_2], {h≠ 0}⊂ [τ_2, T], ∫_τ_2^T h(t)dt ≠ 0 and inf_x∈ [0,τ_1/2] |f'(x)| >0 for some 0<τ_1<τ_2<T, and ψ→ (0,∞) is infinitely differentiable, strictly increasing and satisfies lim_x→∞ψ(x) = ∞. Under these assumptionsthe fourth component of the solution of the SDE (<ref>) at the final time is given by X_4(T)= cos(∫_0^τ_1f(t) dW(t)·ψ(∫_τ_1^τ_2g(t) dW(t)))·∫_τ_2^T h(t) dtand there exist c_1,c_2,c_3∈ (0,∞) such that for every n∈,inf_ s_1,…,s_n∈ [0,T]u^n→ measurable[ | X_4( T ) - u( W( s_1 ), …, W( s_n ) ) |] ≥c_1·exp(-c_2· (ψ^-1(c_3 · n^3/2)^2),see Corollary 4.1 in <cit.>.It follows from (<ref>) that if ∀ q∈ (0,∞)lim_x→∞exp(-q x^2)·ψ (x) = ∞then a polynomialrate of convergence to zero of the left hand side in (<ref>) can not be achieved, see Corollary 4.2 in <cit.>. On the other hand it is straightforward to check that the equidistant Euler-Maruyama scheme for the SDE (<ref>)achieves a polynomial mean error rate if the derivativeψ' of ψ is of at most polynomial growth. The latter two facts are reflected in the growth properties of the first orderderivatives of the coefficients μ and σ of the SDE (<ref>). All of the first orderderivatives of μ and σ are globally bounded, up to thederivatives ∂μ_4/∂ x_2(x) =-h(x_1)·ψ(x_3)·sin(x_2·ψ(x_3)),∂μ_4/∂ x_3(x) =-h(x_1)· x_2·ψ'(x_3)·sin(x_2·ψ(x_3)), which are both of at most polynomial growth if and only if ψ'is of at mostpolynomial growth. For the vast majority of SDEs with locally Lipschitz continuous coefficients used for modelling in applications it holds that the first orderderivatives of the coefficients are of at most polynomial growth. Moreover, a polynomial growth condition on thefirst orderderivatives of the coefficients of an SDE isone of the standing assumptions in the literaturewhen polynomial mean error rates are obtained undermonotonicity conditions, see e.g.<cit.>. Therefore it isimportant to investigate whether a sub-polynomial rate of convergence as in (<ref>) may also happenwhen the first order derivatives of the coefficients are of at most polynomial growth.This question can easily be answered with a yes. For the choice ψ(x) = exp(x^3), which satisfies (<ref>), the random variable X_4(T) in (<ref>) can also be obtained as the fifth component of the solution at the final time of an SDE given by Y(0)=(0,0,0,1,0)anddY_1(t) =dt, dY_2(t) =f(Y_1(t))dW(t), dY_3(t) =g(Y_1(t))dW(t),dY_4(t) =u(Y_1(t)) · Y_3^3 (t)· Y_4(t) dt, dY_5(t)=v(Y_1(t)) ·cos(Y_2(t)· Y_4(t))dtfor t∈ [0,T], where f, g, h satisfy the conditions stated below the SDE (<ref>) and, additionally, f' is bounded, and u,v→ are infinitely differentiable and satisfy {u≠ 0}⊂ [τ_2, τ_3],{v≠ 0}⊂ [τ_3, T], ∫_τ_2^τ_3 u(s)ds=1 and∫_τ_3^T v(s)ds=∫_τ_2^T h(t) dt for some τ_3∈ (τ_2,T). Clearly, the coefficients of the SDE (<ref>) have first order derivatives of at most polynomial growth and Y_5(T) = X_4(T).Note, however, that in contrast to the solution X of the SDE (<ref>), the solution Y of the SDE (<ref>) is not integrable at any timet∈[τ_3, T]. In fact, it is easy to see that[sup_t∈[0, T]|X(t)|] < ∞,inf_t∈[τ_3, T][|Y_4(t)|] = ∞.It therefore seems reasonable to modify the question posed above andto ask whethera sub-polynomial rate of convergence as in (<ref>) may also happen for an SDE (<ref>)that has smooth coefficients with first orderderivatives of at most polynomial growth and asolution X with[sup_t∈[0, T]|X(t)|] < ∞. In the actual paper we show that the answer to this question is positive as well. More precisely, consider the 7-dimensional SDE given by X(0)=(0,0,0,0,1,0,0) and dX_1(t) =dt, dX_2(t) =f(X_1(t))dW(t), dX_3(t)=f^2(X_1(t)) dt + 2 X_2(t)· f(X_1(t))dW(t),dX_4(t) = 14 g'(X_1(t))· X_3(t)dt,dX_5(t) =X_4(t)· X_5(t)dt,dX_6(t) = h'(X_1(t))· X_5(t)(1+X_2^2(t))^1/2·ln^2(2 + X_2^2(t) ), dX_7(t) = X_5(t)· X_6(t)dtfor t∈ [0,T], where f,g,h→ satisfy the conditions stated below the SDE (<ref>) and, additionally,g,h≥ 0, f' is bounded and ∫_0^τ_1 f^2(t)dt = ∫_τ_1^τ_2 g(t)dt = ∫_τ_2^T h(t)dt =1. See Example <ref> for a possible choice of f,g,h. The assumptions on the functions f,g and h imply that all of the first orderderivatives of the coefficients of the SDE (<ref>) are of at most linear growth and the solution X of the SDE (<ref>) satisfies the moment condition (<ref>), see Lemmas <ref> and <ref>. Moreover, as a consequence of Theorem <ref> we obtain that there exists c∈ (0,∞) such that for all n∈,inf_ s_1,…,s_n∈ [0,T]u^n→^7measurable[| X( T ) - u( W( s_1 ), …, W( s_n ) ) |] ≥c·1/ln^2(n+1),and therefore X(T) can not be approximated with a polynomial mean error ratebased onevaluations of the driving Brownian motion W at finitely many fixed time points in [0,T]. To the best of our knowledge this is the first result in the literature, which shows that a sub-polynomial rate of convergence may happen even then when the first order derivatives of the coefficients are of at most polynomial growth. Itimplies in particular that for such SDEseven tamed or projected versions of the Euler-Maruyama scheme or the Milstein scheme, which are specifically designed to cope with the case ofsuperlinearly growing coefficients, see e.g. <cit.>may fail to achieve a polynomial convergence rate.The negative result (<ref>) covers only approximations that are based on n evaluations of the driving Brownian motion W at fixed time points s_1,…,s_n∈[0,T] and leaves it open whether a polynomial mean error rate canbe achieved by employing approximations that mayadaptthe number as well as the location of the evaluation sites of Wto the actual path of W, e.g. bynumerical schemes that adjust the actual step size according to a criterion that is based on the values ofW observed so far, see e.g. <cit.> and the references therein for methods of this type. However, it is well-known that for a huge class of SDEs (<ref>) withglobally Lipschitz continuous coefficients μ and σadaptive approximations of the latter typecan not achieve a betterrate of convergence compared to what is best possible for non-adaptive ones, which at the same time coincides with the best possible rate of convergence that can be achieved by approximations based on evaluatingW at n equidistant times, see <cit.> and the discussion on asymptotic constants therein. Moreover, it has recently been shown in <cit.> that the SDE (<ref>) with ψ satisfying (<ref>)can not be approximated with a polynomial mean error rate even then when adaptive approximations may be used.Up to now there seems to be onlyone example of an SDE known in the literature, for which adaptive approximations are superior to non-adaptive ones with respect to the convergence rate.In <cit.> it has been shown that for the one-dimensional squared Bessel process, i.e. the solution of the SDE (<ref>) with d = m =1, μ=1 and σ(x)=2√(|x|), any non-adaptive approximationof X(T) based on n equidistant evaluations of W can only achieve a mean error rate of order 1/2 in terms of n, while for every α∈ (0,∞) there exist c∈ (0,∞) and a sequence of approximations X_n(T), each based on n sequentially chosen evaluations of W, such that [|X(T)- X_n(T)|] ≤ c· n^-α. Interestingly it turns out that the SDE (<ref>) provides the secondexampleafter <cit.> of an SDE in the literature, for whichthere exist adaptive approximationsthat perform superior to non-adaptive ones with respect to the convergence rate.Indeed,there exists c∈ (0,∞) anda sequence of approximations X_n(T), each based on n sequentially chosen evaluations of W on average, such that for all n∈,[|X(T)- X_n(T)|] ≤ c· n^-1,see Theorem (<ref>). We briefly describe the content of the paper.In Section <ref> we introduce the particular SDE with smooth coefficients that is studied in the present paper and we discuss moment properties of its solution. Our main results on a sub-polynomial lower error bound for non-adaptive methods(Theorem <ref>) and a polynomial upper error bound for a suitable adaptive method (Theorem <ref>) are stated in Section <ref>. The respective proofs are carried out in Sections <ref> and <ref>. Section <ref> is devoted to a discussion of our results and naturally arising open questions. § AN SDE WITH SMOOTH COEFFICIENTS THAT HAVE AT MOST LINEARLY GROWING DERIVATIVES Throughout this article we fix the following setting.LetT∈ (0,∞), let ( Ω, ℱ,) be a probability space with a normal filtration ( ℱ_t )_ t ∈ [0,T] and let W[0,T] ×Ω→ be a standard ( ℱ_t )_ t ∈ [0,T]-Brownian motion on ( Ω, ℱ,).Let0<τ_1<τ_2<T and let f, h, g ∈ C^∞(, ) satisfy {f≠ 0}⊆ ( - ∞, τ_1 ],{g≠ 0}⊆ [ τ_1, τ_2 ],{h≠ 0}⊆ [ τ_2, T] as well as sup_t∈ ( - ∞, τ_1 ]| f(t)| <∞,sup_t∈ ( - ∞, τ_1 ]| f'(t)| <∞,inf_ t∈ [ 0, τ_1/2]| f'(t) | > 0, g≥ 0, h≥ 0and∫_0^τ_1 f^2(t)dt= ∫_τ_1^τ_2 g(t)dt=∫_τ_2 ^ Th(t) dt =1.See the following example for a possible choice of f,g and h. Define f, g, h→ byf(x)= _(-∞,τ_1)(x)·exp(1x - τ_1 ), g(x)= _(τ_1,τ_2)(x)·exp(1 τ_1 - x+1x - τ_2 ), h(x)= _(τ_2,T)(x)·exp(1 τ_2 - x+1x - T ).Then the functions f =(∫_0^τ_1(f(s))^2 ds)^-1/2·f , g= (∫_τ_1^τ_2g(s) ds)^-1·g, h= (∫_τ_1^τ_2h(s) ds)^-1·h satisfy f,g,h∈ C^∞(, ) as well as the conditions (<ref>)-(<ref>). Let p∈ [1, ∞) and define μ, σ^7 →^7 as well as x_0∈^7 byμ(x) = ( 1, 0, f^2(x_1), g'( x_1 )4p· x_3, x_4· x_5,h'( x_1 ) · x_5(1+x_2^2)^1/2p·ln^2/p (2+x_2^2),x_5· x_6 ), σ(x)= (0, f( x_1 ) , 2x_2· f(x_1), 0, 0,0,0),x_0 =(0,0,0,0,1,0,0).We have μ,σ∈ C^∞(^7,^7). Moreover, there exists c∈ (0,∞) such that for all x∈^7,∑_i,j=1^7 (|∂μ_i∂ x_j(x)| +|∂σ_i∂ x_j(x)|)≤ c· (1+|x|).Infinite differentiability of μ and σ is an immediate consequence of the definition of these functions and the fact that f,g,h∈ C^∞(,). Moreover, it is straightforward to check that there exists c∈ (0,∞) such that for all i,j∈{1,…,7}and x∈^7,|∂σ_i∂ x_j(x)| ≤ c·(|f(x_1)|+|f'(x_1)|)· (1+|x|)and|∂μ_i∂ x_j(x)| ≤ c·(|f(x_1)|· |f'(x_1)| + |g'(x_1)| + |g”(x_1)| + |h'(x_1)| + |h”(x_1)| + 1)· (1+|x|),which jointly with the fact that g,h∈ C^∞(,) and the properties (<ref>) and (<ref>)yields at most linear growth for all first order derivatives of μ and σ.We study the SDE (<ref>) with m=1, d=7 and x_0,μ,σ given by (<ref>), i.e. X(0)=(0,0,0,0,1,0,0)^⊤ and dX_1(t) =dt, dX_2(t) =f(X_1(t))dW(t), dX_3(t)=f^2(X_1(t)) dt + 2 X_2(t)· f(X_1(t))dW(t),dX_4(t) = 14p g'(X_1(t))· X_3(t)dt,dX_5(t) =X_4(t)· X_5(t)dt,dX_6(t) = h'(X_1(t))· X_5(t)(1+X_2^2(t))^1/2p·ln^2/p(2 + X_2^2(t) ), dX_7(t) = X_5(t)· X_6(t)dt Observing (<ref>) and using Itô's formula for the component X_3 it is straightforward to see thatthe equation (<ref>) has a unique strong solution given byX_1(t) = t ,X_2(t) = ∫_0^min( t, τ_1 )f(s) dW(s), X_3(t) = X_2^2(t),X_4(t) = 14p X_2^2(τ_1)· g(t),X_5(t)=exp(14pX_2^2(τ_1)·∫_0^min( t , τ_2 )g(s) ds ) , X_6(t) =X_5(τ_2)(1+X_2^2(τ_1))^1/2p·ln^2/p (2+X_2^2(τ_1))·h(t), X_7(t) =X_5^2(τ_2)(1+X_2^2(τ_1))^1/2p·ln^2/p (2+X_2^2(τ_1))·∫_0^t h(s)dsfor all t ∈ [0,T]. In particular, by (<ref>),X_1(T) = T ,X_2(T) =X_2(τ_1) =∫_0^τ_1 f(s) dW(s),X_3(T) = X_2^2(T), X_4(T)= 0,X_5(T)=exp(14pX_2^2(τ_1)),X_6(T) = 0,X_7(T)= exp(1/2p X_2^2(τ_1))(1+X_2^2(τ_1))^1/2p·ln^2/p (2+X_2^2(τ_1)). Next we discuss integrability properties of the solution X.We have X_2(τ_1)∼𝒩( 0,1). Moreover, for all q∈(0, ∞),[sup_t∈[0,T] |X(t)|^q]<∞⇔ q≤ p. The first statement follows immediately from the definition of X_2(τ_1) and the fact that [X_2^2(τ_1)] = ∫_0^τ_1 f^2(t)dt= 1, due to (<ref>). Moreover, applying the Burkholder-Davis-Gundyinequality we obtain that for all q∈(0, ∞) there exists c∈(0, ∞) such that[sup_t∈[0,T] |X_2(t)|^q] ≤ c·(∫_0^T f^2(t)dt)^q2 =c.Employing (<ref>), (<ref>) and the properties of g we conclude that for all q∈(0, ∞), [sup_t∈[0,T] |X_4(t)|^q] = 1(4p)^q·[|X_2(τ_1)|^2q]·sup_t∈ [τ_1,τ_2] (g(t))^q < ∞.Furthermore, (<ref>), (<ref>), (<ref>) and the fact thatX_2(τ_1)∼𝒩( 0,1) imply that for all q ∈ (0,2p), [sup_t∈[0,T] |X_5(t)|^q] = [exp(q4pX_2^2(τ_1))] = √(2p2p-q).By (<ref>), the latterequality and the properties of h we get that for all q ∈ (0,2p), [sup_t∈[0,T] |X_6(t)|^q] ≤1ln^2q/p(2)·[|X_5(τ_2)|^q]·sup_t∈[τ_2,T] h^q(t) <∞.Finally, by (<ref>) we see that for all q∈(0, ∞), [sup_t∈[0,T] |X_7(t)|^q] = [exp(q/2p X_2^2(τ_1))(1+X_2^2(τ_1))^q/2p·ln^2q/p (2+X_2^2(τ_1))]= √(2π)∫_0^∞exp(q-p2p· x^2) (1+x^2)^q/2p·ln^2q/p(2+x^2)dx,and the latter quantity is finite if and only if q≤ p.§ LOWER AND UPPER ERROR BOUNDS We study strong approximation of the solution X of the equation (<ref>) at the final time T. The following result shows that X_7(T) and thus X(T) as wellcan not be approximated in p-th mean sensewith a polynomial error rate in terms of the number of evaluations of the driving Brownian motion W as long as the number and the location of the evaluation nodes for W are not chosen in apath-dependent way. There exists c∈(0, ∞) such that for all n∈,inf_ s_1,…,s_n∈ [0,T]u^n→ measurable([| X_7( T ) -u(W(s_1), …, W(s_n)) |^p])^1p≥ c·1/ln^2/p(n+1). Our next result shows that a polynomial p-th mean error ratefor approximation of X(T)can be achieved if the number of the evaluation nodes for W is adjusted to the current path of W.For n∈ we useW_n[ 0, τ_1 ] ×Ω→ to denote the piecewise linear interpolation of W on [0,τ_1] at the nodes t_i= i/n·τ_1, i=0,…,n, i.e. W_n( t ) =t - t_i-1/τ_1/n · W( t_i) + t_i- t/τ_1/n · W( t_i-1), t∈[t_i-1,t_i],fori∈{1, …, n}. We define approximations of the single components of X(T) in the following way.PutX_n, 1(T)=T, X_n, 2(T)=-∫_0^τ_1f'(t)·W_n(t) dt,X_n, 3(T)= X_n, 2^2(T), X_n, 4(T) = 0,X_n, 5(T)=exp(14p X_n, 2^2),X_n, 6(T)=0.Next, let a_ℓ=2√(lnℓ)forℓ∈ and putX^*_n, 2(T)=∑_ℓ=1^∞X_ℓ n, 2(T)· 1_[a_ℓ, a_ℓ+1)(|X_n, 2(T)|).Finally, defineG→ by G(x)=exp(1/2p x^2)/(1+x^2)^1/2p·ln^2/p (2+x^2), x∈,and putX^∗_n, 7(T)=G(X^*_n, 2(T))as well asX^∗_n(T)=(X_n, 1(T),…,X_n, 6(T), X^∗_n, 7(T) ). Clearly, the random number of evaluations of Wused by the approximation X^∗_n(T) is given bycost(X^∗_n(T)) = n∑_ℓ=1^∞ℓ·_[a_ℓ, a_ℓ+1)(|X_n, 2(T)|). There exists c∈ (0,∞) such that for all n∈,[cost(X^∗_n(T))]≤ c· n and([| X( T ) -X^∗_n(T) |^p])^1p≤c/n.Finally, we show that for q<p a polynomial q-th mean error rate for approximation of X(T)can be achieved with a sequence of non-adaptive approximations. For n∈ put X_n,7(T) = G(X_n,2(T))with G given by (<ref>) and defineX_n(T)=(X_n, i(T))_i=1,…,7.Note that X_n(T) = u_n(W(τ_1/n),W(2τ_1/n),…, W(τ_1)) for some function u_n^n→^7. Let q∈[0,p). Then there exists c∈ (0,∞) such that for all n∈,([| X( T ) -X_n(T) |^q])^1q≤c/n. § PROOF OF THEOREM <REF>For the proof of Theorem <ref> we employ the following lemma, which is a straightforward generalization of Lemma 4.1 in<cit.>. Let (Ω_1, 𝒜_1) and (Ω_2, 𝒜_2) be measurable spaces and let V_1 Ω→Ω_1 and V_2, V_2', V_2” Ω→Ω_2 be random variables such that_ (V_1, V_2)= _ (V_1, V_2')= _ (V_1, V_2”) .Then for all q∈[1, ∞)and for all measurable mappings ΦΩ_1×Ω_2→ and φΩ_1→, ([ |Φ(V_1,V_2)- φ(V_1)|^q ])^1/q≥ 1 / 2([ | Φ( V_1, V_2' ) - Φ( V_1, V_2” ) |^q ])^1/q . Observe that (<ref>) ensures that[ | Φ(V_1, V_2) - φ(V_1) |^q ] = [ | Φ( V_1, V_2' ) - φ(V_1) |^q ] = [ | Φ(V_1,V_2”) - φ(V_1) |^q ] .This and theMinkowski's inequality imply that([ | Φ(V_1,V_2) - φ(V_1) |^q ])^1/q = 1 / 2(([ | Φ(V_1,V_2') - φ(V_1) |^q ])^1/q+([ | Φ(V_1,V_2”) - φ(V_1) |^q ])^1/q) ≥ 1 / 2([ | Φ(V_1,V_2') - Φ(V_1,V_2”) |^q ])^1/q,which finishes the proof of the lemma.We start with the proof of Theorem <ref>.Let n∈ and s_1, …, s_n∈ [0,T]. Clearly, there exist 0≤ t_0 < t_1 ≤ T such that[t_0,t_1]⊂ [0,τ_1/2], (t_0,t_1)∩{s_1,…,s_n} = ∅, t_1-t_0 = τ_12(n+1).Defineprocesses W, B[ t_0, t_1 ] ×Ω→ and W( [ 0, t_0 ] ∪ [ t_1, T ] ) ×Ω→ byW( t ) =(t - t_0) / ( t_1 - t_0 ) · W( t_1 ) +( t_1 - t ) / ( t_1 - t_0 ) · W( t_0 ) ,B( t ) = W( t ) - W( t ) for t ∈ [ t_0, t_1 ]and by W( t ) = W( t ) fort ∈ [ 0, t_0 ] ∪ [ t_1, T ]. Moreover, letY_1 =-∫_0^t_0 f'(s)· W(s)ds-∫_t_0^t_1 f'(s)·W(s)ds-∫_t_1^τ_1 f'(s)· W(s)ds, Y_2 =-∫_t_0^t_1 f'(s)· B(s)ds.By Itô's formula and (<ref>) we have -a.s. Y_1+Y_2= ∫_0^τ_1 f(s)dW(s). Hence, by (<ref>), -a.s.X_7(T) = G(Y_1+Y_2),where G→ is given by (<ref>).Let u^n→ be ameasurable mapping. Using (<ref>) we obtain [|X_7( T )-u(W(s_1), …, W(s_n)) |^p]=[| G(Y_1+Y_2) -u(W(s_1), …, W(s_n)) |^p].The first two statements in (<ref>) imply that there exist measurable functions Φ_1, φ C( [ 0, t_0 ] ∪ [ t_1 , T ] , ) → andΦ_2C( [t_0, t_1] , ) → such that Y_1 = Φ_1( W),Y_2=Φ_2(B),u(W(s_1), …, W(s_n))=φ(W). Moreover, W and B are independent and B has a symmetric distribution, which yields_ ( W, B ) = _ ( W, - B ) .We may thus apply Lemma <ref> with Ω_1 = C( [0, t_0] ∪ [t_1 , T] ,), Ω_2 = C( [t_0, t_1] ,), V_1=W, V_2=V_2'=B, V_2”=-B, Φ(w̃,b)=G(Φ_1(w̃)+Φ_2(b)) for (w̃,b)∈Ω_1×Ω_2 and φ as above, and observing the fact that Φ_2(-B)=-Φ_2(B) we conclude that [| G(Y_1+Y_2) -u(W(s_1), …, W(s_n)) |^p]≥12^p [| G(Y_1+Y_2) -G(Y_1-Y_2) |^p]. For the analysis of the right hand side in (<ref>) we first collect useful properties of the random variables Y_1 and Y_2 and the function G.Clearly, Y_1 and Y_2 are centered normal Gaussian variables. Moreover, independence of W and B implies independence of Y_1 and Y_2.Let σ_1^2 and σ_2^2 denote the variances of Y_1 and Y_2, respectively. Due to (<ref>) and the fact that ∫_0^τ_1f^2(t)dt=1, see (<ref>), we then haveσ_1^2+σ_2^2=1.Put α=inf_t∈[0, τ_1/2] |f'(t)|^2, β=sup_t∈[0, τ_1/2] |f'(t)|^2and note that 0< α≤β <∞, due to (<ref>). Since σ_2^2 = ∫_t_0^t_1∫_t_0^t_1f'(s)· f'(t)·(t_1-max(s,t))(min(s,t)-t_0)t_1-t_0ds dtand [t_0,t_1]⊂ [0,τ_1/2] we conclude thatα·(t_1-t_0)^312≤σ_2^2≤β·(t_1-t_0)^312.Putn_0=⌈τ_12·(β6)^1/3-1⌉.Using (<ref>), (<ref>) and (<ref>) we obtain that if n≥ n_0 then α·τ_1^396(n+1)^3≤σ_2^2 ≤ 1/2 ≤σ_1^2,σ_1^-2-1 =σ_2^2σ_1^2≤β·τ_1^348(n+1)^3. Clearly, for all x≥ 1,G^p(x) =exp( x^2/2)(1+x^2)^1/2·ln^2 (2+x^2)≥exp( x^2/2)√(2)x·ln^2 (3x^2).Moreover, G is differentiable onwithG'(x)=xp· G(x)·(1-1(1+x^2)- 4(2+x^2)·ln(2+x^2)).Hence, for all x≥ 3,G'(x)≥x2p· G(x)>0. Clearly, we may assume that n≥max(3,n_0). Let y_1∈ [n^3/2,2n^3/2] and y_2∈ [0,σ_2]. Then y_1+y_2 ≥ y_1-y_2 ≥ n^3/2-σ_2≥ 3^3/2 - 1 >3, due to (<ref>). Hence, by (<ref>),|G(y_1+y_2)-G(y_1-y_2)|≥∫_y_1^y_1+y_2 G'(x)dx ≥12py_1· y_2· G(y_1),which jointly with (<ref>) yields|G(y_1+y_2)-G(y_1-y_2)|^p ≥1(2p)^py_1^p-1· y_2^p·exp(y_1^2/2)√(2)·ln^2(3y_1^2).Employing (<ref>) and (<ref>)we conclude that[|G(Y_1+Y_2)-G(Y_1-Y_2)|^p] ≥1/(2p)^p 2^3/2π·1/σ_1σ_2∫_n^3/2^2n^3/2∫_0^σ_2y_1^p-1y_2^p/ln^2(3y_1^2)·exp(-y_2^2/2σ_2^2-y_1^2/2(σ_1^-2-1)) dy_2dy_1≥1/(2p)^p 2^3/2π·σ_2^p/(p+1)√(e)∫_n^3/2^2n^3/2y_1^p-1/ln^2(3y_1^2)·exp(-y_1^2/2(σ_1^-2-1))dy_1≥1/(2p)^p 2^3/2π·σ_2^p/(p+1)√(e)·n^3p/2/ln^2(12n^3)·exp(-2n^3(σ_1^-2-1))≥1/(2p)^p (p+1) 2^3/2π√(e)·(τ_1^3α/96)^p/2·(n/n+1)^3p/2·exp(-n^324(n+1)^3·βτ_1^3)/ln^2(12n^3)≥(τ_1^3α)^p/2/2^5p+3/23^p/2 p^p (p+1)π√(e)·exp(-βτ_1^324)·1/ln^2(12n^3).Now combine (<ref>), (<ref>) and (<ref>) to complete the proof of Theorem <ref>. § PROOF OF THEOREMS <REF> AND <REF>Astechnical toolsfor the proof of Theorems <ref> and <ref>we employ the following two results forcentered Gaussian random variables.For every q∈ [0,∞) there exists κ_q∈ (0,∞) such that for everyrandom variable Z∼𝒩( 0,σ^2) with σ^2∈ [0, 14] and everya∈ [0,∞),[|Z|^q ·exp(a·|Z|+|Z|^2)]≤κ_q·σ^q· (1+a^q)·exp(a^2·σ^2). Let q, a∈ [0,∞) and let Z∼𝒩( 0,σ^2) with σ^2∈ [0, 14]. Without loss of generality we may assume thatσ^2>0. Let V∼𝒩( 0, 1).Then[|Z|^q ·exp(a·|Z|+|Z|^2)] =√(2)√(π)σ·∫_0^∞ x^q·exp(a x+x^2-x^22σ^2) dx =√(2)√(π)σ·exp(a^2σ^22-4σ^2)·∫_0^∞ x^q·exp(-1-2σ^22σ^2·(x-aσ^21-2σ^2)^2) dx≤exp(a^2σ^22-4σ^2)·2√(1-2σ^2)·[|σ√(1-2σ^2)· V + aσ^21-2σ^2|^q ]≤exp(a^2σ^22-4σ^2)·2^q+1σ^q (1-2σ^2)^q+1/2·([|V|^q] + a^qσ^q(1-2σ^2)^q/2).Note thatσ^2≤ 1/4 implies 1-2σ^2≥ 1/2 as well as σ^2/(1-2σ^2) ≤ 1/2, which finishes the proof of the lemma. Let q∈[0,∞) and r∈ [0,12q) and let H∈ C^1(,) withsup_x∈|H'(x)|·exp(-q· x^2) < ∞.Then there exists κ∈ (0,∞) such that for allindependent random variables V_1∼𝒩( 0,v_1^2), V_2∼𝒩( 0,v_2^2) with v_1^2+v_1^2 ≤ 1, [|H(V_1+V_2)-H(V_1)|^r]≤κ· v_2^r. Let q∈[0,∞) and r∈ [0,12q), let H∈ C^1(,) satisfy (<ref>) and let V_1∼𝒩( 0,v_1^2), V_2∼𝒩( 0,v_2^2) be independent with v_1^2+v_1^2 ≤ 1. Let U_1 and U_2 be independent standard normal random variables. By the properties of H there exists c∈ (0,∞) such that for all y,z∈,|H(y+z)-H(y)|≤∫_min(y,y+z)^max(y,y+z) |H'(x)| dx ≤ c· |z|·exp(q· (|y|+|z|)^2).By the latter estimate, the Hölder inequality and the fact that v_1^2 + v_2^2 ≤ 1 we get [|H(V_1+V_2)-H(V_1)|^r]≤ c^r· v_2^r·[|U_2|^r·exp(r· q· (v_1·|U_1|+v_2·|U_2|)^2)]≤ c^r· v_2^r ·[|U_2|^r·exp(r· q· (U_1^2+U_2^2))]= c^r· v_2^r·[|U_2|^r·exp(r· q· U_2^2)]·[exp(r· q· U_1^2)]≤ c^r· v_2^r·([(1+|U_2|^r)·exp(r· q· U_2^2)])^2.Note that r q< 1/2 and put v= (1-2rq)^-1/2. Then[(1+|U_2|^r)·exp(r· q· U_2^2)] = ∫_(1+|x|^r)√(2π)·exp(-x^22v^2)dx = v·(1+v^r·[|U_1|^r] ),which completes the proof of the lemma.In the sequel we use the following notation. For n∈ we define B_n [ 0, τ_1] ×Ω→by B_n(t)=W(t)-W_n(t), t∈ [0,τ_1],and we put Y_n=-∫_0^τ_1f'(t) B_n(t) dt, Z_n= X_n, 2(T)as well asσ_n^2 = Var(Y_n),ν_n^2=Var(Z_n).By Itô's formula and (<ref>) we have -a.s. X_2(T)= Z_n+Y_n. Let n∈ and ℓ∈. Then it is easy to check thatZ_n,Z_ℓ n-Z_n,Y_ℓ n areindependent, centered, Gaussian random variables.Moreover, using (<ref>) and Lemma <ref> we getVar(Y_ℓ n) + Var(Z_ℓ n)= σ^2_ℓ n + ν_ℓ n^2 = 1,Var(Z_ℓ n-Z_n)= ν^2_ℓ n - ν^2_n,and, proceeding as in the proof of (<ref>), it is easy to see thatσ_ℓ n^2 ≤γτ_1^312 ℓ^2n^2,where γ=sup_t∈[0, τ_1] |f'(t)|^2.We are ready to establish a p-th mean error estimate for the approximation X^∗_n,7(T). There exists c∈ (0,∞)such that for all n∈,[|X_7( T ) -X^∗_n,7(T)|^p]≤c/n^p. Let n∈. Using (<ref>) we obtain[|X_7( T ) -X^∗_n,7(T)|^p]=[| G(X_2(T)) -G(X^*_n, 2(T)) |^p]=∑_ℓ=1^∞[| G(Z_ℓ n+Y_ℓ n) -G(Z_ℓ n) |^p· 1_[a_ℓ, a_ℓ+1)(|Z_n|)].It follows from (<ref>) that there exists c_1∈ (0,∞) such that |G'(x)| ≤ c_1 ·|x|·exp(x^22p)for all x∈. Hence, for all y,z ∈, |G(z+y)-G(z)|≤∫_min(z,z+y)^max(z,z+y) |G'(x)| dx≤ c_1· |y|· (|z|+|y|)·exp(12p (|z|+|y|)^2),which jointly with (<ref>) implies[|X_7( T ) -X^∗_n,7(T)|^p]≤ c_1^p·∑_ℓ=1^∞[|Y_ℓ n|^p· (|Z_ℓ n|+|Y_ℓ n|)^p·exp(12 (|Z_ℓ n|+|Y_ℓ n|)^2)· 1_[a_ℓ, a_ℓ+1)(|Z_n|)].Note that for all ℓ∈,exp(12 (|Z_ℓ n|+|Y_ℓ n|)^2)· 1_[a_ℓ, a_ℓ+1)(|Z_n|)≤exp(12 |Z_n|^2+a_ℓ+1·(|Z_ℓ n-Z_n|+|Y_ℓ n|)+|Z_ℓ n-Z_n|^2+|Y_ℓ n|^2)· 1_[a_ℓ, a_ℓ+1)(|Z_n|)and(|Z_ℓ n|+|Y_ℓ n|)^p ≤ 3^p· (1+|Z_n|^p)· (1+|Z_ℓ n-Z_n|^p)· (1+|Y_ℓ n|^p).Hence, [|X_7( T ) -X^∗_n,7(T)|^p]≤ (3c_1)^p·∑_ℓ=1^∞[A_ℓ,n· B_ℓ,n· C_ℓ,n],where A_ℓ,n =(1+|Z_n|^p)·exp(12 |Z_n|^2)· 1_[a_ℓ, a_ℓ+1)(|Z_n|),B_ℓ,n =(1+|Z_ℓ n-Z_n|^p)·exp(a_ℓ+1· |Z_ℓ n-Z_n|+|Z_ℓ n-Z_n|^2),C_ℓ,n =|Y_ℓ n|^p· (1+|Y_ℓ n|^p)·exp(a_ℓ+1·|Y_ℓ n|+|Y_ℓ n|^2)for ℓ∈. Observe that (<ref>)implies that for all ℓ∈,[A_ℓ,n· B_ℓ,n· C_ℓ,n] = [A_ℓ,n]·[ B_ℓ,n]·[ C_ℓ,n]. Next, put n_1 = ⌈√(γτ_1^3)⌉.Using (<ref>) and (<ref>) we see thatfor all n≥ n_1 and ℓ∈,ν^2_ℓ n - ν^2_n = σ_n^2- σ_ℓ n^2 ≤σ_n^2 ≤γτ_1^312 n^2≤112.Using (<ref>), (<ref>)and Lemma <ref> we thus obtain that there exist κ_0,κ_p,κ_2p,c_2,c_3∈ (0,∞) such that for all n≥ n_1 and ℓ∈,[ B_ℓ,n]≤(2κ_0 + κ_p· (ν_ℓ n^2- ν_n^2)^p/2· (1+a_ℓ+1^p))·exp(a_ℓ+1^2· (ν_ℓ n^2- ν_n^2)) ≤ c_3·ln^p/2(ℓ+1)and[ C_ℓ,n]≤(κ_p·σ_ℓ n ^p·(1+a_ℓ+1^p) + κ_2p·σ_ℓ n ^2p· (1+a_ℓ+1^2p))·exp(a_ℓ+1^2·σ_ℓ n ^2)≤c_2(ℓ n)^p·ln^p(ℓ+1)·ℓ^13.Furthermore, (<ref>) and (<ref>) jointly imply that 11/12≤ν_n^2 ≤ 1 for all n≥ n_1, and therefore there exists c_4∈ (0,∞) such that for all n≥ n_1 and ℓ∈,[ A_ℓ,n] = 2√(2π)∫_a_ℓ/ν_n^a_ℓ+1/ν_n (1+(ν_n· x)^p)·exp(-x^22· (1-ν_n^2)) dx≤(1+a_ℓ+1^p)· a_ℓ+1-a_ℓν_n≤√(1211)( 1 + 2^pln^p/2(ℓ+1))·2ℓ·ln^1/2(ℓ+1) ≤ c_4 ·ln^p-1/2(ℓ+1) ·1ℓ.Hence, there exists c_5∈ (0,∞) such that for all n≥ n_1 and ℓ∈,[A_ℓ,n]·[ B_ℓ,n]·[ C_ℓ,n] ≤ c_5 ·1n^p·ln^2p-1/2(ℓ+1)ℓ^p+2/3.Combining (<ref>), (<ref>) and (<ref>) we conclude that there exists c_6∈ (0,∞) such that for all n≥ n_1,[|X_7( T ) -X^∗_n,7(T)|^p] ≤ (3c_1)^p · c_5 ·1n^p∑_ℓ=1^∞ln^2p-1/2(ℓ+1)ℓ^p+2/3≤ c_6 ·1n^p. In view of Lemma <ref> it remainsto prove that for all n<n_1,[|G(X^*_n, 2(T)) |^p] < ∞. To this end we define ρ→ [3,∞) by ρ(x) = max(|x|,3). Clearly, ρ is convex. Moreover, by (<ref>) and (<ref>) we obtain that G as well as G' are increasing on [3,∞). In particular, G is convex on [3,∞). Using the monotonicity and convexity of G as well as the convexity of ρ we conclude that G∘ρ is convex.For n,ℓ∈put ℱ_ℓ n = σ({W(iτ_1/(ℓ n)) i = 1,…, ℓ n}) and note that Z_ℓ n = [X_2(T)|ℱ_ℓ n ]. Using the estimate G(x)≤ (ln(2))^-2/pexp(x^2/2p) and the Jentzen inequality we therefore obtain that for all n,ℓ∈|G(Z_ℓ n)|^p ·_[a_ℓ, a_ℓ+1)(|Z_n|)≤(|G(ρ(Z_ℓ n))|^p+(ln(2))^-2exp(9/2)) ·_[a_ℓ, a_ℓ+1)(|Z_n|)≤([|G(ρ(X_2(T)))|^p|ℱ_ℓ n ] +(ln(2))^-2exp(9/2))·_[a_ℓ, a_ℓ+1)(|Z_n|) =[|G(ρ(X_2(T)))|^p _[a_ℓ, a_ℓ+1)(|Z_n|)|ℱ_ℓ n] +(ln(2))^-2exp(9/2)·_[a_ℓ, a_ℓ+1)(|Z_n|).Hence there exists c_7∈ (0,∞) such that for all n∈,|G(X^*_n, 2(T))|^p ≤ c_7·(∑_ℓ=1^∞ [|G(ρ(X_2(T)))|^p _[a_ℓ, a_ℓ+1)(|Z_n|)|ℱ_ℓ n] +1),which implies[|G(X^*_n, 2(T))|^p]≤ c_7·([|G(ρ(X_2(T)))|^p] + 1).Finally, note that G(ρ(X_2(T)))≤ G(X_2(T)) + (ln(2))^-2/pexp(9/2p) and apply Lemma <ref> to complete the proof of (<ref>). Next, we provide error estimates for the approximations X_n,2(T),X_n,3(T),X_n,5(T) and X_n,7(T).Let r_2,r_3∈ (0,∞), r_5∈ (0,2p) and r_7∈ (0,p). Then there exists c∈ (0,∞) such that for every i∈{2,3,5,7} and every n∈,[|X_i(T)-X_n,i(T)|^r_i]≤c/n^r_i. Let r_2,r_3∈ (0,∞), r_5∈ (0,2p) and r_7∈ (0,p). Let n∈.In all of the four cases we apply Lemma <ref> with V_1 = X_n,2(T) = Z_n and V_2=Y_n, see (<ref>). Thus Var(V_1) +Var(V_2) = 1, and according to (<ref>) we have Var(V_2) ≤ (γτ_1^3)/(12n^2). Moreover, for i∈{2,3,5,7} we use the function H=H_i in Lemma <ref>, where H_2,H_3,H_5,H_7→ are given byH_2(x) = x, H_3(x) = x^2, H_5(x) =exp(14px^2), H_7(x) = G(x).Let q_2=0,q_3∈ (0,1/(2r_3)), q_5∈ (1/(4p),1/(2r_5)), q_7∈ (1/(2p),1/(2r_7)).Employing (<ref>) in the case i=7 we then see that there exists c∈ (0,∞) such that for every i∈{2,3,5,7} and every x∈,|H_i'(x)| ≤ c·exp(q_i· x^2),which completes the proof.Clearly, Lemmas <ref> and <ref> jointly yield the error estimates in Theorems <ref> and <ref>. It remains to establish the cost estimate in Theorem <ref>. Let n∈. Clearly, if ν_n=0 then X_n,2(T)=Z_n=0 a.s. and we have cost(X^*_n(T))= n a.s. Next, assume ν_n^2 >0. Using (<ref>) and the fact that ν_n^2≤ 1 we get for every l∈, ({|Z_n|∈[a_ℓ, a_ℓ+1)}) =2√(2π)ν_n∫_a_ℓ^a_ℓ+1exp(-x^22ν_n^2)dx≤2√(2π)ν_n·exp(-a_l^22ν_n^2)· (a_ℓ+1-a_ℓ)≤1ν_n·√(2)√(π)·exp(-2ln(ℓ))·2√(ln(2))·ℓ≤1ν_n·2^3/2√(πln(2))·1ℓ^3.Hence [cost(X^*_n(T))] = ∑_ℓ=1^∞ℓ· n·({|Z_n|∈ [a_ℓ, a_ℓ+1)})≤nν_n·√(2)√(πln(2))·∑_ℓ=1^∞1ℓ^2.Finally note that lim_n→∞ν_n^2 = 1, due to (<ref>) and (<ref>), and therefore inf_nν_n >0ν_n > 0, which completes the proof of the cost estimate and finishes the proof of Theorems <ref> and <ref>. § DISCUSSIONThe key contribution of this paper is to show that even then when an autonomous SDE on [0,T] has smooth coefficients with first order derivatives of at most linear growth and its solution X satisfies [sup_t∈ [0,T] |X(t)|^p]< ∞, where p∈ [1,∞), it may happen that X(T) can not be approximated on the basis of finitely many observations of the driving Brownian motion at fixed times in [0,T] with a polynomial p-th mean error rate, see Theorem <ref>. This result naturally leads to a number of questions related to possible extensions or tightenings with respect to the class of approximations, the speed of convergence, the moment conditions on the solution and the polynomial growth conditions on the first order derivatives of the coefficients. Does there exist an SDE of the above type such that a sub-polynomial rate of convergence holds for any adaptive approximation as well? For the SDE considered in the present paper, there is an adaptive method, which achieves a polynomial error rate, see Theorem <ref>.Does there exist an SDE of the above type such that the smallest possible p-th mean errorthat can be achieved by any non-adaptivemethod based on n evaluations of the driving Brownian motion or evenby any adaptive method based on n evaluations of the driving Brownian motion on average converges to zero slower than a given arbitrarily slow decay in terms of n? A negative result of this type is true for the class ofSDEs with bounded smooth coefficients, see (<ref>) and <cit.>.The first order derivatives of the coefficients of the pathological SDE (<ref>) constructed in the present paper are of at most linear growth.Can a sub-polynomial rate of convergence of the smallest possible p-th mean error also occur when the first order derivatives of the coefficients are of at most polynomial growth with an exponent α∈ (0,1)?Finally it is open, whether a sub-polynomial rate of convergence of the smallest possible p-th mean error can also occur when the solution X has finite moments of some order q>p or even satisfies [sup_t∈[0, T]|X(t)|^q] < ∞ for all q≥ 1. The pathological SDE (<ref>) satisfies [sup_t∈[0, T]|X(t)|^q] < ∞ only for q≤ p, see Lemma <ref>.The key contribution of this paper is to show that a sub-polynomial rate of convergence may happen even then whenthe derivatives of the coefficients are of at most polynomial growth, at least if non-adaptive methods are considered. There are still, however, a number of open questions with respect to negative results for such SDEs, which we would like to address in the future work.Does there exist an SDE with smooth coefficients that have at most polynomially growing derivatives and a solution X satisfying the moment condition (<ref>) such that a sub-polynomial rate of convergence holds for any adaptive approximation as well? Can one choose an SDE from this class in such a way that the smallest possible absolute mean error that can be achieved by any non-adaptive and eventually by any adaptive method converges to zero slower than a given arbitrarily slow rate of convergence, as it is the case for the class ofSDEs with bounded smooth coefficients, see (<ref>) and <cit.>?The derivatives of the coefficients of the actual SDE (<ref>) are of at most linear growth. Can a sub-polynomial rate of convergencehappen when the derivatives of the coefficients are of at most polynomial growth of order α, where α∈ (0,1)? Note that if α=0 then the coefficients are globally Lipschitz continuous. Can a sub-polynomial rate of convergence happen under a stronger moment condition[sup_t∈[0, T]|X(t)|^q] < ∞ for some q>1, which is not satisfied by the solution of the actual SDE (<ref>)? acm | http://arxiv.org/abs/1707.08818v1 | {
"authors": [
"Thomas Müller-Gronbach",
"Larisa Yaroslavtseva"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170727111911",
"title": "A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives"
} |
Wave breaking for the Stochastic Camassa-Holm equation Paper submitted to the Physica D special issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics dedicated to Prof. Edriss S. Titi on the occasion of his 60th birthday. Work partially supported by the EPSRC Standard Grant EP/N023781/1. Dan Crisan Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email: [email protected] Darryl D Holm Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email: [email protected] ========================================================================================================================================================================================================================================================================================================================Raynaud gave a criterion for a branched G-cover of curves defined over a mixed-characteristic discretely valued field K with residue characteristic p to have good reduction in the case of either a three-point cover of ℙ^1 or a one-point cover of an elliptic curve. Specifically, such a cover has potentially good reduction whenever G has a Sylow p-subgroup of order p and the absolute ramification index of K is less than the number of conjugacy classes of order p in G. In the case of an elliptic curve, we generalize this to the case in which G has an arbitrarily large cyclic Sylow p-subgroup.§ INTRODUCTION Let f:Y → X be a G-Galois cover branched at r points. For the case of X = ℙ^1 and r=3, in <cit.>, Raynaud gives a criterion for such covers to have good reduction to characteristic p over mixed-characteristic discretely valued fields. Specifically, f has good reduction whenever p strictly divides the order of the Galois group, G, of this cover and the absolute ramification index of the field is less than the number of conjugacy classes of elements of order p in G. Motivating this was the surjection due to Grothendieck (<cit.>) of tame fundamental groups: pr: π_1(ℙ^1_K ∖{0,1,∞})^p-tame→π_1(ℙ^1_k ∖{0,1,∞})^tame. Understanding the kernel of this map amounts to determining the p-tame three-point covers having good reduction.In the same paper, Raynaud suggested that one may obtain a similar result for both three-point covers and one-point covers (that is to say, covers of elliptic curves branched over one point) whose Galois groups have arbitrarily large cyclic Sylow p-subgroups. Indeed, Raynaud's own result holds also in the one-point cover case, and the two cases often have analogous properties: for instance, both have the same étale fundamental group. Moreover, in <cit.>, Obus obtained this very result in the three-point case. We complete the analogy in the following:Let G be a finite group with cyclic Sylow p-subgroup. Let k be an algebraically closed field of positive characteristic p, K_0 = Frac(W(k)), and K be a finite extension of K_0 such that the absolute ramification index of K is less than the number of conjugacy classes of order p in G. Let f: Y → E be a one-point G-cover defined over K, where E is an elliptic curve having good reduction. Then f has potentially good reduction.Raynaud's proof relied on the study of the stable reduction of such covers. One obtains the stable model after at most a finite base change of the ground field. The Galois group of this extension is called the monodromy group of the cover and its p-part is called the wild monodromy. Raynuad showed that this wild monodromy of such an f is trivial, which sufficed to conclude potentially good reduction.An analogous result holds for covers having cyclic Sylow p-subgroups, which is also due to Obus in <cit.>. He instead, however, concluded good reduction from the torsor structure of the cover at specific components. We follow a similar route. We must account for the fact that a cover of an elliptic curve need not ramify, so that we have a “borderline" case which does not arise in the case of ℙ^1. Such covers, however, are necessarily isogenies of elliptic curves, so this does not present much new difficulty.In 2, we recall some facts regarding stable models of covers and their ramificaiton. In particular, we have Obus' (<cit.>) vanishing cycles formula, which places restrictions on the ramification based on the genus of the base curve and the number of branch points. In 3, we review the notion of multiplicative reduction. One can view reduction to characteristic p as “losing" information and multiplicative reduction as being a way of “recovering" some of this lost information. We also outline the construction of the auxiliary cover, a cover obtained from the original cover but whose Galois group will be of the form ℤ/p^s ⋊ℤ/m, where m is prime to p. We prove an analogue of a result of Wewers (<cit.>) on Jacobians of tame cyclic covers of elliptic curves to detect multiplicative reduction in this auxiliary cover. Finally, in 4, we adapt a result of Obus (<cit.>) to show that a one-point cover defined over a sufficiently small field (for instance, one described in Theorem 1) having bad reduction cannot have multiplicative reduction over certain components, allowing us to conclude good reduction. §.§ Notation Throughout this paper, a one-point cover will refer to a cover of an elliptic curve branched over one point; we typically assume this point to be the origin. A G-cover of K-curves refers to a finite map of curves f: Y → X, where both X and Y are smooth, projective, and geometrically integral K-curves and the extension K(Y)/K(X) is Galois with group G. When G has a cyclic Sylow p-subgroup, P, we denote by m_G the value |N_G(P)|/Z_G(P)|.Given an arbitrary scheme morphism f:Y → X and a finite group G having a subgroup H with H ⊆Aut(Y/X), Ind_H^G f: Ind_H^G Y → X is the map obtained by taking the disjoint union of [G:H] copies of Y and applying f to each copy.§ ACKNOWLEDGEMENTS The author would like to thank Andrew Obus for his invaluable comments, discussions, and patience throughout this project.§ STABLE REDUCTION§.§ The stable model Let R be a complete discrete valuation ring with characteristic 0 fraction field K and and positive characteristic p residue field k. Let X be a curve defined over K having good reduction; we denote a fixed good model of X by X_R. Let f: Y → X be a G-Galois cover defined over K with prime-to-p branching indices, where G is a finite group with cyclic Sylow p-subgroup and Y is a curve of genus at least 2; we suppose that the branch points of f are defined over K and specialize to distinct points of X_k = X_R ×_R k.We know from Deligne and Mumford (<cit.>, Corollary 2.7) that there is a finite extension of K for which we have a stable model Y^st of Y over the valuation ring of this extension; that is to say, the special fiber Y̅ has only ordinary double points as singularities and any irreducible component of Y̅ having genus 0 contains at least 3 marked points n our situation, (that is, ramification points or points of intersection with the rest of Y̅).We combine this with the work of Raynaud in <cit.> and Liu in <cit.> to obtain the stable model of f, f^st: Y^st→ X^st, where Y^st is the stable model of Y, as above, X^st = Y^st / G, and the ramification points on the generic fiber of f specialize to distinct smooth points of Y̅. The cover f^st is defined over the valuation ring R^st of a minimal finite extension K^st of K.We call the special fiber f̅: Y̅→X̅ the stable reduction of f. When Y̅ is smooth, we say that f has potentially good reduction; elsewise, we say that f has bad reduction.We can view X^st as a blow-up of X_R ×_R R^st. We call the strict transform of the special fiber under this blow-up the original component, which we denote X̅_0. §.§ Ramification on the special fiber The action of G on Y^st induces an action on Y̅. By <cit.>, Proposition 2.4.11, the inertia groups of this action at the generic points of Y̅ are p-groups. If V̅ is an irreducible component of Y̅, we will denote by I_V̅ and D_V̅ the inertia and decomposition groups, respectively, at the generic point of V̅.In X̅, the inertia groups at the generic point of an irreducible component U̅ are conjugate p-groups; if they have order p^i, we call U̅ a p^i component. We say that U̅ is étale when i=0 and inseparable otherwise. This comes from the fact that, since Y̅ is reduced, the inertia arises from an inseparable extension of residue fields at this generic point. By <cit.>, Corollary 2.11, if U̅ and U̅^' are two intersecting components of X̅, then either I_U̅⊆ I_U̅^' or I_U̅^'⊆ I_U̅ (we stress that this relies crucially on the Sylow p-subgroup being cyclic).We call an irreducible component U̅ a tail if it intersects the rest of X̅ at exactly one point; otherwise, it is an interior component. If, in addition, a tail contains the specialization of a branch point, we call this tail primitive; otherwise, we say that it is new. This follows the convention established, for example, in <cit.>. We index the new étale tails and primitive étale tails by the sets B_new and B_prim, respectively.Now let x be a point of intersection of two components U̅ and U̅^' with I_U̅^'⊆ I_U̅ and let y be a point lying above x, the intersection point of two components V̅ and V̅^' lying above U̅^' and U̅, respectively. Following <cit.>, we call the conductor of higher ramification (as in <cit.>, for instance) of the extension of complete discrete valuation rings 𝒪̂_U̅,x↪𝒪̂_V̅,y the effective ramification invariant at x, denoted σ_x. If x is on a tail X̅_b we will often simply write σ_b for σ_x.We recall the following, which places some bounds on these invariants; it is originally found in <cit.> in the form of Lemmas 2.20 and 4.2: The effective ramification invariants σ_b are positive and lie in 1/m_Gℤ. If σ_b is a new tail, we have σ_b ≥ 1+1/m_GIn the case in which G = ℤ/p^s ⋊ℤ/m and f: Y → E is a G-cover branched at the points P_1, …, P_r, r > 1, where E is an elliptic curve having good reduction over R, we can be more precise as to the fractional parts of the σ_b, to be denoted ⟨σ_b ⟩. We can decompose f into an étale ℤ/p^n-cover Y → Z and a ℤ/m-cover g: Z → E given, birationally, by z^m = f_r · f_u, where f_r and f_u are rational functions on E corresponding to, respectively, a divisor in the form ∑_i=1^r a_i P_i and an m-divisible divisor on E. The following is a generalization of <cit.>, Proposition 1.8, originally found in <cit.>, Proposition 3.6. It was originally proven in the case in which the base curve is ℙ^1, but can be proven identically in the case in which the base curve is an elliptic curve E. Let f: Y → E be a G = ℤ/p^s ⋊ℤ/m-cover branched in r ≥ 1 points and let f^ss be a fixed semistable model for f; let X̅_b be an étale tail of E̅^ss containing the specialization of a unique branch point x_i and intersecting an inseparable component of E̅^ss. Then ⟨σ_b ⟩ = a_i/m, where the former denotes the fractional part of σ_b.In <cit.>, Corollary 3.4.4, we find the vanishing cycles formula, which relates the invariants σ_b and the genus of X in the case in which G has a Sylow p-subgroup of the form ℤ/p. This formula is generalized to the case in which G has a cyclic Sylow p-subgroup in <cit.>, Theorem 3.14; we recall this version below:Let f: Y → X be a G-Galois cover branched at r points having bad reduction, where G has cyclic Sylow p-subgroup; let g be the genus of X; let B_et be an indexing set for the étale tails, each having effective ramification invariant σ_b. Then we have 2g - 2 + r = ∑_b ∈ B_et (σ_b - 1) In particular, when f is an elliptic curve, we haver = ∑_b ∈ B_new (σ_b - 1) + ∑_b ∈ B_primσ_b§ REDUCTION TYPES We maintain the assumptions of 2; namely, that f: Y → E is a G-cover branched at r ≥ 1 points with prime-to-p branching indices, G having a cyclic Sylow p-subgroup, and Y having genus at least 2. Now we will consider the torsor structure of the reduction of one-point covers f: Y → X = E. These will eventually play a key part in the proof of the main theorem. §.§ Multiplicative reduction Supposing that f has bad reduction, we fix a semistable model f^ss: Y^ss→ E^ss. Let U̅ be an inseparable component of E̅^ss and let V̅ be a component of Y̅^ss lying above W̅ having decomposition and inertia group D_V̅ and I_V̅, respectively. The group D_V̅ has a normal subgroup of order p, since I_V̅ is normal in D_V̅ and I_V̅ is cyclic. Then <cit.>, Corollary 2.4, implies that D_V̅ has a maximal normal prime-to-p subgroup with D_V̅/N ≅ℤ/p^j ⋊ℤ/m_D_V̅ for some j>0; we denote by P the Sylow p-subgroup of D_V̅/N.Let η be the generic point of V̅/N. The group P acts on 𝒪̂_Y_R/N, η; we denote this ring by B and its subring fixed by P by A. The action of P on the residue field of B is trivial, so that B is a totally ramified extension of A. We say that f has multiplicative reduction when B/A has the structure of a μ_p^j-torsor.Following <cit.>, we have an equivalent formulation which will be of use to us. We say that B/A, as above, is of μ_p^j-type if Frac(B) is a Kummer extension of Frac(A) given by the adjunction of a p^j^th root of a unit in B that does not reduce to a p^th power in the residue field of A. Similarly, we say that B/A is potentially of μ_p^j-type if the base-change by some field, K^', is of μ_p^j-type. The former is equivalent to B/A having the structure of a μ_p^j-torsor.Now we assume that G is of the form ℤ/p^s ⋊ℤ/m and is branched at the points P_1, …, P_r, r>1. As we saw in 2, we can decompose f into an étale ℤ/p^s-cover Y → Z and a branched ℤ/m-cover, Z → E.Let χ: ℤ/m →μ_m(K) be a character of order m. We have the reduction χ̅ of χ, which gives a map ℤ→𝔽_p^×. We then may use χ̅ to obtain a semidirect product, ℤ/p ⋊ℤ/m. The p-cyclic cover above Z is étale, so corresponds to a nontrivial class in H^1_et(Z, ℤ/p)_χ. Letting J_Z denote the Jacobian of Z, we have a canonical isomorphism H^1_et(Z, ℤ/p)_χ≅ J_Z[p]_χ(-1). Moreover, we have that J_Z[p]_χ(-1) = Hom_𝔽_p(μ_p((̅K)), J_Z[p]_χ). Choosing a p^th root of unity in a fixed algebraic closure K̅ of K, we identify J_Z[p]_χ(-1) with J_Z[p]_χ. We have the following result concerning the structure of the Jacobian, Jac(Z̅)[p]_χ̅, of the special fiber Z̅, which we denote J̅[p]_χ̅:Let g̅: Z̅→E̅ be an m-cyclic cover given, birationally, by the equation z^m = f_r · f_u, where f_r and f_u are rational functions on E corresponding to, respectively, a divisor in the form ∑_i=1^r a_i P_i and an m-divisible divisor on E, ∑_j=1^ρ m b_j Q_j with branch locus {P_1, …, P_r }, r ≥ 2, satisfying ∑ a_i = m. Then J̅[p]_χ̅≅( ℤ/p )^r-1×μ_p. The action of ℤ/m on g̅_*𝒪_Z̅ arising from that on Z̅ gives us the decompositiong̅_*𝒪_Z̅ = ⊕ℒ_ψ̅, where ψ̅ runs through all characters ℤ→ k^×, into isotypical line bundles.The rational function z will have zeros of order b_j at each Q_j and a pole of order 1/m( ∑ a_i + m ∑ b_j ) = 1+∑ b_j at ∞. This function is a rational section of χ̅, so we get deg(ℒ_χ̅) = ∑ b_j - (1+∑ b_j) = -1. Similarly, z^m-1 has zeros of order a_i - 1 at each x̅_i, zeros of order b_j(m-1) at each P_j and a pole of order (1+∑ b_j)(m-1) at ∞, so that deg(ℒ_χ̅^-1) = (m-r) + (m-1)∑ b_j - (1+∑ b_j)(m-1) = 1-r. So H^1(Z̅, 𝒪_Z̅)_χ̅ and H^1(Z̅, 𝒪_Z̅)_χ̅^-1 have respective dimensions 1 and r-1.Combining <cit.>, Corollary 5.11, and <cit.>, Theorem 2.1, we may compute the rank of J̅[p]_χ̅ as a group scheme as _k(H^1_dR(Z̅)_χ̅) = _k(H^1(Z̅, 𝒪_Z̅)_χ̅) + _k(H^0(Z̅, Ω_Z̅/k)_χ̅) = rsince H^0(Z̅, Ω_Z̅/k)_χ̅ is dual to H^1(Z̅, 𝒪_Z̅)_χ̅^-1.There is a canonical k-linear isomorphism Lie(J[p]) ≅ H^1(Z̅, 𝒪_Z̅); it is compatible with the ℤ/m action, so we also have an isomorphism of χ̅-eigenspaces. So Lie(J[p]_χ̅) has dimension 1 and J[p]_χ̅ is isomorphic to (ℤ/p )^r-1× G, where G is either μ_p or α_p. Since J̅[p]_χ̅^-1 is dual to J̅[p]_χ̅, we also have that J̅[p]_χ̅^-1 is isomorphic to (μ_p )^r-1× G^', where G^' will be dual to G. Since Lie(J[p])_χ̅^-1 has dimension r-1, we must have G^' = ℤ/p, and so J̅[p]_χ̅ = ( ℤ/p )^r-1×μ_p, as we wanted.We say that f: Y → E is of multiplicative type when ∑_i=1^r a_i = m, as is the case in the previous proposition. The following is analogous to <cit.>, Proposition 3.7 and gives the reasoning behind the terminology multiplicative type:Let f: Y → X = E be a G = ℤ/p^s ⋊ℤ/m-Galois cover of multiplicative type. Then any semistable model f^ss: Y^ss→ X^ss of f has multiplicative reduction above every inseparable component of X̅^ss. If s=1, then by Proposition <ref> and <cit.>, p. 190, the p-cyclic cover Y → Z reduces to a μ_p-cover above the generic point of each component, and so will have multiplicative reduction above every component.If s>1, then f has a quotient ℤ/p ⋊ℤ/m-cover, h, which will also be of multiplicative type, so that any semistable model of h will have multiplicative reduction over every component of X̅^ss. In particular, every component of X̅^ss will be a p^s-component for f. Let K^ss be a field of definition for f^ss, a semistable model for f; since K has algebraically closed residue field, <cit.>, Lemma 2.2 implies that f^ss has multiplicative reduction over all of X̅, as we wanted. §.§ The auxiliary coverRaynaud's proof of potentially good reduction in <cit.> the case of a G-cover having a ℤ/p Sylow p-subgroup relies on the construction of the auxiliary cover. Obus (<cit.>) generalized this construction to the case of a cyclic Sylow p-subgroup, whose properties we recall here (cf. <cit.>, <cit.>): Over some finite extension K^' of K, we construct the auxiliary cover of f, a G^aux-Galois cover f^aux: Y^aux→ X^aux, with G^aux≤ G, having the following properties: (i) (X^aux)^ss = X^st and X̅^aux = X̅. (ii) There is an étale neighborhood Z of the union of the inseparable components of X̅ with the property that, as covers, f^st×_X^st Z ≅(Ind_G^aux^G (f^aux)^st)×_X^st Z. (iii) The branch locus of f^aux consists of a branch point x_b of index m_b for each étale tail X̅_b of X̅ for which m_b > 1. If X̅_b is primitive, so that it contains a branch point of f, x_b is this corresponding branch point. If X̅_b is new, x_b specializes to a smooth point. (iv) Given an étale tail X̅_b of X̅ and an irreducible component V̅_b of Y̅^aux above X̅_b with effective ramification invariant σ_b^aux, we have σ_b^aux = σ_b. (v) If N is the maximal prime-to-p normal subgroup of G^aux, then G^aux/N ≅ℤ/p^s ⋊ℤ/m_G^aux with s ≥ 1.We denote by G^st the quotient in (v); we then call the G^str-quotient cover of f^aux the strong auxiliary cover, denoted f^str: Y^str→ X^str. The upper-numbered filtration for thehigher ramification groups is unaffected by taking quotients, so both the auxiliary cover and the strong auxiliary cover will have the same effective ramification invariants as the original cover. By passing to either cover, we allow ourselves the convenience of studying the ramification with a simplified Galois group at the expense of a potentially larger branch locus. Strong auxiliary covers arising from one-point covers have the following property: If the strong auxiliary cover of a one-point cover with prime-to-p branching ramifies, it is of multiplicative type.Let X̅_b be an étale tail of X̅^str with effective ramification invariant σ_b. We have seen that passing to the strong auxiliary cover preserves effective ramification invariants, so by Proposition <ref> and Theorem <ref>, ∑⟨σ_b ⟩ = 1. If there were only one étale tail, this would mean that σ_b = 1 and f^str would be unramified, a contradiction. So we assume there are at least two étale tails. Then by Lemma <ref>, each σ_b is a non-integer, so that each étale tail has m_b > 1. So, applying Proposition <ref>, we see that ∑⟨σ_b ⟩ = ∑a_i/m = 1, so that ∑ a_1 = m and f^str is of multiplicative type. Let f: Y → E be a branched one-point cover with bad reduction and prime-to-p branching indices. Then the stable model of f has multiplicative reduction over the original component.By Propositions <ref> and <ref>, any semistable model of the strong auxiliary cover, f^str, has multiplicative reduction over the original component. Since f^str is a prime-to-p quotient of the auxiliary cover f^aux, any semistable model of f^aux will also have multiplicative reduction over the original component. The cover, f, will be isomorphic to the disjoint union of copies of f^aux over an étale neighborhood of the original component, so that the stable model of f will also have multiplicative reduction over the original component, as we wanted.§ POTENTIALLY GOOD REDUCTION Now let f: Y → X be a cover as in 2 with the additional requirement that X ≅ E, an elliptic curve having good reduction over K. Let f: Y → E be an étale cover defined over K of the elliptic curve E, where E has good reduction over K. Then f has good reduction over K.By the Riemann-Hurwitz formula, Y must also be an elliptic curve, so f is an isogeny. Isogenous elliptic curves either both have good or bad reduction over a given field (see, for instance, <cit.>, Corollary 7.2); since E has good reduction, so must Y, so that f has good reduction, as we wanted. Let f: X → E be one-point cover with abelian Galois group G. Then f is, indeed, an étale cover f: X → E.The étale fundamental group π_1(E ∖{O}) of E ∖{O} has the presentation {a,b,c | aba^-1b^-1c=1}. The map π_1(E ∖{O}) → G factors via the abelianization π_1(E ∖{O}) →π_1(E ∖{O})^ab = {ab | aba^-1b^-1 = 1} = π_1(E), so that f corresponds to an element of π_1(E). So the map f is, indeed, étale, as we wanted.In particular, one-point covers having abelian Galois groups must have good reduction.The following is analogous to <cit.>, Lemma 4.5: Let f: Y → E be a ramified cover, branched at r ≥ 1 points, with tame branching indices; let V̅ be an irreducible component above the original component above the original component with decomposition group D_V̅. Then m_D_V̅ > 1.It suffices to work with the strong auxiliary cover f^str, which, in light of Lemma <ref>, we assume has more than one branch point. It will be enough to show that the decomposition group of a component above the original component is not a p-group.The cover f^str has a ℤ/m_G_str-quotient cover given (as, for instance, in 2) birationally by z^m = f_r · f_u, where the rational function f_r corresponds to the divisor ∑_1^ρ a_i P_i on E, with 0 < a_i < m. It then suffices to show that the reduction of f_r is not an m^th power in k(X).The branch points of f specialize to distinct points on the special fiber, so we have at least r different residue classes among them. By Proposition <ref> and Theorem <ref>, ∑_i=1^ρ a_i = m · r, so that some subset of the a_i satisfy 0 < ∑ a_i < m. So the factor f_r does not reduce to an m^th power, as we wanted.The following is the key step in the proof of good reduction and is analogous to <cit.>, Proposition 5.1. We recall the statement; the proof is identical in our situation, with Lemma <ref> assuming the role of <cit.>, Lemma 4.5. Let G be a finite group with nontrivial cyclic Sylow p-subgroup; let k be an algebraically closed field of characteristic p; let K_0 = Frac(W(k)) and let K be a finite extension with e(K) < p-1/m_G, where e(K) is the absolute ramification index of K; let f: Y → E be a G-cover defined over K, where E is an elliptic curve having good reduction over the valuation ring R of K, branched at the distinct K-points {x_1, …, x_r }, with r ≥ 1; suppose that the branch points of f specialize to distinct points on the special fiber of the good model of E over R. If f has bad reduction, then the stable model of f does not have multiplicative reduction over the original component.We then obtain the following by applying this to the r=1 case.Let G be a finite group with cyclic Sylow p-subgroup. Let K be as in Proposition <ref> and let f: Y → E be a one-point G-cover defined over K. Then f has potentially good reduction.If f is unbranched, then f has good reduction by Lemma <ref>. Otherwise, <cit.>, Lemme 4.2.13, implies that the branching indices of f are tame. If f were to have bad reduction, Corollary <ref> would imply that f has multiplicative reduction over the original component, contrary to Proposition <ref>. So f has potentially good reduction.plain | http://arxiv.org/abs/1707.08996v1 | {
"authors": [
"James Phillips"
],
"categories": [
"math.AG",
"math.NT"
],
"primary_category": "math.AG",
"published": "20170727184349",
"title": "One-point covers of elliptic curves and good reduction"
} |
LetX be a normal noetherian scheme and Z ⊆ X a closed subset of codimension ≥ 2. We consider here the local obstructions to the map π̂_1(X\ Z) →π̂_1(X) being an isomorphism. Assuming X has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-étale cover of X, where the corresponding map on fundamental groups is an isomorphism. Department of Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] Étale Covers and Local Algebraic Fundamental Groups Charlie Stibitz===================================================§ INTRODUCTIONSuppose that X is a normal variety over ℂ and Z ⊆ X is a closed subset of codimension 2 or more. Then a natural question to pose is whether the surjective map of fundamental groups π_1(X Z) →π_1(X) is an isomorphism. For general normal schemes we can ask the same question for étale fundamental groups.For a regular scheme the Zariski-Nagata theorem on purity of the branch locus implies the above map on étale fundamental groups is an isomorphism (see <cit.>, <cit.>). For a general normal scheme however this map need not be an isomorphism so that étale covers of X Z need not extend to all of X. The next question to ask is what are the obstructions to the above map being an isomorphism. Restricting any cover to a neighborhood of a point, we see that in order for it to be étale, it must restrict to an étale cover locally. Hence each point of Z gives rise to a possible obstruction determined by the image of the local étale fundamental group into the étale fundamental group of X Z. Assuming these all vanish, the above map will be an isomorphism.Even if they do not vanish, we can still hope for something in the case where all the obstructions are finite. A first guess might be that this would imply that the kernel of the above map is finite, yet Example 1 of the singular Kummer surfaceshows that this can be far from true in general. What is true however is that after a finite cover that is étale in codimension 1 the corresponding map is an isomorphism. The main theorem here is that this is is fact equivalent to finiteness of the obstructions and a couple other similar conditions: Suppose that X is a normal noetherian scheme of finite type over an excellent base B of dimension ≤ 2. Let Z = (X) ⊆ X. Then the following are equivalent.* For every geometric point x ∈ Z the image G_x := [π_1^(X_x Z_x) →π^_1(X Z)] is finite (see 2 for definitions of X_x and Z_x).* There exists a finite index closed normal subgroup H ⊆π_1^(X Z) such that G_x∩ H is trivial for every geometric point x ∈ X. * For every tower of quasi-étale Galois covers of X X ← X_1← X_2← X_3←⋯X_i+1→ X_i are étale for i sufficiently large. * There exists a finite, Galois, quasi-étale cover Y → X by a normal scheme Y such that any étale cover of Y_ extends to an étale cover of Y.As hinted to above, we can rephrase the question of the π_1^(X Z) →π_1^(X) being an isomorphism as a purity of the branch locus statement of X. Both are equivalent to the fact that any étale cover of X Z extends to an étale cover of X. By purity of the branch locus for X_reg, any étale cover of X Z will at least extend to X_reg. Hence it is enough to consider the case where Z = X_, as we have done in the theorem. From this point of view, (iv) says that we can obtain purity after a finite, Galois, quasi-étale cover. The following example of the singular Kummer surface then elucidates what is going on in the above theorem. Consider the quotient π:A → A/± = X, where A is an abelian surface and X is a singular Kummer surface over ℂ. Away from the 16 2-torsion points this map is étale, but at the 2-torsion points it ramifies. Each of these 2-torsion points gives rise to a nontrivial ℤ/2ℤ obstruction. In particular any étale cover of A will give a cover of X not satisfying purity of the branch locus, and in particular there are infinitely many such covers. On the other hand the fundamental group of X is trivial, which can be seen as X will be diffeomorphic to a standard Kummer surface given as a singular nodal quartic in ℙ^3 with the maximum number of nodes. In particular its étale fundamental group is trivial, and there are no étale covers of X. Note that on the other hand as A is smooth weobtain purity on a finite cover of X that is étale away from a set of codimension 2 as in (iv) of the above theorem. In this sense, although the kernel of the map π̂_1(X_) →π̂_1(X) is large it is not far away from satisfying purity.The recent history of studying these two problems started with a paper of Xu <cit.> showing that the local obstructions are finite for all klt singularities over ℂ. From this Greb, Kebekus, and Peternell <cit.> were able to show the global statements of (iii) and (iv) in the above theorem for klt singularities. Their proof used essentially the existence of a Whitney stratification, which allowed them to check only finitely many strata to prove a cover is étale. Then in positive characteristic, Caravajal-Rojas, Schwede, and Tucker <cit.> proved again that the local obstructions are finite for strongly F-regular singularities (which are considered a close analogue of klt singularities in positive characteristic). Using a bound on the size of the local fundamental groups from this paper, Bhatt, Carvajal-Rojas, Graf, Tucker and Schwede <cit.> were able to construct a stratification that enabled them to run a similar local to global argument to deduce statements of the form (ii), (iii), and (iv) for strongly F-regular varieties. It is also worth noting in the recent preprint of Bhatt, Gabber, and Olsson <cit.> they are able to reprove the results in characteristic 0 by spreading out to characteristic p. The proof of Theorem 1 can be broken down essentially into two parts. One is the construction of a stratification that allows us to deal with only finitely many obstructions. The second is a completely group theoretic fact about profinite groups, namely if we have a finite collection of finite subgroups of a profinite group, then there exists a closed finite index subgroup which intersects all of these groups trivially. Note that the assumption that the covers are Galois and the finite index subgroup is normal is essential. In fact due to the choice of basepoint that we have suppressed above (note that x is not the basepoint of these fundamental groups), G_x is actually only a conjugacy class of a finite subgroup inside of the group π_1^(X Z). Finally it is worth noting that in Xu's paper a different local fundamental group, π_1^(X_x{x}) was shown to be finite. If we instead defined our obstruction groups G_x as Xu did then in fact the theorem is false, as will be shown in the following example. There are two ways around this for klt singularities: either showing the larger fundamental groups π_1^(X_x Z_x) are finite in the klt case, or show that a similar implication (i)(ii),(iii) will hold as long as the smaller local fundamental groups of all covers étale in codimension 1 remain finite, which is the case for klt singularities.We will discuss this issue further in the last section.ConsiderX = CS the cone over a Kummer surface S = A/± where A is an abelian surface. Then there are three types of singular points:First consider the case where x is the generic point of the cone over one of the nodes. Then X_x has a regular double quasi-étale cover ramifying at x (note that since we have localized there is no difference between the two possible fundamental groups). This shows π̂_1^(X_x{x}) = π̂_1^(X_x Z_x) ≅ℤ/2ℤ, and there is no ambiguity in which definition we choose. In general this will work for the generic point of any irreducible component of the singular locus.The next type of point where we start to see a difference is whenx is a closed point in the cone over a node of S. Then in this case π̂_1^(X_x{x}) is trivial while π̂_1^(X_x(X)_x) ≅ℤ/2ℤ. Although they are different they are at least both finite. On the other hand if we desire for these groups to behave well under specialization it is clear that π̂_1^(X_x(X)_x) is the better choice. The last type of point, where the real problem occurs, is the cone point x ∈ CS.First consider the fundamental group π̂_1^(X_x{x}). Then this will be isomorphic to π̂_1(S) ≅ 0 by the Lefschetz hyperplane theorem. In particular all the local fundamental groups defined in this sense are finite. So if this version of the theorem were true then any tower as above would stabilize. On the other hand π̂_1(S_) is infinite, giving an infinite tower of cones X=CS ← CS_1← CS_2←⋯ Galois over X and quasi-étale. In particular finiteness of all the local fundamental groups π̂_1^(X_x{x}) does not imply finiteness of the local fundamental groups π̂_1^(X_x(X)_x). Note that in this case the singularity at the origin is not klt. Given a finite morphism f:Y → X the branch locus, written (f), is the locus over which f fails to be étale. A finite morphism f:Y → X is quasi-étale if it is étale in codimension 1 or in other words the branch locus has codimension ≥ 2. By purity of the branch locus for normal schemes quasi-étale is equivalent to being étale over the regular locus.I would like to thank my advisor János Kollár for his constant support. Also I would like to thank Ziquan Zhuang for several useful discussions.§ LOCAL FUNDAMENTAL GROUPS In this section we review some basic facts and definitions about the fundamental groupswe will be considering.Suppose that (R,m) is a strictly Henselian local normal domain and that Z ⊆(R) is a closed subset of codimension ≥ 2. Then we define the algebraic local fundamental with respect to Z to be π_1^((R) Z). If x is a normal geometric point of an irreducible scheme X and Z ⊆ X has codimension ≥ 2, we define the local space X_x = (𝒪_X,x^), the spectrum of the strict Henselization of the local ring. This comes with a map ι:X_x→ X and we define Z_x = ι^-1(Z). We then define the local fundamental group at x with respect to Z to be the algebraic local fundamental group of the strict Henselization of𝒪_X,x with respect to the closed set Z and use the notation π̂_1^(X_x Z_x). For any geometric point x ∈ Z we define G_x := [π̂_1^(X_x Z_x) →π̂_1(X Z)], the obstructions occurring in theorem 1.The definition above depends on a choice of both strict Henselization (requiring a choice of separable closure of the residue field of x) and a choice of base point. In particular the choice of base point implies that the groups G_x are only defined up to conjugacy. It is for this reason that we need to take Galois morphisms in the main theorem. For counterexamples when the morphisms are not Galois see <cit.>. Also note the difference between the definition here and that used in <cit.>. The following basic lemma shows the purpose in using Henselizations when defining the local fundamental group. Suppose that f:Y → X is a quasi-étale morphism of normal schemes, that is étale away from some subset Z of codimension ≥ 2. Then f is étale over a geometric point x ∈ X if and only if the pull back of the map to U_x: = Xx Z_x is trivial.It is enough to prove that the map is étale once we pull back to the strict Henselization of the local ring. Now in this case if the map f is étale then it induces a trivial cover of (𝒪_X,x^sh) and hence of the open set V. On the other hand if the cover of U_x is trivial, then so is the cover of (𝒪_X,x^sh) since the varieties are normal. Hencethe morphism is étale.§ THE BRANCH LOCUS OF A QUASI-ÉTALE MORPHISM In this section we consider the question of where a quasi-étale cover of a normal variety X branches. We will see that there are finitely many locally closed subsets of X such that any branch locus is the union of some subcollection of these subsets. Moreover we will show that such a stratification is possible to compute in terms of any alteration. We start with the criterion for telling if a morphism is étale from an alteration.Alterations. <cit.> A map π:X̃→ X is an alteration if it is proper, dominant and generically finite. A regular alteration will be an alteration where X̃ is regular. In his work, de Jong showed that regular alterations exist for noetherian schemes of finite type over an excellent base of dimension ≤ 2. We will say that a divisor E ⊂X̃ is exceptional if π(E) has codimension at least 2 on X. Suppose that X is normal Noetherian scheme, with a regular alteration π:X̃→ X. Let f:Y → X be a finite morphism of Noetherian schemes, and denote by f̃:Ỹ→X̃ the normalized fiber product of the maps. Thenf is étale if and only if f̃ is étale and for any geometric point x ∈ X, f̃ induces atrivial cover of π^-1(x).First suppose that f:Y → X is étale. Thenthe base change Y ×_XX̃→X̃ is étale, so that the fiber product was already normal. Hence it follows f̃ is étale. Then since Ỹ is just the fiber product and f is étale,for any of the d points q ∈ Y mapping to p ∈ X we see that σ^-1(q)≅π^-1(p) ×_k(p) k(q). Therefore f̃ is just the trivial degree d cover on every fiber π^-1(p). Now suppose that f̃ is étale and induces a trivial cover on every fiber of π. Then in particular for any x ∈ X,f^-1(x) will have (f) geometric connected components. In particular it must be étale at x (<cit.>, V.7). Each of the two conditions in the lemma above are easily seen to be necessary.For example we can let X be the cone over a smooth conic. This has a quasi-étale double cover f:𝔸^2→ X. Blowing up the origins gives a map of the normalized fiber-products _0𝔸^2→_0X, that will ramify along the exceptional divisors.On the other hand we can take the X to be the cone over an elliptic curve E. Take an étale cover E' → E, which will induce a quasi-étale cover X' → X of their cones. After blowing up the origins we obtain the map of normalized fiber products which is étale, but induces a nontrivial cover of the exceptional divisors. Using the above lemma we can check whether f is étale based on a single regular alteration. Our next goal will be to show that based on this alteration we really only need to check that f is étale at finitely many points. To identify what are the points we need to check we require the following condition, which roughly says that the reduced fibers of a morphism fit together in a flat family. Condition ∗. Suppose that g:Z → S is a propermorphism of Noetherian schemes with S integral. Then g satisfies this condition if there exists a purely inseparable morphism i:S' → S such that if Z' = Z×_S S' → S' is the base change, then Z_red' → S' is flat with geometrically reduced fibers. We now show that there exists a stratification of X such that π will satisfy the abovecondition ∗ over each of the strata. Let π:X̃→ X be a morphism of Noetherian schemes. Then there exists a stratification X = ⋃ S_i, where the S_i are irreducible locally closed subsets,such that π^-1(S_i) → S_i satisfies condition (∗) for all i.We will proceed by Noetherian induction on X. Take an irreducible component Sof X. Consider the map π^-1(S)_red→ S. Taking an irreducible componentW of π^-1(S)_redif W → S is not separable, we can take some high enough power of the Frobenius so that the pullback by the map is separable. Doing this for every irreducible component of π^-1(S)_red we may assume that the general fiber is reduced. Then taking an open subset U of S we may assume that every fiber of π^-1(U)_red→ U is reduced and that this morphism is flat. Continuing on will give the desired stratification.Suppose that g:Z → S is amorphism of Noetherian schemes satisfying condition (∗) with S integral. Then the number of connected components of geometric fibers are constant.We have a purely inseparable morphism S' → S such that Z' → S' is flat with geometrically reduced fibers. Since S' → S is a universal homeomorphism, it follows that Z' → Z is a homeomorphism. Hence the number of connected components remains the same, so we can assume from the beginning that Z → S is flat with geometrically reduced fibers. Now in this case we will show that the Stein factorization of g:Z → S factors as Z →Ŝ→ S where Ŝ→ S is étale. Taking the strict Henselization of the local ring at any point we can reduce to the case where S is the spectrum of a strictly Henselian local ring. In this case Ŝ is a product of finitely many local rings. Our goal is to show that these are isomorphic to S. Now consider a connected component W of Z, so that the map g:W → S is flat and proper, with geometrically reduced fibers. Now since W is connected and S is the spectrum of strictly Henselian ring, the special fiber W_0 is also connected. But then since W_0 is reduced H^0(W_0,𝒪_W_0) = k(0). Hence we see by the theorem of Grauert that 𝒪_S→ g_∗𝒪_W is an isomorphism. This implies that Ŝ→ S is thus étale, so in particular the number of connected components of the geometric fibers are constant. Suppose that X is a normal Noetherian scheme andπ:X̃→ X a regular alteration. Then there exists a stratification X = ⋃_i∈ IZ_i into locally closed subsets such that for any f:Y → X quasi-étale, with Y a normal Noetherian scheme,(f) = ⋃_i ∈ J ⊂ IZ_i.The above lemma gives a stratification X = ⋃_i S_i such that π^-1(S_i) → S_i satisfies condition ∗. Moreover we a finite number of exceptional divisors E_i giving closed subsets π(E_i) on X. Putting these together gives our desired stratification of X. Our goal is then to show that any branch locus of a quasi-étale morphism is a union of these strata. Consider Ỹ = (X̃×_X Y)^n the normalized fiber product which comes with a morphism f̃:Ỹ→X̃ that is étale away from the exceptional locus. Now by purity of the branch locus(f̃) = ⋃_i E_i where the E_i are some subset of the exceptional divisors. In particular the branch locus of f will include B=⋃_iπ(E_i), which will be a union of some strata. Now looking on the complement of B, and replacing X by X B we can assume that f̃ is in fact étale. In particularf̃^-1(π^-1(S_i)) →π^-1(S_i) → S_i satisfies condition ∗. Hencethe number of connected components of the fibers are constant. This implies that for any point s ∈ S_i that if the cover of π^-1(s) is geometrically trivial, then the corresponding cover for other point in S_i is also trivial. Hence we see that the branch locus must be a union of the strata. In the proof of (i) implying (iii) of the main theorem itwould be nice to apply this theorem directly on X. However when we take the normalized pullback of an alteration we may not get another alteration. To remedy this we will need to take alterations of varieties that are further along in the tower.§ PROOF OF THE MAIN THEOREMIn this section we prove the different implications in the main theorem.(i)(ii). Consider a regular alteration π:X̂→ X. This will give us a stratification X = ⋃_i Z_i. Now for each of the finitely many generic points η_i of the different strata consider the finitely many finite groups G_i = G_η_i. Then as π_1^(U) is profinite there exists some finite index closed normal subgroup H intersecting all of these G_i trivially. This corresponds to a quasi-étale cover γ:Y → X that is étale over U. Moreover by our choice of stratification for any geometric point x we will also have that G_x∩ H is trivial. Hence such a finite index normal subgroup H can be taken uniformly for all x ∈ X.(ii)(i). Our assumption (ii) gives a closed finite index normal subgroup H ⊆π_1^(U) such that G_x∩ H = {1}. Then in particular G_x≅ G_x/G_x∩ H ⊆π_1^(U)/H which is finite. Hence G_x is finite as well. (i)(iii).Weproceed by Noetherian induction. Consider our tower of finite morphisms denoted by γ_k:X_k+1→ X_k, and consider the collection 𝒰 of open sets U ⊆ X such that when we restrict the tower over U the morphisms are eventually étale. The assumption that all the morphisms are quasi-étale implies that X_∈𝒰. Since X is assumed to be Noetherian this collection has a maximal element and our goal is to show that this must be all of X. Therefore we need to show that if U ∈𝒰 and U ≠ X then we can find a larger U' ∈𝒰. To do this take any x a generic point of an irreducible component of X U. Consider X_x = (𝒪_X,η^) and restrict the tower of X_i over X_x to get a tower(𝒪_X,η^) = X_x,0← X_x,1← X_x,2← X_x,3←⋯ Now using the assumption (i) applied to the point x, it follows that eventually the covers will be trivial when restricted over the regular locus and hence will be étale. This then shows that there exists some N >> 0 such that γ_n is étale over η for n ≥ N and they are étale over the open set U coming from Noetherian induction.Now take a regular alterationπ:X̂_N→ X_N. Then using π we construct a stratification X_n = ⋃_iZ_i as before. Then any of the maps X_N+k→X_N must be étale over U and η. But because the branch locus must be a union of strata it follows thatthese are all étale over some open set U' ⊃ U with U' ∋η. Hence such a larger U' ∈𝒰 exists and by Noetherian induction we see that X ∈𝒰. This proves property (iii).(iii)(iv). Assuming that no such cover exists, we inductively construct a tower X ← X_1← X_2← X_3←⋯ as in (iii) of the main theorem using Galois closures, such that none of the X_i+1→ X_i are étale. This will contradict our assumption, so eventuallyevery étale cover of one of the X_i, will extend to an étale cover of X_i. This gives the desired cover satisfying purity.(iv)(i).Consider a geometric point x ofX. Take a cover f:Y → X as in (iv), and a geometric point y of Y mapping to x. Denote by U the regular locus of X and Z = X U the singular locus.This gives rise to the following commutative diagram of fundamental groups. π̂_1^(Y_y f^-1(Z)_y) @>>> π̂_1(f^-1(U))@VVV @VVVπ̂_1^(X_x Z_x) @>>> π̂_1(U)Now the assumption on Y implies that the top map is zero. On the other hand, the image of the map on the left is a finite index normal subgroup. Hence looking at the images in π̂_1(U), we see that G_x has a trivial finite index subgroup and hence must be finite. § APPLICATIONS Using our main theorem we can recover the results of <cit.> and <cit.>. Suppose that X is a normal klt variety over ℂ. Then X satisfies the condition (ii).We want to show that X satisfies condition (i). There are two issues to deal with if we wish to apply Xu's result <cit.>. First is the problem that in this paper the local fundamental groups are defined in terms of links instead of the local spaces X_x given by Henselization. The second is that Xu proves the finiteness of π̂_1^(X_x{x}) and we saw that this is not enough to guarantee (ii) in general. There are two ways to get around this. The first is to strengthen the result of Xu to prove the finiteness of algebraic local fundamental groups as considered in this paper. In the proof of his main theorem, Xu cuts down to a surface. It is then possible to consider only quasi-étale covers instead of étale covers of X_x{x}, as these will agree after cutting down. Also you would need an equivalence of the algebraic local fundamental group defined in terms of links and Henselizations. Once this is done though (ii) will follow immediately from the main theorem. Note that also the recent result of <cit.> is strong enough to apply directly. The second way to prove this is to note that we can get around the issue of which fundamental group we consider when we work in a class of normal varieties ℛ satisfying the following. We want for every X ∈ℛ, and every quasi-étale cover Y → X that Y ∈ℛ, and also for every x a geometric point of X ∈ℛ that π̂_1^(X_xx} is finite. In particular klt singularities satisfy both these conditions by <cit.>. Then under these assumptions, the same argument for (i) implies (ii) works with the fundamental groups π̂_1^(X_xx}. This approach is used in <cit.>.Suppose that X is a normal F-finite strongly F-regular variety over a field of characteristic p. Then X satisfies the condition (ii). In this case the result of Carvajal-Rojas, Schwede, and Tucker <cit.> applies directly to the main theorem without any changes. hplain | http://arxiv.org/abs/1707.08611v1 | {
"authors": [
"Charlie Stibitz"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170726190043",
"title": "Étale Covers and Local Algebraic Fundamental Groups"
} |
Unconditional violation of the shot noise limit in photonic quantum metrology Geoff J. Pryde =============================================================================§ INTRODUCTION AND SUMMARY Turbulent fluid flow is prevalent in every-day phenomena. Yet, while common, it is difficult to characterize it in a quantitative manner. Recently, the gauge-gravity duality has provided a novel geometric means to study relativistic fluid flow of strongly coupled, large N gauge theories, <cit.>. Indeed,in a suitable parameter range, and given appropriate initial conditions, such flows have been shown to exhibit (decaying) turbulent behavior <cit.>.This development hints at a fascinating set of connections between turbulence and the dynamics of black hole horizons in general relativity.A connection between turbulence and event horizons of asymptotically AdS black holes opens the possibility of using geometric tools to study turbulence. Indeed, various pioneering works have suggested a connection between the dynamics of the event horizon and a Kolmogorov-type cascade <cit.>. Unfortunately, the structure of time-dependent black hole horizons is not fully understood, which is oneof the reasons this research program has not yet come to fruition. In what follows we continue exploring these connections using newly developed tools for studying gravity in a large number of dimensions. More specifically, we consider the turbulent behavior of event horizons of asymptotically AdS spaces, in the limit where the number of dimensions, d, is very large. There is an extensive, ongoing, program to understand general relativity in the limit where the number of dimensions is large <cit.>. The works of <cit.> in particular address the large d limit of asymptotically AdS spaces. To obtain a large d limit, one considers a metric ansatz in d=n+p+1 spacetime dimensions, where the dynamics depend on p+1 dimensions. The remaining number of dimensions, n, is taken to be infinite. This allows one to relate the large d limit of gravity to gravity in p+1 dimensions.Since our goal is to relate gravity at large dto hydrodynamic flow, we obtain the d→∞ limit through a novel but circuitous path which will eventually allow a direct relation between horizon dynamics and fluid dynamics.We start our analysis by considering the equations of motion of relativistic (conformal) hydrodynamics in the limit where the number of dimensions is large.Generically, relativistic hydrodynamics does not simplify in such a limit.Nevertheless, as we show in section <ref>, if we scale the time and space coordinates appropriately, tune the transport coefficients of the theory to lie in an appropriate subset of parameter space, and work in a suitable fluid frame, the hydrodynamic equations significantly simplify. In appropriate variables they are given by (<ref>).Given the simplicity of the hydrodynamic equations in an appropriately scaled coordinate system, we turn our attention to asymptotically AdS gravity in the same limit. We find that our scaling laws for the coordinates are precisely those introduced in <cit.> to study the Einstein equations in the large d limit. Additionally, the holographic transport coefficients satisfy precisely the relations required for the aforementioned simplification of the large d hydrodynamic equations. Indeed, as we show in section <ref> the large d Einstein equations of <cit.> are precisely the large d hydrodynamic equations of motion, albeit in an unusual, but well motivated, fluid frame.Since we were compelled to scale our coordinate system, the large d limit of the relativistic hydrodynamic equations of motion bear a strikingresemblance to the (non-relativistic) Navier-Stokes equations. More precisely, they are a variant of the compressible version of the Navier-Stokes equations. As such, they may be studied using traditional tools of non relativistic fluid dynamics. As we show in section <ref> when the flow is subsonic the Kolmogorov or Kraichnan scaling laws are expected to emerge and an inverse cascade should be exhibited when the system mimics two dimensional (sustained) turbulent fluid flow. In fact, even when the Mach number for the flow is not small we findevidence for an inverse cascade-like behavior. Following the discussion in section <ref> we turn to a numerical analysis of the equations of motion in section <ref>. By using an appropriate initial condition we generate flow which exhibits decaying turbulence. We then compare various stages of the flow to expectations made in section <ref> for two and three spatial dimensions and to an analysis of holographic (decaying) turbulence in AdS_4 <cit.>.Given our improved understanding and analytic control over the relation of the fluid flow to the dynamics of the event horizon (for subsonic flows in particular), we study, in section <ref>, a proposed relation between the horizon curvature power spectrum and the hydrodynamic energy power spectrum <cit.>. While these quantities are linearly related in the regime of small Mach number, the relation between these two quantities becomes obscure as the Mach number increases.Finally, we present a summary of our findings in section <ref> where we also discuss possible extensions and puzzles associated with out results.§ THE LARGE D LIMIT OF HYDRODYNAMICSHydrodynamics is a universal low energy effective description of many-body systems. Following a Wilsonian type of coarse graining, the equations of motion of relativistic hydrodynamics may be characterized by a handful of fields: a unit normalized velocity field u^μ(x), a temperature field T(x) and, in the presence of charged matter, a chemical potential μ(x). When working in the limit where gradients of the hydrodynamic fields are small compared to the inverse mean free path, all observables may be expressed as local functions of the hydrodynamic fields. These expressions are referred to as constitutive relations.In a d spacetime dimensional conformal theory in flat space (and in the absence of charge, anomalies or parity breaking terms) the constitutive relations for the stress tensor take the formT^μν=T^μν_(0)+T^μν_(1)+T^μν_(2)+𝒪(∂^3) ,whereT^μν_(0) = ϵ(T) u^μu^ν + ϵ(T)/d-1 P^μν ,T^μν_(1)=-2η(T) σ^μν ,T_(2)^μν = λ_1(T) u^λ𝒟_λσ^μν+λ_2(T) (σ^λμσ_λ^ν-σ^αβσ_αβ/d-1P^μν)+λ_3(T) (ω^μλσ_λ^ν+ω^νλσ_λ^μ)+λ_4(T)(ω^μλω^ν_λ-ω^αβω_αβ/d-1P^μν),andP^μν=g^μν+u^μu^ν, σ^μν=1/2P^μαP^νβ(∂_αu_β+∂_βu_α)-1/d-1P^μν∂_αu^α, ω^μν=1/2P^μαP^νβ(∂_αu_β-∂_βu_α),𝒟_λσ^μν=P^μαP^νβ∂_λσ_αβ+∂_αu^α/d-1σ^μν ,and we have set the speed of light to unity.See <cit.> for details.Following the standard reasoning behind the construction of effective field theories, the constitutive relations (<ref>) are the most general ones allowed, compatible with the conformal symmetry of the underlying theory, with a local version of the second law, and taking into account useful features of the derivative expansion. Indeed, in writing down (<ref>) we have used the equations of motion at order n-1 in derivatives to simplify the constitutive relations at order n. In addition, we have used a particular definition of the fluid velocity and energy density, usually referred to as the Landau frame, in order to further simplify the relations (<ref>). Since these features of the derivative expansion will be essential in our study of the large d limit of hydrodynamics, we describe them in some detail in what follows.In the absence of charge, the equations of motion for the hydrodynamic fields are nothing but the conservation equations for the stress tensor ∂_μT^μν=0 ,which need to be solved perturbatively in a derivative expansion. For instance, at leading order in derivatives one hasu^μ∂_μϵ + d ϵ∂_μu^μ= 0 d/d-1ϵu^μ∂_μu^ν+ P^μν∂_μϵ= 0 .Therefore, when constructing T_(1)^μν it is possible to replace u^μ∂_μϵ and u^μ∂_μu^ν with ∂_μu^μ and P^μν∂_μϵ. The same reasoning applies to the construction of T_(2)^μν; at n'th order in the derivative expansion, the available tensor structures from which the stress tensor may be constructed is reduced due to the equations of motion at lower orders.It is possible to modify the relations (<ref>) for T_(1)^μν and T_(2)^μνby an appropriate field redefinition of T and u^μ. The constitutive relations (<ref>) were written in the Landau frame, defined viau_μT^μν = u_μT_(0)^μν= - ϵ(T) u^ν .From a physical standpoint the Landau frame is defined such that u^μ points in the direction of the energy flux, and the relation between ϵ(T) and T does not get modified by derivative corrections.When the system is out of equilibrium, the Landau frame is only one of many choices for possible definitions for T and u^μ. Indeed, working perturbatively in the derivative expansion one may always redefine the temperature and velocity field via, e.g.,ϵ_new(T_new)= ϵ(T) - k_1 u^μ∂_μϵ(T) + 𝒪(∂^2) ϵ(T) u^μ_new= ϵ(T) u^μ - c_1 ∂^μϵ(T) +𝒪(∂^2),where the c's and k's are real numbers.The equations of motion for the velocity field and temperature, which follow from inserting the constitutive relations (<ref>) into the conservation equation (<ref>), are lengthy and cumbersome. While the zeroth order equations of motion (<ref>) are relatively short and simple, writing the first order equations in terms of u^μ is a formidable task, let alone the second order equations. In the remainder of this section we will study the equations of motion in the limit where the number of dimensions d becomes very large. We will see that with some restrictions, to be elaborated on below, the large d limit of these hydrodynamic equations become easier to handle.Before working out the hydrodynamic equations, let us first consider thermodynamic equilibrium in the large d limit. In equilibrium, the temperature and velocity field are constant so that the stress tensor is given exactly by T^μν_(0). Since the velocity is constant, we may conveniently choose a coordinate system where u^μ∂_μ = ∂_t. If we assume, without loss of generality, that the energy density ϵ is 𝒪(d^0) and insist that the pressure term, proportional to P^μν, not be suprressed relative to the energy density, then we must work in a coordinate systems scaled so that the spatial coordinates carry an extra factor of 1/√(d) relative to the time coordinate: ds^2 = -dt^2 + ∑_i (dx^i)^2/d.For the same reasoning, we parameterize the velocity field of a boosted thermally equilibrated system by u^μ = (1, β^i)/√(1-β^iδ_ijβ^j/d). Note that our definition of β^i differs from the conventional definition by a factor of √(d), β^i = √(d)β_conventional^i. With this parameterization in mind, let us now go back to (<ref>). Conservation of energy and momentum provide d equations for the d hydrodynamical variables T and u^μ. In order for the large d limit to provide a useful proxy to a finite p+1 dimensional system we must take the large d limit in such a way that the number of dynamical equations remains p+1. To do so, we use an ansatz where the dynamical variables do not depend on n out of the d=1+p+n coordinates. Thus, we useds^2 = - dt^2 + δ_ab/ndζ^a dζ^b + dχ⃗_⊥^2/nwhere χ⃗_⊥ denotes the n≫ 1 coordinates on which T and u^μ do not depend, and Latin indices run over the remaining p spatial components. The velocity field in this coordinate system is given byu^μ = (1,β^a(t,ζ), 0⃗)/√(1-β^b(t,ζ)β^c(t,ζ)δ_bc/1+p+n) .In the remainder of this work we will raise and lower Latin indices with δ^ab and δ_ab respectively. Keeping with this notation, the tensor structures (<ref>) evaluate toσ^μν=n/2δ^μ aδ^ν b(∂_aβ_b + ∂_bβ_a) + 𝒪(n^0), ω^μν=n/2δ^μ aδ^ν b(∂_aβ_b - ∂_bβ_a) + 𝒪(n^0), u^λ𝒟_λσ^μν=n/2δ^μ aδ^ν b(∂_t+β^c∂_c)(∂_aβ_b+∂_bβ_a)+𝒪(n^0) .Inserting (<ref>) into (<ref>), we find that in order for the shear viscosity and second order transport coefficients to contribute to the dynamics they must both scale as 1/d relative to ϵ. Recall that dimensional analysis and conformal invariance imply that for any finite dϵ∝ T^d , η∝ T^d-1 , λ_i ∝ T^d-2 ,where the proportionality constants are dimensionless numbers. Thus, we may writeη∝1/dϵ^1-1/d = ϵξ/d(ϵξ^d)^-1/d ϵξ/dwhere ξ is an emergent length scale which is determined by requiring that ϵ^1/dξ remain finite in the large d limit. Thus, in what follows we will use the parameterizationη = h_0/nξϵ + 𝒪(n^-2) λ_i = 2 ℓ_i/nξ^2 ϵ + 𝒪(n^-2) ,where h_0 and ℓ_i are dimension independent numbers. In the majority of this work we will set ξ=1. It will be reintroduced where necessary.The large d stress-energy tensor is now given byT^μν = ϵ([1β^b;β^a β^aβ^b+δ^ab+T̃^ab_(1)+T̃^ab_(2)+𝒪(∂^3) ])+𝒪(n^-1)with,T̃^ab_(1) = -h_0(∂^aβ^b+∂^bβ^a), T̃^ab_(2)=ℓ_1(∂_t+β^c∂_c)(∂^aβ^b+∂^bβ^a) +ℓ_2/2(∂^aβ^c+∂^cβ^a)(∂^bβ_c+∂_cβ^b) +ℓ_3/2( (∂^ aβ^c-∂^cβ^ a)(∂^bβ_c+∂_cβ^b) + (a↔b)) +ℓ_4/2(∂^aβ^c-∂^cβ^a)(∂^bβ_c-∂_cβ^b) .The equations of motion (<ref>) which result from the constitutive relations (<ref>) are still somewhat unwieldy. To simplify them further we will use our freedom to perform field redefinitions of the velocity and temperature fields in order to remove all third order derivative terms from the equations of motion. The most general frame transformation we can carry outat second order in derivatives and in the large d limit is given byϵ→ϵ + k_1ϵ∂_bβ^b+k_2∂_b∂^bϵ + k_3ϵ(∂_bβ^b)^2 + k_4ϵ∂_bβ^c∂_cβ^b + k_5∂_bϵ∂^bϵ/ϵ + k_6ϵ∂_bβ_c∂^bβ^c+ 𝒪(∂^3) β^a →β^a + c_1∂^aϵ/ϵ + c_2∂_bβ^b∂^aϵ/ϵ + c_3∂^aβ^b∂_bϵ/ϵ + c_4∂^a∂_bβ^b + c_5∂_bϵ/ϵ∂^bβ^a + c_6∂_b∂^bβ^a + 𝒪(∂^3). Inserting (<ref>) into (<ref>) and utilizing the derivative expansion, c.f., (<ref>), we find that ifℓ_2 + 2 ℓ_3 + ℓ_4 = 0 , andℓ_1 - ℓ_2 - ℓ_3 = 0 ,then the second order terms in the equations of motion vanish (i.e., no contributions from second order terms in the stress tensor), provided we choose a frame wherek_2 = ℓ_1 + ℓ_4-2 c_1 h_0 = c_1^2 + 2 ℓ_1c_5 = 2 (ℓ_1 + ℓ_4) c_6 = ℓ_1 + ℓ_4 , and the remaining c_i's and k_i's vanish.In what follows we will refer to (<ref>) as the large d frame. To iterate our finding in the previous paragraph, when (<ref>) are satisfied, the equations of motion take the form∂_t ϵ + c_1 ∂_a ∂^a ϵ= - ∂_a j^a∂_t j^a - h_0∂_b ∂^b j^a = -∂^a ϵ - ∂_b ( j^a j^b/ϵ) - (c_1/2 - ℓ_1/c_1) ∂_b(j^a/ϵ∂^bϵ - ϵ∂^a(j^b/ϵ))where we have definedj^a = β^a ϵ ,and c_1 is given by the quadratic equation in (<ref>). The stress tensor whose divergence gives these equations of motion is given byT_(0)^μν = [ ϵ j^a; j^b δ^abϵ + j^a j^b/ϵ ]T_(1)^μν = [ 0 c_1 ∂^a ϵ; c_1 ∂^b ϵ (c_1 + h_0) (j^a/ϵ∂^b ϵ + j^b/ϵ∂^a ϵ)-h_0 (∂^a j^b+ ∂^b j^a ) ]T_(2)^μν= [ 0 0; 0 c_1^2 ∂^a ∂^b ϵ ] + (ℓ_1 + ℓ_4) ∂_c ∂^c T_(0)^μν .We emphasize that even though ∂_μ T_(2)^μν≠ 0, the equations of motion are only second order in derivatives due to cancellations coming from the perturbative expansion.§ HOLOGRAPHY IN A LARGE NUMBER OF DIMENSIONS Our final result, equation (<ref>), is, evidently, a simplified version of the full equations of motion. In order to obtain it we had to restrict our transport coefficients to satisfy (<ref>) and (<ref>). Both equations impose restrictions on the transport coefficients when the number of dimensions is large. Equation (<ref>) implies that the four transport coefficients of the theory are restricted to lie on a plane in parameter space while equation (<ref>) implies a particular scaling of the transport coefficients with d when d is large. Since the large d limit is used as a proxy for finite dimensional system one may, naively, enforce by hand a 1/d scaling of transport coefficients for large d so that both (<ref>) and (<ref>) will be satisfied. Notwithstanding, it seems likely that such a behavior will affect the radius of convergence of the large d expansion. It is then somewhat surprising that the relations (<ref>) are valid in a holographic setup. There, we find thatλ_1-λ_2 - λ_3 = 0 λ_2 + 2λ_3 -λ_4 = 𝒪(d^-3)see <cit.>. Note that the first of the two relations is satisfied in any number of dimensions <cit.> even in the presence of matter <cit.> and also in other instances <cit.>. As we will now demonstrate the constraints (<ref>) are also satisfied in a holographic context.The constraints (<ref>) and (<ref>) guarantee that the equations of motion will take the simplified form (<ref>). Therefore, if they are satsified, and we work perturbatively in the derivative expansion, equations (<ref>) must emerge from a holographic analysis <cit.>. We find that not only do (<ref>) emerge, they are also valid to all orders in the derivative expansion. Put differently, the Einstein equations at large d are equivalent to large d hydrodynamics naturally truncated at second order in derivatives. In what follows we will show this property of the Einstein equations explicitly.Consider the D=d+1 space-time dimensional Einstein-Hilbert actionS = ∫√(g)(R + (D-1)(D-2) ) d^Dx ,where we have set the radius of AdS space to unity. Following <cit.>, we wish to take the large D limit of this action while retaining translation invariance in most of the spatial directions. To this end, we use the ansatzds^2=2dt(-A(t,r,z^i)dt+dr-F_a(t,r,z^i)dz^a)+G_ab(t,r,z^i)dz^adz^b+G_⊥(t,r,z^i)dx⃗_⊥^2where a,b=1,…,p, D=p+n+2 and x⃗_⊥ is n dimensional. To obtain a boundary metric as in (<ref>) we make the replcaementsz^a=ζ^a/√(n)x⃗_⊥=χ⃗_⊥/√(n)as well asF̃_a = F_a/√(n)G̃_ab = G_ab/nG̃_⊥ = G_⊥/n ,and choose boundary conditions such that A=r^2(1/2+𝒪(r^-n)) F̃_a=r^2/n(𝒪(r^-n)) G̃_ab=r^2/n(δ_ab+𝒪(r^-n)) G̃_⊥=r^2/n(1+𝒪(r^-n)) .In addition, we rescale the radial coordinate such thatR=(r/r_0)^nis finite in the large d limit. Note that r_0 serves as a reference length scale. In what follows we will set r_0=1 for clarity. In the new coordinate system the line element takes the formds^2=2dt(-A(t,R,ζ^c)dt+R^1/n-1/ndR-F̃_a(t,R,ζ^c)dζ^a)+G̃_ab(t,R,ζ^c)dζ^adζ^b+G̃_⊥(t,R,ζ^c)dχ⃗_⊥^2 . The scaling used to obtain (<ref>) is precisely that of <cit.>. See also <cit.>. Inserting (<ref>) into the Einstein equations and solving order by order in n we find thatA =1/2-a(ζ^μ)/2R + 𝒪(n^-1) F̃_a=f_a(ζ^μ)/nR+ 𝒪(n^-2)G̃_ab=δ_ab/n + 𝒪(n^-2) G̃_⊥= 1/n + 𝒪(n^-2)where the functions a and f_a must satisfy the constraint equations∂_ta-∂_b∂^ba = -∂_bf^b ∂_tf_a-∂_b∂^bf_a= -∂_aa-∂_b(f_af^b/a) . As expected, equations (<ref>) and (<ref>) match as long as we seth_0=1 ,c_1=-1 , ℓ_1= 1/2and make the identificationsa = ϵ f_a = j_a .Thus, in the large D limit the Einstein equations precisely reproduce the large d limit of the equations of motion for relativistic hydrodynamics. The associated stress tensor is given in (<ref>).Before proceeding let us pause to consider the relation between the scale r_0 associated with the horizon, (<ref>), and the scale ξ generated by taking the large d limit of hydrodynamics (<ref>). Recall that for any finite d the (local) Hawking temperature associated with the event horizon is1/a^1/d r_0 = d/4π T ,and that in the absence of charge the entropy density is given bys = ∂ P/∂ T = a/Twhere P is the pressure and ϵ=a the energy density. Using the holographic result for the shear viscosity to entropy density ratio <cit.> we findη = a^1-1/d/d r_0 .Comparing (<ref>) to (<ref>) and using (<ref>) we obtainξ = 1/r_0 . Given (<ref>) the expression for the entropy density (<ref>) takes the forms = 4 πξ/d ain the large d limit and in our current conventions.[Note that using the Bekenstein-Hawking formula S=A/4 G_N we obtain s = r_0^p+n/4 G_N n^n/2 a. Equation (<ref>) agrees with this result up to an overall normalization which was chosen when we set ϵ = a in (<ref>).] Since we have a good handle over the dynamics of the area element of the event horizon we can test various proposals regarding its behavior in the presence of turbulence, and analyze the geometric structure associated with turbulent flows.§ ANALYSIS OF LARGE D FLUID FLOWSThus far we have described hydrodynamics using the variables ϵ and β^a. Since we are working in the large d frame (<ref>) these variables are somewhat different from the energy density and velocity field as described in the Landau frame. Indeed, ϵ and β^a coincide with the energy density and fluid velocity of the Landau frame only in equilibrium. Otherwise, they are related to the Landau frame variables through (<ref>). Nevertheless, if we rewrite the equations of motion (<ref>) in terms of ϵ and β^a = j^a/ϵ, we find∂_tϵ + (β⃗·∇⃗) ϵ + ϵ∇⃗·β⃗= ∇^2 ϵ ∂_tβ⃗+(β⃗·∇)β⃗ + ∇⃗ϵ/ϵ=∇^2 β⃗ +2(∇⃗ϵ/ϵ·∇⃗)β⃗ .In order to understand the dynamics associated with equations (<ref>) and their relation to the compressible Navier-Stokes equations, it is useful to switch to dimensionless variables,u⃗ = β⃗/√(n) U ,p = ϵ/E , ∂/∂ t→L_0/√(n) U∂/∂ t , ∇⃗→ L_0 ∇⃗,where L_0 is a characteristic length scale of the system, √(n)U a characteristic velocity , and E a characteristic energy density. The factor of √(n) in the definition of the characteristic velocity arises from the scaling we used for β^a in (<ref>).In terms of these dimensionless variables, equations (<ref>) take the form∂_tp +(u⃗·∇⃗)p + p∇⃗·u⃗ =1/Re∇^2 p ∂_tu⃗+(u⃗·∇⃗)u⃗ + ∇⃗ p/M^2 p= 1/Re∇^2u⃗ + 2/Re(∇⃗ p/p·∇⃗)u⃗ ,where, in analogy with the compressible Navier-Stokes equations, we have defined a Reynolds number Re=√(n)L_0U and a Mach number M=√(n)U. If we reinsert the dimensionful parameters c and ξ we findRe=√(n)L_0U/ξ c=L_0U/ξ c_s M = √(n)U/c=U/c_s .where c_s is the speed of sound.When the Mach number is small we may expand u⃗ and p in powers of the Mach number, u⃗ = ∑_n=0 M^n u⃗_(n) p = ∑_n=2 M^n p_(n) .On a manifold with no boundary p_(0) and p_(1) are constant and u_(0) and p_(2) satisfy the incompressible Navier-Stokes equation,∇⃗·u⃗_(0) = 0 ∂_tu⃗_(0)+(u⃗_(0)·∇⃗)u⃗_(0) + ∇⃗ p_(2)/ϵ_(0)= 1/Re∇^2 u⃗_(0) .with p_(0) the density andp_(2) the pressure. Equations (<ref>) are precisely the Navier-Stokes equations for an incompressible fluid.Thus, in the limit of small Mach number we may use the exhaustive machinery developed for incompressible flow to study solutions of (<ref>). While this is textbook material (see for instance <cit.>) let us remind the reader of the salient features of such flows. We define the total energy and enstrophy viaE_I =1/2∫|u⃗_(0)|^2 d^px , Ω_I = 1/2∫ω_(0)ijω_(0)^ij d^pxwith ω_(0)ij = ∂_i u_(0) j - ∂_j u_(0) i the vorticity two-form. The evolution equations for E_I and Ω_I are given by∂/∂ t E_I= -1/ReΩ_I∂/∂ tΩ_I= ∫ω_(0)^ijω_(0)jkσ_(0)^ki d^px - 1/Re P_I where we have introduced the non relativistic shear tensor, σ_(0)^ij and the Palinstrophy, P_I,σ_(0)ij = ∂_i u_(0) j + ∂_j u_(0) i ,P_I = 1/2∫∂_k ω_(0)ij∂^k ω_(0)^ij d^dp .We once again emphasize that the energy density is given by T^00 via (<ref>) and that E_I is the would be energy of the analog Navier-Stokes equation in the incompressible limit. Nevertheless, we shall, with some abuse of language, refer to E_I as the energy and to Ω_I as the enstrophy. Note that Ω_I ≥ 0. Thus, the energy E_I can only decrease. On the other hand, Ω_I itself can increase or decrease depending on the sign ofthe first term on the right hand side of (<ref>). The dynamics associated with this term is often referred to as vortex stretching. There is ample phenomenological evidence and various imperfect arguments that vortex stretching increases the enstrophy for turbulent incompressible flow. Indeed, one usually posits that, for sustained turbulent flow, lim_Re→∞Ω_I/Re = e_0 > 0. The appearance of an emergent scale e_0 at large Reynolds number implies an energy cascade. To see this consider the two point functionQ(r⃗) = ∫u⃗_(0)(x⃗) · u_(0)(x⃗ + r⃗) d^p x/∫ d^p xand its Fourier transformQ(k⃗) = 1/(2π)^p∫ Q(r⃗) e^-i k⃗·r⃗ d^pr .We define the energy density E(k) viaE(k) = 1/2∫Q(k⃗) k^p-1 dθ_kwhere dθ_k is a solid angle in momentum space, viz. d^pk = k^p-1 dθ_k dk. With this definition,E_I= ∫_0^∞ E(k) dk ,and alsoΩ_I = ∫_0^∞ k^2 E(k) dk .If E_I is to decrease while Ω_I is to remain constant then E(k) must distribute itself in such a way that energy will flow from lower momentum modes to higher ones. This process is referred to as the Kolmogorov cascade. If we constantly supply energy into the system and assume that E(k) depends on e_0 and k we find the celebrated -5/3 law,E(k) ∼ e_0 k^-5/3 .Various numerical and experimental verifications of (<ref>) can be found in <cit.>.An exception to (<ref>) arises when p=2 <cit.>. In two spatial dimensions we may treat ω_ij as a volume form. It is then straightforward to show that the vortex stretching term vanishes for incompressible flow. In this case the enstrophy is bound from above by its value at t=0 so that energy is conserved when the Reynolds number becomes large. In this case the palinstrophy plays the same role that enstrophy played in three dimensional flow. One then expects that the energy distribute itself towards lower wavenumber. If energy is continuously pumped into large scales via a driving force then in order for the system to remain ina steady state some type of large scale friction needs to be introduced into the system. This friction introduces an energy scale and yields a power law behavior as in (<ref>) referred to as the inverse cascade. In addition to the inverse cacade there is a direct cascade associated with the enstrophy production term. An analysis similar to the one that led to (<ref>) impliesE(k) ∼ w_0 k^-3 .See <cit.> for various discussions.In the current work we will study decaying turbulence where the typical velocity scale U and length scale L_0 vary with time. In the context of the Navier-Stokes equation, a power law behavior associated with a direct cascade, c.f., (<ref>), was observed in numerical simulations of decaying turbulence in p=3 dimensions <cit.>. In p=2 dimensions numerical simulations of the Navier-Stokes equations usually lead to either an inverse or direct cascade. Numerical simulations of decaying turbulence with no slip boundary conditions may exhibit both a direct cascade and an inverse cascade, as in (<ref>) and (<ref>) <cit.>. In periodic domains a direct cascade as in (<ref>) is often observed <cit.>. Recent simulations exhibit both a direct and an inverse cascade in such scenarios <cit.>.Let us return our attention to (<ref>). Using intuition gained from the analysis at small Mach number we focus on the evolution of the energy and enstrophy.We find that our equations of motion (<ref>) imply∂/∂ t E_C= 1/M^2∫ p(∇⃗·u⃗) d^px-1/4 Re∫ p( ω_ijω^ij + σ_ijσ^ij)d^pxwhere we now defineE_C = ∫ p |u⃗|^2 d^pxand ω_ij = ∂_i u_j - ∂_j u_i σ_ij = ∂_i u_j + ∂_j u_i.Note that we have used the letter p for both the number of spatial dimensions and the dimensionless energy density. In the equations above and in the remainder of this section the number of dimensions p will appear only in the measure d^px.Similar to the case of small Reynolds number, we find that E_C is approximately conserved for large M and large Re.Furthermore, similar to (<ref>), we find the following equation for the dynamics of the enstrophy∂/∂ tΩ_C = ∫1/pω_ijω^jk( σ_k^i - δ_k^i∇⃗·u⃗) d^px+ 1/Re∫|∇⃗ p |^2/p^3ω^2- p ∂_k ω_ij/p∂_k ω_ij/p- 2/p^2σ_klω^ lj( ∂_j ∂_k p - ∂_j p ∂_k p/p) - 2/p^2ω_klω^ lj( ∂_j ∂_k p - ∂_j p ∂_k p/p - δ_jk( ∇^ 2p - |∇⃗ p |^2 )) d^px ,withΩ_C = ∫ω_ijω^ij/p d^px .We will refer to the first term on the right hand side of (<ref>) as a vortex stretching term. This term vanishes for p=2 (as does the last term on the right hand side of (<ref>)). Thus Ω_C is conserved at high Reynolds number and we may expect an inverse cascade in such a configuration. When Re is finite, it is difficult to determine the sign of the rate of change of Ω_C and therefore difficult to assess a priori whether a scaling regime exists or not. If it does exist, we may expect the same power law behavior as in (<ref>). Since (<ref>) is less manageable than (<ref>), to make headway we must resort to numerical methods. We carry out such an analysis in the next section.§ ANALYSIS OF TURBULENT FLOWSWe now turn to a numerical analysis of turbulent flows. While the analysis of section <ref> was valid for unbound domains, our simulations focus on flows in a bound, toroidal domains of length L ξ in each direction. We comment on the influence of these boundary conditions on our results when relevant. We solved the equations of motion (<ref>) using a variety of methods including Fourier spectral methods and finite differencing in the spatial directions. We evolved the variables a and f_a= ϵ u_a forward in time using third order Adams-Bashforth or explicit Runge-Kutta. The initial conditions we used for our simulations were “perturbed shear flows", i.e. constant density flows where the velocity field is perpendicular to its gradient.Such flows solve (<ref>) in the limit of infinite Reynolds number.More specifically, for p=2 we usedf_x =E c_s δ f_x(x⃗), f_y =E c_s cos(2π n/L x)+δ f_y(x⃗), a = a_0 Ewhereas for p=3 we usedf_x= E c_s cos(2π n/L y) + δ f_x(x⃗), f_y=E c_s cos(2π n/L z) + δ f_y(x⃗) f_z= E c_s cos(2π n/L x) + δ f_z(x⃗), a= a_0 E . Here E is an energy scale which drops out of the equations of motion. The parameter a_0 in both (<ref>) and (<ref>) is constant in space and the δ f_i denote perturbations of the formδ f_i(x⃗)=∑_m⃗ A_i,m⃗Ec_scos(Δϕ_i,m⃗+2π(m⃗·x⃗)/L)where m⃗, A_i,m⃗ and Δϕ_i,m⃗ are chosen from a random sample. The sum in (<ref>) included 10 to 100 random modes. Typical simulations of two dimensional flow included 20 modes and typical three dimensional flow included 40 or 80 modes. The components of the wavenumber m⃗ was chosen from a uniform distribution of integers running from 1 to 64 for a typical two dimensional flow and 1 to 16 or 1 to 32 for a typical three dimensional flow. The amplitude A_i,m⃗ was chosen from a uniform distribution ranging from 0 to A with A a pre-determined parameter. The phase Δϕ_i,m⃗ was chosen from a uniform distribution ranging from 0 to 2π.Since our setup does not involve a driving force, the flow we generate does not reach steady state turbulence. Nevertheless, we may associate a Reynolds number and a Mach number to the initial conditions of the flow. Choosing U = max(u) - ⟨ u ⟩ and L_0 = L ξ/n, we find thatRe = L/a_0 n , M=1/a_0 .If the initial Reynolds number is sufficiently large we expect a turbulent instability to emerge. From a numerical standpoint these instabilities are trigered by the modes characterized by A_i,m⃗ in (<ref>). For sufficiently large Re we found that numerical roundoff error was sufficient to trigger the instability even for A_i,m⃗=0.In presenting our results we find it convenient to use scaled coordinatesτ = L_0/U = Re/M^2ξ/c_s e=a U^2/L_0^p-1= M^p/Re^p-1E/ξ^p-1 ,for time and energy power spectrum respectively.Typical results of two dimensional and three dimensional fluid flow with relatively high Reynolds number can be found in figures <ref> and <ref>. In both two and three dimensions we found that the dynamical behavior of the fluid can be characterized by three stages. An initial stage where an instability drives the shear flow into a chaotic configuration. Typical flow in this stage can be found in the first row of figures <ref> and <ref>. Once the instability sets in, we observe a turbulent regime where the energy power spectrum exhibits power law behavior. Typical turbulent behavior can be seen in the second row of figures <ref> and <ref>. The distinction between two dimensional flow and three dimensional flow is most apparent (visually) in the turbulent phase: in two dimensional flow large scale structure is formed while in three dimensional flow, vortices break up into smaller ones. In the final stage the flow decays into an equilibrated configuration. This behavior can be observed in the last row of figures <ref> and <ref>. In the remainder of this section we will discuss each of these regimes in detail. §.§ Initial Phase: Onset of InstabilityThe initial phase is characterized by the development of instabilities. To quantify these instabilities, and later also turbulence, we analyze the energy spectrum E_C, defined similar to E_I in (<ref>),E_C(t,k) = ∂/∂ k∫_|k⃗'|≤ kd^pk'/(2π)^p|u⃗_p(t,k⃗')|^2with u⃗_p(t,k⃗) = ∫ d^px√(p(t,x⃗))u⃗(t,x⃗)e^-ik⃗·x⃗. When the Reynolds number is small we find that the shear flow quickly decays to an equilibrated configuration. When the Reynolds number is sufficiently large we find that instabilities dominate the spectrum and lead to turbulent berhavior.Typical behavior of turbulent instabilities in two dimensional fluid flow can be observed in the top right panel of figure <ref>. This particular run was carried out with Re=1562.5, M =0.5, n=32 and an amplitude A=10^-5 for the noise. We have carried out similar runs with Mach numbers ranging from M=0.005 to M=2 and Reynolds number of up to Re∼ 1562. As the Mach number increased the amplitude of the random noise required to generate turbulence became higher; for low Mach number numerical noise was sufficient to generate the instability while for M=2 we had to set A=0.01 to generate a turbulent instability. It is possible that at very high Mach numbers and the initial conditions (<ref>) a turbulent instability will not form.Going back to figure <ref>, around t=10τ the initial disturbance with k=32× 2π/L has decayed and an unstable mode with k=22× 2π/L can be observed. A more detailed analysis of the spectrum shows that the unstable mode is a shear mode orthogonal to the first. Indeed, in figure <ref> we plot the Fourier decomposition of |u_p|^2 at different times. At t=6τ, the instability has not set in and |u_p| receives support from the initial shear mode which we have set as input into the system. At t=12τ some remnants of the initial shear mode are observed (in red), but most of the support for |u_p| comes from a transverse shear mode. The fact that instabilities with lower wavenumber become excited is suggestive of the inverse cascade present in two dimensional flow. In three dimensions, the instabilities are associated with higher wave numbers. Typical behavior for instabilities in three dimensions can be found in the bottom right corner of figure <ref>.The data for the bottom right plot in figure <ref> was obtained for a flow with Re=162.5, M=1 and an initial perturbation associated with the fourth harmonic, n=4. We have carried out similar runs with Mach numbers of up to 10. Here, as opposed to the two dimensional case, a turbulent instability seems to emerge even for high Mach numbers. For the initial conditions associated with figure <ref>, around t=0.1τ we observe a growing unstable mode which is roughly double the wavelength of the original.A typical Fourier decomposition of the velocity field for three dimensional flow can be found in figure <ref> where we show constant k_z slices of the spectrum of |u⃗| at different times. The spectrum of the instabilities suggests that the dominant unstable modes take the form u_i∼cos( 2π n/L(i ± j)) where i,j∈x,y,z, j is the initial mode direction of f_i in (<ref>). (For example: if the initial shear flow was of the form f_x=E c_s cos(2π n/L y), then the associated unstable mode is u_x∼cos( 2π n/L(x ± y)). ) The authors of <cit.> define a critical Reynolds number as the minimal Reynolds number required for a turbulent instability to exist. Put differently, having the Reynolds number above the critical one is a necessary condition for the appearance of turbulence. An analysis similar to that of <cit.> implies that the critical Reynolds number associated with our initial conditions is of the order of 15, roughly two orders of magnitude smaller than the initial Reynolds number needed for turbulence to develop. We provide more details regarding the computation of the critical Reynolds number in appendix <ref>.§.§ Turbulent Phase Once the instability of the initial phase is strong enough, it drives the fluid to a turbulent regime which we characterize using the energy density power spectrum, E_C, defined in (<ref>). We present a detailed analysis of the power spectrum during the turbulent phase in appendix <ref>. Here we confine ourselves to summary of the salient features of that analysis. Representative plots of E_C for two and three dimensional fluid flow can be found in figures <ref> and <ref> respectively.For two dimensional fluid flow we found a consistent k^-4 power law for Mach numbers between 0.005 and 2 and Reynolds number between 781 and 1562. As the Reynolds number decreases the range for which power law behavior is observed becomes smaller and vanishes completely around Re ∼ 390. See appendix <ref>. While it is clear that lower modes get populated as time progresses, indicative of an inverse cascade, the expected k^-5/3 and (or) k^-3 law is absent from the simulations we have studied. Early simulations ofnon- relativistic decaying turbulence displayed similar scaling <cit.>. Perhaps increasing the initial Reynolds number of our flow will ameliorate this problem, as it does in the non-relativistic case. In contrast to two dimensional fluid flow, in three dimensions we found remarkable agreement with an E_C ∼ k^-5/3 power law. Our runs include Mach numbers between M=0.1 and M=10 and Reynolds numbers between Re=81.25 and Re=750, and initial data involving initial modes n=1 and n=4. Simulations with initial modes with n>4 and reasonably high (initial) Reynolds number are expensive. Indeed, for the n=4 run the initial Reynolds number was rather low, Re = 162.5, which apparently manifested itself as a visible λ = L/4 periodic behavior of the flow even in the “turbulent” regime where Kolmogorov scaling was observed. §.§ Final PhaseSince there is no driving force the fluid is expected to reach equilibrium at late times. In two dimensional non-relativistic and incompressible fluid flow on ℝ^2 the late time behavior of the velocity field is given by the Oseen Vortex solution which is an attractor of the Navier-Stokes equation <cit.>. Since we are placing our fluid on a torus the late time behavior of the fluid is somewhat different. In particular, we find that due to the inverse cascade the lowest lying mode dominates the flow, such that at late timesϵ(x,y) =ℰ_0 u_x(x,y) = U_x 0e^- (2π/L)^2tcos((2π/L)y+ϕ_1) u_y(x,y) = U_y 0e^- (2π/L)^2tcos((2π/L)x+ϕ_2) ,where ℰ_0, U_x 0 and U_y 0 are constants. One can check that (<ref>) solves (<ref>) for low Mach number up to exponentially suppressed corrections.Typical flow of the form (<ref>) is depicted in the lower right corner of figure <ref>. A spectral analysis of the flow can be found in figure <ref>.To get a quantitative handle on (<ref>) we have fitted the late time behavior of u_x and u_y to an exponential decay law u_i ∼ e^-α_i(L) t = e^-ν_i (2π/L)^2t. Let us denote the L_2 norm of a quantity X by L_2(X). We evaluate α_i(L) by fitting the dependence of L_2(u_x) and L_2(u_y) on time to a power law fall off, obtainingν_x = 1.004 ± 0.066,ν_y = 1.002 ± 0.053.See figure <ref>. Since simulating three dimensional fluid flow at late times is expensive, we have not carried out a full analysis of its late time behavior as we have done for two dimensional flow. There is some indication that the late time solution will be dominated by the modes associated with the initial condition (<ref>). For the runs we have carried out in three dimensions with n=1 and n=4 (see (<ref>)) we find that the late time behavior takes the formu_x= U_x e^- (2π n/L)^2tcos(2π n/L y), u_y= U_y e^- (2π n/L)^2tcos(2π n/L z) u_z= U_z e^- (2π n/L)^2tcos(2π n/L x),ϵ= ℰ_0 .where n is themode number injected into the system in (<ref>). Typical flow of the form (<ref>) is exhibited on the bottom row of figure <ref>. A k-space view of the velocity and energy density fields matching (<ref>) can be found in figure <ref>. A possible explanation for (<ref>) may be that the initial perturbation still holds most of the energy at late times and due to the direct cascade is slowest to decay. It is also possible that our three dimensional simulations do not exhibit full turbulent behavior which would be in line with the periodic behavior we observe for three dimensional flow at n=4, as mentioned earlier.§ GEOMETRIZING TURBULENCEAs should be clear from our discussion so far, a full understanding of turbulence, even in the incompressible limit, is far from complete. It would be favorable if it were possible to utilize the geometric tools available from numerous studies of black hole dynamics to address turbulence. The AdS/CFT correspondence opens the possibility for such a procedure <cit.> and there are numerous suggestions for identifying an appropriate geometric quantity which captures the turbulent behavior of the dual fluid <cit.>. In what follows we will focus on the work of <cit.> where the authors argue that the horizon power spectrum, 𝒜, to be defined shortly, is proportional to the energy power spectrum defined in (<ref>) and (<ref>). By appealing to the large d limit we obtain an analytic handle over such a relation and can study its regime of validity.Recall that the leading order contribution to the black brane metric is given by ds^2=dt (-(1-a(ζ^μ)/R)dt+2/nRdR - 2f_a(ζ^μ)/nRdζ^a)+δ_ab/ndζ^adζ^b+1/ndχ⃗_⊥^2 .The only null surface associated with the metric (<ref>) which agrees with a black brane topology is given byR=a(t,ζ^a ).which must therefore be identified with the event horizon.The extrinsic curvature on the event horizon is given byΘ_MN≡Π_ M^PΠ_ N^Q∇_Pn_Q , Π_ N^M≡δ_ N^M+ℓ^Mn_Nwhere Latin indices run over all d=n+p+2 dimensions, n_M is the null normal to the horizon, and ℓ_M is an auxiliary null vector, which satisfies ℓ_M n^M=-1.Following <cit.> we consider the rescaled traceless horizon curvature, defined byθ_ j^i≡√(γ/κ^2)Σ_ j^i , Σ_ j^i≡Θ_ j^i-1/d-2Θ_ n^nδ_ j^i,where i,j run over the n+p spatial dimensions of the horizon, √(γ)≡√((g_ij)) is the area element on a spatial slice of the event horizon, and κ is defined by the geodesic equation n^M∇_Mn_Q=κ n_Q.The horizon curvature power spectrum is defined by𝒜( t,k ) ≡∂/∂ k∫_|𝐤'|≤ kd^pk'/(2π)^pθ̃_j^*i( t,𝐤') θ̃_ i^j( t,𝐤') ,whereθ̃_ j^i( t,𝐤) ≡∫d^px θ_ j^i( t,𝐱) e^-i 𝐤· 𝐱and p is the number of spatial dimensions where the dynamics take place. An explicit computation gives usn_M dx^M = dR - ∂_t a dt - ∂_b a dζ^b , ℓ_M d x^M = - 1/nRdt.and √(γ)=a(ζ^μ)n^-(n+p)/2 , κ=a(ζ^μ)n^2/2from whichθ_ j^i = aδ^il/n^(p+n)/2(∂_j(f_l/a-∂_la/a)+∂_l(f_j/a-∂_ja/a))follows. Substituting a=ϵ, f_a=ϵβ_a and taking the incompressible limit, we findθ_ j^i=ϵ/n^(p+n)/2(∂_jβ^i+∂^iβ_j) . The Fourier transform θ̃ will involve a convolution of ϵ̃ and β̃ which are the Fourier transform of ϵ and β respectively, a 'la (<ref>). However, if we are working in the limit of small Mach number then ϵ is approximately constant and we findθ̃_ j^i ϵ/n^(p+n)/2(i k_jβ̃^i+i k^iβ̃_j) .Inserting (<ref>) into (<ref>) and comparing to (<ref>) we find that at low Mach number,𝒜(k)/E(k) = 2 ϵ/n^(p+n) k^2as predicted in <cit.>. If, on the other hand, the Mach number is not small a relation of the form 𝒜∼ k^2 E will not hold. Indeed, in figure <ref> we show typical results for the horizon power spectrum for high and low initial Mach number in two and three dimensions. As expected, 𝒜∼ k^2 E holds only for very low Mach number. A more refined analysis can be found in appendix <ref>. § SUMMARY AND OUTLOOK In this paper we have discussed relativistic hydrodynamics in the limit where the number of dimensions is large,revealing the simplifications that occur in this limit. We have focused our attention particularly on holographic theories whose dynamics surprisingly follow these simplified equations. We have analyzed turbulent flows of these dynamical systems, and their relation to the geometry of black hole horizons.We have seen that three dimensional flows exhibit a Kolmogorov cascade with the expected power law behavior, k^-5/3, for a wide range of Mach numbers. While promising, we remind the reader that the initial conditions for our three dimensional simulations where carried out with a low wavenumber (n=1 and n=4). Higher wave numbers would require more computational resources. Moreover, in the n=4 simulations we observed a periodicity with wavelength λ = L/4 throughout the flow. We expect that such behavior will disappear for sufficiently high initial Reynolds number which we have not yet reached. In contrast to three dimensional flows, two dimensional flows did not exhibit either a k^-5/3 power law or a k^-3 power law, albeit displaying a tendency to form large scale structures, associated with the expected inverse cascade. The behavior we observe is probably due to an insufficiently high Reynolds number. It would be worthwhile to improve on this point; simulations of decaying turbulence which exhibited both a direct and inverse cascade usually require a grid much larger than the one we have used <cit.>. Preliminary runs do indicate that increasing the Reynolds number may result in a canonical power law. Recall that the power law behavior of the energy power spectrum has been evaluated for sustained turbulence in which case a driving force continuously supplies energy into the system. It would be interesting to study sustained turbulence in our setup, which would allow for more robust scaling relations to be observed in the steady state. In order to reach that steady state, one would need to add stochastic force to supply energy at the appropriate range of wave numbers, and for the scenario of inverse cascade to include friction as a sink of energy at large scales (keeping in mind that in the direct cascade energy is dissipated at small scales.)In addition to the Kolmogorov scaling discussed in this work, steady state turbulence would allow us to investigate real space scaling relations of the sort recently described in <cit.>. In the present context of decaying turbulence, those relations are not stable enough to be clearly visible, but we expect that in sustained turbulence we would be able to investigate them more reliably. Similar comments apply to the holographic study of superfluid turbulence. While holographic superfluidity was studied in the large dimension limit <cit.>, the reduction to horizon equations does not hold in the presence of a scalar field. We hope to return to this problem in the future.While in principle these additional elements are possible in the large dimension limit, it seems we are no longer afforded the simplifications of that limit, with those additional elements. The dimensional reduction of the equations, allowing us to discuss the horizon fluid first and deduce the resulting spacetime subsequently, is no longer in effect when adding external dials, since those are imposed at infinity. Nevertheless, perhaps there is another simplifying limit that would allow for investigation of sustained turbulence in the present context.Finally, using the simplicity of the large d black brane metric we were able to compare our analytic expression for the area power spectrum to the energy power spectrum. We have found that the power law behavior of these two quantities is related only at very low Mach number at which point there is very good agreement with <cit.>. It would be interesting to identify a geometric entity which captures the Kolmogorov power law behavior. With such a quantity at hand one may be able to use the powerful tools of general relativity to compute this quantity in a more precise manner.§ ACKNOWLEDGEMENTSThe work of AY and ES is supported by an ISF grant and an ISF-UGC grant. The work of MR is supported by a Discovery grant from NSERC. § CRITICAL REYNOLDS NUMBERFollowing <cit.> one may quantify the critical Reynolds number, above which the instability grows.Given the initial conditions (<ref>), we define a critical Reynolds number, Re^c, for these conditions, as the instantaeous Reynolds number at the moment where the amplitude of the unstable mode reaches a maximal value which is lower than the amplitude of the initial shear mode. That is, starting with (<ref>) we look for a maximum of the amplitude of the growing perturbed mode. We have carried out the analysis for several values of n (mode number), a_0 (Mach number) and L (box size). Our results for two dimensional flow can be found in figure <ref>. We find Re^c = 15.38 ± 0.81.independent of n for L=10^6 and M=0.01 andRe^c = 15.33 ± 1.38.for L=10^4 and M=1. As described in the main text, once the Mach number is much larger than 1, it is difficult to generate a turbulent instability using the initial condition (<ref>).Results for three dimensional flow can be found in figure <ref>.Here, we find that Re^c = 14.7 ± 1.3.for M=0.01 and L=10^4 andRe^c = 12 ± 1.3for M=100 and L=1. While it is relatively simple to generate three dimensional flow with large Mach number, it is difficult to numerically simulate a flow with initial conditions (<ref>) and large n. The main complication in simulating large n is related to the direct cascade where the initial disturbance is pushed to large wave numbers. The fluids tendency to populate modes with large wavenumber require a large number of grid points to simulate. § DETAILED ANALYSIS OF THE POWER SPECTRUMAs discussed in section <ref>, given a sufficiently high Reynolds number the flow eventually reaches a turbulent regime in which the energy power spectrum E_C has power law behavior.In two dimensional decaying turbulence we have observed a k^-4 power law behavior for a variety of initial Mach numbers and initial Reynolds numbers. In figure <ref> we have plotted the energy power spectrum for fixed initial Reynolds number and varying Mach number. In figure <ref> we have plotted the energy power spectrum for fixed Mach number and varying Reynolds number. As the initial Reynolds number decreases, the time t at which the power law behavior may be observed increases, and the size of the inertial range decreases.A spectral analysis of E_C at intermediate times, in three spatial dimensions, can be found in figures <ref> and <ref>. In figure <ref> we have plotted E_C for a variety of initial Reynolds and Mach numbers and an initial disturbance (<ref>) with n=4. Figure <ref> describes fluid flow for the same initial Reynolds and Mach numbers but an initial disturbance with n=1. Somewhat surprisingly, the Kolmogorov power law E_C ∼ k^-5/3, seems to be robust and holds also for large Mach number when the incompressible approximation is no longer valid <cit.>.§ ANALYSIS OF THE HORIZON AREA POWER SPECTRUMIn (<ref>) we have argued that the ratio of the horizon area power law spectrum to the energy spectrum grows like k^2 only for flows with low Mach number. Recall that the traceless part of the expansion, θ^i_jis given by equation (<ref>). Once the Mach number is small enough the flow becomes incompressible and we may approximate (<ref>) by (<ref>)which yields 𝒜/E_C ∼ k^2. A numerical analysis of 𝒜/E_C for various initial Mach and Reynolds number is displayed in figure <ref>. As expected, the dependence of 𝒜/E_C on k seems to deviate from k^2 once the Mach number becomes large.To quantify the deviation of 𝒜/E_c from a k^2 behavior we go back to the compressible contributions to the traceless expansion (<ref>). Let us denote the Fourier transform of a quantity X by ℱ(X). Then, in order for the incompressible terms in (<ref>) to dominate over the compressible ones, we need ∂/∂ k∫ dk^p|ℱ[∂_j(f_ℓ/a)]|^2 to be much larger than ∂/∂ k∫ dk^p|ℱ[∂_j(∂_ℓa/a)]|^2. In order quantify this relation we define the following matrix,M_jℓ = ∂/∂ k∫ dk^p|ℱ[∂_j(∂_ℓa/a)]|^2/∂/∂ k∫ dk^p|ℱ[∂_j(f_ℓ/a)]|^2,which should become negligible whenever 𝒜/E_C ∝ k^2 is valid. In figure <ref> we have plotted M_ij for a two dimensional flow with Mach numbers ranging from 5× 10^-3 to 2. While M_j=ℓ remains small, M_j≠ l becomes non negligible at large Mach number. A similar analysis for three dimensional systems can be found in figures <ref> and <ref> which display 𝒜/E_C for flows with initial conditions (<ref>) and n=4 or n=1 respectively. For both types of flows a k^3 scaling law is observed, even for the lowest Mach number which we could numerically access, M=0.1.Extrapolating the data regarding the ratios of ∂/∂ k∫ dk^p|ℱ[∂_j(f_l/a)]|^2 to ∂/∂ k∫ dk^p|ℱ[∂_j(∂_la/a)]|^2 from figure <ref> we estimate that in order for the flow to become incompressible we need M ∼ 10^-4 with Re=750 and n=1 or Re=162 and n=4. Three dimensional flows with such a low Mach number are expensive and will not be covered in this work.JHEP | http://arxiv.org/abs/1707.08973v1 | {
"authors": [
"Moshe Rozali",
"Evyatar Sabag",
"Amos Yarom"
],
"categories": [
"hep-th",
"gr-qc",
"nlin.CD",
"physics.flu-dyn"
],
"primary_category": "hep-th",
"published": "20170727180011",
"title": "Holographic Turbulence in a Large Number of Dimensions"
} |
positioning LemmaLemma[section] Theorem[Lemma]Theorem Proposition[Lemma]Proposition Corollary[Lemma]Corollary Remark[Lemma]Remark Definition[Lemma]Definition Hypothesis[Lemma]Hypothesis Assumption[Lemma]Assumption Observation[Lemma]Observation ProofProof. .4.4Acknowledgment Acknowledgments.addtoresetfiguresection addtoresetequationsection Re Im #1#1 #1 #1 #1 ∫ #1#2#30=#1#2#3∫ #2#3-.50= - Horizontal patterns from finite speed directional quenchingRafael MonteiroUniversity of Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA December 30, 2023 =======================================================================================================================In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in our case the interface moves with speed c in such a way that the bistable region grows. According to results from<cit.>, several patterns exist when c ≳0, and here we investigate their persistence for finite c>0, clarifying the pattern selection mechanism related to the speed c of the quenching front. Keywords. Phase separation, directional quenching, Allen-Cahn, Spreading speeds.§ INTRODUCTION In the theory of reaction diffusion, the interplay between stable and unstable mechanisms can give rise to spatial patterns, i.e., stationary non-homogeneous structures. In the presence of controllable external parametersthe existence and persistence of these patterns are worth to investigate, both for their mathematical interest and industrial applications; see <cit.> and the survey <cit.>. In this paper we are interested in a directional quenching scenario, where a planar interface (also called the quenching front) moves with constant speed c, across which a phase separation process takes place: ahead of the interface the system is monostable, while in its wake it is bistable. We study this phenomenon inthe scalar model∂_tu= Δ_(ξ,y) u + μ(ξ - ct) u- u^3, where Δ_(ξ,y):= ∂_ξ^2 + ∂_y^2 andμ(ξ - ct ≷ 0) = ∓ 1. Equation (<ref>) is a particular case of the Allen-Cahn model, which describe the behavior of a heterogeneous, binary mixture: the unknown u(ξ,y; t) denotes the relative concentration of one of the two metallic components of the alloy at time t∈^+ :=[0,∞) and point (ξ,y)∈^2.The stationary problem in a moving frame (x,t) = (ξ - ct, t) is written - c∂_x u = Δ_(x,y) u + μ(x) u- u^3 . The most physically relevant scenario to be considered consists of the case c≥ 0. According to results in <cit.>, whenever the speed c of the quenching front is small the equation (<ref>) admits a rich family of patterns , as we now briefly describe.Pure phase selection: 1->0 fronts.This is the simplest, one dimensional case, when (<ref>) reads - c∂_x u(x) = ∂_x^2 u(x) + μ(x)u(x)- u(x)^3,x∈. The quenching trigger generates a pattern θ^(c)(·) solving (<ref>) and satisfying spatial asymptotic conditions lim_x→ -∞θ^(c)(x)= 1andlim_x→ +∞θ^(c)(x)= 0 in the wake and ahead of the quenching front, respectively; see Figure <ref>.Horizontal patterns: H_kappa, pi <k <=infty.In this scenario the patternssought are truly two-dimensional; furthermore, we can take advantage of the odd nonlinearity to reduce the study of (<ref>) to aproblem in×[0,κ]. Whenever κ <∞ the solution Ξ_κ^(c)(·,·) has boundary conditions Ξ_κ^(c)(x,y)|_y=0,κ=0 andspatial asymptotic conditionslim_x→ +∞Ξ_κ^(c)(x,y)= 0, lim_x→ -∞|Ξ_κ^(c)(x,y)-u̅(y;κ)|= 0,whereu̅(y;κ) is a parametrized family of periodic solutions to ∂_y^2u̅+u̅-u̅^3=0, u̅(y;κ)=-u̅(y+κ;κ)=-u̅(-y;κ)≢0, y∈.with half-periodsπ<κ<∞, and normalization ∂_yu̅(0;κ)>0, u̅(0;κ) = 0. In the limiting case κ = ∞ a single interface is created, describing what we call _∞-pattern (see Fig. <ref>); asymptotically we have lim_y→ +∞Ξ_∞^(c)(x,y)= θ^(c)(x), lim_x→ +∞Ξ_∞^(c)(x,y)= 0, lim_x→ -∞Ξ_∞^(c)(x,y)=u(y;∞), where u(·;∞):=tanh(·/√(2)) and θ^(c)(·) is a -front.We summarizebelow the properties of these solutions in the regime c ≳ 0 as given in <cit.>. Their approach is based on a continuation argument from the case in which quenching front has zero speed (c=0), somehow explaining the nature of the smallness on c in their results.[<cit.>;problem] For any c ≥ 0 sufficiently small there exists a function θ^(c)(·) ∈𝒞^(1,α)(; [0,1]), ∀α∈ [0,1), that solves theproblem, i.e., θ^(c) solves the boundary value problem {[ ∂_x^2θ^(c)(x) + c∂_xθ^(c)(x) + μ(x) θ^(c)(x) - (θ^(c)(x))^3 = 0; 0< θ^(c)(x) <1, θ^(c)(-∞)=1, θ^(c)(+∞)=0, ]. where the boundary conditions are attained in the limit sense. Furthermore, the mapping x ↦θ^(c)(x) is non-increasing. The solutions found for c=0 can be continued smoothly to c>0 for sufficiently small c. More precisely, there exist families of solutionsθ^(c)(x) to (<ref>) for 0<c<δ_1, satisfying the same limiting conditions as the solutions at c=0 for x→±∞. Moreover, the solutions depend smoothly on c, uniformly in x.[<cit.>; _κ problem, π < κ≤∞] For any c ≥ 0 sufficiently small equation (<ref>) admits a family of solutions (in the sense of distributions) Ξ_κ^(c)(·,·)∈𝒞^(1,α)(^2; ), ∀α∈ [0,1), κ∈ (π,∞]. In the case π < κ <∞ for the solution Ξ_κ^(c)(·, ·) has the symmetriesΞ_κ^(c)(x,y)=-Ξ_κ^(c)(x,-y)=-Ξ_κ^(c)(x,y+κ)=-Ξ_κ^(c)(x,y+κ) and satisfiesthe asymptotic spatial conditions (<ref>). Moreover, the convergence is exponential, uniformly in y. On the other hand, whenever κ = ∞ the solution Ξ_∞^(c)(·, ·) has the symmetries Ξ_∞^(c)(x,y)=-Ξ_∞^(c)(x,-y) and satisfies the asymptotic spatial conditions (<ref>), where convergence is exponential anduniform. In this paper we give a deeper understanding of the range of validity of these continuations in c. Besides the patterns described above, oblique and vertical structures with respect to the quenching front were also studied in <cit.>, where they were shown to not exist as solutions to (<ref>) when c> 0. Therefore, the only patterns of relevance for us are the ones described above. Throughout this paper we add sub and superscripts to the patterns found in <cit.> under the inconvenience of disagreeing with that paper's notation; this is done because the classification of patterns has become more involved and richer. In this way our notation highlights the dependence of the solutions on the quenching speed c and on theκ-periodicity in the y-direction asx → -∞ (in the multidimensional case; see Fig. <ref>). §.§ Main results Initially we study the problem in one dimensional setting; although less physically relevant, it stands as one of the cornerstone of the construction of the 2D patterns – ℋ_κ and ℋ_∞ (see for instance, <cit.>). [-patterns] For any fixed c ∈ [0,2) there exists a unique solution θ^(c)(·) ∈𝒞^(1,α)(; (0,1)), ∀α∈ [0,1) satisfying{[ ∂_x^2θ^(c)(x) + c ∂_xθ^(c)(x) + μ(x)θ^(c)(x) - (θ^(c))^3(x) = 0; 0< θ^(c)(x) <1, lim_x→ -∞θ^(c)(x)=1, lim_x→∞θ^(c)(x)=0. ].The convergence takes place at exponential rate. Furthermore, the solution solution θ^(c)(·) is strictly positive andsatisfies (θ^(c))'(x) <0 for all x ∈. No solution to (<ref>) exists when c >2. In the higher dimensional setting, boththe magnitude of the speed c of the quenching front and the y-period κ of the end-state u̅(·; κ) play important roles in the analysis. [][htb]< g r a p h i c s >Existence diagramfor parameters c≥0 (speed of the front) and κ> π (y-periodicity of the patterns); the dashed curve represents the critical case 𝒫(c,κ) =1 (see Theorem <ref>). [ℋ_κ, ℋ_∞ patterns] Let π <κ≤∞ be a fixed number.Define the quantity 𝒫(c;κ) := {[ c^2/4 + π^2/κ^2,π<κ <∞;; c^2/4, κ = ∞. ].* (Existence)The solutions Ξ_κ^(0)(·, ·) defined in Prop. <ref> whenc=0 can be continued smoothly in c>0 within the range 𝒫(c;κ) < 1 tosolutionsΞ_κ^(c)(·,·)solving (<ref>) in the sense of distributions and satisfying the asymptotic spatial condition (<ref>) (resp. (<ref>)) when π <κ <∞ (resp., when κ = ∞). The convergence to their spatial asymptotic states takes place at exponential rate, uniformly. Furthermore, for any κ > π the mapping x↦Ξ_κ^(c)(x,y) is non-increasing for any fixed y ∈[0,κ] and 0 <Ξ_κ^(c)(x,y) <u̅(y;κ) in (x,y) ∈×(0,κ), where u̅(·,·) is given by<ref>. The solutions have the symmetries Ξ_κ^(c)(x,y)=-Ξ_κ^(c)(x,-y)=-Ξ_κ^(c)(x,y+κ)=-Ξ_κ^(c)(x,y+κ) (resp., Ξ_∞^(c)(x,y)=-Ξ_∞^(c)(x,-y)). * (Nonexistence) Whenever κ∈ (π, ∞) (resp., κ =∞) and 𝒫(c;κ) > 1 no solution to (<ref>) satisfying 0 <Ξ_κ^(c)(x,y) <u̅(y;κ) in (x,y) ∈×(0,κ) and the asymptotic spatial condition (<ref>) (resp. (<ref>)) can be found. One can see from the previous result that whenever c ≥ 0 the region 𝒫(c;κ) <1 (resp. 𝒫(c;κ) >1) correspondsto c < √(1 - π^2/κ^2) (resp., c > √(1 - π^2/κ^2)) , namely, the linear spreading speed obtained from the linearization of (<ref>) about u≡0 on the region x≤ 0. Overall, we point out that the dependence of the critical spreading speed curve on the parameter κ is a true manifestation of the multidimensionality of the ℋ_κ-patterns; the quantity 𝒫(·;·) describes the maximal speed of spreading of the bistable region and its dependence on they-period of the pattern away from the quenching interface. [Uniqueness results]It is worth to point out that the uniqueness result in Theorems <ref> and <ref> allow us to compare the solutions constructed in <cit.> using perturbation methods for c ≳ 0 with those obtained here.Critical cases; S=1 To the best of the author's knowledge, the cases c = 2 (for the -problem), and 𝒫(c;κ) =1 (for the _κ and _∞-problems) are open. See Sec. <ref>. A summary of our results is given in the Table <ref>. Single interfaces with contact angle. The_∞-patterns obtained in Theorem <ref> are odd functions with respect to the y variable, and as so they satisfy Ξ_∞^(c)(x,0) = 0. Thus, the patterns present a nodal set at the negative part of the x-axis that forms aright angle (φ = π/2) with respect to the quenching front located at a line x- ct=0 (parallel to the y axis). This observation motivates the question of how extra terms added to equation (<ref>) could deform this nodal set. More precisely, we consider the equationΔ u +c_x∂_x u + c_y∂_y u +μ(x) u- u^3 + α g(x,u)=0 , (x,y)∈^2. where μ(x) = ± 1 when x ≶ 0 and g(x,u)={[ g_l(u), x<0; g_r(u), x>0 ]. , for g_{l,r}(·) ∈𝒞^∞(;). Notice that c_y = 0, α =0, the function u(·,·) = Ξ_∞^(c_x)(·,·) solves (<ref>).With regards to (<ref>), we remark that two mechanisms are in play:the growth of the bistable region and new, “unbalancing terms”, that break the odd symmetry of the solutions. [Contact angle]We say (<ref>) possesses a solution u with contact angle φ_* if u possesses the limits lim_x→ +∞ u(x,y)=0, lim_x→ -∞ u(x,(φ)x)={[u_+, φ>φ_*;u_-, φ<φ_* ]. ,for all 0<φ<π. For instance, when α = 0 and c_y=0 the function Ξ_∞^(c_x)(·,·) solves (<ref>) satisfying the limit (<ref>) for φ^* = π/2 and u_± = ±1.It was shown in <cit.> shows thatfor any c_x≳0 fixed, the function Ξ_∞^(c=c_x)(·,·) can be continued in φ^*as a solution to (<ref>) for all |φ^* - π/2| sufficiently small. One of the most important properties usedin their proof concerns to thestrict monotonicity of the mapping y ↦Ξ_∞^(c_x)(x,y) for any x∈, namely, ∂_yΞ_∞^(c_x)(x,y) > 0,(x,y) ∈^2. where Ξ_∞^(c_x)(·,·) is given in Theorem <ref>. An inconvenient in their analysis is the fact that the patterns Ξ_∞^(c_x) for c_x>0 were obtained through perturbation methods (in <cit.>), hence some qualitative information on the patterns are not immediately available. Nevertheless, the authors managed to prove (<ref>) for c_x ≳0 sufficiently small (see <cit.>). Our construction readily gives the validity of (<ref>) for all 0<c_x<2, thus we canmake use of the analysis in <cit.> to conclude the following result:[Unbalanced patterns]For 0<c_x < 2, there exists α_0(c_x)>0 such that for all |α|<α_0(c_x) there exist a speed c_y(α) and a solutionu(x,y;α) to (<ref>) with contact angle φ(α). We have that u(x,y;π/2) = Ξ^(c)(·, ·; ∞). Moreover, c_y(α) and φ(α) are smooth with φ(0)=π/2, c_y(0)=0, and u(x,y;α) is smooth in αin a locally uniform topology, that is, considering the restriction u|_B_R(0) to an arbitrary large ball.§.§ Outline In Section <ref> we focus on theproblem, where we prove Theorem <ref>. Section <ref> is devoted to Theorem <ref> and _κ patterns (π <κ <∞), while the study of the _∞-pattern is left toSection <ref>.A brief discussion and further extensions brings the paper to an end in Section <ref>.Notation. In this paper we write 𝒞^k(X;Y), 𝒞_0^k(X;Y) and 𝒞^(k,α)(X;Y) denote respectively, the space of k-times continuously differentiable functions, the space of k times continuously differentiable functions with compact support in X, the space of (k, α)-Hölder continuously differentiable functions from X to Y. We denote the Sobolev spaces over an open set Ωby H^k(Ω). The inner product of elements in a Hilbert space ℋ is written as ⟨, ⟩_ℋ. Norms on a Banach space ℬ are denoted as ||·||_ℬ. For a given operator ℒ: 𝒟(ℒ) ⊂ X → Y we write Ker( ℒ) :={ u ∈𝒟(ℒ)| ℒu=0 } and Rg(ℒ) :={ f ∈ Y |∃ u ∈𝒟(ℒ), Lu =f }. A distribution T ∈ 𝒟'(Ω) satisfiesT≥ 0 in the sense of distributions if T(ϕ) ≥ 0 for any ϕ(·) ∈𝒞_0^∞(Ω; [0, ∞)). We define a 𝒞^∞(;[0,1]) partition of unity {χ^±(·)} of , of the formχ^-(x) + χ^+(x)=1,χ^-(x) =1 x≤ -2,χ^-(x) =0x≥ -1.Last, we denote the Implicit Function Theorem by IFT.Acknowledgments. The author is grateful to Prof. Arnd Scheel for many interesting discussions and insights throughout the writing of this paper. Many thanks also goto Prof. Yasumasa Nishiura and Prof. Natsuhiko Yoshinaga for sharing their perspective on this work.The author acknowledges partial support throughNSF grants DMS-1612441 and DMS-1311740. § ONE DIMENSIONAL DIRECTIONAL QUENCHING: 1-0 PROBLEM, C>0 The construction of the patternsfollows ideas from<cit.> and <cit.>: initially we solve a family of similar problems in truncated, bounded intervals; later, as we enlarge these intervals and exhaust , we show that these functionsconverge toa solution of problem (<ref>). We begin by setting up a truncated (1 0)^c problem:{[ ∂_x^2u(x) + c∂_xu(x) + μ(x) u(x)- u^3(x) = 0,; u(-M)= 1,u(L)=0,].for 0<M,L, with continuity of u and u_x at x=0. It is shown the existence of a unique solution θ^(c)_(-M,L); later on the section welet M→∞ and, subsequently, L→∞.Roughly speaking, thefront θ^(c)(·)will be given by θ^(c)(·) = sup_L>0{inf_M>0θ^(c)_(-M,L)(·) }.The qualitative properties of the function θ^(c)(·) are proved in this section, where we also show that θ^(c)(x) converges to 1 and 0 as x→∞ and x→-∞, respectively. We finalize with a proof of Theorem <ref>.A substantial part of the techniques and proofs are similar to those in <cit.>; whenever possible we skip details and refer to that article for full proofs. §.§ The 1->0c truncated problem[Existence and uniqueness] The truncated problem (<ref>) has a unique solution θ_(-M,L)^(c)(·) ∈𝒞^(1,α)([-M,L];[0,1]),∀α∈ [0,1). Furthermore, θ_(-M,L)^(c)(x) >0 whenever x ∈ (-M,L).To prove existence of a solution we define an iterative scheme,{[ -U_n+1^”(x) - cU_n+1'(x) +5U_n+1(x)=(5 +μ(x) ) U_n(x) -U_n^3(x), (-M,L); U_n+1(-M)=1,U_n+1(L)=0, ].where we write (·)' = ∂_x(·). Following the reasoning in <cit.>, it is shown that {U_n}_n∈ℕ is pointwisely decreasing and so that θ_(-M,L)^(c)(x) := inf_n ∈ℕU_n(x) is a well defined element of 𝒞^(1,α)([-M,L];[0,1]),∀α∈ [0,1). Furthermore, θ_(-M,L)^(c)(x) >0 for x ∈ (-M,L).The uniqueness proof is a bit different due to thetransport term c ∂_x and we give it here for completeness: assume the existence oftwo solutions, θ(·), θ(·) so that θ (·)≢θ(·). Define the setℐ = {x ∈ [-M,L] | θ(x) ≠θ(x) }; this set is open due to continuity of θ, θ. As an open subset of the real line, we can assume without loss of generality thatℐ = (a,b) where θ(x)> θ(x), x ∈ (a,b), andθ(x)= θ(x), x∈{a,b}. Now, since both θ and θ are solutions, we can integrate against test functions[Recall that distributions of finite order (say, order k) can be extended to the space of 𝒞_0^k functions (cf. <cit.>; see also <cit.>).] e^c xθ(x) and e^c xθ(x) on the interval (a,b): ∫_a^b e^cx(-θ”θ +θθ”)(x) dx - c∫_a^b e^cx(θ' θ - θθ')(x)dx = ∫_a^be^cx(θ^2 -θ^2) (x)θ(x) θ(x)dx, where (·)' denotes ∂_x. Integration by parts gives e^cx(-θ'θ +θθ')(x)|_a^b = ∫_a^be^cx(θ^2 -θ^2)(x) θ(x) θ(x)dx.The term on the left hand side is non-positive, since θ > θ in (a,b), θ(x) = θ(x) for x ∈{a,b}. On the right hand side,the term θθ is strictly positive, thanks to the strict positivity of solutions in (-M,L).Using that θ > θ in (a,b) we conclude that the integral on the right hand side is positive. This contradiction proves the result. In order to compare the families of solutions as M,L vary, we construct trivial extensions of functions u defined on an interval (-M,L) given by theoperator ℰ,ℰ[u](x) = {[ u(x), x ∈ (-M, L);1,x ≤ -M;0,x ≥ L. ]. By construction, ℰ[θ_(-M,L)^(c)(x)] is a continuous function, and this extension is one of the main tools to make (<ref>) meaningful and rigorous. The proof of the next resultfollows the results in <cit.>.[Properties of solutions to the truncated problem] The following properties of ℰ[θ_(-M,L)^(c)](·) hold. * (Monotonicity of ℰ) We have 0 ≤ℰ[θ_(-M,L)^(c)](·) ≤ 1. Furthermore, for w defined on a subset A, (-M,L) ⊂ A ⊂ (-∞,L) with0 ≤ w(·) ≤ 1 and 0 ≤ w(·) ≤θ_(-M,L)^(c)(·) in (-M,L), we have 0 ≤ℰ[w](·) ≤ℰ[θ_(-M,L)^(c)](·)ℝ. * (Monotonicity in M) Let0 ≤ M <M and L ≥ 0 be fixed. Then ℰ[θ_(-M,L)^(c)](x) ≤ℰ[θ_(-M,L)^(c)](x). * (Monotonicity in L) Let0 ≤ L <L and M ≥ 0 be fixed. Then ℰ[θ_(-M,L)^(c)](x) ≤ℰ[θ_(-M,L)^(c)](x).* (Monotonicity in x) For every fixed M and L the mapping x↦ℰ[θ_(-M,L)^(c)](x) is non-increasing. * (Continuous dependence of θ_(-M,L)^(c)(·) on L,M) Let 0<M< ∞, 0 <L< ∞. The mappings L↦ℰ[ θ_(-M,L)^(c) ](·) and M↦ℰ[ θ_(-M,L)^(c) ](·) are continuous in the sup norm.§.§ Passing to the limit We are now ready to pass to the limit M=∞ as a first step towards the proof ofProposition <ref>. Defineθ_(-∞,L)^(c)(x) := inf_M >0ℰ[θ_(-M,L)^(c)](x)= lim_M →∞ℰ[θ_(-M,L)^(c)](x),where the last equality is a consequence of Lemma <ref>(i). The following proposition highlights the role of the front speed c: roughly speaking it says that stretching procedure M →∞ we designed “looses mass” whenever c>2, i.e., θ_(-∞,L)^(c)(x) ≡ 0 when c>2.Although zero is a (trivial) solution to the -truncated problem, one might wonder about the usefulness of the minimax construction we developed in (<ref>), for it seems to be not goodenough to obtain nontrivial solutions to (<ref>) in (-∞,L). It turns out that the limitation is not on the method, but on the nature of the problem: no solution to (<ref>) existswhen c>2, as we will show afterwards in Lem. <ref>.[Dichotomy c ≷ 2] For any fixed c>0 we verify θ_(-∞, L)^(c)(·) ≠ 0,c<2;θ_(-∞, L)^(c)(·) ≡ 0, c>2. Furthermore, whenever 0 ≤ c<2 and L>0, the family θ_(-∞,L)^(c)(·) has the following properties: *The mapping L↦θ_(-∞,L)^(c)(·) is continuous in the sup norm on 0≤ L≤∞. * (Monotonicity)The functions x↦θ_(-∞,L)^(c)(x) are defined for every x ∈ℝ. The mappingL ↦θ_(-∞,L)^(c)(x) is non-decreasing for any fixed x. Furthermore, the mappingx ↦θ_(-∞,L)^(c)(x) is non-increasingfor any fixed L.* The function θ_(-∞,L)^(c)(x) solves (<ref>) on(-∞, L). Furthermore, lim_x → -∞θ_(-∞,L)^(c)(x) = 1.We deal first with the case c<2: from ODE theory (cf. <cit.>) there exists a solution w(·)∈𝒞^∞(;[-1,1]) to ∂_x^2w + c∂_xw + w - w^3=0satisfying lim_x→-∞w(x) =1 and so that w(x) is oscillatory as x → +∞ whenever 0 ≤ c<2; see Fig. <ref>. Translation invariance of solutions to this ODE allow us to assume without loss of generality that 0 = w(0) < w(x) <1 for x <0. Applying classical comparison principles to the problem (<ref>) on the interval [-M,0] we conclude that w(x) ≤θ_(-M,0)^(c)(x) hencew(x) ≤ℰ[θ_(-M,0)^(c)](x)≤ℰ[θ_(-M,L)^(c)](x) for M>0, thanks to Lem. <ref> and to themonotonicity Lemma <ref>. Taking the infimum in M>0 we conclude that θ_(-∞,L)^(c)(·)≠ 0, which proves the first part of the result. As a byproduct we obtain (ii) using a squeezing property, for 1 = lim_x → -∞w(x) ≤lim inf_x→ -∞θ^(c)_(-∞,L)(x)≤ 1.Item (i) is a direct consequence of Lem. <ref>. To show that lim_x→∞θ^(c)(x)=0 we use the function w̅(x) := (x + x_0)/√(2)appropriately shifted so that w̅(0) =1; notice that w̅ satisfies ∂_x^2w̅ - w̅ - (w̅)^3 =0 and that is it monotonic, i.e., ∂_xw(·) ≤ 0. Hence, w(·) is a supersolution on any interval [0,L] and classical comparison principles imply thatθ^(c)_(0,L)(x) ≤w̅(x),x ∈[0,L]. Thus,θ_(-∞,L)^(c)(x) ≤w̅(x) using Lem. <ref>. The result is obtained using that θ^(c)(x) ≥ 0 and w̅(x) → 0 exponentially fast as x →∞. In order to prove the strict monotonicity of the solution, we use Prop. <ref>: the mapping x↦θ^(c)(x) is monotonic in x as the sup of monotonic functions, i.e., ∂_x θ^(c)≤ 0. One obtains ∂_x θ^(c) < 0 by applying Hopf lemma and the maximum principle (notice that the discontinuity of the control parameter μ(·) plays no role here since, by classical regularity theory, we know that θ^(c)(·) is in fact smooth away from the quenching front).We now study the case c>2, showing that θ_(-∞,L)^(c)(·) ≡ 0. We argue bycontradiction: assume the existence of a solution θ_(-∞,L)^(c)(·) satisfying (<ref>) in (-∞,L) and so that θ_(-∞,L)^(c)(x) >0 whenx ∈ (-∞, L). There exists a d ∈ (2, c) and a solution v(·) to ∂_x^2v + d∂_xv + v - v^3=0satisfying v(-∞) =1, v(∞) =0, v(·) >0, ∂_x v(·) <0. Define m(x) = - x/2+ √(x^2/4 +2). Results from the asymptotic theory of ODEs(cf. <cit.>) show that v(x) = 1 + 𝒪(e^m(d)x), θ_(-∞,L)^(c)(x) = 1 + 𝒪(e^m(c)x),x→ -∞.As m(c) <m(d), there exists a R> sufficiently large such thatv(x) ≥θ_(-∞,L)^(c)(x) for all x ≤ -R. Positivity of v(·) impliesthatv(x)≥θ_(-∞,L)^(c)(x) whenever x ≥ L. As θ_(-∞,L)^(c)(·) is monotone, there exists an ϵ >0 such that θ_(-∞,L)^(c)(x) ≤ 1 - ϵ, x ∈ [-R,L].In conclusion, we can make use of monotonicity of v(·)to obtain a τ∈ such thatw(x + τ) ≥θ_(-∞,L)^(c)(x) on x ∈ [-R,L], henceguaranteeing that v(x + τ) ≥θ_(-∞,L)^(c)(x) for all x∈. For such a v(·) we define z(x) := v(x + τ) - θ_(-∞,L)^(c)(x) ≥ 0. Using the properties of v(·) and the assumptionθ_(-∞,L)^(c)(·)≢0 we can find a shift τ in such a way that z(·) vanishes in at least one point. Properties of both θ_(-∞,L)^(c)(·) and v(·) imply that ∂_x^2 z(x) + c ∂_x z(x) + μ(x)z(x) + f[θ_(-∞,L)^(c)(x),w(x)] z(x) = (c-d)∂_x w(x) + [μ(x) - 1]w(x)≤ 0,in x∈ (-∞,L), where f[a,b]:= a^3 - b^3/a-b whenever a≠ b and 3a^2 otherwise. One conclude that z(·) is a supersolution. As z(·) ≥ 0 has an interior minimum point, the maximum principle allied to the Hopf Lemma implies that z ≡ 0. However the latter is equivalent to v(·)≡θ_(-∞,L)^(c)(·), which is a contradiction, for v(·) and θ_(-∞,L)^(c)(·) satisfy different equations. It finishes the proof. [Existence/non-existence;problem] There exists a solution θ^(c)(·) ∈𝒞^(1,α)() [of Theorem <ref>] We begin by proving existence when c<2, following the ideas in <cit.> to which we refer to for further details: define θ^(c)(x) = sup_L>0θ_(-∞,L)^(c)(x) = lim_L→∞θ_(-∞,L)^(c)(x). The asymptotic behavior at x → -∞ is a consequence of Prop. <ref>-<ref>, sincethe mapping L ↦θ_(-∞,L)^(c)(·) is monotonic.Next, we turn to the proof ofnonexistence when c>2: choosing d so that 2<d<c we can find a strictly positive function w(·), ∂_xw(·) < 0, satisfying∂_x^2w(x) + d∂_xw(x) + w(x) - (w(x))^3 = 0, lim_x→ -∞w(x) = 1, lim_x→ -∞w(x) = 0. A direct computation shows that ∂_x^2w(x) + c∂_xw(x) + μ(x)w(x) - (w(x))^3 = (μ(x) -1)w +(c-d)∂_x w≤ 0.A similar analysis to that in (<ref>) shows that w(x) ≥θ^(c)(x) for all x → -∞. In order to understand and compare θ^(c)(x) and w(x) as x →∞ we use an analysis similar to that of <cit.>: we have thatw(·), θ^(c)(·) >0, and bothsatisfy∂_x^2w + d∂_xw - w≤ 0 ≤∂_x^2θ^(c)+c∂_xθ^(c) - θ^(c). We conclude from <cit.> the existence of positive constants M and K such that w(x)≥ K exp(-d/2 - √(d^2/4 -1)),θ^(c)(x)≤ M exp(-c/2 - √(c^2/4 +1)).Reasoning as in the proof of Prop. <ref> one obtain an R>0 so thatw(x) - θ^(c)(x) >0,x∈ (-∞,-R]∪ [R, ∞).Now, as both w(·) and θ^(c)(·) are bounded and non increasing, satisfying respectively the asymptotic properties (<ref>) and (<ref>), we conclude that we can shift w(·) so that w(x - τ) ≥θ^(c)(x) for all x ∈. Now we take the infimum of τ such that w(x - τ) ≥θ^(c)(x) holds with an equality in at least one point (clearly, the asymptotics in w(·) and θ^(c)(·) shows that τ <+∞). Now, defining z(·) = w(· - τ) - θ^(c)(·) we get that z solves an inequality as that of (<ref>), thanks to both the non increasing property of w(·) and its positivity. We conclude from the maximum principle and the Hopf's lemma that z ≡ 0, which is an absurd due to the asymptotic behavior of w(·) and θ^(c)(·), and this contradiction gives the result. From the properties of the subsolution used in the previous proof we readily derive the next result:[Exponential convergence;problem] For any θ^(c)(·) solving (<ref>) when c<2 there exists a C, δ >0 independent of x such that lim_x→∞|θ^(c)(x)| ≤ Ce^-δ |x|, lim_x→∞|θ^(c)(x) - 1 | ≤ Ce^-δ |x|. To finalize this section weshow that the solutions obtained in <cit.> for small c≥0 through perturbation methods agree with those constructed here. In passing we show that their continuity in the parameter c.[Uniqueness of thecontinuation in c;fronts] There exists a unique continuation θ^(c)(·) solving the problem (<ref>) whenever c ∈ [0, 2). Moreover, the mappings c↦θ^(c)(·): [0,2) → L^∞(;) is continuous.The proof consists in showing that for any d∈[0,2) the linearization of the equation (<ref>) at θ^(d)(·) is invertible, i.e.,ℒ_θ^(d)[v] = ∂_x^2 v + d∂_x v + μ(x) v- 3(θ^(d))^2 v , 𝒟(ℒ_θ^(d)) = H_2(). is a boundedly invertible operator from H^2 to L^2. Indeed, assume the latter to be true. Then, plugging θ^(d) + u in (<ref>) we rewrite it as ℒ_θ^(d)[u] = 𝒩[θ^(d),u] + (d-c)∂_x θ^(d). As 𝒩[θ^(d),u] =𝒪(|u|^2) the termon the right hand side is in L^2() and we can apply the IFT to obtain existence andthe uniquenessof solutions in a neighborhood of (θ^(d)(·), d). Furthermore, the mapping d↦θ^(d)(·) is continuous in L^∞(), thanks to the Sobolev embedding H^2() ↪ L^∞() (cf. <cit.>). Keeping the previous discussion in mind, we devote the rest of the proof to showing the invertibility of operator ℒ_θ^(c). Initially, we write the conjugate operatorℒ_θ^(c)[·] := e^d x/2ℒ_θ^(c)[e^-d x/2 ·]: ℒ_θ^(c)[] = ∂_x^2 + (μ(x)-d^2/4)- 3(θ^(d))^2, ṽ∈𝒟(ℒ_θ^(d)) = e^d x/2H^2() =: H_d^2(). Defining ||u||_H_d^2 = ||e^d x/2u||_H^2, we see that the isometry ℐ: H_d^2() → H^2(), ℐ[u]= e^d x/2uimplies that ℒ_θ^(d) is invertible on H^2() if and only ifℒ_θ^(d) is invertible on H_c^2(). For a moment, consider the operator ℒ_1[·] := ℒ_θ^(d)[·] with domain𝒟(ℒ_1)= H^2(); the analysis in <cit.> shows that thisis a self-adjoint, Fredholm operator of index 0, with essential spectrum contained in {z∈| Re(z) < 0}. In order to show invertibility weshow that this operator has a trivial kernel, which is proved as follows: the properties of the operator ℒ_1 imply that the σ(ℒ_1)∩{x ∈| x≥ 0} is either empty or consists of point spectrum only. It is straightforward to show that this set is bounded, therefore assume that there exists a λ_0 ≥ 0 maximal eigenvalue, with corresponding eigenfunction u_0. In the referred paper it was also proven that u_0 ∈ Ker(ℒ_1) is spatially localized, namely,|∇ u_0(x,y)|+|u_0(x,y )| ≤C e^- δ|x| ,(x, y)∈ℝ× [0,κ]a.e. u_0 ∈Ker(ℒ_Ξ). In fact, we know that we can take δ =d/2, thanks to the results in <cit.>. From the self-adjoint properties of ℒ_1 we derive u_0 is a ground state associated to its maximal eigenvalue λ_0∈, therefore it satisfies u_0(·) ≥ 0almost everywhere (cf. <cit.>).We can write the eigenvalue equation ℒ_1[u_0] = λ_0 u_0 asℒ_1[u_0] = ∂_x^2 u_0(x) + (μ(x)-d^2/4)u_0(x)- 3e^-d x(θ^(d)(x))^2 u_0(x) = λ_0 u_0(x) Settingθ^(d)(·) =e^dx/2θ^(d)(·) and using the properties of the function θ^(d)(·), we have ∂_x^2 θ^(d)(x) + (μ(x) - d^2/4) θ^(d)(x) - e^-d x (θ^(d))^3(x) = 0; asymptotic theory of ODEs (cf. <cit.>) implies that lim_|x|→∞θ^(d)(x) =0. Now, multiply (<ref>) by θ^(d)(·) and (<ref>) by u_0(·) subtract both equations and integrate into find ∫_ (θ^(d)(x) ∂_x^2u_0(x) - u_0(x)∂_x^2θ^(d)(x) dx -2 ∫_ e^-d x(θ^(d)(x))^2θ^(d)(x) u_0(x) dx = λ_0 ∫_θ^(d) u_0dx Integration by parts shows that the first integral vanishes, thanks to the decay estimates for θ^(d)(·) and u_0(·). We are left with -2 ∫_ e^-d x(θ^(d)(x))^2θ^(d)(x) u_0(x) dx = -2 ∫_(θ^(d)(x))^2θ^(d)(x) u_0(x) dx= λ_0 ∫_ w u_0dx.We observe that the spatial localization of u_0(·) as asserted in (<ref>) and the fact that θ^(d)(x) = 𝒪(e^-d x/2) as x → -∞ imply that both integrals are finite. Sign considerations of both θ^(d)(·) andu_0(·) show that the right-hand side is non-positive while the left-handis nonnegative (since λ_0 ≥ 0), therefore the integral on the left is zero. Now, invoking the strict positivity of the pattern θ^(d)(·) (or equivalently, that of θ^(d)(·)) we conclude that u_0(·) ≡ 0 almost everywhere, which contradicts the fact that u_0(·) is an eigenfunction. Therefore, no eigenvalue can be found on {z∈ℂ| Re(z) ≥ 0}; in other words, the operator ℒ_1 is boundedly invertible. An intermediate step is necessary in order to go back to the operator ℒ_θ^(d): first, define the family of weighted Sobolev spaces H_(δ,δ)^2() =e^δ<x>H^2(), where <x> := √(1 + x^2). The action of the operator ℒ_1 on these spaces can be studiedby the operators ℒ_1^(δ)[·] = e^-δ<x>ℒ_1[e^δ<x>·], defined as H^2() → L^2() mappings; we point out that the mapping δ↦ℒ_1^(δ) is continuous in the operator norm. Standard Fourier analysis shows that the far field operators ℒ_1^(δ;±∞) = lim_x→±∞ℒ_1^(δ) are boundedly invertible operators fromH^2() to L^2() for any |δ|≤d/2, hencefor any δ in this range the operators ℒ_1^(δ) are Fredholm (cf. <cit.>). Continuity in δ implies that these operators also have index 0. Therefore, proving invertibility is equivalent to showing that the kernel is trivial. In that regard, observe that we have ascale of Banach spaces, i.e., H_(δ',δ')^2() ⊂ H_(δ,δ)^2() whenever δ' > δ. Hence, one can invoke <cit.> or <cit.> to derive the persistence of elements in the kernel, namely, the equality Ker(ℒ_1^(δ)) = Ker(ℒ_1) ={0} holds for any |δ|≤d/2; being Fredholm operators of index 0 this property is equivalent to their invertibility.We finally go back to the family of operators ℒ_θ^(d)[·]. Clearly,e^d/2xH^2()⊂ H_(d,d)^2() = 𝒟(Ker(ℒ_1^(δ))|_δ = d/2). Since the latter set is trivial, the same is also true of the kernel of ℒ_1 taken with domain e^d/2xH^2(), which corresponds to the operator ℒ_θ^(d)[·]. As the latter is a Fredholm operator with index 0, this property immediately impliesbounded invertibility, and we conclude the proof.[of Prop. <ref>] Combine the proofs of Lem. <ref>, <ref> and<ref>. § TWO-DIMENSIONAL QUENCHED PATTERNS – PERIODIC HORIZONTAL INTERFACES: H-KAPPA PATTERNS, PI<KAPPA<INFTYIn this section we prove Theorem <ref> in the case π<κ<∞. As mentioned before, it is important in our approach that the nonlinearity is odd so that we can restrict the study of equation (<ref>) to the stripe (x,y) ∈×[0,κ]; any solution U(·,·) in ×[0,κ]is extended to the whole plane ^2 by successive reflections U(x,-y)=-U(x,y) and U(x,κ+y)=-U(κ-y). The method of proofis similar to the one used in theproblem, although theconstruction of subsolutions is more involved; we mostly follow the arguments in <cit.> by truncating ℋ_κ-problem (with parameter π <κ < ∞) to a strip 𝒮_(-M,L) := (-M,L)×(0,κ), withDirichlet boundary conditions: {[ Δ_x,y U +c∂_x U+μ(x) U - U^3=0, (x,y)∈𝒮_(-M,L) ,; U-h_(-M,L)=0, (x,y)∈∂𝒮_(-M,L), ].where h_(-M,L)(x,y) := θ_(-M,L)^(c)(x)·(y;κ), for θ_(-M,L)^(c)(·) solution to the 1-D problem<ref> and (·, κ) given in (<ref>). We obtain a unique solution Ξ_(-M,L)^(c;κ)∈𝒞^(1,α)(𝒮_(-M,L);[0,1]) by aniteration scheme {[ -ΔΞ_n+1 - c∂_xΞ_n+1 + 5 Ξ_n+1 =(5 +μ(x)) Ξ_n - Ξ_n^3;(Ξ_n+1 - h_(-M,L))|_∂𝒮_(-M,L)=0 ].where Ξ_0(·) is chosen in the classΨ_ℋ_κ := {Ξ∈𝒞^(1,α)(𝒮_(-M,L)) | (Ξ_0 - h_(-M,L))|_∂𝒮_(-M,L)≥ 0, Ξ(x)∈[0,1]x,y }∩{ΔΞ + μ(x)Ξ- Ξ^3 ≤ 0, ΔΞ + μ(x)Ξ - Ξ^3 ≢0}. Throughout this section, we fix κ∈ (π, ∞) and suppress the dependence of Ξ and u̅ on κ for ease of notation. As in the previous section, proofs that are similar to those in <cit.> are only outlined and details are referred to that paper. [Existence and uniqueness; truncated _κ problem] Problem (<ref>) has a unique solutionΞ_(-M,L)^(c)(·,·) ∈𝒞^(1,α)([-M,L]× [0,κ]; [0,1]), ∀α∈ [0,1).The existence is obtained as in<cit.> using the iterative scheme (<ref>); uniqueness follows as in <cit.> and integration by parts, as in Lem. (<ref>). The stated regularity is derived using classical results in elliptic theory, as shown in <cit.>. We define extension operators in order to compare solutions for different values of M,L, namely,ℰ[Ξ_(-M,L)^(c)](x,y) = {[Ξ_(-M,L)^(c)(x,y), (x,y) ∈𝒮_(-M,L); (y) , (x,y) ∈ (-∞, -M)× [0,κ]; 0 ,(x,y) ∈ (L, ∞)× [0,κ], ]. where (·) = (·;κ) is given in (<ref>). We use the same symbols for the one- and two-dimensional extension operators, slightly abusing notation, distinguishing between the two through the domain of definition of the function ℰ is applied to.The proofs of the following Lem. <ref> and Prop. <ref> are obtained asin <cit.>: [Comparison principles; ℋ_κ-problem] Let Ξ_(-M,L)^(c)(·,·) be the solution from Proposition <ref>. * (2D supersolutions) If v satisfies, in the sense of distributions,Δ v+ c∂_x v +μ(x) v- v^3 ≤ 0, (x,y)∈𝒮_(-M,L), (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,0 ≤ v ≤ 1,then v ≥Ξ_(-M,L)^(c) in 𝒮_(-M,L). In particular, v ≥Ξ_(-M,L)^(c) in 𝒮_(-M,L) for any solution of Δ v+ c∂_x v + μ(x) v- v^3 =0, (x,y)∈𝒮_(-M,L), (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,0 ≤ v ≤ 1.* (2D subsolutions)If v satisfies, in the sense of distributions,Δ v+ c∂_x v + μ(x) v- v^3 ≥ 0, (x,y)∈𝒮_(-M,L), (v - h_(-M,L))|_∂𝒮_(-M,L)≥ 0,0 ≤ v ≤ 1,then v ≤Ξ_(-M,L)^(c) in 𝒮_(-M,L). In particular, v ≤Ξ_(-M,L)^(c) in 𝒮_(-M,L) for any solution of Δ v+ c∂_x v +μ(x) v- v^3 =0, (x,y)∈𝒮_(-M,L), (v - h_(-M,L))|_∂𝒮_(-M,L)≤ 0,0 ≤ v ≤ 1.*The functions u̅(·;κ) and θ_(-M,L)^(c)(x) given by (<ref>) and Theorem <ref>, respectively, are supersolutions to (<ref>) in 𝒮_(-M,L). Consequently, for any L, M >0 we have thatΞ_(-M,L)^(c)(x,y)≤min{θ_(-M,L)^(c)(x), u̅(y)}.[Properties of the extension operator, ℋ_κ-problem] The following properties ofℰ[·] hold. * (Monotonicity of ℰ) We have 0 ≤ℰ[Ξ_(-M,L)^(c)](·, ·) ≤ 1. Furthermore, if w is only defined in a subset A ⊂ℝ^2 so that𝒮_(-M,L)⊂ A ⊂Ω_(-∞,L), 0 ≤ w(·) ≤ 1 and w(·) ≤Ξ_(-M,L)^(c)(·, ·),then0 ≤ℰ[w](·) ≤ℰ[Ξ_(-M,L)^(c)(·, ·)](·)ℝ^2. * (Monotonicity in M) Let0 ≤ M <M and L ≥ 0 be fixed. Then M <Mℰ[Ξ_(-M,L)](x,y) ≤ℰ[Ξ_(-M,L)^(c)](x,y).* (Monotonicity in L)Let0 ≤ L <L and M ≥ 0 be fixed. Then L <Lℰ[Ξ_(-M,L)^(c)](x,y) ≤ℰ[Ξ_(-M,L)](x,y).* (Monotonicity in x) LetL,M,ybe fixed. Then the mappingx →ℰ[Ξ_(-M,L)^(c)](x,y) is non-increasing.§.§ Passing to the limit We are now ready to pass to the limit M=∞. Define Ξ_(-∞,L)^(c)(x,y) := inf_M >0 ℰ[Ξ_(-M,L)^(c)](x,y) =lim_M → +∞ℰ[Ξ_(-M,L)^(c)](x,y),where the last equality holds due to monotonicity of the mapping M ↦Ξ_(-M,L)^(c)(x,y), Proposition <ref>(ii). In this section we verify monotonicity properties and limits at spatial infinity of the limits Ξ_(-∞,L)(x,y) constructed in (<ref>). [Monotonicity of Ξ_(-∞,L)(x,y)]The following properties hold:* The function Ξ_(-∞,L)^(c)(·,·) solves the problem (<ref>) in𝒮_(-∞,L):= {(x,y) ∈ℝ^2 | x <L, y ∈ [0,κ] }.* The function Ξ_(-∞,L)^(c)(·,·)is non-increasing in x and non-decreasing in L.*The inequality Ξ_(-∞,L)(x,y)≤min{θ_(-∞,L)^(c)(x), u̅(y)}holds for all (x,y) ∈𝒮_(-M,L), whereθ_(-∞,L)^(c)(·) is given by (<ref>) and u̅(·) by Prop. <ref>.In particular, we havesup_L>0(Ξ_(-∞,L)(x,y) )≤min{θ^(c)(x), θ_(-∞,0)^(c)(y)}.Assertion (i) and (ii) follow as in <cit.>, being a consequence of a comparison method and monotonicityof Ξ_(-M,L)^(c)(·) and Ξ_(-∞,L)^(c)(·) in their arguments. We derive the inequalities in (iii) by passing to the limit L=∞ and using the fact that for all x∈ the mapping L ↦θ_(-∞,L)^(c)(x) is non-decreasing. One can readily conclude from Lem. <ref> that Ξ_(-∞,L)^(c)≡ 0 whenever c >2, L>0. In fact, the non-existence of nontrivial solutions happens in a wider range for theparameter c, as we show next.§.§ Existence for the H-problem: case c2/4 + pi2/kappa2 <1 In order to prove the existence of solutions we construct appropriate subsolutions with the help of the next lemmas: [Properties of the family of periodic solutions u̅(·,κ)] Let u̅(·;κ) be a solution to (<ref>) and κ > π.The following two properties hold:* The quantity M := sup_y∈ [0,κ]u̅(y;κ) satisfies M < √((1 - π^2/κ^2)); *For any 0≤α≤M we havev(y):= αsin(π y/κ) ≤u̅(y;κ), for all y ∈ [0,κ]. To prove the estimate in (i) we use the elliptic integral that gives the relation between amplitude and spatial period given in <cit.>[Lemma 4.1, equation (4.4)], <cit.>, κ := κ(M) = 2 √(2)∫_0^1 dv/√([(1 - v^2)(2 - M^2(1 + v^2))]) = 2 √(2)γ/M∫_0^1 dv/√([(1 - v^2)(1 - (γ v)^2)]),for γ^2 = M^2/2 - M^2. Notice that 0 ≤γ<1. We find a lower bound to the integral on the right hand side:κ(M) >2 √(2)γ/M∫_0^1 dv/√([(1 - v^2)(1 - γ^2)]) =2 √(2)γ/M√(1 - γ^2)∫_0^1 dv/√((1 - v^2))=π√(2)γ/M√(1 - γ^2).Squaring both sides and plugging γ we obtain κ^2 > π^2/1 - M^2⟺ M^2 < 1 - π^2/κ^2, which finishes the proof of (i). In order to prove (ii) weexploits the structure of this ODE in (<ref>), whose Hamiltonian is ℋ(u, u_y) = u_y^2 + u^2 - u^4/2,cf. <cit.>. Indeed, considering u̅(·) a periodic orbit with period κ and maximal amplitude M, we readily obtain that ℋ(u̅, u̅_y) = M^2(2 - M^2)/2. Letv(y) := αsin(πy/κ) and z(x) := u̅(y) - v(y). Whenever 0 ≤α≤ M one can see thatℋ(v, v_y)≤ℋ(u̅, u̅_y). As u̅_y(0) >v_y(0) and (0) = v(0)=0 it is clear that z(x) >0 for and x>0 sufficiently small. By translation invariance of the solutions to the ODE (<ref>),reversibility of the solutions u̅ with respect to x ↦ -x, and the fact that the mapping y ↦sin(y + π/2) is even it suffices to show that z(y) ≥ 0 for 0 ≤ y ≤κ/2. Assume that there exists a 0<x_0<κ/2 such that z(x_0) =0. As z(·)≥ 0 solves the elliptic differential equation, we can find and A>0 sufficiently large so that∂_y^2z(y) + (1 - f[u̅,v](y) - A)z(y) ≤∂_y^2z(y) + (1 - f[u̅,v](y) )z(y)=0, where f[a,b]:= a^3 - b^3/a-b whenever a≠ b and 3a^2 otherwise. We conclude using Hopf's lemma that ∂_y z(x_0) <0, which is absurd, since the inequality (<ref>) prevents it from happening. Therefore z(y) ≥ 0, hence v(y) ≤u̅(y) fory ∈[0,κ/2], and by symmetry, for y ∈[0,κ]. [Sub and supersolutions] Choose d ∈ (c,2)so that c^2/4 + π^2/κ^2 < d^2/4< 1 and letw(·)∈𝒞^∞(;[-1,1]) be a solution to∂_x^2w + d∂_xw + w - w^3=0satisfying w(-∞) =1 and so that 0 = w(0) < w(x) <1 for x <0 (see. Prop. <ref> and Fig. <ref>). Let M>0, α and u̅(·) be given as in Lem. <ref>. DefineV(x,y) := e^-(c-d)x/2w(x)· v_κ(y),v_κ(y) := αsin(πy /κ).Then, V(x,y) ≤Ξ_(-∞,L)^(c)(x,y) [-M,0]×[0,κ]. Furthermore, wheneverθ^(c)(·) is given by (<ref>) and u̅(·) by Prop. <ref>, the following inequality holdsΞ_(-∞,L)^(c)(x,y)≤min{θ^(c)(x), u̅(y)}. Inequality (<ref>) is readily derivedfrom Lem. <ref>, taking infimum in M>0, using the definition of Ξ_(-∞,L)^(c)(·,·) (see (<ref>)) andthe monotonicity of the mappingL ↦θ_(-∞,L)^(c)(·).In order to obtain (<ref>) we use Lem. <ref>, showing that V(·,·)≤Ξ_(-M,0)^(c)(·,·) in𝒮_(-M,0) = [-M,0]×[0,κ] for any fixed M>0. Thanks to Lemma <ref> it is straightforward to show that V(x,y) ≤Ξ_(-M,0)^(c)(x,y) on the boundary ∂𝒮_(-M,0). Now we show that V satisfies Δ_x,yV + c∂_x V + V - V^3 ≥ 0,in 𝒮_(-M,0) = [-M,0]×[0,κ]. Indeed, a direct calculation shows thatΔ_x,yV + c∂_x V + V - V^3= V[d^2/4 - c^2/4- π^2/κ^2] + e^-(c-d) x/2w^3(x)[v_κ(y) - e^-(c-d) xv_κ^3(y)]≥ 0, since V(·,·) is non-negative, x≤ 0 and d >c. It follows that V(x,y) ≤Ξ_(-M,0)^(c)(x,y).We obtain (<ref>) by invoking themonotonicity of the mapping L ↦Ξ_(-M,L)^(c)(·,·) and using the definition (<ref>). [Existence; ℋ_κ-problem, π <κ <∞]Equation (<ref>) hasa solution Ξ^(c)(·,·)∈𝒞^(1,α)(× [0,κ];), for any 0≤α<1, where the latter is defined as Ξ^(c)(x,y) := lim_L→∞Ξ_(-∞,L)^(c)(x,y)Most of the proof goes as in the paper <cit.>. The monotonicity properties of the functions Ξ^(c)_(-∞,L)(·,·) show that the definition (<ref>) makes sense and Lem. <ref> shows that Ξ_(-∞,L)^(c)(·,·)≢0 whenever c^2/4 +π^2/κ^2<1. As 0≤Ξ_(-∞,L)^(c)(·,·) ≤Ξ_κ^(c)(·,·) we conclude that Ξ_κ^(c)(·,·) is also nontrivial. Using Lebesgue dominated convergence we conclude that exists in the pointwise sense and that this sequence converges in L_loc^1 hence in the sense of distribution, solving the equation (<ref>) in the domain × [0,κ]. Now it remains to show that the asymptotic limits are satisfied, namely, thatlim_x → -∞Ξ_κ^(c)(x,y) = u̅(y), lim_x → +∞Ξ_κ^(c)(x,y) = 0.The limit on the right follows easily from inequality (<ref>), for lim_x→∞θ^(c)(x) =0. The proof of the limit on the left is more involved, and our analysis has some similarities to those of <cit.> and <cit.>. Indeed, monotonicity results derived in Lem. <ref> allow us to conclude thatv_L(y) := lim_x→∞Ξ_(-∞,L)^(c)(x,y)≤lim inf_x→∞Ξ_κ^(c)(x,y)≤lim sup_x→∞Ξ_κ^(c)(x,y)≤u̅(y), where the first limit is known to exists thanks to the monotonicity in x of Ξ_(-∞,L)^(c)(·, ·). According to Lem. <ref>we know that Ξ_(-∞,L)^(c)(·, ·) solves the equation (<ref>). Thus, v_L(·) satisfies ∂_y^2v_L(y) + v_L(y) - (v_L(y))^3=0in the sense of distributions in [0,κ], hence in the classical sense.As v_L(y)|_y = 0, κ =0 we conclude that either v≡0 or v is a periodic solution with period τ so that κ/τ∈ℕ. We can readily exclude the first possibility, since Lem. <ref> implies thatαsin(πy/κ)≤ v_L(y)≤(y). The same inequality also implies that τ = 2κ, i.e., u̅(·) and v_L(·) have the same period therefore and obey the same normalization, therefore v_L(·) ≡u̅(·), and the result follows from the equality of(<ref>). Unlike in the previous case, it is not directly clear that the convergence lim_x→ -∞Ξ_κ^(c)(x,y) = u̅(y) has exponential rate of convergence. Our next result implies that. [Exponential convergence; ℋ_κ-problem, π <κ <∞] There exists a C, δ >0 independent of x and y such thatlim_x→∞|Ξ_∞^(c)(x,y)| ≤ Ce^-δ |x|,lim_x→ -∞|Ξ_∞^(c)(x,y) - u̅(x)|≤ Ce^-δ |x|. Initially we show exponential rate of convergence to the far-field as |x|→∞. The result follows if we show that ∂_x(Ξ_∞^(c))∈ e^-δ<x>H^2(×[0,κ]) for some δ >0 and <x> :=√(1 +x^2). Indeed, as we know from Lem. <ref>, lim_x→ -∞Ξ_∞^(c)(x,y) = u̅(y); using the Sobolev embedding H^2(×[0,κ])↪L^∞(×[0,κ]) we have |Ξ_∞^(c)(x,y) - u̅(y)| ≤|∫_∞^x ∂_x(Ξ_∞^(c))(s,y)ds| ≲∫_∞^x e^δ sds ≲ e^δ x,x≤ 0, which givesthe result. The proof requires several tools of Fredholm theory for elliptic operators. The linearization of the equations (<ref>) at Ξ_∞^(c)(·,·) gives ℒ_Ξ[v] :=Δ v + c∂_x v + [μ(x)- 3 (Ξ_∞^(c)(x,y))^2]v, with domain of definition 𝒟(ℒ_Ξ) = H^2(ℝ× [0,κ]) ∩ H_0^1(ℝ× [0,κ]). Although this operator is nonself-adjoint, the limits as |x|→∞ of Ξ^(c) are the same for all c∈ [0,2), therefore the results of <cit.> apply, showing that the operator ℒ_Ξ is Fredholm of index 0, with essential spectrum strictly negative in H^2(ℝ× [0,κ]) ∩ H_0^1(ℝ× [0,κ]). According to <cit.>, we know that elements in the kernel of ℒ_Ξ are spatially exponentially localized, namely,|∇ u_0(x,y)|+|u_0(x,y )| ≤C e^- δ|x| ,(x, y)∈ℝ× [0,κ]a.e. u_0 ∈Ker(ℒ_Ξ)Recall the partition of unity χ^±(·) defined in (<ref>). We know that ∂_x(χ^-(x)Ξ^(c)(x,y)) ∈𝒞^∞(×[0,κ];) solves a problem of the formℒ_Ξ[v] = f, where f∈ L^2(×[0,κ) is spatially localized. Writing H^2(×[0,κ]) = Ker(ℒ_Ξ) ⊕𝒳, we can assume with no loss of generality that v∈𝒳, thanks to property (<ref>) for elements in the kernel. However, as the operator ℒ_Ξ: 𝒳→Rg(ℒ_Ξ) is boundedly invertible, we can apply the same reasoning used in <cit.> to conclude thate^δ<x>v∈ H^2(×[0,κ]) ∩ H_0^1(×[0,κ]) for all δ >0 sufficiently small. A similar analysis can be done by considering w(x,y) = ∂_y(χ^+(y)Ξ^(c)(x,y)), whence exponential rate of convergence to the far field as y→∞ is derived. It finishes the proof.In fact, one can show by following the steps in the proof ofLem. <ref> that the operator ℒ_Ξ in (<ref>) is boundedly invertible from H^2(×[0,κ]) toL^2(×[0,κ]). Once more, using the IFT, we conclude the following result.[Uniqueness of thecontinuation in c;ℋ_κ problem, π < κ < ∞]Recall the definition of 𝒫(c;κ) given in (<ref>). For any fixed κ∈ (π, ∞) the following properties hold:* there exists a unique solution Ξ_κ^(c)(·,·) to the ℋ_κ^(c) problem; * the mapping c↦Ξ_κ^(c)(·):{c≥ 0|𝒫(c;κ) <1}→ L^∞(×[0,κ];) is continuous.The analysis is analogous to that of Lem. <ref> and is outline below, where we point out the necessary modifications. Fix κ∈ (π, ∞). Initially we define the linearized operator about the solutions Ξ_κ^c, obtaining the linearized operatorℒ_Ξ^(c)[v] = ∂_x^2 v + c∂_x v + μ(x) v- 3(Ξ^(c))^2 v , 𝒟(ℒ_θ^(c)) = H_2(×[0,κ])∩ H_0^1(×[0,κ]). Writing v = e^-c x/2 u we rewrite the above operator in a “self-adjoint” form, ℒ_Ξ^(c)[v] = ∂_x^2 u + (μ(x) - c^2/4) u- 3(Ξ^(c))^2 u , 𝒟(ℒ_θ^(c)) = e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])).Notice that the mapping u(·) ↦ e^c x/2u(·) is an isometry between the spaces H_2(×[0,κ])∩ H_0^1(×[0,κ]) and e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])), therefore it suffices tostudy the invertibility of ℒ_Ξ^(c) only. At this point we define ℒ_Ξ^(c)^(δ) as the action of the operator ℒ_Ξ^(c) on the scale of Banach spaces e^δ<x>H^2(×[0,κ]). Continuity of the Fredholm index on the weight δ shows in the range |δ|≤c/2 we have that ℒ_Ξ^(c)^(δ) are Fredholm operators with index zero, with the same kernel. As ℒ_Ξ^(c)^(δ)|_δ =0 is invertible one can argue as in Lem. <ref> to obtain the bounded invertibility of ℒ_Ξ^(c) in e^c x/2(H_2(×[0,κ])∩ H_0^1(×[0,κ])). The results are then obtained from an application of the IFT.§.§ Non-existence for the H-kappa-problem: case c2/4 + pi2/kappa2 >1In this section we prove the non-existence of patterns when c^2/4 +π^2/κ^2>1. The method is standard:roughly speaking, assuming an existing solution Ξ_κ^(c)(·,·), we can obtain a supersolution V(·,·) that is above Ξ_κ^(c)(·,·) and which, under certain conditions, can touch the solution in at least one point. The function z(·,·):= Ξ_κ^(c)(·,·) - V(·,·) solves an elliptic problem and has an interior maximum, which contradicts maximum principle and Hopf's Lemma. We make these words more precise in what follows. We begin with the main ingredients in the construction of the subsolution. [Non-existence; c^2/4 + π^2/κ^2>1]Recall (see Fig. <ref>) the existence of a solution w_d(·)>0 to the problem∂_x^2w_d + d∂_xw_d + w_d - (w_d)^3, lim_x→ -∞w_d(x) = 1, lim_x→∞w_d(x) = 0, where d>max{2,c} is chosen in such a way thatd^2/4 - c^2/4 - π^2/κ^2<0. Likewise, we writew_c(·) for a function with similar spatial asymptotic properties and solving∂_x^2w_c + c∂_xw_c + w_c - (w_c)^3=0. DefineV(x,y) := w_c(x) + e^-(c-d) x/2w_d(x)v_κ(y),v_κ(y) := αsin(πy /κ). Then V(·,·) is a supersolution. Furthermore, V(·,·)≤Ξ_κ^(c)(·,·).Recall that 𝒫(c,κ) := c^2/4 + π^2/κ^2 >1. The first step on our proof is the construction of a supersolution V ≥Ξ_κ^(c). Initially, we fix w_c(·) in such a way that w_c(x) ≥ 0 on x≤ 0. Now, choosing d>max{2,c} in such a way thatd^2/4 - c^2/4 - π^2/κ^2<0 we fix a translated version of w_d in such a way thatd^2/4 - c^2/4 - π^2/κ^2 + w_d^2(x) - 3 w_c^2(x) ≤ 0,x ≤ 0.In fact, we can exploit the translation invariance of solutions to the ODE (<ref>) and the fact that x↦ w_{c,d}(x) is non-increasing forx≤ 0 to conclude that(<ref>) also holds fortranslated version of w_d(·) and w_c(·) of the form w_d(·) ↦ w_d(· - τ) and w_c(·) ↦ w_c(· - τ), whenever τ≥ 0. Set V(x,y) = e^-(c-d) x/2w^(d)(x)v_κ(y) and write V(x,y) = w_c(x) + V(x,y). We claim that V is a supersolution on x≤ 0 (where μ(x) ≡ 1). Indeed, a tedious but straightforward computation shows that Δ_x,yV + c∂_xV + V - V^3=V[d^2/4 - c^2/4 - π^2/κ^2] + e^-(c-d) x/2w_d^3(x)v_κ(y) + w_c^3 - [w_c(x) + e^-(c-d) x/2w_d(x)v_κ(y)]^3 = V[d^2/4 - c^2/4 - π^2/κ^2 +w_d^2(x) - 3 w_c^3(x)] + J_2 where J_2= J_2(w_c, w_d, v_κ) consists of non-positive terms only. Thus, Δ_x,yV + c∂_xV + μ(x)V - V^3= V[d^2/4 - c^2/4 - π^2/κ^2 +w_d^2 - 3 w_c^3] + J_2 ≤ 0,x≤ 0, thanks to (<ref>), which concludes the proof that V is a supersolution in x≤ 0.Our second step consists in proving that V ≥Ξ^(c). Unfortunately, we cannot play with the translation of both w_d and w_c separately without destroying the inequality in (<ref>), so we pursue another route. The asymptotic properties of w_c imply that V(x,y) ≥ w_c(x) ≥Ξ^(c)(x,y) as x → -∞. As w_c(·) can be chosen so that w_c(0) = 0 we need to understand the term V(x,y) for x <0, x finite. Inspecting the calculations that lead to (<ref>), one can see that the parameter α≥ 0 inv_κ(y):= αsin(π y/κ) plays no crucial role. Standard elliptic regularity estimates show that ∂_yΞ^(c)(·,·) is abounded function on ×[0,κ]. Consequently, we can take α>0sufficiently large so thatw_d(0) v_κ(y) = α w_d(0) sin(πy/κ)≥Ξ^(c)(0,y); thanks tow_d(·) >0. Furthermore, the inequality (<ref>) persistswhen we use a translated version of w_d(·), for x ↦w_d(x) is non-increasing, namely, w_d(-τ) v_κ(y) ≥ w_d(0) v_κ(y)≥Ξ^(c)(0,y), for all τ >0. Thus, for a fixed M̃>0 we have thatV(x,y) ≥Ξ^(c)(x,y), x∈ [-M̃,0]× [0,κ]. Translating w_c(·) in such a way that w_c(x) ≥Ξ^(c)(x,y) for x ≤ -M̃ and y ∈ [0,κ] we get that V(x,y)≥Ξ^(c)(x,y) for all (x,y)∈ (-∞,0]×[0,κ]. The third and last part consists of varying the parameter α >0 continuously in order to satisfy V ≥Ξ^(c) with an equality in at least one point. Now the proof goes as in Prop. (<ref>): define Z(x,y) := Ξ(x,y) - V̅(x,y); clearly Z(x,y)≤ 0 in x≤ 0, y ∈ [0,κ]. The function Z satisfies Δ_x,y Z + c∂_x Z + μ(x)Z - f[Ξ,V̅](Z) ≥ 0,x≤ 0. where f[a,b]:= a^3 - b^3/a-b whenever a≠ b and 3a^2 otherwise.We can apply classical maximum principles, since for M>0 chosen large enough in order to give M + f[Ξ,V̅] >0; thus we write the above equation asΔ_x,y Z + c∂_x Z + μ(x)Z - (f[Ξ,V̅] + M)(Z)≥ - M Z ≥ 0 As Z has a maximum point in ×[0,κ] we get that Z≡ 0. This contradiction leads to the non-existence of solutions in the case c^2/4 +π^2/κ^2>1.Finally, we put all these auxiliary results together and prove the main result of this section: [of Theorem <ref>; case π <κ < ∞] Combine the above discussion of the non-existence of solutions with the results of Lemmas <ref>, <ref>, and<ref> § TWO-DIMENSIONAL QUENCHED PATTERNS – SINGLE HORIZONTAL INTERFACES; H INFTY PROBLEM In this section, we shall prove Theorem <ref> in the case κ =∞. To be consistent with the notation introduced in Section <ref> we exploit the fact that the nonlinearity in (<ref>) is odd to solve the problem in the half space × (-∞,0]. Further symmetries of the equation are also exploited: we solve an equivalent ℋ_∞-problem, seeking for a solutionΞ_∞^(c)(·,·) to (<ref>) in ^2, satisfyinglim_x→ -∞Ξ_∞^(c)(x,y)= -tanh(y/√(2)),lim_y→±∞Ξ_∞^(c)(x,y)=∓θ^(c)(x),lim_x→∞Ξ_∞^(c)(x,y)=0,where θ^(c)(·) is the one-dimensional solution to the -problem. Notice that the odd symmetry solutions to (<ref>) with respect to(y, Ξ_∞^(c)(x,y))↦(-y,-Ξ_∞^(c)(x,-y)) readily gives the pattern with the properties stated in Theorem <ref>. Moreover, the Dirichlet boundary conditions at y=0 removes the non-uniqueness of solutions induced by y-translation invariance.Thanks to the results of Sec. <ref> related to the 1-d problem the following observation is readily available.[Restriction to the case c<2]It is clear from (<ref>) that the above problem is meaningless when c>2 for the patterns θ^(c) do not exist. We can readily say that no solution to this problem exists when c>2, immediately restricting our study to the range 0 ≤ c<2.In fact, in this section we prove that for all quenching fronts speeds in the range c∈[0,2) there exists a unique ℋ_∞-pattern (up to translations in the y direction), which corresponds to the statement of Theorem<ref>. The strategy goes as in Sections <ref> and<ref>: first, by reducing the problem to a half plane and truncating it, restricting the problem to a rectangle Ω_(-M,L) := (-M,L)×(-M, 0). Then we let M→∞ and, subsequently, we let L→∞. For the sake of simplicity in this section we will omit any sub-index ∞.The truncated ℋ_∞-problem is set up as{[ Δ_x,y u +c∂_xu + μ(x) u - u^3 =0, (x,y)∈Ω_(-M,L),; u = g_(-M,L),(x,y)∈∂Ω_(-M,L), ]. where g_(-M,L)(x,y):=θ_(-M,L)^(c)(x)·θ_(-M,0)^(c)(y), θ_(-M,L)^(c)(·) the solutions to the truncated one-dimensional problem(<ref>) on the interval (-M, L). Similarly to the results of Lem. <ref> and <cit.>, we construct unique solutions to this truncated problem using an iterative scheme. The solution Θ_(-M,L)^(c)(·,·): Ω_(-M,L)→ [0,1] is shown to be unique; Furthermore, exploiting that 0≤ u≤ 1 andAgmon-Douglis-Nirenberg regularity we readily conclude that u and derivatives are Hölder continuous across x=0, and, in fact, Θ_(-M,L)^(c)∈𝒞^(1,α)(Ω_(-M,L)),for all 0≤α <1.Following the method in Sec. <ref>, we extend these functions to the whole plane ^2: ℰ[Θ_(-M,L)^(c)](x,y) = {[ Θ_(-M,L)^(c)(x,y), (x,y) ∈Ω_(-M,L),; ℰ[θ_(-M,L)^(c)](x)·ℰ[θ_(-M,0)^(c)](y),(x,y) ∈ℝ^2 ∖Ω_(-M,L). ].We summarize the main properties of the functions ℰ[Θ_(-M,L)^(c)](·,·) in the following Proposition, whose proof is similar to that of <cit.>: [Properties of the extension operator, ℋ_∞-problem] The following properties ofℰ[·] hold. * (Monotonicity of ℰ) We have 0 ≤ℰ[Θ_(-M,L)^(c)](·, ·) ≤ 1. Furthermore, if w is only defined in a subset A ⊂ℝ^2 so thatΩ_(-M,L)⊂ A ⊂Ω_(-∞,L), 0 ≤ w(·) ≤ 1 and w(·) ≤Θ_(-M,L)^(c)(·, ·),then0 ≤ℰ[w](·) ≤ℰ[Θ_(-M,L)^(c)(·, ·)](·)ℝ^2. * (Monotonicity in M) Let0 ≤ M <M and L ≥ 0 be fixed. Then M <Mℰ[U_(-M,L)](x,y) ≤ℰ[Θ_(-M,L)^(c)](x,y).* (Monotonicity in L)Let0 ≤ L <L and M ≥ 0 be fixed. Then L <Lℰ[Θ_(-M,L)^(c)](x,y) ≤ℰ[U_(-M,L)](x,y).* (Monotonicity in x) LetL,M,ybe fixed. Then the mappingx →ℰ[Θ_(-M,L)^(c)](x,y) is non-increasing.* (Monotonicity in y) LetL,M,xbe fixed. Then the mappingy →ℰ[Θ_(-M,L)^(c)](x,y) is non-increasing. [Existence; ℋ_∞ problem] There exists a solution Ξ_∞^(c)(·, ·) satisfying(<ref>) in ^2 satisfying the limits (<ref>). Furthermore, ∂_y Ξ_∞^(c)(·, ·) ≤ 0.We obtain a solution in (x,y) ∈×(-∞,0), extending it to the whole plane ^2 as a function satisfying Ξ^(c)(x,-y) = -Ξ^(c)(x,y). The result follows the construction in Lem. <ref>, first defining Θ_(-∞,L)(·,·) :=inf_M>0Θ_(-∞,L)(·,·) and then defining Ξ_∞^(c)(·,·) = sup_L>0Θ_(-∞,L)(·,·). Further properties of Ξ_∞^(c)(·, ·) are derived as in Lem. <ref> (see also <cit.>). The inequality ∂_y Ξ_∞^(c)(·, ·) ≤ 0 is a consequence ofProp. <ref>, because the mapping y ↦Ξ_∞^(c)(x, y) is non-increasing. The next result provides a crucial ingredient in the construction of interfaces with contact angle: [Monotonicity in the y-direction]For all x∈ the mapping y ↦Ξ_∞^(c)(x,y) is strictly monotonic. It suffices to show that ∂_yΞ_∞^(c)(x,·)≠ 0 a.e., thanks to the regularity of Ξ_∞^(c)(·,·). The inequality ∂_yΞ_∞^(c)≤ 0 is obtained without difficulty by making use of Prop. <ref> and the limiting construction in Lem. <ref>. The strict inequality ∂_yΞ_∞^(c) < 0 a.e. then follows using a Harnack inequality (cf. <cit.>; see also <cit.>). [Exponential convergence; ℋ_∞ problem] The limits in (<ref>) take place at exponential rate, i.e., there exists a C, δ >0 independent of x and y such thatlim_x→∞|Ξ_∞^(c)(x,y)| ≤ Ce^-δ |x|, lim_x→∞|Ξ_∞^(c)(x,y) - tanh(y/√(2))| ≤ Ce^-δ |x| ,lim_y→±∞|Ξ_∞^(c)(x,y) ±θ^(c)(x)|≤ Ce^-δ |y|. The proof is similar to that of Lem. <ref>, with small differences. As before, we first trace the essential spectrum ofthe linearized operator (<ref>) at Ξ_∞^(c)(·,·) gives ℒ_Ξ[v] :=Δ v + c∂_x v + [μ(x)- 3 (Ξ_∞^(c)(x,y))^2]v, with domain of definition 𝒟(ℒ_Ξ) = H_odd^2(ℝ^2):= {w ∈ H^2(ℝ^2); w(x,y) = -w(x,-y)}. We remark that <cit.> still applies: indeed, we can define the asymptotic operators ℳ^-[v] := ∂_y^2v(·)+ [1- 3tanh^2(y/√(2))]v(·),ℳ^+[v] :=∂_y^2v(·)-v(·),with domain𝒟(ℳ^±) =H_odd^2([0,κ])∩ H_0^1([0,κ]). We claim that these operators are invertible: indeed,coercivity implies that Ker(ℳ^-) = {0}; The same holds in the case of ℳ^+, for in H^2 its kernel is given by{∂_ytanh(·/√(2))} which is a simple eigenfunction, since it has no nodal points (see <cit.>).Therefore, in the space of odd functions, we have that both operators ℳ^± are invertible.In our next step, we argue as in <cit.>, describing limiting operators associated with ℒ_Ξ[·]:ℒ_Ξ^(x→ +∞)[v]= (∂_x^2 + c∂_x + ℳ^+)[v],ℒ_Ξ^(x→ -∞)[v]= (∂_x^2 + c∂_x + ℳ^-)[v], ℒ_Ξ^(y→ +∞)[v] = (Δ_x,y + c∂_x + μ(x) - 3(θ^(c))^2)[v].Fourier transforming the operators (<ref>)-(<ref>) in x, and (<ref>) in y, shows that these operators are boundedly invertible. Observe that v(x,y) = ∂_x(χ^-(x)Ξ_∞^(c)(x,y)) (resp., v(x,y) = ∂_x(χ^+(x)Ξ_∞^(c)(x,y))) satisfiesℒ_Ξ^(x→ -∞)[v] = f (resp., ℒ_Ξ^(x→ +∞)[v] = f) where the left hand side is spatially localized, it follows from the same reasoning as in Lem. <ref> that e^δ<x>v ∈ H^2(^2)∩ H_0^1(^2). A similar result holds for v = ∂_y (χ^+(y)Ξ), which solvesℒ_Ξ^(y→ +∞)[v] = f for f spatially localized in y, i.e., f∈ e^δ <y>L^2(^2), where <y> = √(1 +y^2). The rest of the analysis is similar to that used in the proof of Lem. <ref>. [Uniqueness up to translation in the y-direction] Whenever c≥ 0 the solutions constructed in Theorem <ref> and the solutions constructed in <cit.> by continuation are the same up to translation in the y-direction. The machinery given in <cit.> can be usedto derive a simple proof: first notice that∂_y Ξ_∞∈ Kerℒ_Ξ_∞[·]. As ∂_yΞ has a sign one can use <cit.> to conclude that ∂_yΞ^∞=C∂_yΘ for a C constant. Upon integration in y and using the fact that both solutions converge to the same limit as y →±∞ and satisfy Θ(x,0) = Ξ^∞(x,0) = 0 we get that C ≡ 1, and the result follows.Our last result concerns the continuity of the patterns Ξ_∞^(c)(·,·) in c.The mapping c↦Ξ_∞^(c)(·):[0,2) → L^∞(^2;) is continuous. As before, we exploit the symmetries of the problem to reduce the analysis to the half-space ×(-∞,0]. A similar analysis to that of Lem. <ref> shows that the linearization of the equation (<ref>) at Ξ_∞^(c) given in (<ref>) is an invertible operator from H^2(×(-∞,0])∩ H_0^1(×(-∞,0]) (i.e., restricted to odd solutions in y) to L^2(×(-∞,0]). However, the result does not readily follows, because Ξ_∞^(c) - Ξ_∞^(d)∉H^2(×(-∞,0])∩ H_0^1(×(-∞,0]).We circumvent this issue with a far-field core decomposition, which we now explain: for a fixed d∈[0,2),we use an Ansatz of the formw(x,y)+ Ξ_∞^(c) + χ^-(y) [θ^(d)(x) - θ^(c)(x)], where w ∈ H^2(×(-∞,0];); notice that Ξ_∞^(c) + χ^-(y) [θ^(d)(x) - θ^(c)(x)] converges exponentially fast to its far-field and asymptotically solves the PDE (<ref>). We rewrite problem (<ref>) as ℒ_Ξ_∞[w] : = Δ_x,y w + c∂_x w + μ(x) w - 3(Ξ_∞^(c))^2 w = 𝒩(w, Ξ_∞^(c),θ^(d)(x), θ^(c)(x) )A reasoning similar to that in the proof of Lem. <ref> shows that ℒ_Ξ_∞ is a bounded, invertible operator from H^2(×(-∞,0])∩ H_0^1(×(-∞,0]) to L^2(×(-∞,0]). Furthermore, the right hand side is in L^2(×(-∞,0]), thanks to the Sobolev embedding H^2(^2) ↪ L^∞(^2). The conclusion then follows from the IFT.[of Theorem <ref>; case κ = ∞] Combine the results of Lemmas <ref>, <ref>, <ref>, and Prop. <ref>.§ DISCUSSIONAmong several possible directions of further investigation, we would like to mention the following: Metastability of patterns. As addressed by the numerical studies of<cit.>, defining theparameter regions of metastability for creation of patterns (either perpendicular or parallel to the quenching front) is a challenging and interesting direction of investigation. From a broader perspective, a numerical, if possible analytical,description of parameter curves on the boundary of different morphological states would be valuable in applications. Selection mechanisms. What are the crucial mechanisms involved in the wavenumber selection in the wake of the front? How relevant are the nonlinearity and the speed of the quenching front in this selection? We refer to <cit.> for a general discussion about wavenumber selection.Critical cases; S=1. The behavior of the patterns in this critical scenario possibly requires a different approach, since one can see in the proofs of Theorem <ref> and <ref> that the speed of the quenching front has to be away from the critical case. The result would be interesting and add valuable knowledge in the classification of patterns obtained from directional quenching. Non-planar quenching fronts and oblique stripes.Is it possible to control the contact angle of the _κ-patterns? 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Helices in the wake of precipitation fronts. Phys. Rev. E, 88:022141, Aug 2013.[Veg93]Vega José M. Vega. Travelling wavefronts of reaction-diffusion equations in cylindrical domains. Comm. Partial Differential Equations, 18(3-4):505–531, 1993.[WL07]Wang_Hui Jin-Liang Wang and Hui-Feng Li. Traveling wave front for the Fisher equation on an infinite band region. Appl. Math. Lett., 20(3):296–300, 2007.[ZWH+16]Langmuir Juan Zhu, Markus Wilczek, Michael Hirtz, Juanyuan Hao, Wenchong Wang, Harald Fuchs, Svetlana V. Gurevich, and Lifeng Chi. Branch suppression and orientation control of Langmuir-Blodgett patterning on prestructured surfaces. Advanced Materials Interfaces, 3(19):1600478–n/a, 2016. 1600478. | http://arxiv.org/abs/1707.09010v1 | {
"authors": [
"Rafael Monteiro"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170727193345",
"title": "Horizontal patterns from finite speed directional quenching"
} |
[icmab] Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spainicmab]Alberto Garcíacor1 [email protected] [cor1]Corresponding authorliege] Matthieu [email protected] [liege]nanomat/Q-MAT/CESAM, Université de Liège, Allée du 6 Août 19 (B5a), B-4000 Liège, Belgiumunican] Yann [email protected] unican] Javier [email protected] [unican] Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Avenida de los Castros s/n, 39005 Santander, SpainNorm-conserving pseudopotentials are used by a significant number of electronic-structure packages, but the practical differences among codes in the handling of the associated data hinder their interoperability and make it difficult to compare their results. At the same time, existing formats lack provenance data, which makes it difficult to track and document computational workflows. To address these problems, we first propose a file format (psml) that maps the basic concepts of the norm-conserving pseudopotential domain in a flexible form and supports the inclusion of provenance information and other important metadata. Second, we provide a software library () that can be used by electronic structure codes to transparently extract the information in the file and adapt it to their own data structures, or to create converters for other formats. Support for the new file format has been already implemented in several pseudopotential generator programs (including atom and oncvpsp), and the library has been linked with siesta and abinit, allowing them to work withthe same pseudopotential operator (with the same local part and fully non-local projectors)thus easing the comparison of their results for the structural and electronic properties, as shown for several example systems. This methodology can be easily transferred to any other package that uses norm-conserving pseudopotentials, and offers a proof-of-concept for a general approach to interoperability.PACS: 71.15.Dx 71.10.-w 31.15.E- Pseudopotential Density functional Electronic StructurePROGRAM SUMMARY Program Title:Licensing provisions: BSD 3-clauseProgramming language: Fortran Distribution format: tar.gzExternal routines/libraries: xmlf90 for XML handling in Fortran (http://launchpad.net/xmlf90)Nature of problem: Enhancing the interoperability of electronic-structure codes by sharing pseudopotential data.Solution method:Create an XML-based pseudopotential format (psml), complete with a formal schema, and a processing library () that transparently connects client codes to the information in the format. § INTRODUCTION Within computational science, reproducibility of research goes beyond using a specific version of a code and the appropriate input files. What is really sought is to replicate a certain physical result with a different code which implements the same basic equations of the domain at hand, but with a different set of approximations or details of implementation. This latter code will most likely have a different input data format, which might not be perfectly mapped to the format used by the original code. Reproducibility is still possible if the input data are curated so that their essential ontological properties are preserved, since it can be assumed that properly implemented codes within a domain will share a basic ontology for this domain, and can properly interpret its elements.Here we are concerned with electronic structure methods in the broad sense <cit.>, and in particular density functional theory (DFT <cit.>), which have become standard theoretical tools to analyze and explain experiments in chemistry, spectroscopy, solid state physics, material science, biology, and geology, among many other fields.Programs which implement DFT fall into two main categories, those which treat all electrons explicitly (linear augmented plane wave, LAPW <cit.>, linear muffin tin orbitals, LMTO <cit.>, or the Korringa-Kohn-Rostoker and related multiple-scattering theory methods, KKR <cit.>), and those which replace the effect of the chemically inert core electrons by a (usually non-local) pseudopotential <cit.>. After some historical initial attempts, seminal work by Hamann, Schlüter and Chiang <cit.> and Kerker <cit.> showed how norm-conserving pseudopotentials achieve a good tradeoff between transferability (a pseudopotential constructed for a specific environment, usually the isolated atom, gives good results when the atom is placed in a different environment) and softness (as measured for example by the number of plane waves that must be used in the representation of the wave functions for a good convergence of the physical properties). Since the 1990s more sophisticated schemes have been developed to treat the basic problem of eliding the core electrons.In particular, ultrasoft pseudopotentials <cit.>, and the Projector Augmented Wave (PAW) method <cit.> are in widespread usefor their improved accuracy and features, although they involve a significantly higher degree of complexity in the implementation and maintenance of the algorithms for electronic structure determination and analysis. Several well-known atomic DFT programs <cit.> generate pseudopotentials in a variety of formats, tailored to the needs of electronic-structure codes. While some generators are now able to output data in different bespoke formats, and some simulation codes are now able to read different pseudopotential formats, the common historical pattern in the design of those formats has been that a generator produced data for a single particular simulation code, most likely maintained by the same group. This implied that a number of implicit assumptions, shared by generator and user, have gone into the formats and fossilized there.Examples include, among others, many flavours of radial grids (from linear <cit.> to logarithmic <cit.>, including all kinds of powers <cit.> or geometrical series <cit.>), different ways of storing radial functions with spherical symmetry (angularly integrated with 4 π r^2 factors included, where r stands for the distance to the nuclei, or multiplication by different powers of r considered), different normalization conditions, etc. This leads to practical problems, not only of programming, but of interoperability and reproducibility, which depend on spelling out quite a number of details which are not well represented for all codes in existing formats.Moreover, pseudopotential information can be produced and used at various levels. The original work was based on a set of semi-local operators (non-local in the angular part and local in the radial part), whereas very soon a computationally more efficient form based on fully non-local projectors plus a local part <cit.> was developed, and is now the standard norm-conserving form used by electronic-structure codes.The transformation of a semi-local pseudopotential into a fully non-local operator is not univocally defined from the semi-local components alone.Therefore, different codes can yield different results even if reading the same semi-local information from an input file. For example, most plane waves codes make the so-called local part equal to one of the semilocal components for reasons of efficiency, whereas siesta optimizes the local part for smoothness, since it is the only pseudopotential part that needs to be represented in a real space grid <cit.>. Modern pseudopotential generators are able to produce directly a set of non-local operators plus a local part.True interoperability can be achieved only if the codes use the same final projector-based pseudopotential.Straightfoward interoperability of codes would allow all to benefit from the best capabilities of each.On the one hand, removing the variability associated with the pseudopotential would enable a direct test of the quality of the basis sets used in the expansion of the one-particle Kohn-Sham eigenstates by different codes (with identical pseudopotentials, and at convergency, one should get the same total energy for the same atomic structure with any code).On the other hand, we can imagine a situation where the most stable atomic configuration of a large system can be found with a given code in a cheap yet accurate way, and then passed to another for further analysis through single-shot expensive calculations for the fixed geometry.The obvious benefits of interoperability have naturally spawned efforts at providing more appropriate data mechanisms. At the most basic level, some pseudopotential generators (e.g. oncvpsp or ape) offer options to write different output files suitable for different electronic-structure codes. While this addresses some of the problems, it falls short of a fully satisfactory solution. More robust are efforts to provide a standard format that can be used universally. Within the PAW domain, the PAW-XML format <cit.> comes close to this goal, being produced by a number of PAW dataset generators and read by most PAW-enabled electronic-structure codes. The UPF (Unified Pseudopotential Format) <cit.> is meant to encompass the full range of pseudopotential options, including (semi)local and fully-non-local norm-conserving, ultrasoft pseudopotentials, and PAW datasets. It is used within the Quantum Espresso suite of codes and converters exist for other codes.We believe that, indeed, the solution to the interoperability problems involves the design a data format that faithfully maps the relevant concepts of the domain's ontology at all the relevant levels (semi-local pseudopotentials, charge densities, non-local projectors, local potentials, etc). But the format must also provide appropriate metadata that represents provenance (generation and any further processing) and documents in a parseable form any details that might be needed downstream. This second aspect has not been properly addressed so far.The need for standardisation of the pseudopotential format and the provision of richer metadata to track provenance and document computational workflows has been made more relevant by the appearance during the last few years of high-throughput simulation schemes for materials design <cit.>, which need well-tested and efficient pseudopotential libraries to draw upon. Examples of the latter are the ultrasoft pseudopotential library by Garrity, Bennett, Rabe and Vanderbilt <cit.>, the PSLibrary associated with QE <cit.>, the Jollet-Torrent-Holzwarth <cit.> PAW library, and the libraries being built <cit.> using multi-projector norm-conserving pseudopotentials generated by the oncvpsp code <cit.>. The latter are proving competitive with ultrasoft pseudopotentials and the PAW method in accuracy. <cit.> Motivated by the all the previous considerations, in this paper we present a file format for pseudopotential data (psml, for PSeudopotential Markup Language) which is designed to encapsulate as much as possible the abstract concepts in the domain's ontology, and to provide appropriate metadata and provenance information.Moreover, we provide a software library () that can be used by electronic structure codes to transparently extract the information in a psml file and adapt it to their own data structures, or to create converters for other formats.Our initial focus is the sub-domain in which norm-conserving pseudopotentials are used, which is not restricted to legacy cases but is set to grow in importance due to the new multi-projector pseudopotentials mentioned above. As the format is based on XML (eXtensible Markup Language) <cit.>, it is very flexible and can serve as a basis for the future accomodation of PAW datasets and ultrasoft pseudopotentials.Our work falls within the scope of CECAM's Electronic Structure Library (ESL) initiative <cit.>, which aims at building a collection of software functionalities, including standards and interfaces, to facilitate the development of electronic structure codes.This paper is organized as follows: Section <ref> describes the basic elements and structure of a psml file, with a more formal XML schema given in Sec. <ref>. Thelibrary is described in Sec. <ref>. Section <ref> discusses the relationship of the current work to previous efforts, and provides a guide to the tools already available in the psml ecosystem and the mechanisms available for development of new ones. Finally, as an example of the interoperability benefits afforded by psml, we present in Sec. <ref> a comparison of actual physical results obtained by abinit and siesta using the same psml files. § THE PSML FILE FORMATThis section contains a design rationale and human-readable description of the format. A formal schema specification can be found in Sec. <ref>. The following documents version 1.1 of the psml, current as of this writing. Any updates and complementary information are available at the ESL psml site: http://esl.cecam.org/PSML.§.§ Root elementThe root element is psml, containing a version attribute for use by parsers, and two attributes to make explicit the (mandatory for now) units used throughout the file: energy-unit (hartree) and length-unit (bohr). In addition, the root element should contain a uuid attribute to hold a universally unique identifier <cit.>. §.§ Provenance element(s) The file should contain metadata concerning its own origin, to aid reproducibility. As a minimum, it should contain information about the program(s) used to generate or transform the pseudopotential,ideally with version numbers and compilation options, and provide a copy of any input files fed into the program(s). The information is contained in provenance elements (anexample is provided in Table <ref>). Its internal structure is subject to a minimal specification:* The attribute creator, where the name of the generator codeis specified, is mandatory.* The attribute date is mandatory.* If input files are provided, they should be included as input-file elements with the single attribute name and no children other than character data, which should be placed in a CDATA section to avoid processing of XML reserved characters.For obvious reasons, the name of the file and its detailed content will depend on the program used to generate the pseudopotential. The name should be a mnemonic reference.* The inclusion of an annotation element, providing arbitrary extra information in the form of key-value pairs, is encouraged. See the description and motivation of the annotation element in Sec. <ref>. There can be an arbitrary number of provenance elements, ordered in temporal sequence in the file, with the most recent first. Since some XML processors might not preserve the order of the elements, it is suggested that a record-number attribute be added to make the temporal ordering explicit, with “1” for the oldest operation, “2” for the second oldest, and so on.This feature supports the documentation of successive actions taken on the file's information. For example, a pseudopotential-generation program might generate a psml file containing only semilocal potentials. This file is then processed by another program that generates a local potential and the corresponding non-local projectors. The psml file produced by the second program should keep the original provenance element, and add another one detailing the extra operations taken.§.§ Pseudo-atom specification The pseudo-atom-spec element contains basic information about the chemical element, the generation configuration, and the type of calculation.Element identification is in principle straightforward, with either the atomic number or the chemical symbol. However, in the interest of generality, and to cover special cases, such as synthetic (alchemical) atoms, the chemical symbol can be an arbitrary label, given by the attribute atomic-label (with the convention that when possible the first two characters give the standard chemical symbol) and the atomic number a real number instead of a simple integer, given by the attribute atomic-number.The atomic calculation can be done non-relativistically, withscalar-relativistic corrections (i.e., with the mass and Darwinterms) <cit.>, or with the full Dirac equation,including spin-orbit effects. In the latter case, the(semilocal) pseudopotentials are typically provided in two sets: thedegeneracy-averaged j=l+1/2 and j=l-1/2 components, appropriatefor scalar-relativistic use, and the spin-orbitcomponents <cit.>. It is also possible to carry out a non-relativistic calculation for aspin-polarized reference configuration (using spin-DFT). Standardpractice is then to keep only the population-averagedpseudopotentials, as a closed-shell frozen core should not berepresented by a spin-dependent potential. (But seeRef. <cit.> for a different point of view.) In practice, then, one might have a primary set ofl-dependent V_l(r) potentials, possibly the result of averaging,and, in the case of fully-relativistic calculations, a spin-orbitset. In other cases, the generation program might output the originallj versions in the Dirac case, or the “up” and “down”components for a spin-DFT calculation. We cover all these possibilities in psml by using theattribute relativity, with possible values “no”,“scalar” or “dirac”, and the optional attribute spin-dft to indicate(with value “yes”) a non-relativistic spin-DFT calculation. Further, as detailedbelow, we provide support for the possible presence of various setsof magnitudes.The pseudopotential construction is fundamentally dependent on the core-valence split. In most cases it is clear which states are to be considered as “valence”, and which ones are to be kept in the frozen core. However, there are borderline cases in which one has the option to treat as “valence” so-called “semicore” states which are relatively shallow and/or exhibit a sizable overlap with proper valence states.This information is provided in the mandatory valence-configuration element, which details the valence configuration used at the time of pseudopotential generation, given by the n and l quantum numbers, and the electronic occupation of each shell.Empty shells can be omitted.In spin-DFT calculations, the spin-up and spin-down occupations are also given.This element has a mandatory attribute total-valence-charge which contains the total integrated valence charge Q_ val for the configuration used to generate the pseudopotential.The core configuration can be determined easily from the knowledge of the valence shells, but for completeness it can be given in the optional core-configuration element, with the same structure. This tag would be useful, for example, if a core-hole pseudopotential has been created <cit.>.The difference between the number of protons in the nucleus and the sum of the populations of the core shells is the effective atomic number of the pseudo-atom Z_ pseudo, which must be given in the mandatory attribute z-pseudo.The “pseudization flavor” or, more properly, a succint identifier for the procedure for pseudopotential generation, is encoded in the optional flavor attribute. If not present, more specific values can be given per pseudopotential block and per pseudization channel (see below).If non-linear core-corrections <cit.> are present, the optional core-corrections attribute must be set to “yes”.One further piece of information is needed to complete the general specification of the computational framework: the type of exchange-correlation (XC) functional(s) used. With the explosive growth in the number of functionals, it is imperative that a robust naming convention be used. In the absence of a general registry of universally-agreed names, we propose a dual naming scheme.The element exchange-correlation contains: * A mandatory element libxc-info that maps whatever XC functional is used in the generation code to the standard set of functionals in the Libxc library <cit.>. This library supports a large number of functionals, and new ones are added promptly as their details are published. We thus require that pseudopotential-generation programs producing psml files, and programs using the psml format, provide internal tables mapping any built-in XC naming schemes to the Libxc one. This element has the attribute number-of-functionals, and as many functional elements as indicated by this attribute, with attributes name, and id,as shown in the example of Table <ref>. These correspond to theLibxc identification standard. The attribute type (with values “exchange”, “correlation”, or “exchange-correlation”) is optional.Further, to support arbitrary mixtures of functionals, the optional attribute weight can also be indicated. * An (optional) element annotation (see Sec. <ref>) that can contain any XC identification used by the creator of the file, in the form of attribute-value pairs. This information can be read in an ad-hoc fashion by client programs, but it is obviously not as complete or robust as that contained in the libxc-info. For maximum interoperability, client programs should thus implement an interface to Libxc. An optional annotation element can also be included inside the pseudo-atom-spec element. Note that the ordering of child elements is significant (see Sec. <ref>). §.§ Radial functions and grid specification At the core of this new format we face the problem of how to store a variety of different radial functions (semilocal pseudopotentials, projectors, pseudo wave-functions, pseudocore and valence charges, etc.) in a radial grid. Most pseudopotential-generation codes export their data for their radial functions f(r) as a tabulation {f(r_i)}, where {r_i} are discrete values of the radial coordinate in an appropriate mesh.A variety of meshes with different functional forms and parametrization details are in common use.Our preferred way to handle this variety of choices is to specify the actual grid point data in the file. This is most extensible to any kind of grid, and avoids problems of interpretation of the parameters, starting and ending points, etc. Furthermore, as explained below, the psml handling library is completely grid-agnostic, since evaluators are provided for the relevant functions f(r), in such a way that a client code can obtain the value of f at any radial coordinate r, in particular at the points of a grid of its own choosing. The precision of the computed value f(r) is of course dependent on the quality of the f(r_i) tabulation in the first place, and producer codes should take this issue seriously. We discuss more points related to the evaluation of tabulated data in Sec. <ref>.The format should be flexible enough to allow each radial function to use its own grid if needed, while providing for the most common case in which all radial functions use a common grid (or a subset of its points).Our solution is to encode the information about each radial function in a radfunc element, which contains the tabulation data f(r_i) in a data element. The grid specification uses a cascade scheme with an (optional but recommended) top-level grid element, optional mid-level grid elements under certain grouping elements, and at the lowest level optional grid elements inside the individual radfunc elements.The grid {r_i} for a function is inherited from the closest grid element at an enclosing level if it is not specified in the local radfunc element. The grouping elements currently allowed to include mid-level grids are those for semilocal potentials, nonlocal projectors, local potential, pseudo-wavefunctions, valence charge, and pseudocore charge.The grid elements should have a mandatory “npts” attribute providing the number of points, and a grid-data element with the grid point data as formatted real numbers with appropriate precision. All radial data must be given in bohr.For convenience, it is allowed to include an annotation element as child of grid, with appropriate attributes, to provide additional information regarding the form of the grid data. The client program can process this information if needed.The data element may contain an optional attribute npts to indicate the number of values that follow. In its absence, the number of values must match the size of the grid. The npts attribute is useful in those cases in which the effective range of a radial function is significatively smaller than the extent of the grid. For example, a psml file might contain a top-level grid with a large range, appropriate for the valence charge density, of which only a subset of points are used for the projectors, which have a much smaller range.A technical point should be kept in mind. When processing a psml file, the radial function information is typically stored internally as a table on which interpolation is performed to obtain values of the function at specific radii. In order to avoid the dangers associated with extrapolation, the radial grid must contain as first point r=0, and any radial magnitudes (pseudopotentials, wave-functions, pseudo-core or valence charges) should be given without extra factors of r that might hamper the calculation of needed values at r=0. In this way the processor can unambiguously determine the function values at all radial points.When evaluating a function at a point r beyond the maximum range of the tabulated data in the psml file, a processor should return:* -Z_ pseudo/r for the semi-local and local pseudopotentials, in keeping with the well-known asymptotic behavior.* Zero for projectors, pseudo-wave-functions and valence and pseudo-core charges.§.§ Semilocal components of the pseudopotentialsWhen available, the l-dependent (or maybe lj-dependent) semilocal components V_l(r) of the pseudopotential are classified under semilocal-potentials elements, with attributes: * set: A string indicating which set (see below) the potentials belong to. If missing, the information is obtained from the records for the individual potentials. * flavor: (optional) The pseudization flavor. If missing, its value is inherited from the value in the pseudo-atom-spec element. It can also be superseded by the records for the individual potentials.The set attribute allows the handling of various sets of pseudopotentials. Its value is normalized as follows, depending on the type of calculation generating the pseudopotential and the way in which the code chooses to present the results:* “non_relativistic” for the non-relativistic, non-spin-DFTcase. * “scalar_relativistic” if the calculation isscalar-relativistic, or if it is fully relativistic and anset of lj potentials averaged over j is provided. * “spin_orbit” if a fully relativistic code provides thiscombination of lj potentials. * “lj” for a fully relativistic calculation with straightoutput of the lj channels. * “spin_average” for the spin-DFT case when thegeneration code outputs a population-averaged pseudopotential. * “up” and “down”,for a spin-DFT calculation with straightoutput of the spin channels. * “spin_difference” for the spin-DFT case when thegeneration code outputs also the difference between the “up” and“down” potentials. This and the previous case are retained forhistorical reasons, but are likely used rarely. Note that a given code might choose to output its semilocal-potential information in two different forms (say, as scalar-relativistic and spin-orbit combinations plus the lj form). The format allows this, although in this particular case the information can easily be converted from the lj form to the other by client programs.For extensibility, the format allows two more values for the set attribute, “user_extension1” and “user_extension2”, which can in principle be used to store custom information while maintaining structural and operative compatibility with the format.The pseudopotentials must be given in hartree.The semilocal-potentials element contains child slps elements,which store the information for the individual semilocal pseudopotential components.The attributes of this element are: * n: principal quantum number of the pseudized shell.* l: angular momentum number of the pseudized shell.* j: (compulsory for “lj” sets) j quantum number* rc: r_c pseudization radius for this shell (in Bohr).* eref: (optional) reference energy (eigenvalue) of the all-electron wavefunction to be pseudized (in hartree).* flavor: (optional) To allow for different schemes for different channels, the value of this attribute, when present, takes precedence over the flavor attributes in the pseudo-atom-spec element and the semilocal-potentials elements.The optional attribute eref might only be meaningful for certain potentials (for example, those that have been directly generated, and are not the product of any extra conversion, such as from lj to scalar-relativistic plus spin-orbit form).Each slps element contains a radfunc element. The order in which the slps elements appear is irrelevant.The semilocal-potentials element can contain an optional grid child applying to all the enclosed elements, as well as an optional annotation element for arbitrary extra information in key-value format.§.§ Pseudopotential in fully non-local form.Most modern electronic-structure codes do not actually use the pseudopotential in its semi-local form, but in a more efficient fully non-local form based on short-range projectors plus a “local” potential:V̂_ps= V̂_ local + ∑_i |χ_i> E_ KB^i <χ_i| proposed originally by Kleinman and Bylander <cit.> and generalized among others by Blöchl <cit.>, Vanderbilt <cit.>,and Hamann <cit.>. The information about this operator form of the pseudopotential is split in two elements, holding the local potential and the nonlocal projectors.§.§.§ Local potential The local-potential element has the attribute * type We suggest the string “l=X” when the local potential is taken to be the semi-local component for channel “X”, or any other succint comment if not.and contains a radfunc element with the actual data for the local pseudopotential.Optionally, the local-potential element can contain a child local-charge element, describing a radial function ρ_ local(r) related to V_ local(r) by Poisson's equation. That is, ρ_ local(r), integrating to Z_ pseudo and localized in the core region, is the effective charge that would generate V_ local(r). The local-charge element is optional because not all V_ local(r) functions are representable as originating from a charge distribution (V'_ local(0) must be zero for this). When it is present, however, it can save some client programs (such as siesta, which uses ρ_ local(r) to generate a very convenient localized neutral-atom potential) the task of computing numerical derivatives of V_ local(r).The local-potential element can also contain an optional grid child applying to all the enclosed elements, as well as an optional annotation element for arbitrary extra information in key-value format.§.§.§ Non-local projectors The information about the non-local projectors is stored in nonlocal-projectors elements, with the optional attribute set, as above, containing proj elements with attributes * ekb: Prefactor of the projector in the corresponding term in Eq. <ref> (in hartree).* eref: (optional) Reference energy used in the generation of the projector (in hartree).* l: angular momentum number * j: (compulsory for “lj” sets) j quantum number* seq: sequence number within a given l (or lj) shell, to support the case of multiple projectors.* type: Succint comment about the kind of projectorand a radfunc element containing the data for the χ_i functions in Eq. <ref>. These functions are formally three-dimensional, including the appropriate spherical harmonic for the angular coordinates, and a radial component: χ_i= χ_i(r)Y_lm(θ,ϕ). What is actually stored in the file is the function χ_i(r), normalized in the one-dimensional sense: ∫_0^∞ r^2| χ_i(r) |^2 dr= ∫_0^∞| rχ_i(r) |^2 = 1, and proportional to r^l near the origin.There can be several nonlocal-projectors elements, using different values for the set attribute, as explained in Sec. <ref>.For projectors in the “dirac” case, the functionality provided by the handling of set attributes can be very useful, as it is not straightforward to convert the lj information into scalar-relativistic and spin-orbit combinations. Unlike in the semi-local case, this conversion is not reversible, so the lj form is more fundamental.For maximum interoperability, producer codes should store both the lj and the combination sets.The optional attribute eref might only be meaningful for certain projectors (for example, those that have been directly generated, and are not the product of any extra conversion, such as from lj to scalar-relativistic plus spin-orbit form). §.§ Pseudo-wave functionsPseudo-wavefunctions are typically produced at an intermediate stage in the generation (and testing) of a pseudopotential, but they are not strictly needed in electronic-structure codes, except in a few cases:* When atomic-like initial wavefunctions are needed to start the electronic-structure calculation.* When the code uses internally a fully-nonlocal form of the pseudopotential which is constructed from the semilocal form and the pseudo-wavefunctions. While these pseudo-wavefunctions could be generated by the client program, we allow for the possibility of including them explicitly in the psml file. Any (optional) wavefunction data must be included in pseudo-wave-functions elements, with a set attribute and as much extra metadata as needed (which we do not try to standardize at this point, so it should be given in the form of annotations).The extra metadata might indicate whether the data is for actual pseudized wave-functions, or for the pseudo-valence wave functions generated with the obtained pseudopotential. There might be subtle differences between them, notably regarding relativistic effects, as some generation codes use a non-relativistic scheme to “test” the pseudopotential and generate pseudo wavefunctions, instead of a scalar or fully relativistic version. Each pswf is given in a pswf element, with attributes that identify the quantum numbers for the shell and the apropriate energy level (which could be the eigenvalue in the original pseudization, or another energy level used in the integration leading to the wavefunction): * n: principal quantum number of the shell. * l: angular momentum number of the shell.* j: (compulsory for “lj” sets) j quantum number* energy_leveland a radfunc element.The data is for the standard radial part of the wave function R_n,l(r), and (for bound states) should be normalized as∫_0^∞ r^2| R_n,l(r) |^2 dr= ∫_0^∞| u_n,l(r) |^2 = 1. Within our psml format R(r) is given, rather than u(r),due to the extrapolation issues detailed in the section covering the grid.The pseudo-wave-functions elements can contain an optional grid child applying to all the enclosed elements. §.§ Valence charge densitySome electronic-structure codes might need information about the valence charge density of the pseudo-atom. This can bethe pseudo-valence charge density used to unscreen the ionic potential during the pseudopotential generation process, or the pseudo-valence charge computed from the pseudopotential itself. In psml it is given under the valence-charge element, whose child radfunc element holds a solid-angle-integratedform q(r) normalized so that:∫_0^∞ r^2 q(r) dr = Q. Here Q is the total charge output, which must be stored in the total-charge attribute. The attributes is-unscreening-charge (with value “yes” or “no”) and rescaled-to-z-pseudo (“yes” or “no”) areoptional. In combination with the information in the valence-configuration element, these attributes will help some client codes process the valence charge density data appropriately, particularly in the case in which the pseudopotential generation used an ionic configuration. Some codes output in this case the unscreening charge rescaled to Z_ pseudo.The valence-charge can also contain an optional annotation element. §.§ Pseudocore-charge densityAn (optional) smoothed charge density matching the density of the core electrons beyond a certain radius, for use with a non-linear core correction scheme <cit.>, is contained within the pseudocore-charge element, with the data in the same solid-angle-integrated form (and implicit units) as the valence charge, and with the extra optional attributes: * matching-radius: The point r_ core at which the true core density is matched to the pseudo-core density.* number-of-continuous-derivatives: In the original scheme by Louie et al. <cit.> the pseudo-core charge was represented by a two-parameter formula, providing continuity of the first derivative only. Other typical schemes provide continuity of the second and even higher derivatives. It is expected that the number of continuous derivatives, rather than the detailed form of the matching, is of more interest to a client program.An optional annotation element with appropriate attributes might be given to document any extra details of the model-core generation.The pseudocore-charge element must appear if the core-corrections attribute of the pseudo-atom-spec element has the value “yes”. §.§ The handling of annotationsXML provides for built-in extensibility and client programs can use as much or as little information as needed. For the actual mapping of a domain ontology to a XML-based format, however, clients and producers have to agree on the terms used. What has been described in the above sections is a minimal form of such a mapping, containing the basic concepts and functions needed. The extension of the format with new fixed-meaning elements and attributes would involve an updated schema and re-coding of parsers and other programs. A more light-weight solution to the extensibility issue is provided by the use of annotations, which have the morphology of XML empty elements (containing only attributes) but can appear in various places and contain arbitrary key-value pairs. Annotations provide immediate information to human readers of the psml files, and can be exploited informally by client programs to extract additional information. For the latter use, it is clear that some degree of permanence and agreed meaning should be given to annotations, but this task falls not on some central authority, but on specific codes.Annotations are currently allowed within the following elements: provenance, pseudo-atom-spec, exchange-correlation, valence-configuration, core-configuration, semilocal-potentials, local-potential, nonlocal-projectors, pseudo-wave-functions, valence-charge, core-charge, and grid. Top-level annotations are not allowed. They properly belong in the provenance elements.§ FORMAL SPECIFICATION OF THE PSML FORMATWe provide a formal XML schema for the psml format, given in the very readable RELAX NGcompact form <cit.>. This schema can be used directly for validation of psml files,or converted to an W3C schema file (using respectively theandtools of the RELAX NG project).This is the overall structure of a psml document, showing the main building blocks.The definitions of the grammar elements are given in the next section, when the closely related API is discussed.[style=customrnc] default namespace = "http://esl.cecam.org/PSML/ns/1.1"PSML =element psml Root.Attributes, Provenance+ # One or more provenance elements , PseudoAtomSpec , Grid? # Optional top-level grid , ValenceCharge, CoreCharge? # Optional pseudo-core charge , ( (SemiLocalPotentials+ , PSOperator?) | (SemiLocalPotentials* , PSOperator ) ) , PseudoWaveFunctions*# Zero or more Pseudo Wavefunction groupsPseudoAtomSpec =element pseudo-atom-spec PseudoAtomSpec.Attributes , Annotation? , ExchangeCorrelation , ValenceConfiguration , CoreConfiguration?PSOperator = ( LocalPotential# Local potential, NonLocalProjectors* ) # Zero or more fully nonlocal groups The quantifiers '*', '+', and '?' mean “zero or more”, “one or more”, and “at most one”, and the '|' sign expresses an exclusive “or” (choice) operation. Even though RELAX NG can accommodate interleaved elements, this feature is not fully representable in W3C schema, and the ordering of the elements above is strict. We use a URI in the “esl.cecam.org” domain to identify the schema namespace, but note that it is not a resolvable location.The main feature not discussed above in Sec. <ref> is the constraint of non-emptiness of the psml file: it must contain at least either a set of semilocal-potentials, or a complete pseudopotential operator consisting of a local potential and a set of nonlocal projectors. It is also possible to have a single local potential as a degenerate form of pseudopotential operator. Beyond the minimal requirements, a psml file can contain multiple occurrences of any of these elements.§ THE PSML LIBRARY We provide a companion library to the psml format, , that provides transparent parsing of and data extraction from psml files, as well as basic editing and data conversion capabilities.The library is built around a data structure of type ps_t that maps the information in a psml file. Instances of this structure are populated by psml parsers, processed by intermediate utility programs, and used as handles for information retrieval by client codes through accessor routines. The library provides, in essence:* A routine to parse a psml file and produce a ps object of type ps_t.* A routine to dump the information in a ps object to a psml file.* Accessor routines to extract information from ps objects.* Some setter routines to insert specific blocks of information into ps objects. These might be used by intermediate processors or by high-level parsers. The library is written in modern Fortran and provides a high-level Fortran interface. A C/C++ interface is in preparation.An example of use of the library is provided in Table <ref>.In what follows we describe the basic exported data structures and procedures. Full documentation for the library, as well as general information about the psml format ecosystem, is available at the psml reference page under the Electronic Structure Library project website: http://esl.cecam.org/PSML.§.§ Exported typesThe library exports a few fortran derived types to represent opaque handles in the routines: * ps_tThis is the type for the handle which should be passed to most routines in the API* ps_annotation_tThe associated handle is used in the routines that create annotations or extract the data in them (see Section <ref>). * ps_radfunc_tThis corresponds to the internal implementation of a radialfunction. Its use is mostly reserved for low-level operations, as the API provides convenience evaluators for most functions.Additionally, and to avoid ambiguities in real types, the library exports the integer parameter ps_real_kind that represents the kind of the real numbers accepted and returned by the library. §.§ Parsing * psml_reader (filename, ps, debug)parses the psml file filename and populates the data structures in the handle ps. An optional debug argument determines whether the library issues debugging messages while parsing. * ps_destroy (ps)is a low-level routine provided for completeness in cases where a pristine ps is needed for further use. §.§ Library identification * function ps_GetLibPSMLVersion()result(version)The version is returned as an integer with the two least significant digits associated to the patch level (for example: 1106 would correspond to the typical dot form 1.1.6).§.§ Data accessors The API follows closely the element structure of the psml format. Each section in the high-level document structure of Sec. <ref> is mapped to a group of routines in the API. Within each, there are routines to query any internal structure (attributes, existence, number, or selection of child elements) and routines to obtain specific data items (attributes, content of child elements).§.§.§ Root attributes[style=customrnc] Root.Attributes =attribute energy_unit"hartree" , attribute length_unit"bohr" , attribute uuidxsd:NMTOKEN , attribute versionxsd:decimal* ps_RootAttributes_Get (ps,uuid,version,namespace)As in all the routines that follow, the handle ps is mandatory. All other arguments are optional, with “out” intent, and of type character(len=*).version returns the psml version of the file being processed. A given version of the library is able to process files with lower version numbers, up to a reasonable limit. For our purposes, the NMTOKEN specification refers to a string without spaces or commas. §.§.§ Provenance data[style=customrnc] Provenance =element provenanceattribute record-numberxsd:positiveInteger ? , attribute creatorxsd:string, attribute datexsd:string , Annotation? , InputFile*# zero or more input files InputFile =element input-file attribute namexsd:NMTOKEN , # No spaces or commas allowedtextAs there can be several provenance elements, the API provides a function to enquire about their number (depth of provenance information), and a routine to get the information from a given level: * function ps_Provenance_Depth(ps)result(depth)The argument depth returns an integer number. * ps_Provenance_Get(ps,level,creator,date,annotation,number_of_input_files)The integer argument level selects the provenance depth level (1 is the deepest, or older, so to get the latest record the routine should be called with level=depth as returned from the previous routine). All other arguments are optional with “out” intent. creator and date are strings. Here and in what follows, annotation arguments are of the opaque type(see Section <ref>). If there is no annotation, an empty structure is returned. The information in an annotation object can be accessed using routines described in Section <ref>.§.§.§ Pseudo-atom specification attributes and annotation [style=customrnc] PseudoAtomSpec.Attributes = attribute atomic-labelxsd:NMTOKEN ,attribute atomic-numberxsd:double ,attribute z-pseudoxsd:double ,attribute core-corrections"yes" | "no" ,attribute relativity"no" | "scalar" | "dirac" ,attribute spin-dft"yes" | "no" ?,attribute flavorxsd:string ? * ps_PseudoAtomSpec_Get (ps, atomic_symbol, atomic_label, atomic_number, z_pseudo, pseudo_flavor, relativity, spin_dft, core_corrections, annotation)The arguments spin_dft and core_corrections are boolean, and the routine returns an empty string in flavor if the attribute is not present (recall that flavor is a cascading attribute that can be set at multiple levels). The arguments atomic_number and z_pseudo are reals of kind ps_real_kind (see Sec. <ref>). §.§.§ Valence configuration[style=customrnc] ValenceConfiguration =element valence-configurationattribute total-valence-chargexsd:double , Annotation?, ValenceShell+ ValenceShell = ShellShell =element shell attribute_l,attribute_n,attribute occupationxsd:double ,attribute occupation-upxsd:double ?,attribute occupation-downxsd:double ?attribute_l = attribute l"s" | "p" | "d" | "f" | "g"attribute_n = attribute n"1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"* ps_ValenceConfiguration_Get(ps,nshells,charge,annotation)This routine returns (as always, in optional arguments), the values of the top-level attributes, any annotation, and the number ofelements, which serves as upper limit for the index i in the following routine, which extracts shell information: * ps_ValenceShell_Get(ps,i,n,l,occupation,occ_up,occ_down)The n and l quantum number arguments are integers (despite the use of spectroscopic symbols for the angular momentum in the format), and the occupations real.§.§.§ Exchange and correlation[style=customrnc] ExchangeCorrelation =element exchange-correlation Annotation?, element libxc-infoattribute number-of-functionalsxsd:positiveInteger , LibxcFunctional+ LibxcFunctional = element functionalattribute idxsd:positiveInteger , attribute namexsd:string , attribute weightxsd:double ?,# allow canonical names and libxc-style symbols attribute type"exchange" | "correlation" | "exchange-correlation" | "XC_EXCHANGE" | "XC_CORRELATION" | "XC_EXCHANGE_CORRELATION" ?The routines follow the same structure as those in the previous section. * ps_ExchangeCorrelation_Get(ps,annotation,n_libxc_functionals) * ps_LibxcFunctional_Get(ps,i,name,code,type,weight)The argument type corresponds to the type of Libxc functional, and code to the id number.§.§.§ Valence and Core Charges[style=customrnc] ValenceCharge =element valence-charge attribute total-chargexsd:double ,attribute is-unscreening-charge"yes" | "no" ?,attribute rescaled-to-z-pseudo"yes" | "no" ?,Annotation?,Radfunc# ========= CoreCharge =element pseudocore-chargeattribute matching-radiusxsd:double , attribute number-of-continuous-derivativesxsd:nonNegativeInteger , Annotation?, RadfuncThese are radial functions with some metadata in the form of attributes, an optional annotation, and a Radfunc child. The accessors have the extra optional argument func that returns a handle to aobject, which can later be used to get extra information.* ps_ValenceCharge_Get(ps,total_charge, is_unscreening_charge, rescaled_to_z_pseudo, annotation,func)The routine returns an emtpy string in is_unscreening_charge and rescaled_to_z_pseudo if the attributes are not present in the psml file.* ps_CoreCharge_Get(ps,rc,nderivs,annotation,func)rc corresponds to the matching radius and nderivs to the continuity information. Negative values are returned if the corresponding attributes are not present in the file.The func object can be used to evaluate the radial functions at a particular point r:* function ps_GetValue(func,r) result(val) but the API offers some convenience functions * function ps_ValenceCharge_Value(ps,r) result(val)* function ps_CoreCharge_Value(ps,r) result(val) §.§.§ Local Potential and Local Charge Density [style=customrnc] LocalPotential =element local-potentialattribute typexsd:string , Annotation?, Grid?, Radfunc, LocalCharge?# Optional local-charge element LocalCharge =element local-charge Radfunc * ps_LocalPotential_Get(ps,type,annotation,func,has_local_charge,func_local_charge)In this version of the API, the optional local-charge element is not given a first-class status. To evaluate it (if the boolean argument has_local_charge is true), the func_local_charge argument has to be used in the ps_GetValue routine above. The local potential can be evaluated via the func object or with the convenience function * function ps_LocalPotential_Value(ps,r) result(val)§.§.§ Semilocal potentials[style=customrnc] SemiLocalPotentials =element semilocal-potentials attribute_set,attribute flavor xsd:string ?,Annotation?, Grid?,Potential+ Potential = element slpsattribute flavorxsd:string ?, attribute_l, attribute_j ?, attribute_n, attribute rcxsd:double , attribute erefxsd:double ?, Radfunc As explained in Sec. <ref>, there can be several semilocal-potentials elements corresponding to different sets. Internally, the data is built up in linked lists during the parsing stage and later all the data for the slps child elements are re-arranged into flat tables, which can be queried like a simple database. The table indexes for the potentials with specific quantum numbers, or set membership, can be obtained with the routine * ps_SemilocalPotentials_Filter(ps,indexes_in,l,j,n,set,indexes,number)Here, the optional argument (of intent “in”) indexes_in is an integer array containing a set of indexes on which to perform the filtering operation. If not present, the full table is used. The optional arguments l,j,n,set are the values corresponding to the filtering criteria. Upon return, the optional argument indexes would contain the set of indexes which match all the specified criteria, and number the total number of matches.The set argument has to be given using special integer symbols exported by the API: SET_SREL, SET_NONREL, SET_SO, SET_LJ, SET_UP, SET_DOWN, SET_SPINAVE, SET_SPINDIFF, or the wildcard specifier SET_ALL. An example of the use of this routine has already been given in Table <ref>.The appropriate indexes can then be fed into the following routines to get specific information:* ps_Potential_Get(ps,i,l,j,n,rc,eref,set,flavor,annotation,func)All arguments except ps and i are optional. The value returned in set is an integer which can be converted to a mnemonic string through the str_of_set convenience function. The annotation returned corresponds to the optional annotation element of the parent block of the slps element.The routine returns a very large positive value in eref if the corresponding attribute is not present in the file. * function ps_Potential_Value(ps,i,r) result(val)§.§.§ Nonlocal Projectors[style=customrnc] NonLocalProjectors =element nonlocal-projectorsattribute_set, Annotation?, Grid?, Projector+ Projector = element projattribute ekbxsd:double , attribute erefxsd:double ?, attribute_l, attribute_j ?, attribute seqxsd:positiveInteger , attribute typexsd:string , Radfunc + The ideas are exactly the same as for the semilocal potentials. The relevant routines are:* ps_NonlocalProjectors_Filter(ps,indexes_in,l,j,seq,set,indexes,number)* ps_Projector_Get(ps,i,l,j,seq,set,ekb,eref,type,annotation,func)The routine returns a very large positive value in eref if the corresponding attribute is not present in the file. * function ps_Projector_Value(ps,i,r) result(val) §.§.§ Pseudo Wavefunctions[style=customrnc] PseudoWaveFunctions =element pseudo-wave-functions attribute_set,Annotation?,Grid?,PseudoWf+PseudoWf =element pswf attribute_l, attribute_j ?, attribute_n, attribute energy_levelxsd:double ?, RadfuncAgain, the same strategy:* ps_PseudoWavefunctions_Filter(ps,indexes_in,l,j,set,indexes,number)* ps_PseudoWf_Get(ps,i,l,j,n,set,energy_level,annotation,func)The routine returns a very large positive value in energy_level if the corresponding attribute is not present in the file. * function ps_PseudoWf_Value(ps,i,r) result(val)§.§ Radial function and grid information[style=customrnc] Radfunc =element radfunc Grid?,# Optional grid elementelement data listxsd:double+# One or more floating point numbersGrid =element gridattribute nptsxsd:positiveInteger , Annotation?, element grid-data listxsd:double+# One or more floating point numbers In keeping with the psml philosophy of being grid-agnostic, the basic API tries to discourage the direct access to the data used in the tabulation of the radial functions. The values of the functions at a particular point r can be generally obtained through the ps_(Name)_Value interfaces, or through the ps_GetValue interface using func objects of type ps_radfunc_t.It is nevertheless possible to get annotation data for the grid of a particular radial function, or for the top-level grid, through the function* function ps_GridAnnotation(ps,func) result(annotation)If a radial function handle func is given, the annotation for that radial function's grid is returned. Otherwise, the return value is the annotation for the top-level grid. §.§ The evaluation engineIn the current version of the library the evaluation of tabulated functions is performed by default with polynomial interpolation, using a slightly modified version of an algorithm borrowed (with permission) from the oncvpsp program by D.R. Hamann <cit.>.By default seventh-order interpolation, as in oncvpsp, is used.If the library is compiled with the appropriate pre-processor symbols, the interpolator and/or its order can be chosen at runtime, but we note that this should be considered a debugging feature, as the reproducibility of results would be hampered if client codes change the interpolation parameters at will. Generator codes should instead strive to produce data tabulations that will guarantee a given level of precision when interpolated with the default scheme, using appropriate output grids on which to sample their internal data sets. For example, our own work on enabling psml output in oncvpsp (see below) includes diagnostic tools to check the interpolation accuracy.Most codes use internally a non-uniform grid (e.g. logarithmic). We have found that a good choice of output grid is a subset of the producer's working grid points that leaves out most of the very close points near the origin but maintains the rest. This can be achieved by imposing a minimum inter-point separation δ. This parameter δ can be smaller than the typical linear-grid step used currently by most codes, and still lead to smaller grids (in terms of number of points) that preserve the accuracy of the output.High-order interpolation can lead to ringing effects (oscillations of the interpolating polynomial between points), notably near edge regions when the shape of the function changes abruptly. This is the case, for example, if the function drops to zero within the interpolation range as a result of cutting off a tail. The actual interpolated values will typically be very small, but might cause undesirable effects in the client code. To avoid this problem, theevaluator works internally with an effective end-of-range that is determined by analyzing the data values after parsing.If needed for debugging purposes, the evaluator engine can be configured by the routine:* ps_SetEvaluatorOptions(quality_level,debug, use_effective_range, custom_interpolator)All arguments are optional, and apply globally to the operation of the library. The custom_interpolator argument is not allowed if the underlying Fortran compiler does not support procedure pointers. quality_level (an integer) is by default and will typically be the interpolation order, but its meaning can change with the interpolator in use. The evaluator uses an effective range by default, as discussed above, but this feature can be turned off by setting use_effective_range to . The debug argument will turn on any extra printing configured in the evaluator. By default, no extra printing is produced.Finally, in case it is necessary to look at the raw tabular data for debuggingpurposes, the library also provides a low-level routine:* ps_GetRawData(func,rg,data)It accepts a radial function handle func, and the grid points and the actual tabulated data are returned in rg and data, which must be passed as allocatable real arrays.§.§ Editing of ps structures The psml library has currently some limited support for editing the content of ps_t objects from user programs. For example, such an editing might be done by a KB-projector generator to insert a new provenance record (and KB and local-potential data) in the ps_t object, prior to dumping to a new psml file. Editing operations not yet supported directly by a given version can still be carried out by a direct handling of the internal structure of the ps_t object, which is for now also visible to client programs. * ps_RootAttributes_Set(ps,version,uuid,namespace)* ps_Provenance_Add(ps,creator,date,annotation)Annotations can be created by client programs using routines exported by the psml API (see Section <ref>). * ps_NonlocalProjectors_Delete(ps)* ps_LocalPotential_Delete(ps)Only “deletion” operations are supported as yet. §.§ Dump of ps structures The contents of a (possibly edited) ps_t object can be dumped to a psml file using the routine* ps_DumpToPSMLFile (ps,fname,indent)Here fname is the output file name, and indent is a logical variable that determines whether automatic indenting of elements is turned on (by default it is not).In principle, there could be dumpers for other file formats, but their implementation is better left to specialized programs that are clients of the library. §.§ Annotation APITo support the annotation functionality (see Sec. <ref>), the library contains a module with a basic implementation of an association list (a data structure holding key-value pairs). The library exports the ps_annotation_t type, an empty annotation object EMPTY_ANNOTATION, and the following routines:* reset_annotation(annotation)Cleans the contents of the ps_annotation_t object annotation so that it can be reused. * insert_annotation_pair(annotation,key,value,stat)Inserts the key, value pair of character variables in the ps_annotation_t object annotation. Internally, annotation can grow as much as needed. * function nitems_annotation(annotation) result(nitems)Returns the number of key-value pairs in the annotation object * get_annotation_value(annotation,key,value,stat)* get_annotation_value(annotation,i,value,stat)This routine has two interfaces. The first gets the value associated to the key, and the second gets the value associated to the i'th entry in the annotation object. * get_annotation_key(annotation,i,key,stat)Gets the key of the i'th entry in the annotation object.Together with the second form of get_annotation_value, this routine can be used to scan the complete annotation object. The first form of get_annotation_value is appropriate if the key(s) are known.In all the above routines a non-zero stat signals an error condition. § DISCUSSION: THE PSML MODEL AND ECOSYSTEMOur vision for the role of psml in addressing the interoperability and documentation problems is as follows:* Databases offer psml files (with smart searching made possible by the clear internal structure) and most codes use them directly. As they have full provenance and a uuid tag built in, calculations can properly document the pseudopotentials used. In some cases, specific legacy formats can also be produced from psml files.* psml files are produced directly by most pseudopotential generators, either de novo or through conversion of existingformats (with extra metadata added).* The psml ideas and technology are extended to include PAW datasets and ultra-soft pseudopotentials.The psml format is able to span a wide range of uses for norm-conserving pseudos, from semilocal-only data files, up to full-operator datasets, with flexibility (e.g.: cascading grids), robustness (e.g.: comprehensive exchange-correlation specification) and full provenance. It does so in an extensible manner, and we plan to adapt it to include PAW and USPP in the future.To appreciate the place of psml in relation to similar work, we can list its strengths compared to the corresponding subset of Quantum Espresso's UPF format (which however already supports PAW datasets and ultra-soft pseudopotentials):* The psml format is accompanied by a complete stand-alone processing library that eases its adoption by client codes.* Support for alternative forms of datasets in the same file (i.e., “scalar-relativistic” plus “spin-orbit” and/or “lj” form).* Support for different grids for different radial functions.* API based on interpolation that can suit any client grid, without having to adapt the client code to the dataset grid.* Full provenance specification, even when various codes are involved in different stages of the pseudopotential generation (i.e., semilocal generation followed by KB transformation).* A more complete specification of the exchange-correlation functional(s) used through a libxc-compatible scheme.* A complete and parseable specification of the valence configuration used for generation, and of the reference energies used for the projectors. The above list could serve also in a comparison of psml with the QSO XML-based norm-conserving pseudopotential format (See Ref. <cit.>).As a proof of concept of the above vision, we have modified two different atomic pseudopotential generation codes to generate psml files, and interfacedto two electronic-structure programs.The first generator is the open-source oncvpsp code implemented by D. Hamann <cit.> to generate optimized multiple-projector norm-conserving pseudopotentials. The projectors are directly stored in the psml format together with the local potential. In addition, a set of semi-local potentials, a by-product of the oncvpsp algorithm, is also included in the psml file. The patches needed to produce psml output in oncvpsp are available inthe Launchpad code development platform <cit.>. To ease the production of XML, a special library (wxml, part of the xmlf90 project maintained by one of the authors (A.G.)) <cit.> is used.The second generator enabled for psml output is the atom code, originally developed by S. Froyen, later modified by N. Troullier and J. L. Martins, and currently maintained by one of us (A. G.)within the siesta project.atom, freely distributed to the academic community, generates norm-conserving pseudopotentials in the semilocal form. We have developed a post-processing tool (psop) which takes as input the semilocal components and computes a smooth local potential and the KB projector functions in the same way as it is done within the Siesta code. These new elements, together with a new provenance record, are incorporated in a new psml file, which describes a well-defined, client-code independent and unique operator.We have thus already two different generators of psml files, their specific idiosyncrasies being describable by a common standard. Our plans are to enable psml output in other pseudopotential-generation codes. On the client side, we have incorporated thelibrary in Siesta (version 4.2, soon to be released) and Abinit (version 8.2 and higher). psml files can then be directly read and digested by these codes as described in Sec. <ref>, achieving pseudopotential interoperability between these codes, as exemplified in Sec. <ref> below.Our immediate plans include the development of a UPF-to-psml converter and to encourage the adoption ofto enable the interoperability of Siesta and the codes in the Quantum Espresso suite.We are aware that a complete deployment of the psml road-map will require the development of interfaces toin other languages, such as C and Python, which are in progress.§ INTEROPERABILITY EXAMPLE: LOCAL-ORBITAL AND PW CALCULATIONS WITH THESAME PSEUDOPOTENTIAL We present in this section two examples of interoperability between siesta and abinit using psml files: (i) the test of the convergence of a numerical atomic orbital basis setwith respect to the asymptotic limit of a converged basis of plane waves, and (ii) the equation-of-state (energy versus volume profiles) for elemental crystals, a test that has been proposed as a benchmark for the comparison of different codes <cit.>.Four paradigmatic systems are chosen as a testbed: a standard semiconductor (bulk Si in the diamond structure),an sp-metal (bulk Al in the fcc structure),a noble metal (bulk Au in the fcc structure),and a 3d ferromagnetic transition metal (bulk Fe in the bcc structure). The reference electronic configurations, cutoff radii, corecorrections flags,and other parameters required to generate the pseudopotentialswith the oncvpsp code (version 3.3.0) are taken from the Pseudo-Dojo database <cit.>, and summarized inTable <ref>. The pseudopotentials for Au and Fe include the semicore states (5s and 5p for Au, and 3s and 3p for Fe) in the valence. For the Troullier-Martins pseudopotentials generated with the atomcode, we use the parameters reported in Table <ref>.The rest of the technical details of the calculations, that are common toboth siesta and abinit simulations, are given inTable <ref>. In siesta, the electronic density, Hartree, and exchangecorrelation potentials, as well as the corresponding matrix elementsbetween the basis orbitals, were calculated on a uniform real spacegrid, controlled by an energy cutoff <cit.> which was wellconverged at values of 300 Ry for Si, 400 Ry for Al and Fe, and 600Ry for Au. The reader has to keep in mind that these cutoffs are notdirectly comparable to plane-wave cutoffs, and that they controlthe operation count of stages of the calculation which typicallyrepresent a small fraction of the total computation time. Figure <ref> shows the convergence of the total energyfor Al, and Fe, using atom-generated pseudopotentials, and for Au using oncvpsp pseudopotentials, as afunction of the basis-set quality. For abinit, the latter can simply be measured by the plane-wave energy cutoff. For siesta,the hierarchy and the nomenclature of the basis sets of numerical atomicorbitals is described in Ref. <cit.>. For a given tier within the basis hierarchy, the NAO basis sets used to produce the results of Fig. <ref> were generated using the defaultparameters implemented in siesta. The only exceptions are those marked with an“opt” suffix for the atom pseudopotentials,that were optimized following the recipe given in Ref. <cit.>, and are available on the siesta web page <cit.>. The “optimized” NAO basis sets of Au and Fe with the oncvpsp pseudopotentials were generated following the automaticprocedure implemented in siesta with the cutoff radii for all the shells defined by a unique parameter: the energy shift <cit.>(0.005 Ry for Au and 0.002 Ry for Fe). The calculations are made at lattice constants of 3.97 Å for Al, 2.87 Å for Fe, and 4.16 Å for Au. Although the convergence of NAO results is not a priori systematicwith respect to the size of the basis, the sequence of bases presentedin Fig. <ref> shows a uniform convergence of thetotal energy with respect the basis size.This is specially remarkable for fully optimized basis sets, such as the one used withthe DZP quality for Fe and Al, where several eV can be gained. But significant reductions in the total energy can be obtained simplyby tuning a reduced subset of the parameters that define the atomic orbitals. This is exemplified here in the case of metallic bulk Au, where the total energy is lowered by almost 2 eV increasing the range of the atomic orbitalsfor the same DZP size of the basis. In any case, we can observe how the polarization orbitalsare important for convergence, much more so than doubling the basis set. A basis of relatively modest size (DZP) is equivalent, from thetotal energy point of view, to the PW basis cutoff which would be used in realistic calculations (9 Ha for Al, 35 Ha for Fe, or 23 Ha for Au). It is important to stress that, when using the same pseudopotential,the total energies are given with respect to the same reference and,therefore, can be directly compared. The use of a commonpseudopotential format thus allows a more straightforward and detailed analysisof convergence.In related work, a comparison of energy differences, ionic forces andaverage pressures for water monomers, dimers, two phases of ice andliquid water at ambient and high density have been presented inRef. <cit.>. They were obtained with siesta andabinit using the same pseudopotentials (with an early versionof the psml framework).Highest order bases are shown to giveaccuracies comparable to a plane-wave kinetic energy cutoff of around1000 eV.Figure <ref> shows the comparison of the equationsof state for bulk Si, Al, Fe and Au, computed with a basis set ofdouble-zeta polarized NAOs and for a PW basis set of comparable quality(see Fig. <ref>). Both the position of the minimumenergy and the curvature of the energy as a function of volume are verysimilar, indicating that for the same quality of the basis we canobtain essentially the same structural information.The distance between the two curves, for the volume range plotted in Fig. <ref>, can be further quantified using the delta-factor<cit.>. Taking the plane waveequation of state as a reference, we quantify the delta valuesthat are reported in Table <ref>.In every case,the delta factor is smaller than 1 meV/atom, demonstratingthe excellent agreement obtained between the two codes, andhighlighting the level of interoperability achievable. § CONCLUSIONSWe have presented the psml norm-conserving pseudopotential file format and the associated open sourcelibrary for parsing and data handling. psml is based on XML and implements provenance and flexibility in a widely applicable and extensible format. We demonstrate its potential for enablinginteroperability among electronic-structure codes by comparing results from a plane wave (abinit) and an atomic orbital code (siesta), using the same input. We find a systematic convergence in absolute values of energies, and a delta factor of less than 1 meV.§ ACKNOWLEDGEMENTSWe thank Xavier Gonze in particular for backing this project and providing useful comments. Many constructive discussions are acknowledged with Don Hamann, Matteo Giantomassi, Michiel Van Setten, Paolo Giannozzi, Gian-Marco Rignanese, and François Gygi.This work was supported by CECAM through the Electronic Structure Library (ESL) initiative and the ETSF through the libpspio project.MJV acknowledges support from ULg and CfWB through ARC projects AIMED and TheMoTherm (GA 15/19-09 and 10/15-03) and a FNRS PDR project (GA T.1077.15-1/7).A.G. was funded by EU H2020 grant 676598 (“MaX: Materials at the eXascale” CoE), Spain's MINECO (grants FIS2012-37549-C05-05 and FIS2015-64886-C5-4-P, and the “Severo Ochoa” Program grant SEV-2015-0496), and GenCat (2014 SGR 301).JJ and YP acknowledge support from Spain's MINECO (grants RTC-2016-5681-7 and FIS2015-64886-C5-2-P).10 url<#>1urlprefixURL href#1#2#2 #1#1Martin R. M. Martin, Electronic Structure. Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 2004.Kohanoff J. 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Solid State 39 (2014) 1–24.Kittel C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1986. | http://arxiv.org/abs/1707.08938v1 | {
"authors": [
"Alberto García",
"Matthieu Verstraete",
"Yann Pouillon",
"Javier Junquera"
],
"categories": [
"physics.comp-ph",
"cond-mat.mtrl-sci"
],
"primary_category": "physics.comp-ph",
"published": "20170727171908",
"title": "The PSML format and library for norm-conserving pseudopotential data curation and interoperability"
} |
M. I. Krivoruchenko^1,2,3 December 30, 2023 =============================§ INTRODUCTION Being a common tools to model complex systems with randomness, stochastic partial differential equations have attracted a considerable attention. The number of articles in this area is constantly growing, and we mention only few of them which consider subjects closely related to our work; further references are available therein. The sample properties of wave equations with Gaussian random noise were studied in <cit.>. Several authors investigated equation with square integrable Lévy noise, where the same martingale methods are applicable, see <cit.> and references therein. However, in the heavy-tailed situation, there are only few results. Articles <cit.> are devoted to heat equations with stable noise. To the best of our knowledge, there are no articles studying the wave equation with a heavy-tailed noise, neither there are attempts to consider equations with coloured stable noise.It is worth to mention that some authors investigated stochastic partial differential equations with a general stochastic measure, where, in particular, no assumptions on the integrability of measure are made. For example, the heat equation with a general stochastic measure was considered in <cit.>, the wave equation, in <cit.>. However, in these works only few results are proved and in rather restricted cases; it is probably the generality of exposition what leads to such restrictions. In this article we continue our research started in <cit.>, where a planar wave equation was studied, and <cit.>, which, as the present work, was devoted to wave equation in ^3. The main object of our study is the wave equation{( ∂ ^2/∂ t^2- a^2 Δ)U(x,t)=Ż^H(x), x∈^3, t>0,U(x,0)=0, ∂ U/∂ t(x,0)= 0, .where the source is a spatial random noise with symmetric α-stable distribution: it is a derivative Ż^H(x) of a real anisotropic harmonizable fractional stable field Z^H. Thus, the random noise is “coloured” in the sense that its increments are not independent. To give a meaning to this equation requires defining a random measure generated by Z^H and an integral with respect to this measure. As far as we know, such questions did not appear in the previous literature. The rest of article is organized as follows. Section 2 contains preliminary information on stable random variables, measures and integrals. In Section 3, we recall the definition and properties of real anisotropic harmonizable fractional stable field, construct a random measure generated by this field, define integrals with respect to this measure and study their properties. In Section 4 we prove results on the sample path properties of the solution to (<ref>). § PRELIMINARIESThroughout the paper, C will denote a generic constant; its value may vary between lines. Random constants will be denoted by C(ω).In this article we will consider symmetric α-stable (S α S) random variables. In this section wewill provide essential information about them; additional detail may be found in<cit.>. For α∈(0,2), a random variable ξ is called Sα S with scale parameter ||ξ||_α if its characteristic function is [e^iλξ]=e^-λ||ξ||_α^α, λ∈. A crucial role in the construction of processes and fields with stable distribution is played by independently scattered Sα S random measure; in this article it is enough to consider a measure on ^3. This is a function M: ℬ_f(^3)×Ω→, where ℬ_f(^3) is a family of Borel sets of finite Lebesgue measure, having the following properties: * for any A ∈ℬ_f(^3), the random variable M (A) isSα Swith scale parameter equal to λ(A) the Lebesgue measure of A; * for any disjoint sets A_1, … ,A_n ∈ℬ_f(^3), the values M(A_1),… ,M(A_n)are independent; * for any disjoint sets A_1, A_2,…∈ℬ_f(^3) such that ⋃_n=1^∞ A_n∈ℬ_f(^3), the series ∑_n=1^∞ M(A_n) converges almost surely and M(⋃_n=1^∞ A_n)=∑_n=1^∞ M(A_n) almost surely.For a function f ∈ L^α(^3), the integral I(f)=_^3f(x)M(dx)is defined as a limit in probability of integrals of finitely supported simple functions, and there is an isometric property:I(f)^α_α=_^3|f(x,t)|^α dx. A convenient tool to study stable random variables is the LePage series representation, defined for the measure M as follows. Let φ be any positive continuous probability density function on ^3, and {Γ_k,k≥1}, {ξ_k,k≥1}, {g_k,k≥1} be independent families of random variables satisfying * {Γ_k,k≥1} is a sequence of arrivals of a Poisson process with unit intensity; * {ξ_k,k≥1} are iid vectors in ^3 with density φ; * {g_k,k≥1} are iid centered Gaussian variables with[|g_k|^α]=1.Then {M(A),A∈ℬ_f(^3)} has the same distribution as M'(A)=C_α∑_k≥1Γ_k^-1/αφ(ξ_k)^-1/α1_A(ξ_k)g_k, A∈ℬ_f(^3),where C_α=(Γ(2-α)cosπα/2/1-α)^1/α; the series convergesalmost surely for any A∈ℬ_f(^3).There is also the LePage series representation for integrals: for any f_1,f_2,…, f_n ∈ L^α(^3) the vector (I(f_1), I(f_2), … , I(f_n)) has the same distribution as (I'(f_1), I'(f_2), … , I'(f_n)), whereI'(f)=C_α∑_k≥1Γ_k^-1/αφ(ξ_k)^-1/α f(ξ_k)g_k.Further we will assume without loss of generality that M is given by(<ref>) with φ(x)=∏_l=1^3K/|x_l| (|log|x_l| |+1)^1+η,where η>0 is a fixed number, and K=(∫_-∞^+∞|x|^-1(|log|x| |+1)^-1-ηdx)^-1 is the normalizing constant. Respectively, the integral I(f)=_^3f(x)M(dx) is given by (<ref>).We will need several auxiliary results. For convenience, we reformulate them to suit our needs. [<cit.>] Let X_n = I(f_n) with some f_n∈ L^α(^3), n≥ 1. Then I(f_n)𝖯⟶ I(f), n→∞,if and only if _^3 |f_n(x) - f(x)|^α dx→ 0, n→∞.Assume that α∈ (1,2). For𝐓∈ℬ (^m) and measurablef_t(x): 𝐓×^3→defineX(t)=_^3f_t(x)M(dx), t∈𝐓.If∫_𝐓|X(t)|dt<∞almost surely, then∫_𝐓X(t)dt=_^3∫_𝐓f_t(x)dtM(dx)almost surely. A sufficient condition for (<ref>) to hold is∫_𝐓(_^3|f_t(x)|^α dx)^1/αdt<∞.This is a compilation of Theorem 3.3 and Theorem 4.1 from <cit.>. The measurability of X, assumed in these theorems, in our case is a consequence of the series representation <ref>.§ REAL ANISOTROPIC HARMONIZABLE FRACTIONAL STABLE FIELD The real anisotropic harmonizable fractional stable field with Hurst parameter H∈(0,1) is defined asZ^H(x)=_^3∏_l=1^3e^ix_ly_l-1/|y_l|^H+1/αM(dy)for technical reasons we will restrict ourselves to the case α∈(1,2), H∈ (1/2,1). The properties of Z^H were established in <cit.>, in <cit.> its multifractional was defined and investigated. In particular, it was proved in the cited articles that this field has a modification, which is locally Hölder continuous of any order β∈ (0,H). Thanks to our assumptions about M, Z^H can be represented as its LePage series: Z^H(x)=C_α∑_k=1^∞Γ_k^-1/α∏_l=1^3e^ix_lξ_k,l-1/|ξ_k,l|^H+1/αφ(ξ_k)^-1/αg_k, φ(x)=∏_l=1^3K/|x_l| (|log|x_l| |+1)^1+η.Our aim is to define a random measure corresponding to Z^H(x). Towards this end, we will use the following heuristic reasoning. DifferentiateZ^H(x) informally:∂^3/∂ x_1∂ x_2 ∂ x_3Z^H(x)=(-i_^3e^i(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy))=_^3sin(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy). Note that the integral is not well defined in general, but it allows to define the random measure corresponding to Z^H as a result of formal differentiation and change of order of integration:Z^H(A)=_A∂^3/∂ x_1∂ x_2 ∂ x_3Z^H(x) dx=_A_^3sin(x,y)∏_l=1^3 y_l /|y_l|^H+1/α-1M(dy)dx=_^3∏_l=1^3 y_l /|y_l|^H+1/α-1_Asin(x,y) dyM(dz).In other words, we setZ^H(A)=_^3_Asin(x,y)dx∏_l=1^3 y_l /|y_l|^H+1/α-1M(dy)by definition.For any A∈ B_f(^3), the integral in (<ref>) is well defined.Denotef_A(y)=_Asin(x,y)dx∏_l=1^3 y_l /|y_l|^H+1/α-1.We need to show that _^3|f_A(y)|^α dy <∞.Note that f_A(y)=A^H1_A(y), where A^H is defined in (<ref>). Since 1_A(y) ∈ L^1(^3)∩ L^2(^3), the finiteness of integral follows from Proposition <ref>.In <cit.> we have proposed to define the integral with respect to Z^H as follows. For ε>0 define smooth approximations of Z^H byZ^H,ε(x)=_^3sin(x,y)e^-ε^2|y|^2/2∏_l=1^3 y_k,l/|y_k,l|^H+1/α-1M(dy).It was proved in <cit.> that there exists a weak derivative ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H, ε(x) and, in view of this,the following definition was proposed:_^3f(x)Z^H(dx)=lim_ε→ 0+_^3f(x)∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x) dx,provided that the limit in probability exists. Let us show that (<ref>) agrees with this definition.For any A∈ℬ_f(^3), the following convergence in probability holds:Z^H(A)=lim_ε→ 0+_AZ^H,ε(x)dx.Denotef_A,ε(y)= e^-ε^2|y|^2/2 f_A(y)= e^-ε^2|y|^2/2∏_l=1^31/|y_l|^H+1/α-1_Asin(x,y)dxandZ^H,ε(A)=_^3_Asin(x,y)e^-ε^2|y|^2/2∏_l=1^31/|y_l|^H+1/α-1dx M(dy)=_^3f_A,ε(y)M(dy).Let us first prove the possibility to change the order of integration in(<ref>). To this end, according to Theorem <ref>, it is enough to show that_A(_^3|sin(x,y)|^α e^-αε^2|y|^2/2∏_l=1^31/|y_l|^α(H-1)+1dy)^1/αdx<∞.Estimate_^3|sin(x,y)|^α e^-αε^2|y|^2/2∏_l=1^3dy/|y_l|^α(H-1)+1 ≤_^3(1∧|x||y|)^α e^-αε^2|y|^2/2∏_l=1^3dy/|y_l|^α(H-1)+1:=I.Through the spherical change of variables y_1=ρsinθcosν; y_2=ρsinθsinν; y_3=ρcosθ; ρ>0, θ∈ [0,π], ν∈ [0,2π],we getI=∫_0^∞∫_0^π∫_0^2π(1∧|x|ρ)^αe^-αε^2ρ^2/2ρ^2(sinθ)^2α(H-1)-1dρdθ dφ/ρ^3α(H-1)+3(cosφsinφcosθ)^1-α(H-1) ≤ C∫_0^∞(1∧|x|ρ)^α e^-αε^2ρ^2/2ρ^3α(1-H)-1dρ<∞.Therefore, _A ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x)dx=_^3f_A,ε(y)M(dy).Now _^3|f_A(y)-f_A,ε(y)|^α dy ≤_^3|e^-αε^2|y|^2/2-1||f_A(y)|^α dy→ 0, ε→ 0,in view of the dominated convergence theorem, since ∫_^3|f_A(y)|^α dy<∞ by Theorem <ref>. Therefore, by Proposition <ref>, _A ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x)dx 𝖯⟶Z^H(A), n→∞.Now we will prove that Z^H(A) is a stochastic measure, that is, that is a σ-additive in probability function of A∈ℬ_f(^3). From the definition and linearity of the integral w.r.t. M it is clear that Z^H(A) is additive. Therefore, it suffices to prove only continuity at zero, which is done in the following proposition.Let {A_n, n≥ 1}⊂ B_f(^3) be such that for any n ≥ 1, A_n+1⊂ A_n, and ∩_n=1^∞ A_n=∅. Then Z^H(A_n)𝖯⟶ 0, n→∞.Thanks to Proposition <ref>, we need to show that_^3|f_A_n(y)|^α dy→ 0, n→∞.This follows immediately from Proposition <ref> and the continuity of the Lebesgue measure, as||f_A_n||_L^1(^3) + ||f_A_n||_L^2(^3)≤λ(A_n) + λ(A_n)^1/2. {Z^H(A), A ∈ B_f(^3)} is a stochastic measure. We turn now to integration with respect to the measure Z^H. Since its increments are dependent, the standard integration theory, as described in Section 2, is not available. Nevertheless, we can proceed in a standard way. For a simple functiong(x)=∑_k=1^na_kI_A_k(x)with A_k∈ B_f(^3), k=1,…,n, defineI^H(g)=_^3g(x)Z^H(dx)=∑_k=1^na_kZ^H(A).Observe that in this caseI^H(g)=_^3∑_k=1^na_k f_A_k(y)M(dy)=_^3 A^Hg(y)M(dy),where A^H is given by (<ref>). Then it is natural to define for arbitrary gI^H(g)=_^3f(x)Z^H(dx)=_^3A^Hg(y)M(dy).Thanks to Proposition <ref>,this is well defined for any g ∈ L^1(^3) ∩ L^2(^3). Moreover, the map I^H is continuous in the following sense.Let {g_n,n≥ 1}⊂ L^1(^3)∩ L^2(^3) be such that g_n→ g, n→∞, both in L^1(^3) and in L^2(^3). ThenI^H(g_n)𝖯⟶I^H(g), n→∞.From linearity of operator A^H and that of integral w.r.t. M we haveI^H(g_n)-I^H(g)=I^H(g_n-g).Therefore, it is enough to prove thatI^H(g_n-g)→ 0, n𝖯⟶∞,which follows from Propositions <ref> and <ref>.In particular, the integral I^H(f) = ∫_^3 f(x)Z^H(dx) may be equivalently defined as a limit in probability of integrals of simple functions approximating f in L^1(^3)∩ L^2(^2). Another important observation is that our definition agrees with that given in <cit.>. For any f∈ L^1(^3)∩L^2(^3), the convergence in probability (<ref>) holds. The proof is the same as in Theorem <ref>.§ WAVE EQUATION WITH COLOURED STABLE NOISELet us return to the wave equation with coloured Sα S noise:{( ∂ ^2/∂ t^2- a^2 Δ)U(x,t)=Ż^H(x), x∈^3, t>0,U(x,0)=0, ∂ U/∂ t(x,0)= 0. .This equation was already studied in <cit.>, wherein we proved that its candidate solution given by Kirchhoff's formulaU(x,t)=1/4π a_y:|x-y|<at1/|x-y|Z^H(dy)is a weak (generalized) solution, that is for any θ (x,t)∈ C_fin^∞(^3×^+) almost surely it holds that∫_0^∞_^3U(x,t)(∂^2/∂ t^2θ (x,t)-a^2Δθ (x,t))dx dt=∫_0^∞_^3θ (x,t)Z^H(dx)dt.In <cit.>, the integral in (<ref>) was understood in the sense (<ref>). Nevertheless, thanks to Theorem <ref>, this agrees with the definition (<ref>) taken in the present paper. Indeed, definingf_t,x(y)=1/4π a|x-y|I_|x-y|<atso thatU(x,t)=_^3f_x,t(y)Z^H(dy),we have f_t,x∈ L^1(^3)∩ L^2(^3). As a result, U(x,t)=_^3A^H f_x,t(y) M(dy)=1/a_^3sin(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-11-cos at|y|/|y|^2M(dy)(the expression for A_H f_x,t(y) is computed in <cit.>).§ SAMPLE PROPERTIES OF SOLUTION TO WAVE EQUATION WITH COLOURED STABLE NOISE Write the random field U(x,t) as its LePage series: U(x,t)=C_α/a∑_k=1^∞Γ_k^-1/αφ(ξ_k)^-1/α∏_l=1^3ξ_k,l/|ξ_k,l|^H+1/α-1sin(x,ξ_k)1-cos at|ξ_k|/|ξ_k|^2g_k.For notational simplicity assume that the underlying probability space has the following structure:(Ω, ℱ, 𝖯)=(Ω_Γ×Ω_ξ×Ω_g, ℱ_Γ⊗ℱ_ξ⊗ℱ_g, 𝖯_Γ⊗𝖯_ξ⊗𝖯_g),and for all ω= (ω_Γ,ω_ξ,ωg), k≥ 1 : Γ_k(ω)=Γ_k(ω_Γ), ξ_k(ω)=ξ_k(ω_ξ), g_k(ω)=g_k(ω_g).1. The random field U has a modification, which is γ-Hölder continuous in t and locally in X for any γ∈ (0,(3H-1)∧ 1). Moreover,for any δ>0 this modification satisfiessup_|x'|, |x”|≤ R,t',t”∈ [0,T]|x'-x”|≤ h, |t'-t”|≤ h|U(x',t')-U(x”,t”)|≤ C(ω)h^(3H-1)∧ 1|log h|^3/α-1/2+δfor all R>0 and all h>0 small enough.2. If additionally H ∈(2/3, 1), then this modification is absolutely continuous in each variable. 1. Estimate_g[|U(x',t')-U(x”,t”)|^2] ≤ 2 (_g[|U(x',t')-U(x”,t')|^2]+_g[|U(x”,t')-U(x”,t”)|^2])Let h ∈(0,1/2). Definea_1(h)=sup_ t ∈ [0,T], x',x”∈^3|x'-x”|≤ h _g[|U(x',t)-U(x”,t)|^2],a_2(h)=sup_|x|≤ R, t',t”∈ [0,T] |t'-t”|≤ h_g[|U(x,t')-U(x,t”)|^2].Using the LePage representation, estimate_g[|U(x',t)-U(x”,t)|^2] ≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α|1-cos at|ξ_k||^2/|ξ_k|^4|sin (x',ξ_k) - sin(x”,ξ_k)|^2× ×∏_l=1^31/|ξ_k,l|^2H+2/α-2.From simple inequalities|1-cos at|y||/|y|^2≤ 1∧|y|^-2, |sin (x',ξ_k) - sin(x”,ξ_k)|≤ 2∧(|y|· |x'-x”|)we get the following estimate:a_1(h)≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α(1∧|ξ_k|^-4)(4 ∧ (|ξ_k|^2 h^2)) ×∏_l=1^31/|ξ_k,l|^2H+2/α-2 := a^-2 C_α^2∑_k=1^∞Γ_k^-2/αQ(h,ξ_k).Consider_ξ[Q(h,ξ_k)]=_^3φ(y)^1-2/α(1∧|y|^-4) (4∧ (|y|^2 h^2))∏_l=1^31/|y_l|^2H+2/α-2dy=K^3-6/α_^3(1∧|y|^-4) (4∧ (|y|^2 h^2))∏_l=1^3dy/|y_l|^2H-1(|log|y_l||+1)^(1+η)(1-2/α).Now we make the spherical change of variables (<ref>) and estimate the logarithms as(|log|ρ a(ν,θ)||+1)^d≤(|logρ|+ |log|a(ν,θ)||+1)^d ≤(|logρ|+1)^d(|log|a(ν,θ)||+1)^d,where d=(1+η)(2/α-1), and a(ν,θ) is one of the functions sinθcosν, sinθsinν, cosθ. Thus we get_ξ[Q(h,ξ_k)]≤ C ∫_0^∞(1∧ρ^-4)(ρ^2h^2∧ 1)(|logρ|+1)^3d/ρ^6H-5 ×∫_0^2π∫_0^π(|log|sinθcosν||+1)^d(|log|sinθsinν||+1)^d×(|log|cosθ||+1)^d|sinθ|dθdνdρ ≤ C ∫_0^∞(1∧ρ^-4)(ρ^2h^2∧ 1)(|logρ|+1)^3d/ρ^6H-5dρ= C(h^2∫_0^1ρ^7-6H(|logρ|+1)^3d dρ +∫_1^h^-1h^2ρ^3-6H(|logρ|+1)^3ddρ +∫_h^-1^∞ρ^1-6H(|logρ|+1)^3d dρ):=C(I_1+I_2+I_3).Let us estimate the integrals individually. Obviously, I_1 ≤ C h^2. Further, for ρ∈[1,h^-1], (|logρ|+1)^3d≤(|log h|+1)^3d,whence I_2 ≤ C (|log h|+1)^3d∫_1^1/hρ^3-6H dρ≤C h^(6H-2)(|log h|+1)^3d.Finally,I_3=h^-1∫_1^∞(ρ/h)^1-6H(logρ/h+1)^3ddρ ≤ h^6H-2∫_1^∞ρ^1-6H(logρ+|log h|+1)^3d dρ ≤ h^6H-2(|log h|+1)^3d∫_1^∞ρ^1-6H(logρ+1)^3d dρ≤ C h^6H-2(|log h|+1)^3d.Collecting the estimates, we get that_ξ[ Q(h,ξ_k)]≤ Ch^(6H-2)∧ 2(|log h|+1)^3d.Define b(h)=h^(6H-2)∧ 2|log h|^3d, h ∈(0,1/2). Taking into account (<ref>), (<ref>) and the almost sure convergence of the series ∑_k=1^∞Γ_k^-2/α we get that for any ε>0_ξ[∑_n=1^∞a_1(2^-n)/b(2^-n)n^1+ε]<∞𝖯_Γ-almost surely. Hence∑_n=1^∞a_1(2^-n)/b(2^-n)n^1+ε<∞𝖯_ξ⊗𝖯_Γ-almost surely. It follows in particular that a_1(2^-n)/b(2^-n)n^1+ε→ 0, n→∞,𝖯_ξ⊗𝖯_Γ-almost surely.Note that, for positive h small enough, b(h) increases and satisfies b(2h)≤ C b(h). Consequently, (<ref>) implies that for almost all (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γa_1(h)≤ C(ω_ξ,ω_Γ) h^(6H-2)∧ 2|log h|^3d+1+εholds for all sufficiently small h>0.In order to estimate a_2(h), use the following inequality:|(1-cos at'|y|)-(1-cos at”|y|)|≤ 2∧(a|y||t'-t”|).Then a_2(h)≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α4 ∧(a^2|ξ_k|^2h^2)/|ξ_k|^4 ×(1∧ (|x|^2|ξ_k|^2))∏_l=1^31/|ξ_k,l|^2H+2/α-2.Using the same reasoning as above, we get that for almost all (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γa_2(h)≤ C (ω_ξ,ω_Γ)h^(6H-2)∧ 2|log h|^3d+1+ε.As a result, we get for almost all ω_ξ∈Ω_ξ,ω_Γ∈Ω_Γsup_A_R,h_g[|U(x',t')-U(x”,t”)|^2]≤ C(ω_ξ,ω_Γ)h^(6H-2)∧ 2|log h|^3d+1+ε,for all h small enough,whereA_R,h = {(x',x”,t',t”)∈^6×[0,T]^2: |x'|, |x”|≤ R,|x'-x”|≤ h, |t'-t”|≤ h}.For a fixed (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γ,U has a centered Gaussian distribution. Therefore (see e.g. <cit.>) there is a modification of U satisfyingsup_A_R,h|U(x',t')-U(x”,t”)|≤ C(ω)h^(3H-1)∧ 1|log h|^3d/2+1+ε/2.Recalling that d = (1+η)(2/α -1), we get the required statement by choosing ε and η sufficiently small.2. Assuming that the continuous modification of U is chosen, define g(x,y,t)=∂/∂ tA^Hf_x,t=sin(x,y)sin at|y|/|y|^2|y| ∏_l=1^3 y_l/|y_l|^H+1/α-1and V(x,t)=_^3g(x,y,t)M(dy)=_^3sin(x,y)sin at|y|/|y| ∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy).With the help of spherical change of variables (<ref>), estimate_^3|g(x,y,t)|^α dy≤_^3((1∧|x||y|)(1∧|a||t||y|)/|y| ∏_l=1^3|y_l|^H+1/α-1|y|)^α dy=∫_0^∞∫_0^π∫_0^2π(1∧|x|ρ)^α(1∧|a||t|ρ)^αρ^2sinθ/ρ^3α H+3-2α|sinθ|^2α H+2-2α ×(cosφsinφcosθ)^α-1-α Hdθdφdρ=∫_0^∞∫_0^π∫_0^2π(1∧|x|^αρ^α)(1∧|a|^α|t|^αρ^α)×(cosφsinφcosθ)^α-1-α H(sinθ)^2α-2α H-1ρ^2α-3α H-1dθdφdρ ≤ C ∫_0^∞(1∧|x|^αρ^α)(1∧|a|^α|t|^αρ^α)ρ^2α-3α H-1dρ<∞;the integral is convergent since 4α-3α H -1>-1 and2α-3α H-1<-1 for α∈ (1,2) andH∈ (2/3,1). As a result,∫_0^t(_^3|g(x,y,s)|^α dy)^1/αds<∞,so by Theorem <ref>∫_0^t V(x,s)ds=_^3∫_0^tg(x,y,s)dsM(dy)=_^3A^H f_x,t(y)M(dy)=U(x,t)almost surely. Since both sides of the last equality are continuous, and the left-hand side is absolutely continuous, then we get that, almost surely, U is absolutely continuous in t. The absolute continuity in other variables is shown similarly.§ DEFINITION AND PROPERTIES OFA^H Let g∈ L^1(^3)∩ L^2(^3). DefineA^Hg(y)=g(y)∏_l=1^3 y_l /|y_l|^H+1/α-1,where g(y)=_^3e^i(x,y)g(x)dx is the Fourier transform of g.For any α∈ (0,2), H∈(1/2,1), there exists a constant C(H,α) such that for any g ∈ L^1(^3)∩ L^2(^3), A^Hg_L^α(^3)≤ C(H,α) (g_L^1(^3) + g_L^2(^3)).We need to estimate _^3|A^Hg(y)|^α dy. Let us split the integral into two: over the unit ball B(0,1) and over its complement B(0,1)^c. To estimate the former, write _B(0,1)|A^Hg(y)|^α dy =_ B(0,1)|g(y)|^α∏_l=1^31/|y_l|^α(H-1)+1dy_l ≤g^α_L^1(^3)_ B(0,1)∏_l=1^31/|y_l|^α(H-1)+1dy_l,and the integral is finite, since H<1. To estimate the integral over B(0,1)^c, use the Hölder inequality with p=2/α, q = 2/(2-α) to get_B(0,1)^c|A^Hg(y)|^α dy =_ B(0,1)^c|g(y)|^α∏_l=1^31/|y_l|^α(H-1)+1dy_l ≤(_ B(0,1)^c|g(y)|^2dy)^α/2(_ B(0,1)^c∏_l=1^3|y_l|^α q(1-H)-qdy_l)^1/q.In the second integral make the spherical change of variables (<ref>) to obtain_B(0,1)^c∏_l=1^3|y_l|^α q(1-H)-qdy_l≤ C ∫_1^∞ρ^3α q(1-H)-3q+2dρ <∞,since the inequality 3α q(1-H)-3q+2<-1 easily transforms to α (1-2H)<0. For the first integral in (<ref>) we use the Parseval identity: _ B(0,1)^c|g(y)|^2dy ≤_^3|g(y)|^2dy=1/(2π)^3_^3(g(y))^2dy.This implies the required estimate.degruyter-plain | http://arxiv.org/abs/1707.08415v1 | {
"authors": [
"Larysa Pryhara",
"Georgiy Shevchenko"
],
"categories": [
"math.PR",
"60H15, 35L05, 35R60, 60G52"
],
"primary_category": "math.PR",
"published": "20170726125754",
"title": "Wave equation with a coloured stable noise"
} |
Max K-armed bandit: On the ExtremeHunter algorithm and beyond Mastane Achab1 Stephan Clémençon1 Aurélien Garivier2 Anne Sabourin1 Claire Vernade1 December 30, 2023 ======================================================================================= Theoriginal ImageNet dataset is a popular large-scale benchmark for training Deep Neural Networks. Since the cost of performing experiments (e.g, algorithm design, architecture search, and hyperparameter tuning) on the original dataset might be prohibitive, we propose to consider a downsampled version of ImageNet. In contrast to the CIFAR datasets and earlier downsampled versions of ImageNet, our proposed ImageNet32x32 (and its variants ImageNet64x64 and ImageNet16x16) contains exactly the same number of classes and images as ImageNet, with the only difference that the images are downsampled to 32×32 pixels per image (64×64 and 16×16 pixels for the variants, respectively). Experiments on these downsampled variants are dramatically faster than on the original ImageNet and the characteristics of the downsampled datasets with respect to optimal hyperparameters appear to remain similar. The proposed datasets and scripts to reproduce our results are available at<http://image-net.org/download-images> and<https://github.com/PatrykChrabaszcz/Imagenet32_Scripts> § INTRODUCTION Deep learning research has been substantially facilitated by the availability of realistic and accessible benchmark datasets, such as CIFAR-10 and CIFAR-100 <cit.> (and MNIST <cit.> in the 1990s). With the progress of machine learning, simple datasets lose some of their relevance, and more complex datasets/tasks become more important. While good results can be achieved on more complex datasets, such as ImageNet <cit.>, this incurs a large computational burden, making it intractable to achieve state-of-the-art performance without massive compute resources (training a strong ImageNet model typically requires several GPU months). Due to this computational expense of running experiments on the original ImageNet dataset we propose to explore cheaper alternatives that preserve the dataset's complexity.In order to check the scalability of new methods, neural architectures and hyperparameters associated with them, one might be interested in a downscaled version of ImageNet which allows for cheaper experimentation. Moreover, a lower resolution of the images would make the classification task much more difficult and would thus postpone the saturation of benchmarking results currently observed on CIFAR-10, e.g., 3% error obtained by <cit.> compared to roughly 6% obtained by a trained human <cit.>. To address this issue, we provide downsampled variants of the original ImageNet dataset and analyze results on them w.r.t. different hyperparameter settings and network sizes. We obtain surprisingly strong classification results on our downsampled variants and find qualitative results to be very similar across downsampling sizes. This suggests that these downsampled datasets are useful for facilitating cheap experimentation.The basic contributions of this report are as follows: * We make available downsampled versions of ImageNet (64× 64, 32× 32, and 16×16 pixels) to facilitate fast experimentation with different network architectures, training algorithms, and hyperparameters.* We show that different downsampling techniques yield similar results, except for a nearest neighbor approach, which performed worse in all our experiments.* Using Wide ResNets <cit.>, we obtain surprisingly good performance, matching the baseline by the pioneering AlexNet <cit.> (18.2% top-5 error) while using ImageNet32x32 (whose images have roughly 50× less pixels per image than the original ones). * We show that the range of optimal learning rates does not change much across ImageNet16x16, ImageNet32x32, and ImageNet64x64, as well as across different network widths. This could be exploited by multi-fidelity methods for architecture and hyperparameter search <cit.>. § DOWNSAMPLING IMAGENET The original ImageNet dataset consists of images released as a part of the ILSVRC-2012 classification dataset <cit.>. Each image belongs to one of 1000 object classes, with the number of training images per class varying from 732 to 1300; there are 50 validation images per class. The size of the original images varies; therefore, a preprocessing step is usually applied to scale and crop images to the size of 224 × 224 pixels.We are aware of two datasets that contain low resolution images derived from the ImageNet dataset:* Downsampled ImageNet <cit.>, like our datasets, contains all images in ImageNet, but since it was constructed for unsupervised learning, it does not provide the actual image labels and can therefore not be used for supervised learning.* TinyImageNet (available at <https://tiny-imagenet.herokuapp.com/>) contains a subset of 200 classes with 500 images per class. <cit.> suggested to use 128x128 pixelsImageNet images to evaluate various deep learningtechniques, but their dataset is not available. We downsample / resize the original images to smaller images of 32x32 pixels to form ImageNet32x32, to images of 64x64 pixels to form ImageNet64x64 and to images of 16x16 pixels to form ImageNet16x16. In contrast to TinyImageNet, we do not reduce the number of classes and number of images. All images are shuffled and then divided into 10 different files so that each file is expected to have images from all classes. The validation data is stored in a separate file, both the training and validation data points are labeled (e.g., indexed starting from 1) according to the mapping file of theImageNet devkit. Each file contains images, labels and the mean image computed over the whole training set. We keep the same format of files as the one that is commonly used for the CIFAR datasets. ImageNet16x16, ImageNet32x32 and ImageNet64x64 take 1 GB, 4 GBand 16 GB of disk space, respectively. We consider 6 different downsampling techniques available in the Pillow library[Pillow version 4.1 available at <https://python-pillow.org>]: lanczos, nearest, bilinear, bicubic, hamming, box (see Figure <ref>). In order to check the quality of the downsampled images we use them to train Wide Residual Networks (WRNs) by <cit.>, expecting that better validation errors will tend to be achieved with downsampling techniques that lose less information. § EXPERIMENTAL SETUPWe train Wide Residual Networks WRN-N-k by <cit.>, where N is the number of layers and k is a multiplicative factor for the number of filters, with k=1 corresponding to 16 filters in the first residual block; increasing k makes the network wider. We use Stochastic Gradient Descent with momentum factor 0.9, drop the learning rate by a factor of 5.0 every 10 epochs, and train up to a total budget of 40 epochs. Throughout, we show validation error rates obtained after training for 31 epochs (right after the last drop of the learning rate). Our experiments on ImageNet32x32 employ the original WRNs designed for the CIFAR datasets with 32 × 32 pixels per image. To adapt WRNs for images with 64 × 64 pixels per image as used in ImageNet64x64, we add an additional stack of residual blocks to reduce the spatial resolution of the last feature map from 16 × 16 to 8 × 8 and thus double the number of features. Analogously, for ImageNet16x16, we remove the last stack of residual blocks. For data augmentation, weflip images horizontally and concatenate them to the original images, effectively doubling the number of images per epoch. We also use random image shifts (up to 4 pixels horizontally and vertically). § RESULTSDoes the downsampling technique matter?We evaluated the six downsampling techniques described in Section <ref> using a small WRN-28-2 network and various initial learning rates LR ∈{0.001, 0.0025, 0.005, 0.01, 0.025, 0.05}. The results in Figure <ref> show that all downsampling techniques performed very similarly, except for the nearest neighbour technique which yielded the worst results for all learning rates. This observation is in line with Figure <ref> and also holds for ImageNet16x16 and ImageNet64x64 (results not shown for brevity). For all remaining experiments in this paper, we used the box method.Do conclusions drawn for cheap evaluations carry over to expensive ones?Next, we studied to which extent conclusions drawn for small networks and downsampled images carry over to larger networks and higher resolution images. This in turn determines the usefulness of these techniques for speeding up the experimental loop of architecture design and hyperparameter optimization. For this, we performed three experiments: * We studied how the results scale with the network size, more specifically, network width, defined by k. Table <ref> shows that larger k yielded better results independently of the downsampling size.Performance on our downsampled datasets was surprisingly strong; for example, on ImageNet32x32, using k=10 achieved 40.96% Top-1 validation error and 18.87% Top-5 validation error. Interestingly, this matches the original results by AlexNets <cit.> (40.7% and 18.2%, respectively) on full-sized ImageNet (which has roughly 50 times more pixels per image). Clearly, greater image resolution yielded better results (e.g., 12.64% top-5 performance for ImageNet64x64).* We studied how optimal learning rates changed across different combinations of downsampling sizes and network widths. Figure <ref> shows that the region of optimal learning rates remained similar across all our experimental setups, including networks whose space and time complexity differed by up to a factor of 100. Additionally, Figure <ref> compares performance as a function of both learning rate and width multiplier k for downsampling sizes of 32x32 and 16x16, showing qualitatively very similar results in both cases, including the interaction effect that larger values of k favor somewhat larger learning rates than smaller k. This suggests that small networks anddownsampled images may indeed facilitate faster experimentation. * We also investigated the tradeoffs of performance vs. training time resulting from different downsampling and network sizes. Figure <ref> and Table <ref> show that both mechanisms for reducing the computational cost should be considered simultaneously to achieve optimal anytime performance.An additional mechanism could be to perform warm restarts <cit.>, which was shown to substantially improve anytime performance over reductions of the learning rate at regular intervals.Since the relative ranking of learning rates was consistent across different downsampling and network sizes, we also envision that architecture and hyperparameter search methods could exploit cheap proxies of computationally more expensive setups based on varying these degrees of freedom. Possible methods for exploiting these include <cit.>.§ DISCUSSION AND CONCLUSION Our proposed downsampled versions of the original ImageNet dataset might represent a viable alternative to the CIFAR datasets while dealing with more complex data and classes. Quite surprisingly, even by greatly reducing the resolution of images to 32 × 32 pixels, one can predict image labels quite well (see also Figure <ref> and Figure <ref>). Classification of low resolution images might also be of interest when (i) data storage is important (the original ImageNet dataset is 145GB), (ii) the input images are corrupted by noise, or (iii) a small subpart of a high resolution image must be classified. We hope that the provided datasets will fill the gap between the CIFAR datasets and the full ImageNet dataset, representing a good benchmark for experimental studies, such as algorithm design, neural network architecture search and hyperparameter optimization. Our preliminary experiments support the hypothesis that findings obtained on smaller networks for lower resolution images may transfer to larger networks for higher resolution images, while being up to 100 times cheaper to obtain. This could be exploited by multi-fidelity methods for architecture and hyperparameter search <cit.>.§ ACKNOWLEDGEMENTThis work has partly been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant no. 716721 and by the German Research Foundation (DFG), under the BrainLinksBrainTools Cluster of Excellence (grant number EXC 1086).The authors acknowledge support by the High Performance and Cloud Computing Group at the Zentrum für Datenverarbeitung of the University of Tübingen, the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 37/935-1 FUGG. plainnaticlr2017_conference | http://arxiv.org/abs/1707.08819v3 | {
"authors": [
"Patryk Chrabaszcz",
"Ilya Loshchilov",
"Frank Hutter"
],
"categories": [
"cs.CV",
"cs.LG"
],
"primary_category": "cs.CV",
"published": "20170727112222",
"title": "A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets"
} |
RPA theory for two charged sequences]Charge Pattern Matchingas a “Fuzzy” Mode of Molecular Recognition for the Functional Phase Separations of Intrinsically Disordered Proteins Department of Biochemistry,University of Toronto, and Molecular Medicine, Hospital for Sick Children, Toronto, Ontario, CanadaDepartments of Molecular Genetics, Biochemistry, and Chemistry, University of Toronto, Toronto, Ontario, CanadaMolecular Medicine, Hospital for Sick Children,and Department of Biochemistry,University of Toronto, Toronto, Ontario, CanadaDepartments of Biochemistry and Molecular Genetics,University of Toronto, Toronto, Ontario, Canada [email protected] 30, 2023 Biologically functional liquid-liquid phase separation of intrinsically disordered proteins (IDPs) is driven by interactions encoded by their aminoacid sequences. Little is currently known about the molecular recognitionmechanisms for distributing different IDP sequences intovarious cellular membraneless compartments. Pertinent physicswas addressed recently by applyingrandom-phase-approximation (RPA) polymer theory to electrostatics, which is a major energetic component governing IDP phase properties.RPA accounts for charge patterns and thus has advantages over Flory-Huggins and Overbeek-Voorn mean-field theories. To make progress toward deciphering the phase behaviors of multiple IDPsequences, the RPA formulation for one IDP species plus solvent is herebyextended to treat polyampholyte solutions containing two IDP species.The new formulation generally allows for binarycoexistence of two phases, each containing a different set of volume fractions (ϕ_1,ϕ_2) for the two different IDP sequences.The asymmetry between the two predicted coexisting phases withregard to their ϕ_1/ϕ_2 ratios for the two sequences increases with increasing mismatch between their chargepatterns.This finding points to amultivalent, stochastic, “fuzzy” modeof molecular recognition that helps populate various IDP sequences differentially into separate phase compartments. An intuitiveillustration of this trend is provided by Flory-Hugginsmodels, whereby a hypothetical case of ternary coexistence is also explored.Augmentations of the present RPA theory with a relative permittivityϵ_ r(ϕ) that depends on IDP volume fractionϕ=ϕ_1+ϕ_2lead to higher propensities to phase separate, in line with the case with one IDP species we studied previously.Notably, the cooperative, phase-separation-enhancing effects predicted bythe prescriptions for ϵ_ r(ϕ) we deem physically plausibleare much more prominent than that entailed by common effective mediumapproximations based on Maxwell Garnett and Bruggeman mixing formulas.Ramifications of our findings on further theoretical developmentfor IDP phase separation are discussed.§ INTRODUCTION Nearly two decades of increasingly intensive research established that intrinsically disordered proteins/protein regions (abbreviated collectively as IDPs here) serve many important biological functions, and are especially critical for signaling and regulationin multicelluar organisms <cit.>. Recently, it was discovered that IDPs function not only at the level ofindividual molecules. An expanding repertoire of IDPs have been seen to undergoliquid-liquid phase separation in vitro, intriguingly parallelling theformation of many types of condensed liquid/gel-like bodies/organizations inliving organisms, including extracellular materials, transcription complexes, nucleating sites of intermediate filament organization, and various membraneless organelles. It is apparent from these experimental observations that IDP condensation constitutes a majorphysical underpinning of these condensed bodies, which serve ashubs for specifically regulated sets of biomolecules to interact.As such, IDP phase separation is one of Nature's means to achieve the spatial and temporal compartmentalizationnecessary for the organization of vital processes <cit.>. Although much detail remains to be ascertained, examples of membraneless organelles and an IDP species whose phase separation has been found to be a likely contributor totheir assembly include chromatoid bodies, nuage orgerm granules in mammalian male germ cells <cit.> and the DEAD-box RNA helicase Ddx4 <cit.>,the Caenorhabditis elegans germline-specific perinuclear RNA granules known as P-granules <cit.> and the Ddx3 RNA helicase LAF-1 <cit.>, as well as the stress granules triggered by integrated stressresponse <cit.> and the RNA-binding heterogeneousnuclear ribonucleoprotein A1 (hnRNPA1) <cit.>. Because of the importance of these bodies to biological regulation, malfunctioning of the corresponding IDP phase separationprocesses can lead to deregulation and diseases, including cancer due to loss of regulation of stressgranules <cit.>, proteinfibrillization and thus amyloid diseases <cit.>, and various forms of neurological disorder <cit.>. §.§ Seeking “sequence-phase” relationships With the advent of IDPs, the molecular biology paradigm of seeking“sequence-structure” relationships for globular proteins has to be expanded to encompass “sequence-ensemble” relationships for individualIDPs <cit.>.Now, the additional question we need to ask is: How do the phase behaviors of IDPs depend on their amino acid sequences? In otherwords, what are the “sequence-phase” relationships <cit.>?Although computational study of IDPs is still in its infancy,much insight into the conformational properties and binding energetics ofindividual IDPs has been gained by explicit-chain simulations <cit.>. In contrast, because IDP phase separation is a multiple-chainproperty, computationally it is extremely costly to simulate using fully atomic explicit-chain models <cit.>, notwithstanding promising progress made by coarse-grained approaches that treat groupsof amino acid residues of IDPs as interaction modules in continuum space <cit.> or on lattices <cit.> andsimulation algorithms developed recently <cit.>for phase separations of globular proteins <cit.>. In this context, analytical theories of IDP phase separation are valuable not only because of their tractability, but also—and more importantly—for the conceptual framework they offer for understanding a highly complexphenomenon. §.§ Mean-field and random-phase-approximation (RPA) theories of phase coexistence Developed mainly for synthetic polymers at its inception, the basic statistical mechanical framework of Flory-Huggins (FH)theory <cit.>is useful for describing phase separation in the biomolecularcontext <cit.>. FH assumes that the interactions among the monomers (residues) of the polymers have short spatial ranges. In view of the fact that not only short-range interactions such as solvent-mediated hydrophobic effects butlong-range Coulomb forcesare important for driving the phase separation of certain IDPs, it has beensuggested <cit.> that the Overbeek-Voorn (OV)theory <cit.> should be more appropriate inthose cases. Inasmuch as sequence dependence is concerned, however, both FH and OV are mean-field theories that account only for compositionbut not sequence information. In these theories, all residuesbelonging to any given set of chain sequences are allowed to interact on equalfooting irrespective of the correlation dictated by chain connectivity. Therefore, to address sequence specificity of IDP phase separation, one needs to go beyond FH and OV. Accordingly, we recently put forth arandom-phase approximation (RPA) theory that approximately accountsfor the effects of arraying different charge patterns along the IDP chainsequence <cit.>. The term “RPA” was first introduced by Bohm and Pines in their quantum mechanical collective description of electron interactions <cit.>. The analogy of this approach with the approximate polymer theory that considersterms up to quadratic in particle density was recognized by de Gennes<cit.>. As detailed elsewhere <cit.>,the RPA formulation we developed <cit.>, which follows largely thatof Olvera de la Cruz<cit.>,is successful in providing a physicalrationalization <cit.> for the experimental saltdependence of Ddx4 phase separation as well as the difference in phasebehavior between the wildtype and a charge-scrambled variant ofDdx4 <cit.>. Our RPA theory suggests further that the tendency for acollection of IDP chains with a given charge sequence to phase separate isstrongly—but negatively—correlated with the conformationaldimensions of individual IDP molecules of the sequence <cit.>, and that both of these properties are well correlated with thecharge pattern parameter κ of Das and Pappu <cit.> and the “sequence charge decoration” parameter SCD of Sawle andGhosh <cit.>. In principle, these predictions are now testableusing experimental techniques similar to thoseemployed to study “IDP polymers” <cit.>. §.§ Sequence-dependent multiple-component IDP phase separation Membraneless organelles are complex functional units consisting of many protein and nucleic acid components <cit.>.Different types of such units are enriched with different varieties ofproteins and nucleic acids. Some individual membraneless organelleshave mesoscopic substructures with different degrees of fluidity, as inthe case of stress granules <cit.>. In a similar vein, immiscibleliquid phases have been shown to contribute to subcompartmentalizationof the nucleolus <cit.>. Clearly, a viable spatial organization of cellular processes necessitates a heterogeneous distribution of differentbiomolecular components into different membraneless organelles andtheir substructures, rather than having all IDP species condensing into a big gemisch. How is this achieved physically?One aspect of this question was addressed recently using a simple cubic lattice model of multicomponent mixtures confined to a 6× 6× 6box, with each component represented by a bead on thelattice <cit.>. By considering hypothetical intercomponentinteraction strengths (contact energies), the authors found that withsufficient heterogeneity in contact energies, demixed domains are likelyto segregate. This and other results of this big-picture study suggest that phase separation into cellular compartments with different compositions is a robust consequence of interaction heterogeneity among biomolecules <cit.>.With this in mind, the logical next step in our pursuit of sequence-phase relationships is to ascertain how interaction heterogeneity is encoded genetically. For globular protein folding, the fact that the amino acidalphabet is finite <cit.> implies that there are physical limits tointeraction heterogeneity and structural encodability, as has been illustrated by simple exact models <cit.>. Similarly, physical limits should exist in the ability of different IDP sequencesto demix. Taking a step toward deciphering what is physically achievable, here we present an RPA formulation for the phase behavior of twocharged sequences as models for two IDP species.Consistent with physical intuition, we found that the tendency for thetwo IDP species to demix in two coexisting phases increases with increasingdifference in their charge patterns. This phenomenon represents a statistical,multivalent mode of molecular recognition for cellular organizationthat differ from the structurally highly specific form of recognitionamong folded proteins but share similarities with the“fuzzy complexes” <cit.> involving individual IDP molecules <cit.>.Delving deeper into the role of electrostatics in IDP phase separation, we have also extended the two-sequence RPA formulation to address how arelative permittivity ϵ_ r(ϕ) that depends on IDPvolume fraction ϕ may affect IDP phase properties. Several commoneffective medium approximations <cit.> posit a gradual decrease inϵ_ r from the ϵ_ r(ϕ=0)≈ 80 value for pure water with increasing ϕ. But physical consideration <cit.>and experimental volumetric measurements suggest a much sharper decrease,with ϵ_ r(ϕ=0.2)≈ 20, leading to largecooperative effects that enhance phase separation significantly.These findings and their ramifications are detailed below.§ METHODS §.§ Theoretical development of the RPA formulation The development of RPA theory for a pair of charged sequences constitutes the bulk of the results presented in subsequent sectionsof this article. This effort is based on an extension of the RPAformulation for a single sequence that we put forthrecently <cit.>. §.§ Experimental determination of dissolved protein volumes To address the effect of volume of dissolved proteins on the relativepermittivity of the resulting aqueous solution, nuclear magnetic resonance (NMR) andabsorbance measurements were performed on two foldedglobular proteins bovine serum albumin (BSA) and hen egg white lysozyme(HEWL) as well as two IDPs and , which are, respectively, amutant of Ddx4 in which all 14 phenylalanines are mutated to alanines and the concentrated phase of phase-separated wildtype Ddx4. §.§ Measurement of water content of protein samples NMR spectra were recorded using a BrukerAscend III spectrometer at 14.0 T equipped with a cryogenically cooledtriple resonance gradient probe. Spectra were processed using NMRPipe <cit.>. 1D ^1H spectra were recorded on protein samples over a range ofconcentrations (5-400 mg mL^-1) and the integrated water signalswere compared with the corresponding integrals obtained from a spectrumrecorded of buffer (the same buffer composition as used for eachprotein sample). BSA was purchased from Sigma and samples wereprepared in 20 mM sodium phosphate (NaPi), 100 mM sodium chloride (NaCl),10 % ^2H_2O/90 % ^1H_2O, pH 6.5. HEWL was purchased from BioBasic and sampleswere dissolved in 20 mM sodium citrate, 100 mM NaCl,10 % ^2H_2O/90 % ^1H_2O, pH 5 (lower pH was used due tolimited solubility of HEWL in NaPi at pH 6.5). samples were prepared according to <cit.> and dialysedagainst 20 mM NaPi, 100 mM NaCl, 5 mM tris(2-carboxyethyl)phosphine (TCEP),10 % ^2H_2O/90 % ^1H_2O, pH 6.5. For the same bufferwas used but the NaCl concentrationwas varied between 100-400 mM inorder to generate samples with protein concentrations between 200and 400 mg mL^-1. For phase-separated samples, it was ensured that the entirety of theprobe coil was occupied by the condensed phase, thus avoidingcontaminating signals from the more hydrated dilute phase. Spectra were recorded using both small flip angle(θ < 10^∘) and θ = 90^∘ pulses withvery similar results in both cases. §.§ Measurement of protein concentrationProtein concentrations were determined by absorbance at 280 nm (A_280)after dilution into 6 M guanidinium HCl, 20 mM NaPi, pH 6.5 using theBeer-Lambert law with extinction coefficients of 23 950, 23 950, 36 000, and44 309 M^-1cm^-1 for wildtype Ddx4, , HEWL, and BSA,respectively <cit.>. § OVERVIEW OF THREE-COMPONENT PHASE BEHAVIORS Possible phase behaviors of a three-component liquid system are outlined in Fig. 1. In general, the system can be a homogeneoussolution [Fig. 1(b], or it can separate into two coexisting phases [binarycoexistence; Fig. 1(c)–(e)], or separate into three coexistingphases [ternary coexistence; Fig. 1(f)]. Fundamentally, phase behavior is governed by the intra- and inter-component interactions as wellas environmental conditions such as temperature and pressure. Our theories below provide a rudimentary physical account ofhow interactions among IDP chains with two different amino acid sequences affect the conditions under which binary and ternary coexistence emerge. The theories presented here are for solution systems with an effective infinite volume. As such, our theoriesaccount for the differences among scenarios typified by the leftmostdrawings in Fig. 1(c)–(f) but they are not equipped to address detailssuch as droplet size and geometry. In other words,they provide no discrimination among different droplet geometriesalong a given horizontal row in Fig. 1. Accounting for the latter wouldrequire additional modeling of the interfacial tensions between differentsolution phases <cit.>.§ RPA THEORY FOR TWO CHARGED SEQUENCES Based on our previous RPA formulation for a singlesequence <cit.>, the approach is now extended to considertwo model polypeptide (IDP) sequences s_1 and s_2 in a salt-free aqueoussolution. Using notation similar to before <cit.>, the electriccharges along the sequences are written as{σ_1}≡{σ_1^(1), σ_1^(2), σ_1^(3),…, σ_1^(N_1)} and{σ_2 }≡{σ_2^(1), σ_2^(2), σ_2^(3),…, σ_2^(N_2)}, where N_1 and N_2 are the numbers of residues in s_1 and s_2, respectively. The corresponding volume fractions of the IDPs in solution are denoted as ϕ_1 and ϕ_2. The present study is restricted to IDP sequences with zero net charge. Counterions are not considered. §.§ Free energy as a function of two IDP volume fractions Following the FH lattice argument <cit.>, we partition thespatial volume V of the solution system into lattice units, a^3, that corresponds to the volume of a solvent molecule. Accordingly, theRPA free energy per unit volume and in units of the product of Boltzmann constant k_ B and absolute temperature T is castas the per-lattice-site quantityf(ϕ_1,ϕ_2) ≡F_ RPAa^3/VT= -s(ϕ_1,ϕ_2)+ f_ el(ϕ_1,ϕ_2) ,where the negative entropy -s is the entropic contribution tofree energy in units of k_ BT. This term is given by the standardFH entropy of mixing for a system comprising of s_1, s_2, and solvent:-s(ϕ_1,ϕ_2)= ϕ_1/N_1lnϕ_1 +ϕ_2/N_2lnϕ_2 + (1-ϕ_1-ϕ_2)ln(1-ϕ_1-ϕ_2) ,where 1-ϕ_1-ϕ_2 is the volume fraction of solvent. The electrostatic contribution f_ el is calculated byRPA <cit.>, viz.,f_ el(ϕ_1,ϕ_2)= ∫_0^∞d k k^2/4π^2{ln[ 1+G(k) ] -G(k) },where k is the reduced wave number that absorbs the virtual bond length b≃ a of the polypeptide backbone by re-defining kb=k̃ in Ref. <cit.> as k (i.e., kb → k),such thatG(k) = 4π/k^2(1+k^2)T^*⟨ q | Ĝ_k | q ⟩,where 4π/[k^2(1+k^2)] is from the Fourier transformation ofCoulomb interaction with a short-range cutoff <cit.> in units of T,U_ el(r) = e^2/4πϵ_0ϵ_ r T1-e^-r/b/r,ϵ_0 is vacuum permittivity and ϵ_ r is relativepermittivity, T^* ≡ b/l_B is the reduced temperature defined byBjerrum lengthl_B = e^2/(4πϵ_0ϵ_ r T). Here | q ⟩ is the(N_1+N_2)-dimensional column vector representing the two chargesequences, namely q_i = σ_1^(i) for 1≤ i≤ N_1and q_i = σ_2^(i-N_1) for N_1+1 ≤ i≤ N_1+N_2,⟨ q | is the transposed row vector,⟨ q | Ĝ_k |q⟩≡∑_ij q_i(Ĝ_k)_ijq_jwith (Ĝ_k)_ijbeing the i,j element of the bare two-body correlation matrix Ĝ_k of all possible sequence-sequence correlations <cit.>,Ĝ_k = ( [Ĝ_11(k) Ĝ_12(k),;Ĝ_21(k)Ĝ_22(k) ]),and Ĝ_12(k) = Ĝ_21(k).As in our previous studies <cit.>, we consider a simpleformulation in which all IDPs are modeled as Gaussian chains withoutexcluded volume within the RPA formalism <cit.>. In this approximation, there is no correlation between the positions of differentchains; hence Ĝ_12(k) = Ĝ_21(k) =0 andĜ_11(k)_ij =ϕ_1/N_1exp( -1/6k^2|i-j|)Ĝ_22(k)_ij =ϕ_2/N_2exp( -1/6k^2|i-j|)follow from the average of exp(i k· R_ij) over a Gaussian chain ensemble wherein R_ij is the vector between chain positions i,j andk^2= k· k (Eq. (IX.59) of <cit.>). Eq. (<ref>) then becomesG(k) = 4π/k^2(1+k^2)T^*[⟨σ_1 | Ĝ_11(k) | σ_1 ⟩+⟨σ_2 | Ĝ_22(k) | σ_2 ⟩] ,where ⟨σ_1 | Ĝ_11(k) | σ_1 ⟩ =ϕ_1/N_1∑_i,j=1^N_1σ_1^(i)σ_1^(j)exp( -1/6k^2|i-j|) ,⟨σ_2 | Ĝ_22(k) | σ_2 ⟩ =ϕ_2/N_2∑_i,j=1^N_2σ_2^(i)σ_2^(j)exp( -1/6k^2|i-j|) . §.§ Free energy landscape and spinodal instability As an example, we first apply this formulation to two N_1=N_2=50charged IDP sequences corresponding to sv28 and sv24 inDas and Pappu <cit.>. The sequences are labeled here as seq1 and seq2 respectively. The RPA free energy function for the two sequences (Fig. <ref>)consists of regions of different curvatures: parts of the landscape are convex downward (i.e., has a convex downward curvature) whereas someother parts are convex upward (concave downward).If a given IDP solution has overall (bulk-state) IDP volume fractions(ϕ_1^0, ϕ_2^0) situated in a convex-downward region, the state isthermodynamically stable and thus the bulk-state volume fractions are maintained. In contrast, if the bulk state is in a concave-downwardregion, the state is thermodynamically unstable because a phase-separated state allows for a lower free energy. Accordingly, the system undergoes phase separation to multiple coexisting phaseswith different protein volume fractions <cit.>.The boundary between the convex and concave regions is determined bythe saddle point condition <cit.> F̂≡= 0 .This boundary defines a spinodal instability region inwhich F̂ < 0. Because the determinant of a matrixis equal to the product of all its eigenvalues, this instabilitycondition indicates that one, but not both, of the eigenvaluesof F̂ is negative. This means that second-order perturbationsof free energy with respect to ϕ_1,ϕ_2 along the direction ofthe corresponding eigenvector diverges, signaling that the systemcannot maintain a homogeneous phase. Note that although the opposite of the F̂ < 0 condition, viz., F̂ > 0, does not by itself exclude the possibility that both eigenvalues of F̂ are negative and thus the system is thermodynamically unstable despite not satisfying F̂ < 0 (e.g. at a local maximum),exhaustive numerical searches (ϕ_1,ϕ_2=0.001,0.002,…,0.999) did not find any such instance for all RPA and FH systems studied in this work.Hence, for these systems, Eq. (<ref>) is thevalid condition for spinodal boundaries, examples of which are shown asdashed curves in Fig. <ref> for the (seq1, seq2) system.In two-component systems such as those consisting of a single IDP species and solvent molecules, spinodal instability necessarily leads tobinary coexistence <cit.>. In systemswith more than two components, more coexisting phases are allowed. According to the Gibbs phase rule, the maximum number ofcoexisting phases undera given set of environmental conditions is equal to the numberof components in the system <cit.> but, depending on system specifics, the actual number of coexisting phases can be smaller.Spinodal instability always implies thatthe system existing as a single phase is thermodynamically untenableand thus phase separation must occur. However, when the number of components is larger than two,the precise number ofcoexisting phases is governed by how total free energy varies with changes in the component volume fractions in differentcombination of phases or, equivalently, by the balance of chemicalpotentials for each component across different phases <cit.>.As examples, three instances of binary coexistence in the(seq1, seq2) system are depicted in Fig. <ref> as pairs of dots connected by solid tie lines.The general procedure for determining the conditions for suchcoexistence is as follows. §.§ Binary coexistence in three-component systems: Applications toIDPs with two different sequences plus solvent As mentioned, we use two methods to determine phase equilibrium: by ascertaining the minimum free energy among single- andmultiple-phase states <cit.>, andby balancing the chemical potentials for each of the components across different phases <cit.>. The two approachesare mathematically equivalent. They can be applied simultaneously to yield more accurate numerical results. Here we describe in detail howthe two approaches are applied to study three-component systems thatundergo binary phase separation.A system is dictated by thermodynamics to seek its lowest free energy state. Whether a system phase separates can be ascertained by comparing its overall free energy with and without phase separation. For a system with bulk IDP volume fractions (ϕ_1^0, ϕ_2^0), the free energy f_ bulk without phase separation and the free energy f_ sep for separating into two phases (labeled as α and β) that take up fractional volumes v_α=v (0≤ v≤ 1) and v_β=1-v of the total system volume and withIDP volume fractions (ϕ_1^α,ϕ_2^α) and(ϕ_1^β,ϕ_2^β), respectively, are given byf_ bulk = f(ϕ_1^0,ϕ_2^0),f_ sep = v f(ϕ_1^α,ϕ_2^α) +(1-v) f(ϕ_1^β,ϕ_2^β), wherein conservation of volume of each of the components implies thatv ϕ_1^α + (1-v)ϕ_1^β= ϕ_1^0 , v ϕ_2^α + (1-v)ϕ_2^β= ϕ_2^0 . By rewriting Eq. (<ref>) to express IDP volume fractions in β as functions of v with 0<v<1 and volume fractions in α: ϕ_1^β= ϕ_1^0 - v ϕ_1^α/1-v, ϕ_2^β= ϕ_2^0 - v ϕ_2^α/1-v, f_ sep is seen as a function of ϕ_1^α, ϕ_2^α, and v,f_ sep(v,ϕ_1^α, ϕ_2^α) = v f(ϕ_1^α,ϕ_2^α) + (1-v) f(ϕ_1^0- v ϕ_1^α/1-v, ϕ_2^0 - v ϕ_2^α/1-v) . Note that lim_v→ 0 f_ sep(v,ϕ_1^α, ϕ_2^α) =lim_v→ 1f_ sep(v,ϕ_1^α, ϕ_2^α)=f(ϕ_1^0,ϕ_2^0), where(ϕ_1^β,ϕ_2^β)→ (ϕ_1^0,ϕ_2^0) for v→ 0 and (ϕ_1^α,ϕ_2^α)→ (ϕ_1^0,ϕ_2^0) for v→ 1.To find the set of variables that yields the global minimum of f_ sep,we numerically search the three-dimensional space of(v,ϕ_1^α,ϕ_2^α) by implementing the sequential least squares programming (SLSQP) algorithm <cit.> using the scipy.optimize.minimize function in Scipy, a Python-based numerical package for scientific computation <cit.>. If a given(v,ϕ_1^α,ϕ_2^α) is foundto yield a minimum of f_ sep among computed f_ sep values and also satisfiesf_ sep(v, ϕ_1^α,ϕ_2^α) < f_ bulk(ϕ_1^0, ϕ_2^0), the system is judged to be in a state of binary phase separation to the two phases α and β. Incontrast, if all f_ sep for a given (ϕ_1^0, ϕ_2^0) arelarger than f_ bulk, the bulk state (ϕ_1^0, ϕ_2^0) isthermodynamically stable and the system does not phase separate. Unlike in the two-component case in which the two separated phases are unique and independent of the bulk IDP concentration/volume fraction insofar as it isin the phase-separated regime, in three-component systems different bulkIDP concentrations/volume fractions can result in different α, βphases. Thus a complete binary phase diagram is generated by consideringall possible (ϕ_1^0, ϕ_2^0) combinations.For two-component systems (one IDP sequence plus solvent), we have shown that linear terms of ϕ in the system free energy do not affect the determination of phase equilibrium <cit.>. In the same vein, here we demonstrate that the same principle applies also tothree-component (two IDP sequences plus solvent) systems.As described above, binary coexistence is governed by the free energydifferenceΔ f ≡ f_ sep - f_ bulk = v f(ϕ_1^α,ϕ_2^α)+ (1-v) f(ϕ_1^β,ϕ_2^β) - f(ϕ_1^0,ϕ_2^0) .Consider a modified free energy g(ϕ_1,ϕ_2) with an additional arbitrary linear function a_0+a_1ϕ_1+a_2ϕ_2 of ϕ's where a_0, a_1, and a_2 are constants: g(ϕ_1, ϕ_2) = f(ϕ_1,ϕ_2) + a_0+a_1ϕ_1+a_2ϕ_2 .Now, the free energy difference between phase-separated and bulk phases becomesΔ g ≡ g_ sep - g_ bulk =Δ f + a_1 [ v ϕ_1^α + (1-v) ϕ_1^β -ϕ_1^0 ] + a_2 [ v ϕ_2^α + (1-v)ϕ_2^β -ϕ_2^0 ]because a_0 in g_ sep and g_ bulk cancel.The two bracketed terms in Eq. (<ref>) are identically zero because of Eq. (<ref>). Hence Δ g = Δ f, meaning that any linear function of ϕ's added to f, such as the one utilized in Fig. <ref> for graphical clarity, has no impact on phase separation.We apply the above-described minimization procedure for three-component (two IDP sequences plus solvent) systems to determine (α, β)for selected pairs of sequences in Table <ref>. To minimizepossible numerical errors, every set of {ϕ_1^α,ϕ_2^α,ϕ_1^β,ϕ_2^β} obtained by minimizing f_ sep is subject to further testing by comparing the chemical potentials in α and β.As described in Eq. (A.5) of Ref. <cit.>, phase equilibriumimplies the following equalities, f_1^'α= f_1^'βf_2^'α= f_2^'β μ_ w^α= μ_ w^β wheref^y ≡f(ϕ_1^y, ϕ_2^y) , f_x^'y≡ .∂ f(ϕ_1,ϕ_2)/∂ϕ_x|_(ϕ_1,ϕ_2)=(ϕ_1^y, ϕ_2^y),x = 1,2; y = α,β,and μ_ w^y ≡ f^y - ϕ_1^y f_1^'y - ϕ_2^y f_2^'yis the chemical potential of water <cit.>.Making use of the volume conservation conditions in Eq. (<ref>)and substituting Eq. (<ref>) for ϕ_1^β and Eq. (<ref>) for ϕ_2^β, Eq. (<ref>) becomes three equalities for three variables ϕ_1^α, ϕ_2^α, and v. It follows that a unique determination of thephase-separated volume fractions ϕ_1^α, ϕ_2^α,ϕ_1^β, and ϕ_2^β is afforded by Eq. (<ref>).It is straightforward to show that the set of phase-separated volume fractions {ϕ_1^α,ϕ_2^α,ϕ_1^β,ϕ_2^β}determined by Eq. (<ref>) are identical to that obtained by minimizing f_ sep in Eq. (<ref>). A necessarycondition for the minimization of f_ sep is that its Jacobian vector J_ sep of first-order partial derivatives of independent variables vanishes:J_ sep(v, ϕ_1^α, ϕ_2^α) ≡( ∂ f_ sep/∂ϕ_1^α ∂ f_ sep/∂ϕ_2^α ∂ f_ sep/∂ v) =0. In other words, ∂ f_ sep/∂ϕ_1^α = v ( f_1^'α - f_1^'β) = 0, ∂ f_ sep/∂ϕ_2^α = v ( f_2^'α - f_2^'β) = 0 , ∂ f_ sep/∂ v= f^α - f^β +(ϕ_1^β-ϕ_1^α) f_1^'β + (ϕ_2^β-ϕ_2^α) f_2^'β = 0, wherein we have utilized Eq. (<ref>) for ϕ_1^β and Eq. (<ref>) for ϕ_2^β. Clearly, Eqs. (<ref>) and (<ref>)are equivalent to Eqs. (<ref>) and (<ref>),respectively, and Eq (<ref>) is equivalent toEq. (<ref>) by virtue of Eq. (<ref>). Q.E.D. Starting with (α, β) obtained by minimizing f_ sep in Eq. (<ref>), only those that deviate less than 0.1% from the chemical-potential-balancingequalities in Eq. (<ref>)are accepted as valid binary pairs in our analysis. For thesequence pairs (seq1, seq5) and (seq1, seq6), a smaller threshold of 0.01% is used to ensure accuracy of the computed phase-separated ϕ's because for these sequence pairs the chemical potential balancing conditionsare quite insensitive to variations of the ϕ's. §.§ Binary coexistence of two charged sequences Using RPA, we investigated previously how the phase separation behaviorsof charged IDP sequences are affected by their chargepatterns <cit.>. In particular, for the set of thirtyKE sequences of Das and Pappu <cit.> with zero net charge but anequal number of 25 positively charged lysine (K) and 25 negatively chargedaspartic acids (E) in different permutations, the critical temperature T^*_ cr of phase separation was found <cit.> to be correlatedwith charge pattern parameters κ <cit.> andSCD≡1/N∑_i=1^N∑_j=i+1^N σ_i σ_j√(j-i),where i,j label the residues with charges σ_i,σ_j along a chainof length N <cit.>. The κ and SCD parameters exhibitsimilar correlations with single-chain radius of gyrationR_ g <cit.>.The correlation of T^*_ cr and R_ g with SCD is strongerthan that with κ. A likely reason is that SCD accounts for nonlocal effects between charges far apart along the chain sequence whereas κ does not <cit.>. Although we use only SCD in our analysis below, an equivalent analysis using κ is expected to produce a similar trend.How does the phase behavior of an IDP solution with two sequences depend on the sequences' difference in charge patterns? Intuitively, whentwo sequences with different SCD values are present together, their different propensities to phase separate are expected to interfere. Indeed, such an effect of inter-sequence interference is seen clearly in the Taylor expansion of the integrand of the RPA expression f_ el for the electrostatic contributionto free energy in Eq. (<ref>), ln[1 +G(k)] -G(k) = -1/2 G(k)^2 + 1/3 G(k)^3 + ... = -1/2⟨σ_1 | Ĝ_11^*(k) | σ_1 ⟩^2-1/2⟨σ_2 | Ĝ_22^*(k) | σ_2 ⟩^2 - ⟨σ_1 | Ĝ_11^*(k) | σ_1 ⟩⟨σ_2 | Ĝ_22^*(k) | σ_2 ⟩ + O( G(k)^3) ,where Ĝ_11^*(k) and Ĝ_22^*(k) are the product of 4π/[k^2(1+k^2)T^*] with, respectively, the Ĝ_11(k) and Ĝ_22(k) in Eq. (<ref>). The first two terms after the second equality in Eq. (<ref>) are self-interactions of the two sequences, identical to those in one-sequenceRPA theory (see, e.g. Eq. (1) in Ref. <cit.>). The thirdterm represents the interference effect in RPA. Since it is the product of square roots of the two self-interaction terms,its strength is intermediate between them, suggesting that phase behaviors of two-sequence systems are sensitive to the similarity/dissimilarity in charge pattern between the two sequences.To investigate this sensitivity, we use the sequences inTable <ref> to compute the phase diagrams of six pairs of sequences, namely seq1 with each of the six other sequences. The pairs are selected to represent a broad range ofsimilarity/dissimilarity in charge pattern as quantified by the difference in SCD values: from the (seq1, seq2) pair withSCD =(-15.99,-17.00) to (seq1, seq7) with SCD =(-15.99,-0.41).To compare the phase behaviors of the six sequence pairs on an equal footing, all phase diagrams in Fig. <ref> are computed at the same reduced temperature T^*=4. Noting that condensed-phase volume fractions tend to decrease with increasing T^*, this temperature is chosen because it falls in the mid-rangeof the broad span of T^*_ cr's for the sequences in Table <ref>. T^*=4 is much higher thanthe T^* = 0.55 equivalent of room temperature (T = 300 K) when an aqueous ϵ_ r=80 is assumed <cit.>. This seemingly unphysicalcondition in our calculation has little impact, however, on the presentgoal of ascertaining general principles and behavioral trends. Although we do not aim for direct, detailed comparison with experiment here, T^*=4 is experimentally relevant, for example, to IDPs with charge patterns similar to those considered here butwith their electrostatic interaction strength significantly scaled down for various physical reasons such as screening or a more sparse charge distribution along the IDP sequence.For each of the six phase diagrams in Fig. <ref>, the areasurrounded by blue dots and “shaded” by black dashed lines is the region of binary coexistence. In other words, bulk-state volume fractions falling within this region will phase separate into two coexisting liquid phases [as in Fig. 1(c)–(e)], whereas bulk-state volumefractions residing outside this region will be stable as a singleliquid phase [as in Fig. 1(b)].Every black dashed line is the tie line connecting apair of blue dots representing separated phases (α, β) forany bulk-state volume fractions lying on the given tie line [corresponding to the arrows in the schematic phase digrams for Fig. 1(c)–(e)]. The average slope of the tie lines changes from positive for similarly patterned sequences [Fig. <ref>(a), (b)] to negative forvery differently patterned sequences [Fig. <ref>(e), (f)]. This trend may be understood as follows. When both sequences of a given pair can undergo phase separation individually (T^*_ cr>4 for both), the tie lines near the ϕ_1 and ϕ_2 axes must be close to being parallel to the axes because when either ϕ_1^0→ 0 or ϕ_2^0→ 0,the two-sequence system reduces to the corresponding single-sequence system that phase separates. This situation applies to (seq1, seq2) and(seq1, seq3), resulting in positive tie-line slopes, indicating thatthe populations of the two sequences in each pair are well mixed even when they undergo phase separation. They prefer to stay together after they phase separate,with similar population ratios for the two sequencesin the “both-dilute” (small ϕ's) as well as the “both-condensed” (larger ϕ's) phases [as in Fig. 1(c)].Because T^*_ cr < 4 for the other four sequences (seq4–seq7),they do not phase separate by themselves individuallyand therefore tie lines near the ϕ_2-axis need not be approximately parallel to it. Nonetheless, tie lines close to the ϕ_1-axis are still required to essentially line up with the axis. For tie lines that possess large ϕ_2 values, thevolume conservation condition ϕ_1+ϕ_2=1 enforces negative tie-line slopes. The combined effect of these constraints lead to tie-line slopes that gradually change from ≈ 0 near the ϕ_1-axisto ≈ -1 near the ϕ_1+ϕ_2=1 boundary, as exemplified by the case of (seq1, seq7) in Fig. <ref>(f). As shown in Fig. <ref>(d) and (e) for (seq1, seq5) and(seq1, seq6), this trend is apparent even when the phase-separatedregime does not extend all the way to the ϕ_1+ϕ_2=1 boundary. Negative tie-line slopes imply various degrees of demixing of the populations of the two sequences: the phase-separated state (α,β) now comprises oneϕ_1-enriched (ϕ_1^α≫ϕ_2^α) phase coexisting with one ϕ_2-enriched (ϕ_2^β≫ϕ_1^β) phase. The degree of population demixing depends on how dissimilar are the charge patterns of the two sequences in the pair.For large differences in SCD as in Fig. <ref>(f), one of the coexisting phases can have a very low population of seq1but a substantial seq7 volume fraction, whereas the other phase has a relatively low population of seq7 but a substantial seq1 volume fraction [as in Fig. 1(e)].The (seq1, seq4) pair in Fig. <ref>(c) is at the crossover between the well-mixed and demixed extremes. Tie-line slopes in this case are all ≈ 0, indicating that although increasing seq4 volume fraction decreases the phaseseparation tendency of seq1, even in the phase-separated regime theconcentration of seq4 is essentially identical in the two coexisting phases, i.e., ϕ_2^α≃ϕ_2^β≃ϕ_2^0 [as in Fig. 1(d)].The difference in the ratio of component populations in coexisting phases (α,β) may be quantified by comparing thevolume ratio of the two sequences in the two phases. We first consider a rather intuitive measure Δ_αβ( ϕ_1/ϕ_2 ) ≡⟨| ϕ_1^α/ϕ_2^α - ϕ_1^β/ϕ_2^β|⟩of compositional asymmetry between coexisting phases, where the absolute value ensures that Δ_αβ(ϕ_1/ϕ_2) is α↔β symmetric, and the bracket⟨…⟩ denotes averaging over all(α, β) pairs of coexisting phases.One disadvantage of this measure, however, is that the average is strongly dominated by those pairs of (α, β) with large ϕ_1 butsmall ϕ_2, i.e., coexisting pairs that are close to ϕ_1-axisin Fig. <ref>. Therefore, we also consider another composition asymmetry measure A_αβ≡⟨2/π|tan^-1(ϕ_1^α/ϕ_2^α) -tan^-1(ϕ_1^β/ϕ_2^β) | ⟩that avoids this potentially problematic feature by replacing the ratio ϕ_1/ϕ_2 with its arctangent value normalized by π/2 such that 0≤ A_αβ≤ 1. Summarizing our findings using two-sequence RPA theory, Fig. <ref> shows the variation of Δ_αβ(ϕ_1/ϕ_2) as well as A_αβ with the difference in SCD values of the six sequence pairs in Fig. <ref>.A reasonable correlation is seen for both composition asymmetry measures,with the A_αβ measure exhibiting a better correlationby varying monotonically with SCD difference, indicating that compositionalasymmetry or degree of demixing of phase-separated populationsas quantified by A_αβis positively correlated with the difference in charge patterns asquantified by difference in SCD values. This plot illustrates graphically how a stochastic, multivalent form of molecular recognition that arises from the interactions among the diverse conformations in a multiple-chain ensemble can lead todemixing of different IDP species into different coexisting phases. § COMPARISON WITH PHASE SEPARATION IN FLORY-HUGGINS (FH) MODELS We next seek a deeper understanding of the RPA results and their ramifications by comparing them with the predictions ofa variety of FH models. As emphasized, unlike RPA, FH by itself does not address the physicsof sequence dependence <cit.>.Accordingly, FH χ interaction parameters for IDP sequences have to be provided phenomenologically by experimentor theoretically by microscopic physicaltheory.[In the caption describing the FH results in Fig. 8 of Ref. <cit.>, r_d is in fact the symbol r for the equilibrium spacing in Eq.(S19) of Ref. <cit.>. This typographical error does not affect the results.It should also be noted that because Ref. <cit.> equatesthe ionic strength I with [NaCl] but not 2[NaCl], their effective Debye length is 4.3 Å instead of the correct value of 3.04 Å. To facilitate comparisonwith Ref. <cit.>, however, the effective Debye length in Fig. 8 of Ref. <cit.> was alsoset to 4.3 Å.]For example, as will bediscussed further below, an intuitive and semi-quantitative connectionbetween RPA and FH is provided by the expansion in Eq. (<ref>). It should also be noted that FH neglects interaction terms that are higher than quadratic order in IDP volume fractions/concentrations(ϕ's) such as the O( G(k)^3) terms in Eq. (<ref>) because G(k)∝ϕ [Eqs. (<ref>) and (<ref>)].This approximation can be problematic when IDP concentrations are high. Nonetheless, by treating the three parameters χ_11, χ_22,and χ_12 in the three-component FH interaction term for two IDP species plus solventf_ int^ FH = -( χ_11ϕ_1^2 + χ_22ϕ_2^2 +2χ_12ϕ_1ϕ_2 )as free (arbitrary) variables, we can either match FH behavior to that of RPA to gain conceptual insights or explore other interactionscenarios that might be physically plausible when interactionsother than the rudimentary electrostatics embodied in RPA are includedin the physical picture. §.§ FH models that imitate RPA theory by having two independentχ's We begin this analysis by first constructing FH models with interactionschemes similar to RPA, then tuning the interaction parameters to producephase behaviors similar to those predicted by RPA in Fig. <ref>. If we identify the ∫ dk k^2/4π^2 integral [Eq. (<ref>)]of the order G(k)^2 terms in the RPA expansion Eq. (<ref>) with the FH interaction term inEq. (<ref>),we may define χ̃_11(k)ϕ_1^2≡ 2{⟨σ_1 | Ĝ_11(k)|σ_1⟩/[ k (1+k^2) T^*]}^2 and χ̃_22(k)ϕ_2^2≡ 2{⟨σ_2 | Ĝ_22(k)|σ_2⟩/[ k (1+k^2)T^*]}^2 such that χ_11=∫ dk χ̃_11(k), χ_22=∫ dk χ̃_22(k), and χ_12= ∫ dk √(χ̃_11(k))√(χ̃_22(k)), from which it is clear that only two set of interaction parameters χ̃_11(k) and χ̃_22(k) as functions of k are independent. Here we approximate this dependence byconstraining χ_12 = √(χ_11)√(χ_22).We then construct three FH systems that have χ_11 = χ_22, χ_11≳χ_22, andχ_11≫χ_22, corresponding respectively tosequence pairs with small, intermediate, and large charge pattern(SCD) differences. The phase diagrams of these models (Fig. <ref>) exhibit a trend similar to that seen in the RPA-predicted Fig. <ref>. Specifically, Fig. <ref>(a) is similar toFig. <ref>(a), Fig. <ref>(b)to Fig. <ref>(c), and Fig. <ref>(c)to Fig. <ref>(f). This correspondence offers conceptual clarity because the degree to which the interaction betweenthe two IDP species is favorable is explicit in FH. When χ_11 = χ_22=χ_12, the two species are miscible and their phase separation propensities are identical, resulting in the coexistence of one both-dilute phase and one both-condensed phase. The similarity between Fig. <ref>(a)Fig. <ref>(a) indicates that this behavior can be achieved physically by two IDP species with similar charge patterns. In contrast, when χ_11≫χ_12≫χ_22, miscibility of the two species is poor and their phase separation propensities arequite different, resulting in a high degree of population demixing. The similarity of this behavior shown in Fig. <ref>(c) with that in Fig. <ref>(f) underscores once again that charge-pattern mismatches between IDPs can lead to substantially weakening of attractiveinteractions. §.§ FH models with three independent χ's We next consider the general case in which the three χ's in Eq. (<ref>) are independent. Although this modeling setup does not have a simple correspondence withIDP sequences interacting via physical forces like that describedabove, the expanded variety of scenarios explored here would be valuablewhen behaviors much more complex than those allowed by our current RPA formulation are considered in more comprehensive and detailed physical theories. As simple examples of the rich possibilities, here we focus on FH models with χ_11=χ_22 (≡χ) and variable χ_12 values that are not related to χ. Fig. <ref> shows the phase diagrams of three representativemodels with (a) χ_12≳χ, (b) χ_12≲χ, and(c) χ_12 < χ.Compared to the χ≡χ_11=χ_22=χ_12case in Fig. <ref>(a), it is clear that a stronger inter-component attraction (larger χ_12) makes the phase-separated region bulge [Fig. <ref>(a)], whereas a weaker inter-component attraction (smaller χ_12) shrinks it [Fig. <ref>(b)].Nonetheless, the tie-line slopes are positive in both situations, indicating that the two components are largely miscible.Fig. <ref>(b) indicates that a weakened χ_12shrinks the phase-separated region most around ϕ_1=ϕ_2. As χ_12 decreases further, inter-component attraction all but vanishes, micibility disappears, resulting in the phase-separated region being broken into two parts,one for ϕ_1 ≫ϕ_2 and the other for ϕ_1 ≪ϕ_2 [Fig. <ref>(c), shaded area and inset]. The tie lines in these two regions are almost parallel to either the ϕ_1- or the ϕ_2-axis, implying that one component is dominant while the concentration of the other component barely changes upon phase separation.With such an effective inter-component repulsion (i.e., less favorable inter-component attraction vis-à-vis the strengths of intra-component cohesion), anadditional phase-separated regime of poor miscibility similar to that in Figs. <ref>(f) and <ref>(c) is induced, wherein all tie-line slopes are negative,signaling substantial demixing [Fig. <ref>(c), region close to ϕ_1+ϕ_2=1with tie-line slopes =-1].§.§ Ternary coexistence in FH model If χ_12 is made even weaker than that inFig. <ref>(c), the region of poor miscibilitybelow the ϕ_1+ϕ_2=1 boundary would grow. Finally, the three phase-separated regions intersect and a new ternarycoexistence region emerges in-between.According to the Gibbs phase rule, an n-component system can separate into atmost n coexistingphases, when all other environmental conditions, e.g. temperature and pressure, are kept constant <cit.>. In our system of n=3 (two sequences plus solvent), whether the system will separate to two or three coexisting phases may be deduced by observing the variation of tie-line slopes inputative regions of binary coexistence: If the tie lines have to become parallel to the ϕ_1-axis, ϕ_2-axis, or the ϕ_1+ϕ_2=1 line when they approach these boundaries respectively, the tie-line slopes have to be able to vary smoothly to satisfy theseconstraints in order for binary coexistence to be stable. In that case, ternary coexistence is unlikely. Conversely, if there are conflicts that prevent a smooth change of tie-line slope, a ternary coexistenceregion ensues.An example is provided by using Fig. <ref>(c) as starting point. Here the three phase-separated regions are close to the three boundaries, and their tie lines are essentially parallel to the respective boundaries. Under this circumstance, when effective inter-component repulsion is enhanced by weakening χ_12 to cause the three regions to evolve toward merging, the conflict among the three different trends of tie-line slopes necessitates reconcilation by a region of ternary phase separation (Fig. <ref>). In order to determine the three phases in ternary coexistence mathematically,we extend the phase-separated free energy expression in Eq. (<ref>) for phases (α,β) to including oneadditional phase γ, viz.,f_ ternary =v_α f(ϕ_1^α,ϕ_2^α)+ v_β f(ϕ_1^β,ϕ_2^β) + (1-v_α-v_β) f(ϕ_1^γ,ϕ_2^γ) ,where the fractional volumes v_α, v_β are functions ofthe six ϕ's by virtue of volume conservation,∑_y v_y ϕ_x^y + (1- ∑_y v_y) ϕ_x^γ=ϕ_x^0,where x=1,2 and ∑_y is over y=α,β.Now the equalities in Eq. (<ref>) have to include theaddition phase to become f_1^'α= f_1^'β = f_1^'γ,f_2^'α= f_2^'β = f_2^'γ,μ_ w^α= μ_ w^β = μ_ w^γ. Because here we have six equalities in Eq. (<ref>)for six phase-separated ϕ_x^y's, the solution for ternary coexistence of (α, β, γ) is unique irrespective of the bulk-state volume fractions insofar as they fall within the ternary coexistence region. This situation is different from that of binarycoexistence in which the separated phases (α, β) can be different for different bulk-state (ϕ_1^0,ϕ_2^0)'s when they are on different tie lines.Similar to the binary coexistence case in Sec. <ref>, we proceed to demonstrate that minimizing Eq. (<ref>)is equivalent to solving the equations in Eq. (<ref>).Using essentially the same approach, we rewrite f_ ternary asa function of six independent variables: v_α, v_β,ϕ_1^α, ϕ_2^α, ϕ_1^β, and ϕ_2^β by first utilizing Eq. (<ref>) to express ϕ_1^γand ϕ_2^γ asϕ_x^γ = ϕ_x^0-_y v_y ϕ_x^y/1- _y v_y,where x,y and ∑_y have the same meanings as above. In this notation, Eq. (<ref>) becomesf_ ternary(v_α,v_β,ϕ_1^α,ϕ_2^α,ϕ_1^β,ϕ_2^β) = ∑_y v_y f^y + (1-∑_y v_y) f^γ.Similarly to Eq. (<ref>) for binary phase separation, wecalculate the six derivatives of f_ ternary and set them to zero as in Eq. (<ref>) as necessary conditions for the minimization of f_ ternary, resulting in ∂ f_ ternary/∂ϕ_x^y = v_y f^' y_x- (1-∑_y^' v_y^')f_x^'γ·-v_y/ 1-_y^' v_y^'= v_y ( f^' y_x - f_x^'γ) = 0,∂ f_ ternary/∂ v_y = f^y - f^γ+ (1-∑_y^' v_y^') [ ∑_x f_x^'γ·-ϕ_x^y(1-_y^' v_y^') +ϕ_y^0- _y^' v_y^'ϕ_x^y^'/(1-_y^' v_y^') ^2]= f^y -f^γ + ∑_x f^' γ_x (ϕ^γ_x - ϕ^y_x)=μ_w^y - μ_w^γ = 0 , where ∑_y^' sums over y^'=α,β. Substituting x=1,2 in Eq. (<ref>) yieldsEqs. (<ref>a) and (<ref>b),whereas substituting y=α, β in Eq. (<ref>) yieldsEq. (<ref>c). Q.E.D.Fig. <ref> provides an FH phase diagram withboth binary and ternary coexistence. In this example, χ_12 is significantly smaller than χ_11=χ_22, resulting in strong effective repulsion between the two sequences.Consequently, the two islands of binary coexistence around theϕ_1- and ϕ_2-axes intersect with the top-right binary region with (ϕ_1^α, ϕ_2^α) = (ϕ_2^β, ϕ_1^β), resulting in a ternary phase separation region corresponding to the αβγ triangle and its interior [marked byblue lines in Fig. <ref>(a) and turquoise linesin Fig. <ref>(b) and (c)]. The thermodynamic stability of the ternary phase-separated state within this region is illustrated by the Δ f quantity plotted in Fig. <ref>(b) and (c); Δ f(ϕ_1,ϕ_2) is the bulk (not-phase-separated) free energyf_ bulk minus the free energy value for the same ϕ_1,ϕ_2 on a plane defined by the three ternary phases (i.e., Δ f = 0 forthe three points corresponding to the α, β, γ phases). Because Δ f > 0 forany other point within the triangular region, the ternary state is morestable than the bulk state in this region.Moreover, the free energy f_ sep of any putative binarycoexistence state of a bulk state within the triangular region must lieon a tie line joining two points on the landscape inFig. <ref>(b) and (c). Because Δ f>0 for any point other than the three ternary phases in the entire plotted region—including points outside the triangular region, Δ f > 0holds also for any putative binary coexistence state for the bulk state within the triangular region, implying that they are lessstable than the ternary phase-separated state in the region.For any given bulk-state (ϕ_1^0,ϕ_2^0) in the ternary region, fractional volumes v_α, v_β,and v_γ in the respective coexisting phases α, β,and γ are determined by solving Eq. <ref> and setting v_γ=1-v_α-v_β. In terms of the three-dimensional vectors Φ_0≡(ϕ_1^0,ϕ_2^0,0), Φ_z≡(ϕ_1^z,ϕ_2^z,0), Φ_0z≡Φ_z-Φ_0, and Φ_z_1z_2≡Φ_z_2-Φ_z_1 where z,z_1,z_2=α,β,γ,v_α=|Φ_0γ×Φ_βγ| /|Φ_αγ×Φ_βγ|, v_β=|Φ_0α×Φ_αγ| /|Φ_αγ×Φ_βγ| and v_γ=|Φ_0β×Φ_αβ| /|Φ_αγ×Φ_βγ|. Because the area of a triangle defined by two vectors is equal tohalf of the magnitude of their cross product, these fractional volumes correspond to specific ratios of triangular areas as described in the caption for Fig. <ref>. § DISCUSSION§.§ Insights into cellular binary and ternary IDP phase coexistence The present theoretical development bears on the sequence dependence of multicomponent phase separation in the cell. However, because the cellular processes involve many species of biomolecules and are extremely complex <cit.>, development of treatments much more elaborated than our simple theories will be needed for quantitativecomparison with experiments. Nonetheless, it is instructiveto explore whether our RPA and FH results are qualitatively consistent with what has been observed experimentally.Of interest are fibrillarin FIB1 (323 residues) andnucleophosmin NPM1 (299 residues) from frog (Xenopus laevis)oocytes. These IDPs tend to demix, exhibiting phase behaviors thatlikely underpin the assembly of nucleolar subcompartments <cit.>.Treating histidine sidechains at pH ≳ 7 as neutral, the net charge of FIB1 is 19 and of NPM1 is -22.Their charge patterns, as quantified by SCD[Eq. (<ref>)] =4.126 and -0.119, respectively, are substantially different. Thus, the tendency for FIB1 and NPM1 to demix is qualitatively in line with the RPA-predictedtrend in Figs. <ref> and <ref>.Aqueous solutions with both FIB1 and NPM1 undergo both binary and ternary liquid-liquid phase separations. In this respect, their experimental phase diagram in Fig. 4D of Feric et al. <cit.> is similar to our FH phase diagram in Fig. <ref>. The two regions of binary coexistence of one condensed (around α or around β) and one dilute (around γ) phases inFig. <ref> correspond to their “FIB1 rich/NPM1 lean”and the “NPM1 rich/FIB1 lean” areas, whereas the ternary coexistence region in Fig. <ref> with two condensed (α,β) and one dilute (γ) phases [as in Fig. 1(f)] corresponds to their “3 Phase”area <cit.>.It is noteworthy that our attempts to seek numerical solutionsto ternary coexistence in the RPA models studied in Fig. <ref>by minimizing Eq. (<ref>) either ended in failure orresulted in solutions with two (among three) phases essentially identical and thus reduces the solution to that of binary coexistence. Apparently, ternary coexistence requires an effective intercomponent repulsion that is substantially stronger, as in Fig. <ref>,than that posited by RPA. The reason is that RPA constrains the intercomponent interaction strength χ_12 to approximatelythe geometric mean of the two intracomponent interaction strengths χ_11 and χ_22 (Sec. <ref>) and thereforeχ_12≪χ_11,χ_22 is highly unlikely if not impossible.This consideration suggests that difference in charge pattern alone maybe insufficient to account for the rather strong effective repulsion between FIB1 and NPM1, although the impact of them beingnot very close to being neutral remains to be investigated. (Unlike the model KE sequences in Fig. <ref> with zero net charge or Ddx4 with a charge ratio = (net charge/chain length)=-1.7% <cit.>, their charged ratios are, respectively, +5.9% and -7.4%). In addition to the electrostatic interactions among FIB1 and NPM1, otherdriving forces surely also contribute to their phase behaviors.For example, the presence of ribosomal RNA (rRNA) appears to be important; and the role of rRNA and inter-phase surface tensions have been modeledin lattice simulations to rationalize the FIB1/NPM1 droplet-in-droplet organization <cit.>. Building on our findings, much effort will be required toascertain the precise role played by charge pattern mismatch in thisintriguing phenomenon. §.§ Cooperativity driven by concentration-dependent relativepermittivity Most analytical formulations for charged polymer solutions, including commonRPA theories, treat the relative permittivity ϵ_ r of thesolution as a constant independent of polymer concentration.However, as wenoted recently <cit.>, because of the significantly differentpermittivities of water (≈ 80) and protein(≈ 2–4) <cit.>, the effective ϵ_ r of a protein solution can change dramatically with protein concentration.Indeed, protein-dependent variations of dielectric properties of theaqueous medium have been shown to be relevant to globular protein stability in thermophilic species <cit.>.Because of the anticipated importance of dielectric properties to IDPenergetics such as enabling a greatly enhanced propensity to phaseseparate <cit.>, here we expand our theoretical explorationto two-IDP systems and consider in more detail the physical basis of various effective medium approximations that may be applied to estimate ϵ_ r of IDP solutions. Since the scope ofthis exploration is limited to establishing certain general principles, for simplicity of the discussion we letbe the relative permittivity of both IDP species in the system such thatϵ_ r(ϕ_1,ϕ_2)=ϵ_ r(ϕ) where ϕ=ϕ_1+ϕ_2.Previously we considered two models for ϵ_ r <cit.>, namely the “slab model” derived by considering a planar electric capacitor, whereinϵ_ r^ Slab(ϕ) = /ϕ + (1-ϕ),which corresponds to Eq. (3) of Bragg and Pippard when the depolarizingcoefficient L=0 <cit.>, and the Clausius-Mossotti (CM) model predictingϵ_ r^ CM(ϕ) = 1+2[(1-ϕ) +ϕ ]/1-[(1-ϕ) + ϕ ],where≡ -1/ + 2, ≡ -1/ + 2are proportional to the CM expression for molecularpolarizability <cit.>. The ϕ-dependences ofϵ_ r^ Slab(ϕ) and ϵ_ r^ CM(ϕ) are very similar (Fig. 11 of <cit.>). Both models recognize amino acid residues and water as materials with excludedvolume such that the contributions of their molecular polarizabilities to the overall dielectric property of the medium are against a vacuum background.An alternate approach to effective medium is the Maxwell Garnett (MG)model <cit.> that pictures the effective dielectric property of a composite material as arising from embedding a component (IDP in our case) into an all-permeating background medium (water in our case). IDP excluded volume is not taken into account in this approach. Applying this method to our IDP solution yieldsϵ_r^ MG(ϕ) =1+2ϕγ_ MG/1-ϕγ_ MG,whereγ_ MG≡ -/ +2.Comparing the γ_ p expression in Eq. (<ref>) withEq. (<ref>) indicates that γ_ MG corresponds—up to a constant—to aneffective molecular polarizability of IDP material, not against vacuum but rather in a water background(γ_ p is mathematically equal to γ_ MG when → 1, thus MG is related to CM in this respect <cit.>). Another approach known as the Bruggeman (BG) model <cit.> is an ↔ symmetrized form of MG. The BG ϕ-dependent permittivity is given byϵ_r^ BG(ϕ) ≡b_ BG(ϕ) +√(8+b_ BG(ϕ)^2)/4,whereb_ BG(ϕ) = [2ϕ-(1-ϕ)] + [2(1-ϕ)-ϕ].A graphical illustration of the different physical pictures assumed bythe slab/CM versus the MG/BG approaches is provided by Fig. <ref>. Leaving aside questions about the appropriateness oftheir respective physical pictures for the moment, we firstexamine the properties of these ϵ_ r(ϕ)'s and their implications for IDP phase separation.Variations of several properties of the ϵ_ r(ϕ)'sare shown in Fig. <ref>.Although all four modelsgive ϵ_ r(ϕ=0)= and ϵ_ r(ϕ=1)= asthey should, the ϕ-dependences of the slab/CM models are very different from that of the MG/BG models. Here we are more interested in 1/ϵ_ r than ϵ_ r itself because 1/ϵ_ r is directly proportional to Coulomb energy.Fig. <ref>(a) shows that 1/ϵ_ rof the slab model is linear in ϕ, that of CM is approximately linear; but the 1/ϵ_ r's of MG and BG increase very little when ϕ is small and exhibit rapid increases toward the 1/ value for ϕ=1 only for ϕ≳ 0.8 and 0.6, respectively.The linear and near-linear ϕ-dependences of the 1/ϵ_ r'sfor the slab and CM models are underscored by their smallfirst and second derivatives in ϕ, whereas the sharp rises of1/ϵ_ r near ϕ≈ 1 for the MG and BG modelsare illustrated by their large ϕ-derivatives for ϕ≳ 0.7 [Fig. <ref>(b) and (c)]. As described in our previous work <cit.>, when permittivitybecomes a function of IDP concentration, the last subtraction ofG(k) in Eq. (<ref>) has to be modified because G(k) is no longer linear in ϕ's <cit.>.A straightforward generalization of Eqs. (68) and (69) of Ref. <cit.> to the present case with two IDP sequences (but now with neither salt nor counterions) leads to the following replacement for the RPA expressionin Eq. (<ref>) to accommodate a ϕ-dependentϵ_ r:f_ el(ϕ_1,ϕ_2)= ∫_0^∞d k k^2/4π^2{ln[ 1+G_1(k) ] -G_2 (k) },whereG_1(k) =4π/k^2(1+k^2)T^*_0ϵ_r(ϕ)[⟨σ_1 | Ĝ_11(k) | σ_1 ⟩+⟨σ_2 | Ĝ_22(k) | σ_2 ⟩], G_2(k) =4π/k^2(1+k^2)T^*_0ϵ_r(ϕ)[ϕ_1/N_1∑_i| σ_1^(i) | + ϕ_2/N_2∑_i| σ_2^(i) | ] ,and, following Eq. (67) of Ref. <cit.>,T_0^* ≡ 4πϵ_0 k_ BTb/e^2 = T^*/ϵ_ r.The resulting spinodal phase diagrams predicted by the fourϵ_ r(ϕ) models are shown inFig. <ref> forthe (seq1, seq2) pair at T_0^* = 0.05. This temperature is chosen to facilitate comparison with constant-ϵ_ r results because T_0^* = 0.05 corresponds to T^*= 4 when ==80. Fig. <ref>(a)–(d) shows that all fourϵ_ r(ϕ) models have large spinodal regions extending to ϕ_1+ϕ_2≈ 0.8. These spinodal regions are much larger than those predicted by constant-ϵ_ r theories [Fig. <ref>(e), (f)], pointing once again toa significant cooperative effect arising from a decrease in permittivity upon IDP condensation which in turn increases electrostatic attraction and hence more IDP condensation <cit.>. For instance, the condensed-phase volume fractions of the slab and CMmodels ≈ 0.8, which represents a >20-fold increasefrom the condensed-phase volume fraction of ≲ 0.03 for a constantϵ_ r= =80 [Fig. <ref>(e)]. The corresponding enhancement in the MG and BG models are even more prominent—their condensed-phase volume fraction almost reaching the ϕ_1+ϕ_2=1 limit [Fig. <ref>(c), (d)]—because of the sharp rises of their ϵ_ r(ϕ)'swhen ϕ→ 1 (Fig. <ref>).Although the precise quantitative impact of ϵ_ r(ϕ) remains to be ascertained experimentally, our theoretical results suggest strongly that ϕ-dependent relative permittivity should play a significant role in IDP phase separation, and that such a physical cooperative effect may help rectify some of the RPA-predicted condensed-phase volume fractions[e.g. those in Fig. <ref> (a)–(d)] that seemunrealistically low.Interestingly, while the slab and CM models enlarge a single spinodal region vis-à-vis that for constant-ϵ_ r, the MG and BG models produce an additional spinodal region close to the ϕ-ϕ_2 origin. This region is similar in scope tothe constant-ϵ_ r spinodal region, and is well separated from the MG and BG models' lower boundaries at ϕ_1+ϕ_2 ≈ 0.6 (MG)or 0.4 (BG) for their main (much larger) spinodal regions [Fig. <ref>(c)–(f)]. This feature likely arises from the fact that1/ϵ_ r's for MG and BG barely change for ϕ≲ 0.2 [Fig. <ref>(a)], thus the behaviors of these models at small ϕ's should be similar to those of constant-ϵ_ r models. For larger ϕ's, however, because of the rapid increase of their 1/ϵ_ r's with ϕ,the MG and BG models become similar to the slab and CM models.Although the slab/CM and MG/BG models produce similarly expanded spinodal regions for the example in Fig. <ref>, the difference in their predicted ϵ_ r(ϕ)'s do have important implications on the energetics of IDP phase separation. For example, the condensed-phase volume fraction of Ddx4 was recently determined to be ≈ 0.15–0.2 <cit.>. At ϕ≈ 0.2, the slab/CM models posit a significant cooperative effect in favor of phase separation, but the MG/BG models suggest that such cooperativity is negligible unless ϕ≳ 0.6 (Fig. <ref>). Issues related to effective medium approximations can be intricate, as witnessed by the extensivematerials-science literature on the topic <cit.>. Nevertheless, for IDP solutions, our intuition is that the slab/CM scenariois more physically plausible than the MG/BG scenario.Consistent with theslab/CM picture in Fig. <ref>(a), just like dissolved folded proteins, dissolved IDPs occupy volumes excluded to water.Dissolved IDPs and folded proteins have similar densities(Fig. <ref>) of 1.32 – 1.52 g cm^-3 <cit.> and hence similar partial molar volumes on a per-gram basis. In contrast, the MG/BG picture in Fig. <ref>(b)invokes a negative effective molecular polarizabilityfor IDP that counteracts an all-permeating water medium(γ_ MG<0 because <). This scenario is apparently at odds with the reality of IDP excluded volume, suggesting that while the MG/BG models may be applicable in certain solid-state situations <cit.>, they may be problematic for IDP solutions. A definitive resolution of this questionawaits further theoretical and experimental investigation. § CONCLUSIONS To recapitulate, we have taken a step to address the sequence-phase relationship in regard to how mixing/demixing of IDP components in membraneless organelles are governed by the IDPs' amino acid sequences. Going beyond mean-field FH and OV approaches, RPA provides an approximate physical account of sequence effects <cit.>. Our findings point to a multivalent, stochastic, “fuzzy” mode of molecular recognition in that mixing the populations of a pair of IDP sequences is favorable if their charge patterns are similar whereas population demixing is promoted when their charge patterns are very different. For the examples studied, a quantitative correlation is observed betweenthe RPA-predicted tendency for a pair of sequences to demix in two (binary)coexisting phases and the difference in theirSCD parameters. This predicted trend is qualitatively in line with theobserved demixing of the nucleolar IDPs FIB1 and NPM1 because they have verydifferent SCD's, although a comparison of the experimental FIB1/NPM1phase diagram <cit.> with our RPA and FH results indicates thatinclusion of non-electrostatic interactions as well as more biomolecularspecies in the analysis will be necessary for a quantitative theoreticalaccount of sequence-specific ternary coexistence. It should also be noted that our current RPA formulation does not consider counterion condensation,which can be important for IDP sequences with high net charges. A recent transfer matrix theory <cit.> should be helpful in tackling such situations, although as it stands this theory does not addresssequence dependence. Despite our theory's limitations, the simple principles of sequence dependence suggested bythe present effort should already be testable by in vitro experimentson IDP polymers <cit.>. As illustrated by our consideration of IDP-concentration-dependentpermittivity, theoretical study of IDP phase separation is only in its infancy. The logical next steps in the development of RPA theory include extending to systems with more than two sequences and sequences with non-zero net charges. Much biophysics of IDP phaseseparation remains to be discovered. § ACKNOWLEDGEMENTSWe thank Alaji Bah, Robert Vernon (Hospital for Sick Children),Lewis Kay (University of Toronto), Che-Ting Chan (Hong Kong Universityof Science and Technology), Kingshuk Ghosh (University of Denver),Pak-Ming Hui (Chinese University of Hong Kong), and Huan-Xiang Zhou (University of Illinois at Chicago) for helpfuldiscussions. We are also grateful to an anonymous referee for suggestions that led to the pedagogical graphics in Fig. 1.This work was supported by Canadian Cancer Society ResearchInstitute grant no. 703477, Canadian Institutes of Health Research grantMOP-84281, and computational resources provided by SciNet ofCompute/Calcul Canada.§ REFERENCESiopnum | http://arxiv.org/abs/1707.08990v2 | {
"authors": [
"Yi-Hsuan Lin",
"Jacob P. Brady",
"Julie D. Forman-Kay",
"Hue Sun Chan"
],
"categories": [
"q-bio.BM"
],
"primary_category": "q-bio.BM",
"published": "20170727182804",
"title": "Charge Pattern Matching as a \"Fuzzy\" Mode of Molecular Recognition for the Functional Phase Separations of Intrinsically Disordered Proteins"
} |
2017/07/21 2017/07/27 1Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland 2SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands 3Institute for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601 4Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA 5Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA 6SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA 7NASA, Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA 8Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290 Versoix, Switzerland 9Department of Physics, Ehime University, Bunkyo-cho, Matsuyama, Ehime 790-8577 10Department of Physics and Oskar Klein Center, Stockholm University, 106 91 Stockholm, Sweden 11Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 12Research Center for the Early Universe, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 13Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 14Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 15Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA 16Department of Physics and Astronomy, Wayne State University,666 W. Hancock St, Detroit, MI 48201, USA 17Department of Physics, Yale University, New Haven, CT 06520-8120, USA 18Department of Astronomy, Yale University, New Haven, CT 06520-8101, USA 19Centre for Extragalactic Astronomy, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK 20Japan Aerospace Exploration Agency, Institute of Space and Astronautical Science, 3-1-1 Yoshino-dai, Chuo-ku, Sagamihara, Kanagawa 252-5210 21Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502 22The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8302 23Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397 24Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK 25Faculty of Mathematics and Physics, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192 26School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 27Fujita Health University, Toyoake, Aichi 470-1192 28Physics Department, University of Miami, 1320 Campo Sano Dr., Coral Gables, FL 33146, USA 29Department of Astronomy and Physics, Saint Mary's University, 923 Robie Street, Halifax, NS, B3H 3C3, Canada 30Department of Physics and Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK 31Laboratoire APC, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France 32CEA Saclay, 91191 Gif sur Yvette, France 33European Space Research and Technology Center, Keplerlaan 1 2201 AZ Noordwijk, The Netherlands 34Department of Physics and Astronomy, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi 448-8543 35Department of Physics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore,MD 21250, USA 36Department of Applied Physics and Electronic Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-Nishi, Miyazaki, 889-2192 37Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602 38Department of Earth and Space Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043 39Department of Physics, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337 40Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501 41Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA 42Meisei University, 2-1-1 Hodokubo, Hino, Tokyo 191-8506 43Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands 44Research Institute for Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169-8555 45Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo 112-8551 46Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550 47Department of Physics, Toho University,2-2-1 Miyama, Funabashi, Chiba 274-8510 48Department of Physics, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510 49Department of Physics, Kyoto University, Kitashirakawa-Oiwake-Cho, Sakyo, Kyoto 606-8502 50European Space Astronomy Center, Camino Bajo del Castillo, s/n.,28692 Villanueva de la Cañada, Madrid, Spain 51Universities Space Research Association, 7178 Columbia Gateway Drive, Columbia, MD 21046, USA 52National Science Foundation, 4201 Wilson Blvd, Arlington, VA 22230, USA 53Department of Electronic Information Systems, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, Saitama 337-8570 54Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 55Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama 351-0198 56Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 57Department of Physics, University of Wisconsin, Madison, WI 53706, USA 58Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada 59Department of Astronomy, University of Michigan, 1085 South University Avenue, Ann Arbor, MI 48109, USA 60Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son Okinawa, 904-0495 61Faculty of Liberal Arts, Tohoku Gakuin University, 2-1-1 Tenjinzawa, Izumi-ku, Sendai, Miyagi 981-3193 62Department of Astronomy, University of Maryland, College Park, MD 20742, USA 63Faculty of Science, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata, Yamagata 990-8560 64Department of Physics, Nara Women's University, Kitauoyanishi-machi, Nara, Nara 630-8506 65Department of Teacher Training and School Education, Nara University of Education, Takabatake-cho, Nara, Nara 630-8528 66Frontier Research Institute for Interdisciplinary Sciences, Tohoku University,6-3 Aramakiazaaoba, Aoba-ku, Sendai, Miyagi 980-8578 67Astronomical Institute, Tohoku University, 6-3 Aramakiazaaoba, Aoba-ku, Sendai, Miyagi 980-8578 68Department of Physics, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570 69Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027, USA 70Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada 71Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258 72Astronomical Observatory of Jagiellonian University, ul. Orla 171, 30-244 Kraków, Poland 73Max Planck Institute for extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching , Germany 74Faculty of Education, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529 75Faculty of Health Sciences, Nihon Fukushi University , 26-2 Higashi Haemi-cho, Handa, Aichi 475-0012 76MTA-Eötvös University Lendület Hot Universe Research Group, Pázmány Péter sétány 1/A, Budapest, 1117, Hungary 77Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic 78Kashima Space Technology Center, National Institute of Information and Communications Technology, Kashima, Ibaraki 314-8501 79Planetary Plasma and Atmospheric Research Center, Tohoku University, Sendai, Miyagi 980-8578 80The Research Institute for Time Studies, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8511 [email protected]:individual:B0531+21 — radio continuum:stars – X-rays:stars – Giant radio pulsesHitomi X-ray studies of Giant Radio Pulses from the Crab pulsarCorresponding authors are Yukikatsu Terada, Teruaki Enoto, Shu Koyama, Aya Bamba, Toshio Terasawa, Shinya Nakashima,Tahir Yaqoob, Hiromitsu Takahashi, and Shin Watanabe. Hitomi Collaboration, Felix Aharonian1, Hiroki Akamatsu2, Fumie Akimoto3, Steven W. Allen4,5,6, Lorella Angelini7, Marc Audard8, Hisamitsu Awaki9, Magnus Axelsson10, Aya Bamba11,12, Marshall W. Bautz13, Roger Blandford4,5,6, Laura W. Brenneman14, Gregory V. Brown15, Esra Bulbul13, Edward M. Cackett16, Maria Chernyakova1, Meng P. Chiao7, Paolo S. Coppi17,18, Elisa Costantini2, Jelle de Plaa2, Cor P. de Vries2, Jan-Willem den Herder2, Chris Done19, Tadayasu Dotani20, Ken Ebisawa20, Megan E. Eckart7, Teruaki Enoto21,22, Yuichiro Ezoe23, Andrew C. Fabian24, Carlo Ferrigno8, Adam R. Foster14, Ryuichi Fujimoto25, Yasushi Fukazawa26, Akihiro Furuzawa27, Massimiliano Galeazzi28, Luigi C. Gallo29, Poshak Gandhi30, Margherita Giustini2, Andrea Goldwurm31,32, Liyi Gu2, Matteo Guainazzi33, Yoshito Haba34, Kouichi Hagino20, Kenji Hamaguchi7,35, Ilana M. Harrus7,35, Isamu Hatsukade36, Katsuhiro Hayashi20, Takayuki Hayashi37, Kiyoshi Hayashida38, Junko S. Hiraga39, Ann Hornschemeier7, Akio Hoshino40, John P. Hughes41, Yuto Ichinohe23, Ryo Iizuka20, Hajime Inoue42, Yoshiyuki Inoue20, Manabu Ishida20, Kumi Ishikawa20, Yoshitaka Ishisaki23, Masachika Iwai20, Jelle Kaastra2,43, Tim Kallman7, Tsuneyoshi Kamae11, Jun Kataoka44, Satoru Katsuda45, Nobuyuki Kawai46, Richard L. Kelley7, Caroline A. Kilbourne7, Takao Kitaguchi26, Shunji Kitamoto40, Tetsu Kitayama47, Takayoshi Kohmura48, Motohide Kokubun20, Katsuji Koyama49, Shu Koyama20, Peter Kretschmar50, Hans A. Krimm51,52, Aya Kubota53, Hideyo Kunieda37, Philippe Laurent31,32, Shiu-Hang Lee21, Maurice A. Leutenegger7, Olivier O. Limousin32, Michael Loewenstein7, Knox S. Long54, David Lumb33, Greg Madejski4, Yoshitomo Maeda20, Daniel Maier31,32, Kazuo Makishima55, Maxim Markevitch7, Hironori Matsumoto38, Kyoko Matsushita56, Dan McCammon57, Brian R. McNamara58, Missagh Mehdipour2, Eric D. Miller13, Jon M. Miller59, Shin Mineshige21, Kazuhisa Mitsuda20, Ikuyuki Mitsuishi37, Takuya Miyazawa60, Tsunefumi Mizuno26, Hideyuki Mori7, Koji Mori36, Koji Mukai7,35, Hiroshi Murakami61, Richard F. Mushotzky62, Takao Nakagawa20, Hiroshi Nakajima38, Takeshi Nakamori63, Shinya Nakashima55, Kazuhiro Nakazawa11, Kumiko K. Nobukawa64, Masayoshi Nobukawa65, Hirofumi Noda66,67, Hirokazu Odaka6, Takaya Ohashi23, Masanori Ohno26, Takashi Okajima7, Kenya Oshimizu68, Naomi Ota64, Masanobu Ozaki20, Frits Paerels69, Stéphane Paltani8, Robert Petre7, Ciro Pinto24, Frederick S. Porter7, Katja Pottschmidt7,35, Christopher S. Reynolds62, Samar Safi-Harb70, Shinya Saito40, Kazuhiro Sakai7, Toru Sasaki56, Goro Sato20, Kosuke Sato56, Rie Sato20, Makoto Sawada71, Norbert Schartel50, Peter J. Serlemtsos7, Hiromi Seta23, Megumi Shidatsu55, Aurora Simionescu20, Randall K. Smith14, Yang Soong7, Łukasz Stawarz72, Yasuharu Sugawara20, Satoshi Sugita46, Andrew Szymkowiak17, Hiroyasu Tajima3, Hiromitsu Takahashi26, Tadayuki Takahashi20, Shiníchiro Takeda60, Yoh Takei20, Toru Tamagawa55, Takayuki Tamura20, Takaaki Tanaka49, Yasuo Tanaka73, Yasuyuki T. Tanaka26, Makoto S. Tashiro68, Yuzuru Tawara37, Yukikatsu Terada68, Yuichi Terashima9, Francesco Tombesi7,62, Hiroshi Tomida20, Yohko Tsuboi45, Masahiro Tsujimoto20, Hiroshi Tsunemi38, Takeshi Go Tsuru49, Hiroyuki Uchida49, Hideki Uchiyama74, Yasunobu Uchiyama40, Shutaro Ueda20, Yoshihiro Ueda21, Shiníchiro Uno75, C. Megan Urry17, Eugenio Ursino28, Shin Watanabe20, Norbert Werner76,77,26, Dan R. Wilkins4, Brian J. Williams54, Shinya Yamada23, Hiroya Yamaguchi7, Kazutaka Yamaoka3, Noriko Y. Yamasaki20, Makoto Yamauchi36, Shigeo Yamauchi64, Tahir Yaqoob35, Yoichi Yatsu46, Daisuke Yonetoku25, Irina Zhuravleva4,5, Abderahmen Zoghbi59, Toshio Terasawa55, Mamoru Sekido78, Kazuhiro Takefuji78, Eiji Kawai78, Hiroaki Misawa79, Fuminori Tsuchiya79, Ryo Yamazaki71, Eiji Kobayashi71, Shota Kisaka71, Takahiro Aoki80,December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== To search for giant X-ray pulses correlated with the giant radio pulses (GRPs) from the Crab pulsar, we performed a simultaneous observation of the Crab pulsar with the X-ray satellite Hitomi in the 2 – 300 keV band and the Kashima NICT radio observatory in the 1.4 – 1.7 GHz band with a net exposure of about 2 ks on 25 March 2016, just before the loss of the Hitomi mission. The timing performance of the Hitomi instruments was confirmed to meet the timing requirement and about 1,000 and 100 GRPs were simultaneously observed at the main and inter-pulse phases, respectively, and we found no apparent correlation between the giant radio pulses and the X-ray emission in either the main or inter-pulse phases. All variations are within the 2 sigma fluctuations of the X-ray fluxes at the pulse peaks, and the 3 sigma upper limits of variations of main- or inter- pulse GRPs are 22% or 80% of the peak flux in a 0.20 phase width, respectively, in the 2 – 300 keV band. The values become 25% or 110% for main or inter-pulse GRPs, respectively,when the phase width is restricted into the 0.03 phase. Among the upper limits from the Hitomi satellite, those in the 4.5-10 keV and the 70-300 keV are obtained for the first time, and those in other bands are consistent with previous reports. Numerically, the upper limits of main- and inter-pulse GRPs in the 0.20 phase width are about (2.4 and 9.3) × 10^-11 erg cm^-2, respectively.No significant variability in pulse profiles implies that the GRPs originated from a local place within the magnetosphere and the number of photon-emitting particles temporally increases. However, the results do not statistically rule out variations correlated with the GRPs, because the possible X-ray enhancement may appear due to a >0.02% brightening of the pulse-peak flux under such conditions.§ INTRODUCTIONGiant Radio Pulses (GRPs) consist of sporadic and short-lived radiation, during which time the radio flux density becomes 2–3 orders of magnitudes brighter than the regular, averaged pulse flux density.So far, this phenomenon has been discovered in ∼14 radio pulsars (for a review, see <cit.> and references therein), including both “traditional” rotation-powered pulsars (e.g., the Crab pulsar) and millisecond pulsars (e.g., PSR B1937+21). Although the emission mechanism of the GRPs is still unknown, previous radio studies have shown some distinctive properties of the GRPs. The typical temporal width of individual GRPs is narrow, spanning a range from a few nanoseconds to a few microseconds <cit.>. GRPs occur in certain pulse phases with no clear periodicity.The energy spectrum of GRPs follows a power-law distribution <cit.>, different from the Gaussian or log-normal distribution of the normal pulses <cit.>. Since studies of the ordinary pulses can only provide average information from the pulsar magnetosphere, observations of GRPs are imperative for furthering our understanding of the pulsar radiation mechanism.More recently, a hypothetical proposal of GRPs from young pulsars as candidates for the origin of fast radio bursts (FRBs; <cit.>) have been attracting more and more attention. These phenomena areextragalactic bright radio transients with ∼1 msec duration <cit.>. Seeking to reveal properties of known GRPs, such as identifying their counterparts in other wavelengths, is also a key line of investigation to examine the young pulsar model that could explain FRBs (e.g., <cit.>). The Crab pulsar (PSR B0531+21) is one of the most intensively studied rotation-powered pulsars since the initial discovery of its GRPs <cit.>.This famous pulsar exhibits GRPs occurring both in the main pulse and the interpulse, which have been mainly studied at radio wavelengths <cit.>. Since the pulsed energy spectrum of the Crab pulsar covers a wide range, from the coherent radio emission to the incoherent high-energy radiation at optical, X-rays, and gamma-rays, there have been multi-wavelength campaigns to attempt to search for enhancements at higher energy bands, simultaneous with the GRPs. In the optical band, a significant 3% optical enhancement was discovered with 7.2σ significance from the main pulse peak phase by the Westerbork Synthesis Radio Telescope and the 4.2-m William Herschel Telescope <cit.>. This result was further confirmed by the Green Bank Telescope and the Hale telescope <cit.>.These detections imply that the coherent radio emission is somehow linked to the incoherent higher energy (optical) radiation.Despite intensive efforts to search at even higher energy bands, so far there are only upper-limits in soft X-rays and higher energy bands. Reports of these upper limits can be found for soft X-rays (Chandra, 1.5–4.5 keV; <cit.>), for soft gamma-rays (CGRO/OSSE, 50–220 keV; <cit.>), for gamma-rays (Fermi/LAT, 0.1–5 GeV; <cit.>), and for very high energy gamma-rays (VERITAS, >150GeV; <cit.>). The X-ray astronomical satellite Hitomi (ASTRO-H) was launched on February 17, 2016 via a H-IIA launch vehicle from Tanegashima Space Center in Japan, and successfully entered into a low Earth orbit at an altitude of 575 km <cit.>. The satellite is designed to cover a wide energy range from 0.3 keV up to 600 keV with four new X-ray instruments: the microcalorimeter (Soft X-ray Spectrometer, SXS; <cit.>), a wide field-of-view X-ray CCD detector (Soft X-ray Imager, SXI; <cit.>), two Si/CdTe hybrid hard X-ray imagers (Hard X-ray Imager, HXI; <cit.>), and a Compton telescope (Soft Gamma-ray Detector, SGD; <cit.>). Such a wide energy coverage with high time resolution at a few microseconds <cit.> made Hitomi suitable for the search for X-ray enhancement simultaneously with the GRPs.After initial operations before opening the gate valve of the SXS,including successful observations of the Perseus cluster of galaxies<cit.> and other supernova remnants (e.g., N132D and G21.5-0.9)the spacecraft lost communications with the ground stations on March 26,and eventually the mission was terminated. Therefore, the energy coverage below 2 keV was lost for the SXS.On March 25, just before the satellite loss, we observed the Crab pulsar with Hitomi for onboard instrumental calibration activity.The requirement and goal of the absolute timing accuracy of the Hitomi satellite are 350 μs and 35 μs, respectively <cit.>. In order to verify the timing tag accuracy, we compared arrival times of the main pulse peak of the Crab pulsar with the radio or X-ray ephemeris provided by other observatories <cit.>. The archival monthly radio ephemeris of the Crab pulsar has been regulary provided by the Jodrell-Bank observatory[http://www.jb.man.ac.uk/ pulsar/crab/crab2.txt] on every 15th monitoring <cit.>. Interpolating from this, the predicted ephemeris of the Crab pulsar timingin Barycentric Dynamical Time (TDB) is tabulated in table <ref>.Arrival times of the radio pulses are known to be delayed from X-rays in proportion to the interstellar dispersion measure (DM).The long-term, averaged DM of the Crab pulsar is ∼56.8 pc cm^-3, corresponding to ∼120 ms delay of the 1.4 GHz radio pulses relative to X-rays. Although this radio delay is corrected via radio analyses (de-dispersion), this DM is known to show fluctuation in time with ∼0.028 pc cm^-3 (1σ) of a Gaussian distribution. This corresponds to an intrinsic uncertainty of ∼60 μs timing accuracy, higher than our goal for the timing accuracy (35 μs).Therefore, we coordinated follow-up radio observations simultaneous with our X-ray observations to reduce uncertainties due to this fluctuating DM. In this paper, we report X-ray studies of GRPs from the Crab pulsar based on simultaneous X-ray and radio observations. Detailed investigations of the instrumental timing calibration will be summarized in a different paper <cit.>.§ OBSERVATION AND DATA REDUCTION§.§ X-ray and radio simultaneous observation of the Crab pulsarThe X-ray observation of the Crab pulsar was made with all the instruments on board the Hitomi satellite, starting from 12:17 on March 25, 2016 until 18:01 (UT) [TDB] with a total on-source duration of 9.7 ks. The radio observations of the Crab pulsar were made in two frequency bands, (a) 1.4 – 1.7GHz at the Kashima observatoryfrom 03:00:00 to 14:00:00 UTC, and(b) 323.1 – 327.1MHz at the Iitate observatory from 09:30:00 to 13:00:00 UTC. The locations of the observatories are listed in table 1 of <cit.>.After the start time of the X-ray observation we idenfitied 3350 GRPs (section <ref>) for (a), but only 94 GRPs for (b).In terms of the occurence probability (number of GRPs per minute), the ratio between (a) and (b) was ∼19:1. <cit.> reported, on the other hand, that the ratio was ∼3:1 on 6 – 7 September 2014. The marked difference between these ratios seems to be caused byrefractive interstellar scintilation (RISS; e.g. Lundgren et al. (1995)): While the RISS condition for 1.4 – 1.7GHz would have correspondedto a phase of the intensity larger than the average,the RISS condition for 325MHz would have corresponded to a phase of the intensity smaller than the average.Therefore we concentrate on observation (a) in what follows.§.§ Data Reduction of Radio observationsData reduction radio (TBA). Table <ref> is prepared here.comment(YT-20170419): Section 2.2 and 2.3 were swapped. - how the radio data were cleaned up? - how the GRPs are identified? - statistics of GRPs of main pulse / inter pulse. - (A/I) ask Terasawa-san et al to fill this section. §.§.§ Frequency assignmentThe radio observation in the 1.4 – 1.7GHz band was made with the 34m telescope at the Kashima Space Technology Center <cit.> operated by the NICT (National Institute of Information and Communications Technology). We used the ADS3000+ recorder <cit.> which has a capability of 8 individual channels with 4-bit 64MHz Nyquist-rate sampling. (The sampling time step δ t is 1/64 MHz=15.625ns, and the data rate 2Gbit/s.) table <ref> shows the frequency assignment for 8 channels. Channel 7, the backup for channel 6 with a slight frequency shift, was not used for the following data analysis.§.§.§ Determination of DM We first determined the dispersion measure (DM) appropriate for the epoch of the observation, 25 March 2016 (MJD 57472). While the Jodrell Bank Crab pulsar monthly ephemerisreports the values of 56.7657 pc cm^-3 (=DM_ JB) for 15 March 2016 (MJD 57462), and 56.7685 pc cm^-3 for 15 April 2016 (MJD 57493), we should take into account possible intra-month variations of DMwhich are sometimes very erratic (e.g., <cit.>). With DM_ JB as a trial value, we coherently dedispersed<cit.> the ch0 data and found several bright GRPs. We then extended the dedispersion analysis to all the channels for ∼50 ms intervals including these GRPs. The best value of DM, 56.7731± 0.0001pc cm^-3 (=DM_ best), was obtained so as to get the alignment of the substructures of these GRPs (e.g., <cit.>) in all channels with ∼0.1μs accuracy. An example of a successful alignment can be seen in Figure 2 of<cit.>. The frequency bands LL and LH approximately correspond to ch1 and ch4-6 here.§.§.§ Frequency-domain RFI rejection During the process of finding DM_ best,we noticed that two channels, ch2 and ch3, were severly contaminated byradio frequency interferences (RFI). Since RFI occurred intermittently, we could still use these channels for the bright GRP search. However, for weaker GRPs search, we excluded ch2 and ch3 from the following analysis. We further noticed that other channelswere weakly contaminated by RFI in some limited frequency ranges. To minimize the effect of RFI, we filtered out these contaminated frequency ranges.The numerical filter was applied at the first stage of the coherent dedispersion process, where the time series of antenna voltage data are subjected to FFT (fast Fourier transformation) and decomposed into Fourier components. For the RFI-contaminated frequency ranges, we set their Fourier components to zero. The overlapping frequency ranges for ch4-ch5 and ch5-ch6 are also filtered out at this stage.The rightmost column of table <ref> gives the resultant effective bandwidths after filtering. The total effective bandwidth, Δν_ sum=Δν_0 + Δν_1 + ...+ Δν_6, is 106.36MHz.§.§.§ GRP selection In the main panel of figure <ref> dots show all GRP candidates with a S/N (signal to noise ratio) >5.5 (or pulse energy >∼2.2 kJy μs,see Appendix <ref>), where dots are sized and color-coded with the values of S/N.The abscissa and ordinate of the panel representthe time in TDB and the pulsar rotation phase φ respectively.The top panel of figure <ref> shows the time intervals subjected to the time-domain RFI rejection (see Appendix <ref>)in black (OFF), and the time intervals kept, in red (ON). In total, 571s (5729s) in time intervals were rejected (kept).As can be seen in figure <ref> there are two clusters of GRP candidates in φ. We adjusted the initial value of φ at 00:00:00TDB (y_0 in (<ref>)) so as to locate the peak of the main cluster at φ=0, which is the main pulse GRPs(hereafter we call them the “MP-GRP”). The second cluster found around φ=0.4056 corresponds to the inter-pulse GRPs (hereafter “IP-GRP”). Scattered points also seen in figure <ref> are due to the noise component. With the selection criteria, (1) -0.0167 ≤φ≤ +0.0167 for the MP-GRPs and (2) 0.3889 ≤φ≤ 0.4222 for the IP-GRPs, we identified 3090 MP-GRPs and 260 IP-GRPs during the interval between 12:15:00TDB and 14:00:00TDB. We estimated the noise contributions in terms of fake GRPs to be 11 ± 3 (0.4 ± 0.1% and 4 ± 1%for the MP- and IP-GRPs, respectively.) The pulse energy distributions of GRPs have power-law shapes with the spectral indices -2.88 ± 0.52 for the MP-GRPs and -2.91 ± 1.13 for the IP-GRPs.§.§ Data Reduction of Hitomi observation The X-ray data obtained with the Hitomi satellite were processed by the standard Hitomi pipelineversion 03.01.005.005 (Angelini et al 2016) with the pre-pipeline version 003.012.004p2.004using the hitomi ftools in the HEAsoft version 6.20, withCALDB versions gen20161122, hxi20161122, sgd20160614, sxi20161122, and sxs20161122. In the timing analyses of Hitomi data in the following sections,the SXI and SGD-2 data were not used,because the timing resolution of the SXI was insufficient for the analyses andthe SGD-2 was not in the nominal operation mode during the simultaneous epochwith the radio observation. The standard cleaned events were used for the SXS and HXI analyses;the low resolution events (ITYPE == 3 or 4; <cit.>) of the SXS events were not excludedin order to maximize the statistics,although the time resolution of the low resolution events was worse (80 μs)than those of high or med resolution events (5 μs).The HXI data were extracted using a sky image region around the Crab pulsar,out to 70 arcsec radius from the image centroid. On the analyses of the SGD-1 data, the photo-absorption events were extractedas described in Appendix <ref>. At this stage, the total exposure times of the Hitomi Crab observation were 9.7, 8.0, and 8.6 ksec for the SXS, HXI, and SGD, respectively. The background-inclusive light curves of these data were shown in Fig.<ref> black. Note that no energy selection were applied to the events; the rough energy band for the SXS, HXI, and SGD-1 photo absorption events were 2 – 10 keV, 2 – 80 keV, and 10 – 300 keV, respectively.The TIME columns of all the event lists of SXS, HXI, and SGD-1 were convertedinto a barycentric position using the “barycor”ftool in the hitomi packageof HEAsoft 6.20 and the hitomi orbital file <cit.>.The target position for the barycentric correction was(R.A., DEC) = (83D.633218, +22D.014464) for this analyses. The period and period derivatives determined only with the Hitomi data were consistent with the ephemerisfrom the radio summarized in table <ref>. As described in <cit.> and <cit.>, the time differences between instrumentswere negligible for the timing analyses of the giant radio pulses reported here.Finally, all the good time intervals of the radio observation were applied to theHitomi Crab data, which then results in a shorter duration, as shown in Fig.<ref> (red).Consequently, the total exposure times for the SXS, HXI, and SGD-1 that were simultaneously observedwith the radio observatories become 2.7, 1.7, and 2.1 ks, respectively. About 10^3 GRP cycles were exposed among (5 – 6)× 10^4 cyclesby each instrument, as summarized in table <ref>.§ ANALYSES AND RESULTSIn this section, we used the cleaned events of Hitomi SXS/HXI/SGD instruments obtained at the end of the section <ref>and the pulsar ephemeris in table <ref>. §.§ Variation of the Pulse Profiles in GRPs Most significant MP- and IP-GRPs, shown as large circles in figure <ref>, were detected at 12:46:44 and 12:54:10 (TDB) on 25 May 2016, respectively, but no significant variations were seen in the X-ray photons before and after the GRPs. Therefore, we then try to stack X-ray events which were correlated with MP- or IP-GRPs to see a possible enhancement in the X-ray band. X-ray events within one cycle of each MP-GRP (hereafter we call them the “MP-GRP cycles”) were accumulated between φ= -0.5 to +0.5 phasesfrom the arrival time of the main pulse of the radio-defined MP-GRPs (i.e., φ=0).The events outside the MP-GRP cycles were defined as “NORMAL cycles” and were accumulated for comparison. Both groups of events were folded by the radio ephemeris in table <ref>to see the pulse profiles of MP-GRP and NORMAL cycles.As shown in the top panels of the left-hand plots in figure <ref>,no major enhancements could be seen between the two profiles. The difference between the two, shown in the bottom panels, was consistent with being statistically constantamong all the instruments and along all of the phases (-0.5 ≤φ≤ 0.5).Note that the pulse profile of the Crab pulsar with the SXS is free from a possible distortion by the dead time, which occurs in > 5 s on the SXS. Similarly, the distortion of the profiles of the HXI and SGD can be also ignored in comparison between the GRP and NORMAL shapes, although the absolute fractions of the dead to live times were about 75% (Section <ref>).The same analyses were performed for the inter-pulse GRPs (hereafter, “IP-GRP cycles”), and no significant enhancement between pulse profiles at IP-GRP and another NORMAL cycles was found,as seen in figure <ref> (right). The statistical errors were very high on the Hitomi datasets, both in MP- and IP-GRPs. In order to see some possible enhancements in several cycles around the GRPs in a wider time range,we then accumulated the events from 2 cycles before, to 2 cycles after the MP-GRPs; i.e., five pulses -2.5 < φ < 2.5 were plotted where -0.5 ≤φ≤ 0.5 corresponds to the MP-GRP cycle.Similar to the previous single-pulse analyses,the NORMAL cycles, here, were defined outside the 5 cycles around the MP-GRPs.The results were shown in figure <ref>. According to the time intervals between GRPs, about 0.7 % and 2.4% of MP-GRPs were contaminated within ± 1 or ± 2 cycles from the GRP, respectively. To estimate the statistical errors on the pure-pulsed components,the non-pulsed counts accumulated from the OFF phase (φ= 0.6 – 0.8) were subtracted from the pulse profiles of the MP-GRP and NORMAL cycles.Several possible enhancements could be seen in several main pulses in the soft energy band by the SXSin the top panels of figure <ref>,however, the significance was all below 2 σ as indicated in the bottom panels,and no corresponding enhancement was seen in the hard X-ray band by the HXI. The same study could be performed for IP-GRPs but the statistical errors were very highand the results were the same as for the MP-GRP cases.Therefore, no enhancements were detected in all phases among five cycles around GRPs from the Hitomi data. §.§ Pulse peak enhancement at GRPsSince no significant enhancement found in five cycles before/after GRPs (section <ref>), we then concentrated on the statistical tests of possible enhancements at the peak of pulses. Here, we compared the non-pulse subtracted peak-counts (C_ grp) of main- or inter-pulses of MP-GRP or IP-GRPs with those of corresponding NORMAL cycles (C_ nor).In this comparison, we defined four types of phase widths (Δφ) to accumulate the peak counts; i.e., Δφ = 0.20 phases (covering main- or inter pulses), 1/11, 1/31, and 1/128 phases. The enhancement of C_ grp from C_ nor accumulated within Δφ can be defined as ξ(Δφ) ≡C_ grp(Δφ) - C_ nor(Δφ)/C_ nor(Δφ).The table <ref> summarize ξ(Δφ) of each Δφ, shown in the percentage, for each instrument, with the significance to the statistical errors. As a result, no larger than a 2 sigma enhancement was detected around GRPs in all cases. The fluctuation got smaller when we restricted the phase width for MP-GRPsdue to the sharp pulse profile of the main pulse, except for the Δφ = 1/128 ∼ 0.008 phase-width cases with poorer photon statistics, although such a trend could not be seen for the inter-pulses that had a shallower shape. To test the enhancement ξ(Δφ) at the snapshot on GRP (φ=0),same trials were repeated for 29 cycles around the GRP,i.e. the 14 cycles before to the 14 cycles after the MP-GRP or IP-GRPs(-14.5 ≤φ≤ 14.5) as plotted in figure <ref>. Therefore, a possible enhancement at φ= was within thefluctuations of ξ(Δφ) in other cyclesto within 2 σ variations for 28+1 cycles. Numerically, the 3 σ upper limits of the variations at the MP-GRP during the main-pulse phases(i.e., φ= -0.1 – 0.1, with 0.200 phase-width in figure <ref>)will be ξ_ MPGRP(0.200 phase) = 40, 30, and 110 % of the X-ray flux in theNORMAL cycles, with the SXS, HXI, and SGD, corresponding roughly to the 2 – 10, 2 – 80, 10 – 300 keV bands. Similarly, the 3 σ upper limits for the IP-GRP during inter-pulse phases (φ= 0.3 – 0.5)were ξ_ IPGRP (0.200 phase) = 130, 90, and 420 % in the same energy bands listed above, respectively.When all of the instruments (i.e., the SXS, HXI, and SGD-1) were used for this study,the upper-limit values become tighter at ξ(0.200 phase) = 22% and 80% of the NORMAL cycles for the MP- and IP-GRPs, respectively.In addition, in order to see a possible enhancement on a short-time scale around the peaks of pulses, as hadbeen seen in the optical observations <cit.>,the enhancements of MP- and IP-GRPs accumulated within the Δφ = 1/31 ∼ 0.03 phase-width were also numerically checked, ξ(0.03 phase) = 25% and 110% for MP- and IP-GRPs were obtained.The 3-σ upper limits of ξ from the 29-cycles study were summarized in table <ref>.§.§ Upper limit of Enhanced Peak fluxTo convert the enhancement of GRP in count rate into an X-ray flux, the X-ray spectra of purely pulsed components (i.e., main and inter pulses) were numerically tested.First, the SXS and HXI events were extracted by phases,φ= -0.1 – 0.1, φ= 0.3 E 0.5, and φ= 0.6 - E0.8, corresponding to the main-pulse (MP), inter-pulse (IP), and off (OFF) phases, respectively, and the pulse-height distributions were accumulated. The dead time correction was applied to the HXI data with the Hitomi ftools, hxisgddtime; the live time of the HXI-1 and HXI-2 were 73.9 % and 76.6 % for this observation. Only the high-primary and the medium-primary grades (Hp and Mp grades, respectively, defined in <cit.>) were accumulated in the SXS spectral analyses here in order to reduce systematic errors in the response matrix.The X-ray spectra of the pure pulsed components were calculated by subtraction of the OFF-phase spectrum from the MP or IP spectra. Thanks to the fine timing resolutions of the SXS, HXI, and SGD <cit.>, the X-ray spectra of the pure-pulsed components were clearly demonstrated in figure <ref>. To perform spectral fitting of the MP and IP spectra, the spectral response matrices were generated with the Hitomi ftools sxsmkrmf and aharfgen, with the exposure map calculated for the HXI and SXS using the ftools sxsregext and ahexpmap, respectively.The result was that the MP and IP spectra were well reproduced by a single power-law model with a photon indices of 1.94 ± 0.02 and 1.87 ± 0.02 and X-ray fluxes of (4.7 ± 0.1) × 10^-9 and (4.4 ± 0.5) × 10^-9 ergs cm^-2 s^-1 in the 2 – 10 keV band with the reduced χ^2 of 305.85 and 301.18 for 250 degrees of freedom, respectively, as shown in the figure <ref>. The pulsed flux obtained with Hitomi accumulated within the time interval of Δφ phase was summarized in table <ref>.Therefore, the 3 σ upper limit values of enhancement in terms of flux can be obtained by multiplying the values in table <ref> (section <ref>) with those in table <ref>. The upper limits of enhancements of MP-GRPs in the X-ray flux in the 2 – 300 keV band within the phases of 0.20 or 0.03 become (24 or 3.3) × 10^-12 erg cm^-2, respectively, and the same for IP-GRPs were (93 or 9.9) × 10^-12 erg cm^-2, respectively. § DISCUSSIONWith the simultaneous observations of the Crab with Hitomi and Kashima radio observatory, the correlation studies in the X-ray band with about 1,000 GRPs have been performed (Section <ref>).No significant changes in the X-ray pulse profiles were detected along all the phase bins at the GRP cycles (Section <ref>), and the 3-σ upper-limit values for the MP-GRPs in the 2 – 300 keV band with Hitomi were ξ = 22% and 80% within the time interval of Δφ = 0.200 phase, as summarized in table <ref>. The upper limits in the 4.5 – 10 keV and the 70 – 300 keV were obtained for the first time with Hitomi, and those in other bands were consistent with previous works <cit.> as shown in Fig.<ref>. Our result constitutes the second case of a study in the hard X-ray bandwhere the flux (in ν F_ν space) of the pulsed component of the Crab pulsar became the highest,following a previous report of a marginal detection at the 2.7 σ level with Suzaku <cit.>. Our results were mainly limited by the photon numbers in the X-ray band,and the statistical errors dominated the results.The pulse shape in the X-ray band was observationally confirmed to be stable with the 1 σ fluctuation of ∼ 0.7 % level by RXTE showing about-two-times intensive pulses <cit.>, which could be shallow “giant X-ray pulses” (GXPs) but the timing correlations between these shallow GXPs and the GRPs were unknown.In our X-ray correlation study using the Hitomi satellite, we could not identify such GXPs due to a poor effective area. As described in section <ref>, no significant variabilities were detected in the X-ray pulse profiles of 14 cycles before and after the GRPs. Similarly, in the optical band,the enhancements related to the GRPs only happenin narrow time intervals (∼ 100 μs) at the pulse peak,and the pulse profiles in other phases were stable <cit.>. These facts indicated that the magnetosphere is stable during the GRPs, which should originate from a local place within the magnetosphere.What happens during the GRP on the pulsar, when the structure of the magnetosphere does not change? Here, we assumed the emission mechanism of the optical pulses issynchrotron emission, like X-ray pulses,because the optical emission seems to have the same originas that of the X-rays from the multi-wavelength spectrum ofthe pulsed component of the pulsar (e.g. see <cit.>).To increase the synchrotron emission temporarily on a short time scaleof μs, whilst maintaining the structure of the magnetosphere,only two candidates can be considered;a) an increase in the number of particles for radiation, orb) a change in the local magnetic field strength. However, case b) would be considered difficult to achieve normally, and the pulse phase needs not be aligned to the main or inter pulses,and therefore it is straightforward to think thatthe case a) is the origin of the optical enhancement at GRPs. Such occasion might occur after a magnetic reconnection near the light cylinder <cit.> resulting a higher density plasma than the Goldreich-Julian density in the GRP region <cit.>. In the Crab pulsar, the emission regions for the radio, optical, and X-rays arenormally considered to be close to each otherbecause the pulses are well aligned in these energy bands,although the pulse profile in the X-ray band is wider than that in radio. If the number of particles for synchrotron emissionincreases in a local region that emits very short GRPs,a possible X-ray enhancement should also occur very shortly, within about 10 μs just on the pulse peak,like the optical and radio cases. If such short enhancement will be detected in the X-ray band, we are able to reinforce the idea a),but the fine pulse profiles divided by 1/128 phases with Hitomi(Fig. <ref>) do not show such enhancement at the peak statistically.Finally, we discuss the energy balance between the X-ray upper limit of the pulse-peak flux (ξ) and the radiation energy of GRP in the radio band, E_ radio. As described in section <ref>, the enhancement at the pulse peak ξ were within the 2 σ fluctuation among the 29 cycles around GRPs, and the upper limit of ξ in flux were obtained at about 3.3 × 10^-12 erg cm^-2 accumulated in the time interval of Δφ = 0.03 phase. The threshold of detection of GRPs in our radio observation was 2.2 kJy·μs(Section <ref>),which corresponds to a total emission-energy par area ofE_ radio > 2.2 × 10^-17 erg cm^-2in a 10 μs accumulation under the assumption that the GRP pulses emitwith 1 GHz-width in the radio <cit.>. Interestingly, the optical enhancement of ξ_ opt(0.003 phase) = 3% reported by <cit.> within 100 μs bin corresponds to a roughly-equivalent energy with E_ radio. Therefore, here we assume that the radiation energy of possible X-ray enhancement is almost equivalent to E_ radio, although the pulsed energy spectrum of Crab (e.g. <cit.>) indicates that the optical light and X-rays have different origins.If an X-ray detection is performed with rather wide phase-bins (Δφ = 0.20 phase around the peak), it appears at ξ_ X(0.20 phase) = 2 × 10^-5 % of the X-ray normal pulses in the 2 – 300 keV band within the same phase-bins.This enhancement appears better at the ξ_ X (0.0003 phase) = 0.02 % of normal pulse flux when the X-ray observation can resolve the 10 μs time-bin at the pulse peak. But the value is still undetectable under the poor statistics of our Hitomi data and the timing accuracy <cit.>. Therefore, the results did not statistically rule out variations correlated with the GRPs,because the possible X-ray enhancement may appear as >0.02% brightening of the pulse peak under such conditions. We can expect future X-ray missions with larger effective areaand better timing capability,such as the recently-launched NICER mission <cit.>,for continuing the X-ray correlation studies of GRPs. If the GXPs appears in short-phase bins and are correlated with GRPs, NICER may detect the enhancement in X-ray band, althoufh the sensitivity peak of NICER is at a softer energy band thanthat of the previous, larger area mission RXTE, so the count rate expected for Crab pulses is comparable.§ ACKNOWLEDGEMENTS We thank the support from the JSPS Core-to-Core Program.We acknowledge all the JAXA members who have contributed to the ASTRO-H (Hitomi) project.All U.S. members gratefully acknowledge support through the NASA Science Mission Directorate. Stanford and SLAC members acknowledge support via DoE contract to SLAC National Accelerator Laboratory DE-AC3-76SF00515. Part of this work was performed under the auspices of the U.S. DoE by LLNL under Contract DE-AC52-07NA27344.Support from the European Space Agency is gratefully acknowledged.French members acknowledge support from CNES, the Centre National d'Études Spatiales.SRON is supported by NWO, the Netherlands Organization for Scientific Research.Swiss team acknowledges support of the Swiss Secretariat for Education, Research and Innovation (SERI).The Canadian Space Agency is acknowledged for the support of Canadian members.We acknowledge support from JSPS/MEXT KAKENHI grant numbers 15J02737, 15H00773, 15H00785, 15H02090, 15H03639,15K05088,15K05069,15H05438, 15K05107, 15K17610, 15K17657, 16J00548, 16J02333,16J06773,16H00949, 16H06342, 16K05295, 16K05296, 16K05300, 16K13787, 16K17672, 16K17673, 21659292, 23340055, 23340071, 23540280, 24105007, 24244014, 24540232, 25105516, 25109004, 25247028, 25287042, 25400236, 25800119, 26109506, 26220703, 26400228, 26610047, 26800102, JP15H02070, JP15H03641, JP15H03642, JP15H06896, JP16H03983, JP16K05296, JP16K05309, JP16K17667, and JP16K05296.The following NASA grants are acknowledged: NNX15AC76G, NNX15AE16G, NNX15AK71G, NNX15AU54G, NNX15AW94G, and NNG15PP48P to Eureka Scientific.H. Akamatsu acknowledges support of NWO via Veni grant.C. Done acknowledges STFC funding under grant ST/L00075X/1.A. Fabian and C. Pinto acknowledge ERC Advanced Grant 340442.P. Gandhi acknowledges JAXA International Top Young Fellowship and UK Science and Technology Funding Council (STFC) grant ST/J003697/2. Y. Ichinohe, K. Nobukawa, and H. Seta are supported by the Research Fellow of JSPS for Young Scientists.N. Kawai is supported by the Grant-in-Aid for Scientific Research on Innovative Areas “New Developments in Astrophysics Through Multi-Messenger Observations of Gravitational Wave Sources”.S. Kitamoto is partially supported by the MEXT Supported Program for the Strategic Research Foundation at Private Universities, 2014-2018.B. McNamara and S. Safi-Harb acknowledge support from NSERC.T. Dotani, T. Takahashi, T. Tamagawa, M. Tsujimoto and Y. Uchiyama acknowledge support from the Grant-in-Aid for Scientific Research on Innovative Areas “Nuclear Matter in Neutron Stars Investigated by Experiments and Astronomical Observations”.N. Werner is supported by the Lendület LP2016-11 grant from the Hungarian Academy of Sciences.D. Wilkins is supported by NASA through Einstein Fellowship grant number PF6-170160, awarded by the Chandra X-ray Center, operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.We thank contributions by many companies, including in particular, NEC, Mitsubishi Heavy Industries, Sumitomo Heavy Industries, and Japan Aviation Electronics Industry. Finally, we acknowledge strong support from the following engineers.JAXA/ISAS: Chris Baluta, Nobutaka Bando, Atsushi Harayama, Kazuyuki Hirose, Kosei Ishimura, Naoko Iwata, Taro Kawano, Shigeo Kawasaki, Kenji Minesugi, Chikara Natsukari, Hiroyuki Ogawa, Mina Ogawa, Masayuki Ohta, Tsuyoshi Okazaki, Shin-ichiro Sakai, Yasuko Shibano, Maki Shida, Takanobu Shimada, Atsushi Wada, Takahiro Yamada; JAXA/TKSC: Atsushi Okamoto, Yoichi Sato, Keisuke Shinozaki, Hiroyuki Sugita; Chubu U: Yoshiharu Namba; Ehime U: Keiji Ogi; Kochi U of Technology: Tatsuro Kosaka; Miyazaki U: Yusuke Nishioka; Nagoya U: Housei Nagano; NASA/GSFC: Thomas Bialas, Kevin Boyce, Edgar Canavan, Michael DiPirro, Mark Kimball, Candace Masters, Daniel Mcguinness, Joseph Miko, Theodore Muench, James Pontius, Peter Shirron, Cynthia Simmons, Gary Sneiderman, Tomomi Watanabe; ADNET Systems: Michael Witthoeft, Kristin Rutkowski, Robert S. Hill, Joseph Eggen; Wyle Information Systems: Andrew Sargent, Michael Dutka; Noqsi Aerospace Ltd: John Doty; Stanford U/KIPAC: Makoto Asai, Kirk Gilmore; ESA (Netherlands): Chris Jewell; SRON: Daniel Haas, Martin Frericks, Philippe Laubert, Paul Lowes; U of Geneva: Philipp Azzarello; CSA: Alex Koujelev, Franco Moroso. Y. Terada led this study in data analysis and writing drafts, in addition to the Hitomi timing Calibration and software preparation. X-ray data analyses and calibrations were done withT. Enoto, S. Koyama, A. Bamba S. Nakashima, T. Yaqoob,H. Takahashi, S. Watanabe, and K. Oshimizu. T. Terasawa led the radio data analysis with M. Sekido, K. Takehuji, E. Kawai, H. Misawa, F. Tsuchiya, R. Yamazaki, E. Kobayashi, S. Kisaka, and T. Aoki.T. Dotani, L. Gallo, R. Mushotzky, C. Ferrigno, K. Pottschmidt, M. Loewenstein,M. Tsujimoto, and S. 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TEMPO2 also gives the signal frequency ν_ ISMin the rest frame of the interstellar matter (ISM) corresponding to the observation frequency ν_ obs, which is Doppler shifted by the revolutionary+rotationary motion of the earth with respect to the solar system barycenter. We defined the Doppler factor η≡ν_ obs/ν_ ISM.§.§ Calculations of S/NWe dedispersed the raw antenna voltage data of the channel k, V^ raw_k(t),to obtain V_k(t).We integrated[ For simplicity we use the term, 'integrate'. In reality, of course, we calculated equation (<ref>) as finite sums over t_n ≤ t <t_n+1 with an original sampling time step δ t =15.625ns. ]|V_k(t)|^2 over the time interval of Δ t=10 μs,E_k (t_n) = 1/Δ t∫_t_n^t_n+1 |V_k(t')|^2 dt'where we defined the binned times as t_n=t_ start + n Δ t (Here we represent t and t_n by UTC. t_ start=12:15:00 UT.) Between the channel k and6, there is an arrival time difference owing to the propagation group delay,τ_k,6 = e^2/2π m_e c DM {1/ν_k^2 - 1/ν_6^2}ηwhere (e, m_e, c) are usual physical quantities, DM the dispersion measure, and ν_k and ν_6 the highest frequencies of the bands k and 6. The Doppler factor ηappears in equation (<ref>) since τ_k,6, ν_k, and ν_6 are defined in the observer's frame. We combined incoherently E_0 (t_n), E_1 (t_n),..,E_6 (t_n), as,E_ sum (t_n) = ∑_k=0,1,4,5,6 E_k (t_n-τ_k,6)where appropriate interpolations were taken to calculate the RHS of equation (<ref>).We calculated the average E̅_ sum and standard deviation σ_ sum of E_ sum(t_n) over an appropriate longer time interval (Δ T, for which we take 1s). Ideally E̅_ sum and σ_ sum are constant in time. In reality, however, they showed slight and gradual variations in time T_M (= t_ start + M Δ T; M=0,1,2...). We calculate the signal-to-noise ratio (S/N) at t_n for T_M ≤ t_n < T_M+1 as S_ sum(t_n) = 1/σ_ sum(T_M) ( E_ sum(t_n) - E̅_ sum(T_M))With a given threshold S_ sum,thr, we selected `GRP candidates' for each one in which there was an enhancement[ A strong GRP gives enhancements of the average and standard deviation, so that the S/N obtained by equation (<ref>) is reduced. To avoid this effect, we replacethe average and standard deviation in equation (<ref>) with the values interpolated from those obtained in the surrounding time intervals that are unaffected by GRPs. ], S_ sum(t_n) >= S_ sum,thr.§.§ Time-domain RFI rejectionTo eliminate RFI further, we conducted the following auxiliary process: We calculated the squared antenna voltages E^ raw_k (t_n)following equation (<ref>) except that we used V^ raw_k(t) instead of V_k(t). From E^ raw_k (t_n) we then calculatedE^ raw_ sum (t_n) ≡∑_k=0,1,4,5,6 E^ raw_k (t_n)and their average and standard deviation, E̅^ raw_ sum(T_M) and σ^ raw_ sum(T_M). We watched the behaviors of σ_ sum(T_M) and σ^ raw_ sum(T_M) throughout the observation interval. When σ_ sum(T_M) < σ^ raw_ sum(T_M), we reject the dataS_ sum(t_n) for T_M ≤ t_n <T_M+1, as affected by RFIs. §.§ Rotation phase and GRP identificationWe calculated the phase φ_n of a GRP candidate at the time in UTC t_n, φ_n = frac(y) with y = y_0 + ν_ rott̃_n + 0.5 ν̇ _ rott̃_n^2,wheret̃_n = f̃(t_n) is the time in TDB, frac(y) the fractional part of y, y_0 the initial phase at 00:00:00 TDB,ν_ rot and ν̇_ rot the rotation frequency and its time derivative fromthe Jodrell bank monthly ephemeris (table <ref>). In the operation of equation (<ref>),we also recorded the integer part of y as the sequential pulse number of the day, N_ pulse, which is to be used for the GRP and X-ray photon comparison.As discussed in section <ref> we classify the GRP candidates according to their values of φ by setting two selection ranges,(φ_ MP,1, φ_ MP,2) for main pulse GRPs, and (φ_ IP,1, φ_ IP,2) for interpulse GRPs. With the choice of Δ t=10μs, two thirds of GPR candidates in the 1.4-1.7GHz band are isolated in the φ space. However, for the remaining one third of (stronger) GRPs, 2∼4 GRP candidates of the same N_ pulse are found in the same selection range for φ. For such cases, we count them as one GRP, either main pulse or interpulse.For a GRP occurring near the binned time boundary t=t_n, its contribution is divided into E_ sum (t_n) and E_ sum (t_n+1), and the corresponding S_ sum(t_n) and S_ sum(t_n+1) are artificially lowered (sometimes both are less than S_ sum ,thr). To avoid mal-counting of GRPs caused by this effect, we repeated the procedure from equation (<ref>) to equation (<ref>) for the binned time with a shift of Δ t/2: Firstly, we calculateE_k (t_n+1/2) = 1/Δ t∫_t_n+Δ t/2^t_n+1+Δ t/2 |V_k(t')|^2 dt' equation (<ref>')If the resultant S_ sum(t_n+1/2) exceeds S_ sum, thr, this GRP is `rescued from the sea of the noise'. We found that about 10% of GRPs are thus rescued. §.§ Radiometer equationThe flux density F is calculated as C × (S/N) with C given by the radiometer equation (Dicke, 1946; Lorimer and Kramer, 2004),C=SEFD+S_ CN/√(Δν_ sumΔ t) [Jy]where SEFD is the system equivalent flux density, and S_ CN is the flux density of the Crab nebula. With the representative values, SEFD= 500Jy and S_ CN= 810Jy (Mikami et al., 2016), we get C = 40.2Jy for Δν_ sum=106.36MHz (section <ref>) and Δ t=10μs. The GRP threshold S/N=5.5 used in section <ref> corresponds to a flux density threshold of F= 220Jy, or a pulse energy threshold FΔ t=2.2kJy μs. § EXTRACTION OF THE SGD PHOTO ABSORPTION MODEOn the timing analyses of the SGD-1 datain section <ref>,the photo-absorption events were extractedfrom the unscreened event files to have more effective areas than those of thestandard Compton scattered events;an expression that "FLAG_LCHKMIO==b0 && FLAG_CCBUSY[1]==b0 && FLAG_CCBUSY[2]==b0 && FLAG_CCBUSY[3]==b0 && FLAG_HITPAT[1]==b0 && FLAG_HITPAT[2]==b0 && FLAG_HITPAT[3]==b0 && FLAG_HITPAT[4]==b0 && FLAG_FASTBGO[1]==b0 && FLAG_FASTBGO[2]==b0 && FLAG_FASTBGO[3]==b0 && FLAG_FASTBGO[4]==b0 && FLAG_SEU==b0 && FLAG_LCHK==b0 && FLAG_CALMODE==b0 && FLAG_TRIGPAT[29]==b0 && CATEGORY==85 && MATTYPE==1 && NUMSIGNAL==1 " were applied to the ufa event and standard GTI in the 2nd extension of the cleaned events were also applied. | http://arxiv.org/abs/1707.08801v2 | {
"authors": [
"Hitomi Collaboration",
"Felix Aharonian",
"Hiroki Akamatsu",
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We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.A Locally Adapting Technique for Boundary Detection using Image Segmentation Marylesa Howard [email protected] Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USAMargaret C. Hock [email protected] Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USADepartment of Mathematical Sciences University of Alabama in Huntsville 301 Sparkman Drive, 258A, Huntsville, AL 35899, USAB. T. Meehan [email protected] Signal Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USALeora Dresselhaus-Cooper [email protected] Department of Physical Chemistry, Institute for Soldier Nanotechnology Massachusetts Institute of Technology 77 Massachusetts Avenue, NE47-593, Cambridge, MA 02139, USA December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION Abstract elementary classes (AECs) <cit.> were introduced by Shelah in the late seventies as a purely semantic framework in which to study the model theory of infinitary logics such as _∞, ω. At about the same time, Lair <cit.> introduced the notion of an accessible category (which he called a “catégorie modelable”) and proved that accessible categories are exactly those that can be sketched. Independently, the second author showed in his doctoral thesis <cit.> that accessible categories correspond to classes of models of an _∞, ∞-sentence, thus exhibiting an explicit connection with model theory. Later Makkai and Paré <cit.> independently rediscovered these results and showed that they say essentially the same thing in different languages. The connection between AECs and accessible categories was then studied more closely by the first author <cit.> and independently by Beke and the second author <cit.>. These works characterized AECs as special kinds of accessible categories with all directed colimits and whose morphisms are monomorphisms.In recent joint work with Boney and Grossberg <cit.> the authors introduced μ-AECs, a generalization of AECs where only closure under μ-directed colimits is required. It was shown that μ-AECs correspond exactly to accessible categories whose morphisms are monomorphisms (the correspondence is given in terms of an equivalence of categories: see Fact <ref> for a more precise statement). This gives evidence that these frameworks are natural.One of the questions that motivated Shelah's introduction of AECs is the eventual categoricity conjecture: an AEC categorical in a single high-enough cardinal (i.e. with a unique model of that cardinality, up to isomorphism) should be categorical in all high-enough cardinals. This conjecture has profoundly shaped the development of the field (see the introduction of <cit.> for a survey and history of the conjecture). Note that Shelah's eventual categoricity conjecture can be rendered as a purely category-theoretic statement, with cardinalities replaced by presentability ranks—see <cit.>)—and hence can be posed in relation to general accessible categories. Recently, the third author <cit.> proved that the conjecture holds in the more restricted framework of universal classes: classes of structures closed under isomorphism, substructures, and union of chains. Universal classes had previously been studied by, among others, Tarski <cit.> and Shelah <cit.>.In this paper, we show (Theorem <ref>) that universal classes have a natural category-theoretic analog: the locally ℵ_0-multipresentable categories (whose morphisms are monomorphisms) introduced by Diers <cit.>. They can be characterized as the ℵ_0-accessible categories that have all connected limits, i.e. those generated from equalizers and wide pullbacks (see Fact <ref>). More generally, we prove that μ-universal classes (where we allow the relations and functions to have arity strictly less than μ) correspond to locally μ-multipresentable categories.After the initial submission of this paper, Hyttinen and Kangas <cit.> have continued the work of the third author by proving that, in a technical model-theoretic sense, the big-enough models in an eventually categorical universal classes look like either vector spaces or sets. In light of the present paper, it would be interesting to know whether their result has a category-theoretic analog.Another type of AEC that was studied in the third author's proof of the categoricity conjecture for universal classes are those which admit intersections. They were introduced by Baldwin and Shelah <cit.> and can be characterized as the AECs admitting a certain closure operator. Any universal class admits intersections and the third author has shown <cit.> that the categoricity conjecture holds in AECs admitting intersections if we assume a large cardinal axiom. We suggest that such AECs are also natural by characterizing them (Theorem <ref>) as the locally ℵ_0-polypresentable categories of Lamarche <cit.>. These are the ℵ_0-accessible categories with wide pullbacks (see Fact <ref>).Indeed, this result generalizes as well: per Theorem <ref>, any μ-AEC admitting intersections is a locally μ-polypresentable category. Examples of locally multi- and polypresentable categories arise naturally in areas such as functional analysis, algebra, and even computer science, see Example <ref>. In the final section of this paper, we present a generalization of Shelah's presentation theorem <cit.> to all accessible categories with arbitrary μ-directed colimits whose morphisms are monomorphisms: we show (Theorem <ref>) that any such category is the essential image by a faithful functor of a μ-universal class. This generalizes the category-theoretic presentation theorem of the first and second authors <cit.> as well as Boney's presentation theorem for metric AECs <cit.>.(Recall that metric AECs are an analogue of AECs in which the structures have underlying complete metric spaces, rather than sets: see <cit.> or <cit.>.) We give one more characterization of the μ-AECs admitting intersections as those classes for which the functor is, in a sense, close to being full (see Definition <ref> and Theorem <ref>).Throughout this paper, we assume the reader is familiar with basic category theory as presented e.g. in <cit.>. We also assume some familiarity with μ-AECs and their relationship with accessible categories <cit.>.We use the following notational conventions: we write K for a class of τ-structures and(boldface) for a pair (K, ), whereis a partial order. We write(script) for a category. We will abuse notation and write M ∈ instead of M ∈ K. For a structure M, we write U M for its universe, and |U M| for the cardinality of its universe. We write M ⊆ N to mean that M is a substructure of N. For α an ordinal, we let <αA [resp. αA] denote the set of sequences of length less than [resp. exactly] α with elements from the set A. We will abuse this notation as well, writing <αM in place of <αU M.The authors would like to thank Will Boney for comments that helped improve the presentation of this paper.§ UNIVERSAL CLASSES OR CLASSES WITH INTERSECTIONS The following concept was introduced by Tarski for finite vocabularies <cit.> and by Shelah for infinite (but still finitary) vocabularies <cit.>. In this paper, we study it for potentially infinitary vocabularies: Let μ be a regular cardinal. K is a μ-universal class if:* K is a class of structures in a fixed μ-ary vocabulary τ = τ (K).* K is closed under isomorphism, unions of μ-directed systems of τ-substructure inclusions, and τ-substructures. When μ = ℵ_0, we omit it and just say that K is a universal class. We say that an _∞, ∞-sentence is universal if it has the form ∀ϕ, where ϕ is quantifier-free. We say that a theory is universal if it consists only of universal sentences. Tarski has shown <cit.> that a universal class in a finite (and finitary) vocabulary is the class of models of a universal _ω, ω-theory. The proof generalizes without difficulties to infinitary logics:[Tarski's presentation theorem] Let μ be a regular cardinal and K be a class of structures in some μ-ary vocabulary τ. The following are equivalent:*There is a set Γ of quantifier-free _∞, μ-types such that K is the class of all τ-structures omitting Γ.*K is the class of models of a universal _∞, μ theory.*K is a μ-universal class.That (<ref>) implies (<ref>) and (<ref>) implies (<ref>) can be easily verified. To see that (<ref>) implies (<ref>), let K_0 be the class of τ-structures that are generated by a set of size less than μ and are not contained in any member of K. Let Γ be the set of types that code each structure in K_0, and use a μ-directed system argument to see that K is the set of τ-structures omitting Γ. Recall from <cit.> that a (μ-ary) abstract class is a pair = (K, ≤) such that K is a class of structures is a fixed μ-ary vocabulary τ = τ (), and ≤ is a partial order on K that respects isomorphisms and extends the τ-substructure relation. We say that such ais a μ-universal class if K is a μ-universal class and = ⊆.In any abstract class , there is a natural notion of morphism: we say that f: M → N is a -embedding if f is an isomorphism from f onto f[M] and f[M]N. We can see an abstract class and its -embeddings as a category. In fact (see <cit.>), an abstract class is a replete and iso-full subcategory of the category of τ-structures with injective homomorphisms.We now recall the definition of a μ-AEC from <cit.>: Let μ be a regular cardinal. An abstract classis a μ-abstract elementary class (or μ-AEC for short) if it satisfies the following three axioms:* Coherence: for any M_0, M_1, M_2 ∈, if M_0 ⊆ M_1M_2 and M_0M_2, then M_0M_1.* Chain axioms: if M_i : i ∈ I is a μ-directed system in , then: * M := ⋃_i ∈ I M_i is in .* M_iM for all i ∈ I.* If M_iN for all i ∈ I, then MN. * Löwenheim-Skolem-Tarski (LST) axiom: there exists a cardinal λ = λ^<μ≥ |τ ()| + μ such that for any M ∈ and any A ⊆ U M, there exists M_0 ∈ with M_0M, A ⊆ U M_0, and |U M_0| ≤ |A|^<μ + λ. We write () for the least such λ.Note that when μ = ℵ_0, we recover Shelah's definition of an AEC from <cit.>. Note, too, that a μ-universal class, equipped with the τ-substructure relation, is a μ-AEC. In this case, we may call it a universal μ-AEC. Equivalently, a universal μ-AEC is a μ-AECsuch that for any N ∈, M ⊆ N implies that M ∈ and MN.More general than μ-universal classes are μ-AECs admitting intersections. AECs admitting intersections were introduced in <cit.> and further studied in <cit.>. A μ-AECadmits intersections if for any N ∈ and any A ⊆ U N, the set ^N (A) = ^N (A) := ⋂{M ∈| MN, A ⊆ U M}is the universe of a -substructure of M. In this case, we abuse notation and write ^N (A) for this substructure as well. Note that any universal μ-AECadmits intersection, since ^N (A) is a substructure of N andis closed under substructures. On the other hand, the AEC of algebraically closed fields admits intersections but is not a universal class (ℚ is a subfield of an algebraically closed field, but it is not algebraically closed).Many of the properties of AECs admitting intersections proven in <cit.> can be generalized to μ-AECs: it suffices to replace ℵ_0 by μ in the proofs. For example: Letbe a μ-AEC that admits intersection.* If MN and A ⊆ U M, then ^M (A) = ^N (A).* Local character: If B ⊆^N (A) and |B| < μ, then there exists A_0 ⊆ A such that |A_0| < μ and B ⊆^N (A_0).We will use these facts without much comment in the sequel.Similarly to <cit.>, one can define the notion of a pseudo-universal μ-AEC: A μ-AECis pseudo-universal if it admits intersections and whenever f, g: M → N are -embeddings and A ⊆ U M is such that fA = gA, then f ^M (A) = g ^M (A). A universal μ-AEC is pseudo-universal: since ^M (A) is just the closure of A under the functions of M, any -embedding with domain ^M (A) is determined by its restriction to A. Some motivation may be in order: consider the class of groups in the language containing a symbol for multiplication, identity, and inverse. Then this is a universal class. On the other hand, if we look at the class of groups in the language containing only multiplication and identity, then it is not universal: substructures need not be closed under inverses. However it will still be pseudo-universal, since inverses are definable from the rest. In fact, we will see shortly (Theorem <ref>) that this is how all pseudo-universal classes look: any such class admits a functorial expansion to a universal one.Although we will not need it in this paper, we also note that we can imitate the proof of <cit.> to show that any pseudo-universal μ-AEC is fully (<μ)-tame and short over ∅.We now show that pseudo-universal μ-AECs are the same as universal μ-AECs, up to extension of the vocabulary. Recall from <cit.> the notion of a functorial expansion of an abstract class : roughly, it is an abstract classin an expansion of τ () such that the reduct map is an isomorphism of concrete categories. We show that every pseudo-universal class (that does not contain the empty structure) can be functorially expanded to a universal class. This result is not needed in the rest of this paper, where we deal with equivalence of categories, but shows that every pseudo-universal class is isomorphic (not just equivalent) to a universal class. Letbe a pseudo-universal μ-AEC that does not contain the empty structure. Then there is a functorial expansionofwhich is a universal μ-AEC. Moreover, |τ ()| ≤ 2^ (). Any abstract class admits a notion of type, called Galois (or orbital) types in the literature. They can be defined as the finest notion of type preserving -embeddings (see <cit.> for a formal definition). The proof, in short, is to take the (<μ)-Galois Morleyization (i.e. expand the language with a symbol for every type of length less than μ, see <cit.>) and then add Skolem functions. This works since types are able to code the closure operator. We now give a self-contained implementation of this proof.We will write ^M () instead of ^M () (heredenotes the range of , i.e. the set of elements in the sequence ) and b for the concatenation of the sequencewith the element b. For M_1, M_2 ∈, _ℓ∈<μM_ℓ, ℓ = 1,2, write (_1, M_1) ≡ (_2, M_2) if there exists an isomorphism f: ^M_1 (_1) ≅^M_2 (_2) such that f (_1) = _2. Note that ≡ is an equivalence relation with at most 2^ ()-many classes. We say that an equivalence class C codes closure if C = [( b, M)]_≡ implies that b ∈^M (). Note that if C codes closure, C = [( b, M)]_≡, and ( b', M) ≡ ( b, M), then b' = b. Indeed, by definition there is an isomorphism f: ^M ( b) ≅^M ( b') such that f ( b) = f ( b'). This is in particular an isomorphism f: ^M () ≅^M () sendingto . The identity is another such isomorphism, hence by pseudo-universality f must be the identity.Fix a well-ordering ≼ on the set of equivalence classes coding closure. Letconsist of τ () together with a new α-ary function symbol f_C for each C coding closure with C = [( b, M)]_≡, ℓ () = α. For each M ∈, define a -expansionby setting f_C^ () to be the unique b ∈^M () such that C = [( b, M)]_≡, provided it exists. Otherwise, let D be ≼-least coding closure such that there exists b' ∈ M with D = [(b', M)]_≡ and set f_C^ () = b'. Note that such a D always exists as ^M (∅) ≠∅ (we are assuming the empty structure is not part of ).Let K := {| M ∈}. It is easy to check that K is a functorial expansion of(see also <cit.>), hence we can let := (K, ), where MN if and only if M ⊆ N and M τ ()N τ (). We claim thatis a universal μ-AEC. It is enough to show that for any N ∈, M ⊆ N implies that MN. Indeed, noting that , and hence , admits intersections, it suffices to show that ^N (M) = M. In what follows, we do not distinguish between ≡ computed inandand between the closure operations _ and _: sinceis a functorial expansion of , they are the same up to expansion of the language.Clearly, M ⊆^N (M), so let us check that ^N (M) ⊆ M. Let b ∈^N (M). By local character (Fact <ref>), there exists A ⊆ U M of size less than μ such that b ∈^N (A). Letbe an enumeration of A. Let C := [( b, N)]_≡. Then C codes closure and b ∈ N, hence we must have that f_C^N () = b. Since M ⊆ N, b = f_C^N () = f_C^M (), so b ∈ U M, as desired.The restriction thatdoes not contain the empty structure is essential, as otherwise one would not be able to add constant symbols to the vocabulary to code the closure of the empty set. This is, nonetheless, not a serious restriction: given any μ-AECthere is a μ-AEC ' that is isomorphic toas a category and does not contain the empty structure: for each M ∈, expand M to a (τ () ∪{c})-structure M' ∈', where c is a new constant symbol, such that c^M'∉ U M, U M' = {c^M'}∪ U M, and a relation R^M' () holds if and only if either c^M'∈ran () or c^M'∉ran () and R^M ().For any function symbol f∈τ (), define f^M' () = c^M' if c^M'∈ran () and f^M' () = f^M () otherwise. Finally, for M_1', M_2' ∈', let M_1 ' M_2 if and only if M_1M_2 and c^M_1' = c^M_2'. § ACCESSIBLE CATEGORIES To present category-theoretic equivalents of μ-universal classes and μ-AECs admitting intersections, we require the notion of an accessible category (see <cit.> or <cit.>). Letbe a category and let λ be a regular cardinal.* An object M is λ-presentable if its hom-functor (M,-):→ preserves λ-directed colimits. Put another way, M is λ-presentable if for any morphism f:M→ N with N a λ-directed colimit ⟨ϕ_α:N_α→ N⟩, f factors essentially uniquely through one of the N_α, i.e. f=ϕ_α f_α for some f_α:M→ N_α.*is λ-accessible if it has λ-directed colimits andcontains a set S of λ-presentable objects such that every object ofis a λ-directed colimit of objects in S.*is accessible if it is λ'-accessible for some regular cardinal λ'.Intuitively, an accessible category is a category with allsufficiently directed colimits and such that every object can be written as a highly directed colimit of “small” objects.Here “small” is interpreted in terms of presentability, a notion of size that makes sense in any (possibly non-concrete) category. In the category of sets, of course, a set is λ-presentable if and only if its cardinality is less than λ; in an AEC , the same is true for all λ>().From <cit.>, we have that μ-AECs are the same as accessible categories whose morphisms are monomorphisms, up to equivalence of categories: Ifis a μ-AEC, then it is an ()^+-accessible category with all μ-directed colimits whose morphisms are monomorphisms. Conversely, any μ-accessible category whose morphisms are monomorphisms is equivalent to a μ-AEC. In general, a μ-AEC need not be a μ-accessible category and, in fact, the least cardinal λ such thatis λ-accessible cannot be bounded by a function of μ: this is the case, in short, because the index of accessibility is determined as much by the parameter () as by μ. For example, given a regular cardinal λ, the (ℵ_0-)AECof sets of cardinality at least λ (ordered by subset inclusion) contains no λ-presentable objects, hence cannot be ℵ_0-accessible or even λ-accessible.It is, however, λ^+-accessible, as ()=λ. We can show, though, that μ-AECs admitting intersections are μ-accessible: Ifis a μ-AEC admitting intersections, thenis μ-accessible. By definition of a μ-AEC,has all μ-directed colimits. It remains to exhibit a set of μ-presentable objects insuch that every element ofis a μ-directed colimit thereof.Let us call M ∈ μ-generated if there exists a set A ⊆ U M with |A| < μ such that M = ^M (A). Clearly, there is only a set of isomorphism types of μ-generated objects. Moreover by the local character property of the closure operator (Fact <ref>), it follows that any N ∈ can be written as ⋃_A ⊆ U N, |A| < μ^N (A). In other words, every N is a μ-directed colimit of μ-generated objects. Moreover it is easy to check that μ-generated objects are μ-presentable. This completes the proof. § AXIOMATIZABILITY OF ACCESSIBLE CATEGORIES For completeness, we recall that accessible categories can be presented syntactically. For an _∞, ∞-sentence ϕ, we denote by (ϕ) the category of models of ϕ with homomorphisms (i.e. mappings preserving all functions and relations). We say that a categoryis axiomatizable by ϕ ifis equivalent to the category (ϕ). We call ϕ basic if it is a conjunction of sentences of the form ∀ (ϕ_1 →ϕ_2), where ϕ_1 and ϕ_2 are positive existential formulas. Following <cit.>, let us call a category (∞, μ)-elementary if it is axiomatizable by a basic _∞, μ-sentence (this is the same as the category of models of an arbitrary _∞, μ-sentence ordered by elementarity according to a suitable fragment, see <cit.>). A theorem of the second author <cit.> asserts that accessible categories are precisely the (∞, ∞)-elementary ones. More precisely, by <cit.> we have: * Any (∞, μ)-elementary category is accessible and has all μ-directed colimits.* Any μ-accessible category is (∞, μ)-elementary.In this paper, we are interested in abstract classes where the morphisms are substructure embeddings. For ϕ an _∞, ∞-sentence, denote by (ϕ) the category of models of T with substructure embeddings (i.e. injective mappings preserving all functions and relations and reflecting all relations). Thus the difference betweenandwhen all morphisms are mono is the same as the difference between graphs ordered by the subgraph relation and graphs ordered by the induced subgraph relation. From the point of view of category theory, this is not a serious difference: Let μ be an infinite cardinals and letbe a category. The following are equivalent:* is equivalent to (ϕ), for ϕ a basic _∞, μ-sentence, and all the morphisms ofare monomorphisms.* is equivalent to (ϕ), for ϕ a basic _∞, μ-sentence.Assume (<ref>). Add a sort for each relation symbol and code relations as sets of tuples. Let ϕ' describe this. Then (ϕ) and (ϕ') are isomorphic. Therefore (<ref>) holds. Conversely, assume (<ref>). For each relation symbol R, add a relation symbol R coding the negation of R. Also add a relation symbol ≠ coding non-equality. Once again, we obtain a sentence ϕ' such that (ϕ') is isomorphic to (ϕ). Therefore (<ref>) holds. It follows, for example, (see Lemma <ref> and Fact <ref>) that μ-AECs admitting intersections are equivalent to the category of models of an _∞, μ-sentence. Note that the stronger statement that every μ-AEC admitting intersections is directly axiomatizable (without changing the category) by an _∞, μ-sentence is false. Indeed, the AEC whose models are equivalence relations all of whose classes are countably infinite, ordered by the relation “equivalence classes do not grow,” is not the class of models of an _∞, ω-sentence.Similarly, it is not known whether any AEC is directly _∞,()^+-axiomatizable, but by Fact <ref>, it is equivalent to a category of models of an _∞,()^+-sentence.It is also not known whether there is a natural category-theoretic definition of the (∞, μ)-elementary categories (see the discussion after <cit.>). In fact, it is not known whether all μ-AECs are (∞, μ)-elementary (this question was posed in <cit.> in relation to accessible categories with all morphisms monomorphisms).§ LOCALLY MULTI- AND POLYPRESENTABLE CATEGORIES The concept of a locally presentable category is originally due to Gabriel and Ulmer <cit.>. We will use the following definition:For μ a regular cardinal, we say that a categoryis locally μ-presentable if it is μ-accessible and has all (small) colimits. We say it is locally presentable if it is μ'-locally presentable for some μ'. Examples include the category of groups with group homomorphisms or any Grothendieck topos. Although we will move quickly to other characterizations, the initial definitions of, and motivations for, locally multipresentable and locally polypresentable categories arise from mathematically natural weakenings of the notion of colimit.Just as colimits go back to initial objects, one can define a notion of multicolimit that goes back to a multiinitial object. This is a setof objects of a categorysuch that for every object M ofthere is a unique i ∈ and a unique morphism f_i : i → M. For example, the category of fields and field homormorphisms does not have an initial object, but does have a multiinitial object: the prime fields in each characteristic. There is also the still weaker notion of a polyinitial object. This is a setof objects of a categorysuch that for every object M in :* There is a unique i ∈ having a morphism i → M.* For each i ∈, given f,g:i → M, there is a unique (isomorphism) h:i → i with fh=g. For example, the algebraic closures of the prime fields form a polyinitial object in the category of algebraically closed fields.The multicolimit of a diagram D in a categoryis a multiinitial object in the category of cones on D, i.e. a set of cones such that for any cone on D, there will be a unique induced map from exactly one of the members of the set.The polycolimit of a diagram is defined similarly.One then defines locally multi- and polypresentable categories by replacing colimits by multicolimits and polycolimits, respectively: For μ a regular cardinal, we say that a categoryis locally μ-[multi/poly]presentable if it is μ-accessible and has all [multi/poly]colimits. Locally [multi/poly]presentable means locally μ'-[multi/poly]presentable for some μ'. The setin the definition of a multiinitial object is allowed to be empty. Thus in a locally μ-multipresentable category, a diagram of the form @=3pc M_1M_0[u][r] M_2may not even be completable: there might not exist morphismsM_ℓ→ N making the diagram commute. In model-theoretic terms, the category may not have the amalgamation property (see <cit.> for examples of universal classes failing the amalgamation property in non-trivial ways). * As noted above, the category of all fields with field homomorphisms is locally ℵ_0-multipresentable.It is not locally ℵ_0-presentable as it cannot have an initial object—a consideration of characteristics makes clear that no single field can map into all of the others.There is, however, an initial object relative to the fields of each fixed characteristic, which together form the multiinitial family (<cit.>).* The category of linearly ordered sets with order-preserving maps is locally ℵ_0-multipresentable, with multiinitial family consisting of two objects, one for the connected linear orders and one for the disconnected (<cit.>).* The category of pre-Hilbert spaces (i.e. we do not require completeness) with linear orthogonal maps is locally ℵ_0-multipresentable, while the category of Hilbert spaces with linear orthogonal maps is locally ℵ_1-multipresentable (<cit.>).* As noted above, the category of algebraically closed fields with fields homomorphisms is locally ℵ_0-polypresentable (<cit.>).Here the polyinitial family consists of the closures of the prime fields.* Lamarche's categories of aggregates <cit.> and Coquand's categories of embeddings <cit.> are examples of, respectively, locally ℵ_0-multipresentable and locally ℵ_0-polypresentable categories whose morphisms are monomorphisms.Both notions arise in theoretical computer science, as efforts to model the phenomenon of polymorphism in type theory.We can also characterize each of these categories in terms of limits: Letbe a μ-accessible category.* <cit.>is locally μ-polypresentable if and only ifhas wide pullbacks (that is, limits over diagrams A_i→ A, i∈ I, where I is a set).* <cit.>, <cit.>is locally μ-multipresentable if and only ifhas all connected limits.* <cit.>, <cit.>is locally μ-presentable if and only ifhas all limits.Note that the locally presentable categories whose morphisms are monomorphisms are not very interesting: since they have coequalizers, it must be the case that any two morphisms f, g : M → N are equal! In other words, between any two objects there is at most one morphism. Thus locally presentable categories whose morphisms are monomorphisms are exactly the complete lattices. In particular, they are small and their objects are rigid.However, the situation is different for locally multi- and polypresentable categories: Letbe a μ-AEC. Ifadmits intersections, thenis locally μ-polypresentable. If in additionis pseudo-universal, thenis locally μ-multipresentable.By Lemma <ref>,is μ-accessible. It is easy to check that having wide pullbacks in a μ-AEC is the same as admitting intersections. Thereforehas wide pullbacks and hence by Fact <ref> is locally μ-polypresentable.Assume now thatis pseudo-universal. We check thathas connected limits, which is enough by Fact <ref>. To see thathas connected limits, it suffices to check thathas wide pullbacks and equalizers (this is similar to the proof that having arbitrary limits is the same as having products and equalizers <cit.>). We have already checked thathas wide pullbacks. As for equalizers this is implied by the definition of a pseudo-universal class: assume we have two -embeddings f, g: M → N. Let A := {x ∈ U M | f (x) = g (x)}. By definition of A, fA = gA. By the definining condition of pseudo-universality, f ^M (A) = g ^M (A). Therefore A = ^M (A), so AM. Therefore h := fA is a -embedding and hence the equalizer of f and g. We obtain the following characterization of μ-AECs admitting intersections: Letbe a category and let μ be a regular cardinal. The following are equivalent:*is locally μ-polypresentable and all its morphisms are monomorphisms.*is equivalent to a μ-AEC which admits intersections.Ifis equivalent to a μ-AEC which admits intersections, then by Lemma <ref>,is locally μ-polypresentable. Conversely, assume thatis locally μ-polypresentable and all its morphisms are mono. By Fact <ref>,is equivalent to a μ-AEC . Sincehas wide pullbacks,has wide pullbacks, hencemust admit intersections. For locally μ-multipresentable categories, we give a syntactic description. Johnstone <cit.> has shown (for μ = ℵ_0) that they can be axiomatized by disjunctive theories (see the proof below), and we can then Skolemize to obtain a universal class. Ifis a locally μ-multipresentable category with all morphisms monomorphisms, thenis axiomatizable by a universal _∞, μ-theory. Consider the canonical embedding E:→^^ whereis the representative full subcategory of μ-presentable objects in . Then E preserves μ-directed colimits and μ-presentable objects. Following <cit.>,is equivalent to a μ-cone-orthogonality class in ^^ and the latter category can be viewed as an equational variety of many-sorted universal algebras. Thuscan be axiomatized by a disjunctive theory in the corresponding unary vocabulary τ (see <cit.>). Recall that this theory consists of sentences(∀)(φ()→(∃ !) ⋁_i∈ Iψ_i(,))∧(∀,)⋁_i≠ j∈ I(ψ_i(,)∧ψ_j(,)where φ and ψ_i are conjunctions of atomic formulas and , are strings of variables. The use of ∃ ! can be eliminated by introducing a new relation symbol R (,) such that R(,) if and only if⋁_i∈ Iψ_i(,) and then adding Skolem functions. Thuscan be axiomatized by a universal _∞, μ-theory. We obtain the following characterization of universal classes: Letbe a category and let μ be a regular cardinal. The following are equivalent:* is locally μ-multipresentable and all its morphisms are monomorphisms.* is equivalent to (T), for some universal _∞, μ-theory T.* is equivalent to a universal μ-AEC.* is equivalent to a pseudo-universal μ-AEC.(<ref>) implies (<ref>) is Lemma <ref> combined with Lemma <ref>. (<ref>) implies (<ref>) is Fact <ref>. (<ref>) implies (<ref>) is Remark <ref>. Finally, (<ref>) implies (<ref>) is Lemma <ref>. Note that a syntactic characterization of locally polypresentable classes (in terms of models of a pullback theory) is also known <cit.> (see also Rabin's characterization of first-order theories with intersections <cit.>). §.§ SummaryWe have the following hierarchy of categories whose morphisms are monomorphisms, for μ a fixed regular cardinal:* Locally μ-multipresentable.* Locally μ-polypresentable.* Accessible. Each level is properly contained in the next and each level admits a characterization in terms of abstract classes (see Theorem <ref>, Theorem <ref>, and Fact <ref>):* Universal μ-AEC.* μ-AEC admitting intersections.* μ'-AEC, for some μ'. Each level also has a known syntactic characterization (see Theorem <ref>, <cit.>, and Fact <ref>):* (T), for T a universal _∞, μ-theory.* (T), for T a pullback _∞, μ-theory.* (T), for T a basic _∞, ∞-theory. We do not know where μ-AECs (for fixed μ) stand in this hierarchy: Is every accessible category with μ-directed colimits and all morphisms monomorphisms equivalent to a μ-AEC? § MORE ON SHELAH'S PRESENTATION THEOREM Shelah's presentation theorem <cit.> says that any AEC is the reduct of a class of models of an _ω, ω theory omitting a set of types. Moreover the reduct map is functorial. The proof generalizes to μ-AECs <cit.>. In <cit.>, the following category-theoretic analog was proven (for μ = ℵ_0, but the proof generalizes). A functor is called essentially surjective if every object of its codomain is isomorphic to an object in its range.Letbe an accessible category with μ-directed colimits and all morphisms monomorphisms. Then there is a μ-accessible categorywhose morphisms are monomorphisms and an essentially surjective faithful functor F: → preserving μ-directed colimits. Fact <ref> says that accessible categories with μ-directed colimits and all morphisms mono are, while not necessarily μ-accessible, functorial images of μ-accessible categories. This cannot be obtained directly from Shelah's presentation theorem, since we do not know whether accessible categories with μ-directed colimits are μ-AECs (the problem is that we are fixing μ, see Question <ref>). In this section, we combine Fact <ref> with the proof of Shelah's presentation theorem <cit.> to obtain the following common generalization: Letbe an accessible category with μ-directed colimits whose morphisms are monomorphisms. Then there is a universal μ-AECand an essentially surjective faithful functor F: → preserving μ-directed colimits. Note that Theorem <ref> is indeed a generalization of Fact <ref>, since universal μ-AECs are μ-accessible (Lemma <ref>). Moreover, Theorem <ref> is a generalization of Shelah's presentation theorem: by Tarski's presentation theorem (Fact <ref>) every universal μ-AEC is a class of models omitting a set of quantifier-free _∞, μ-types. Theorem <ref> also generalizes Boney's presentation theorem for metric AECs <cit.>.In Shelah's presentation theorem the functor F from Theorem <ref> is the reduct map, as we now establish by imitating Shelah's proof: Letbe a μ-AEC which does not contain the empty structure. There exists an expansion τ' of τ := τ () and a universal μ-AEC ' in the vocabulary τ' such that the reduct map ' → is a faithful functor preserving μ-directed colimits which is surjective on objects. Moreover, |τ'| = |τ| +(). Let τ' := τ∪{f_i^α : i <(), α < μ}, where each f_i^α is a new α-ary function symbol. Let: ' := {M' ∈ (τ') | M' τ∈∀ A ⊆ U M' . ^M' (A) τ M' τ} Here, (τ') is the class of τ'-structure and ^M' (A) denotes the closure of A under the functions of M'. It is easy to check that ' is a μ-universal class and that the reduct map is a faithful functor from ' topreserving μ-directed colimits. We show that it is onto: let M ∈, and pick a μ-directed system {M_s : s ∈ [M]^<μ} (where [M]^<μ denotes the set of subsets of U M of cardinality strictly less than μ) such that |U M_s| ≤ (), s ⊆ U M_s, and s ⊆ t implies that M_sM_t. For each s ∈ [M]^<μ, let {c_i^s : i <()} be an enumeration (possibly with repetitions) of U M_s. Finally, for ∈αM, α < μ, and i <(), define (f_i^α)^M' () := c_i^ (). This works: check that for any A ⊆ U M', ^M' (A) = ⋃_s ∈ [^M' (A)]^<μ M_s and use the chain axioms of μ-AECs.By Fact <ref>, there is a μ-accessible category _0 (whose morphisms are monomorphisms) and an essentially surjective faithful functor F_0 : _0 → preserving μ-directed colimits. By Fact <ref>, _0 is equivalent via a functor G to a μ-AEC _0 (G may not be surjective but it will be essentially surjective) and without loss of generality (see Remark <ref>) _0 does not contain the empty structure. By Lemma <ref>, there is a universal μ-AECsuch that the reduct functor F_1 : →_0 is a faithful functor preserving μ-directed colimits. Let F := F_1 G F_0.If we ask for the functor F from Theorem <ref> to be full, then it will be an equivalence of categories, sowill be equivalent to a μ-universal class. On the other hand, the reduct functor F from Lemma <ref> is always surjective on morphisms (that is, if f: A → B, then there exists f̅: A' → B' such that F (f̅) = f). Consider now the following intermediate weakening of fullness: A functor F: → is pullback-full if for any sequence f̅_i: B_i → C | i ∈ I of morphisms inand any sequence g_i : A → F B_i | i ∈ I of morphisms in , if (F f̅_i) g_i =(F f̅_j) g_j for all i, j ∈ I, then there exists an object A' inand morphisms g̅_i: A' → B_i | i ∈ I such that F g̅_i = g_i for all i ∈ I. When is the functor F from Theorem <ref> pullback-full? The answer yields yet another characterization of μ-AECs admitting intersections: Letbe a μ-AEC. The following are equivalent:* admits intersections.*There is a universal μ-AECand an essentially surjective pullback-full and faithful functor F: → preserving μ-directed colimits.*There is a μ-AEC admitting intersectionsand an essentially surjective pullback-full and faithful functor F: → preserving μ-directed colimits.* (<ref>) implies (<ref>): Trivial.* (<ref>) implies (<ref>): By Fact <ref>, it suffices to check thathas wide pullbacks. Observe that the definition of pullback-fullness (taken with I a singleton set and f̅_i = 𝕀_C) implies that whenever g : A → F C, there exists g̅ : A' → C such that F g̅ = g. The proof is now routine using the fact thathas wide pullbacks, together with the pullback-fullness and faithfulness of F.* (<ref>) implies (<ref>): In the proof of Lemma <ref>, let ” be the class of members M' of ' such that ^M' (M_0) = M_0 for all M_0M' τ. The reduct map from ” tois a faithful functor from ” topreserving μ-directed colimits. As in the proof of <cit.>, this functor is onto, and we show that it is also pullback-full.Let A, C, B_i, f̅_i, g_i | i ∈ I be as in the definition of being pullback-full. For i ∈ I, let h_i := (F f̅_i) g_i. By hypothesis, h_i = h_j for all i, j ∈ I, so let h := h_i. First observe that h [U A] induces a substructure C_0 of C, because h[A]C τ = F C and so by definition of ”, ^C (h [U A]) = h[U A]. We can find a τ'-expansion A' of A such that h induces a τ'-isomorphism from A' onto C_0. Thus h will induce a ”-embedding h̅ from A' into C.We now claim that for all i ∈ I, g_i induces a ”-embedding g̅_i from A' into B_i. This is enough to prove what we want. To prove the claim, it is enough to see that for any α-ary τ'-function symbol ρ and any ∈αA', g_i (ρ^A' ()) = ρ^B_i (g_i ()). Now by the previous paragraph h (ρ^A' ()) = ρ^C (h ()). Since C_0 = h[U A] = h[U A'], ρ^C (h ()) = ρ^C' (h ()) = ρ^f̅_i[B_i] (h ()). By definition of h, the latter is equal to ρ^f̅_i[B_i] (f̅_i g_i ()) = f̅_i (ρ^B_i (g_i ()). Putting these calculations together, we have obtained the equality h (ρ^A' ()) = f̅_i (ρ^B_i (g_i ())). Since (as a concrete function) h = f̅_i g_i and f̅_i is a monomorphism, we conclude that g_i (ρ^A' ()) = ρ^B_i (g_i ()), as desired.Note that (<ref>) implies (<ref>) holds more generally when we start with an accessible category with μ-directed colimits. We do not know whether (<ref>) implies (<ref>) also holds at that level of generality.amsalpha | http://arxiv.org/abs/1707.09005v4 | {
"authors": [
"Michael Lieberman",
"Jiří Rosický",
"Sebastien Vasey"
],
"categories": [
"math.LO",
"math.CT",
"03C48 (Primary), 18C35, 03C52, 03C55, 03C75 (Secondary)"
],
"primary_category": "math.LO",
"published": "20170727191334",
"title": "Universal abstract elementary classes and locally multipresentable categories"
} |
Corresponding author: [email protected] Department of Engineering and Applied Sciences, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, JapanDepartment of Applied Physics, Tohoku University, 6-6-05 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, JapanIn this review article, we show our recent results relating to the undoped (Ce-free) superconductivity in the electron-doped high-T_ c cuprates with the so-called T' structure. For an introduction, we briefly mention the characteristics of the electron-doped T'-cuprates, including the reduction annealing, conventional phase diagram and undoped superconductivity. Then, our transport and magnetic results and results relating to the superconducting pairing symmetry of the undoped and underdoped T'-cuprates are shown. Collaborating spectroscopic and nuclear magnetic resonance results are also shown briefly. It has been found that, through the reduction annealing, a strongly localized state of carriers accompanied by an antiferromagnetic pseudogap in the as-grown samples changes to a metallic and superconducting state with a short-range magnetic order in the reduced superconducting samples.The formation of the short-range magnetic order due to a very small amount of excess oxygen in the reduced superconducting samples suggests that the T'-cuprates exhibiting the undoped superconductivity in the parent compounds are regarded as strongly correlated electron systems, as well as the hole-doped high-T_ c cuprates.We show our proposed electronic structure model to understand the undoped superconductivity. Finally, unsolved future issues of the T'-cuprates are discussed. PACS numbers:Novel Electronic State and Superconductivity in the Electron-Doped High-T_ c T'-Superconductors Y. Koike May 29, 2018 ===============================================================================================§ INTRODUCTIONIn the past three decades, cuprates exhibiting high-temperature superconductivity have been intensively studied. The superconducting (SC) transition temperature T_ c ∼ 134 K <cit.> in the Hg-1223 cuprate is the highest among all SC materials at ambient pressure. Although new high-T_ c superconductors of iron arsenides with the maximum T_ c of ∼ 55 K <cit.> and of sulfur hydrides with T_ c ∼ 203 K at 200 GPa <cit.> have been observed, the cuprates have been standing for potential materials of both exotic physics and future applications.All series of high-T_ c cuprates, La-214, Y-123, Bi-2212, etc., have the parent compounds, which are antiferromagnetic (AF) Mott insulators. As shown in the phase diagram of Figure <ref>a, in the hole-doped cuprates, doping of hole carriers into the parent compound destroys an AF order quickly, and then, the superconductivity appears. With increasing hole doping, T_ c is raised in the underdoped regime and exhibits the maximum in the optimally doped regime. With further doping, T_ c turns into a decrease in the overdoped regime, and finally, the superconductivity disappears in the heavily-overdoped regime where the system becomes a metal. Comprehensive neutron scattering <cit.> and muon spin relaxation (μSR) <cit.> experiments have uncovered AF fluctuations in the SC doping regime, suggesting that AF fluctuations are a glue to form SC electron pairs. In the non-SC heavily-overdoped regime, recent theories <cit.> and experiments <cit.> have suggested the existence of ferromagnetic fluctuations of itinerant carriers.Electron-doped high-T_ c cuprates were first discovered by Tokura et al. in 1989 <cit.>. Pronounced features different from those in the hole-doped cuprates are not only the electron carrier, but also the reduction annealing, which is absolutely essential to obtain the superconductivity in the electron-doped cuprates. The phase diagram depicted by the normally reduced samples is shown in Figure <ref>a, which roughly resembles that of the hole-doped cuprates.The doping of electrons into the parent compounds weakens the AF order, but the AF order is maintained around the optimally doped regime. The superconductivity with the maximum T_ c appears around the optimally doped regime and is monotonically suppressed with over-doping.Since AF fluctuations were observed around the optimally doped regime <cit.>, the electron pairing mediated by AF fluctuations has been believed to be the case also in the electron-doped cuprates. In 2005, a surprising result was obtained in thin films of the electron-doped cuprates with the Nd_2CuO_4-type structure (the so-called T' structure), in which the superconductivity appears without electron doping <cit.>.Following observations of the undoped (Ce-free) superconductivity in the parent compounds <cit.> and the suggestion of a new phase diagram shown in Figure <ref>b <cit.> opened a new era of research in the high-T_ c superconductivity.Yet to date, the mechanism of the undoped superconductivity is unclear.In this review article, we show our recent results relating to the undoped superconductivity in the T'-cuprates <cit.>. First, we present the characteristics of the electron-doped T'-cuprates including the reduction annealing, conventional phase diagram and undoped superconductivity. The next three sections consist of our transport and magnetic results and results relating to the SC pairing symmetry. Collaborating spectroscopic and nuclear magnetic resonance (NMR) results are also shown briefly. Then, we discuss our proposed electronic structure model to understand the undoped superconductivity. In the final section, unsolved future issues of the T'-cuprates are discussed. Note that there are already excellent and complete reviews on the materials and physics focusing on the conventional electron-doped cuprates <cit.>. § ELECTRON-DOPED T'-CUPRATE Electron-doped T'-cuprates were discovered in Nd_2-xCe_xCuO_4 <cit.>. This class of materials are expressed as RE_2CuO_4 (RE = Pr, Nd, Sm, Eu) with the T' structure. The electron doping is achieved by the substitution of the tetravalent Ce^4+ for the trivalent RE^3+. The T' structure consists of the CuO_2 plane and fluorite-type blocking layer. Because as-grown T'-samples are insulating and AF, the reduction annealing for the as-grown T'-cuprates is indispensable to obtain superconductivity. One of the unsolved puzzles for the T'-cuprates is the exact mechanism of the reduction process. To date, there are mainly two candidate mechanisms for this puzzle. One is the removal of excess oxygen from the as-grown sample, which is mostly believed from the early stage of the research of the T'-cuprates. This idea originated from early neutron-diffraction experiments in Nd_2-xCe_xCuO_4 <cit.>. Because of the size mismatch between the CuO_2 plane and blocking layer, excess oxygen tends to be included right above Cu, resulting in the shrinkage of the blocking layer in the ab-plane. Since the excess oxygen is understood to induce disorder of the electrostatic potential in the CuO_2 plane, electron pairs are destroyed, and the system shows no SC behaviors <cit.>.Therefore, the removal of excess oxygen from the as-grown sample is crucial for the appearance of superconductivity and the investigation of intrinsic properties of the T'-cuprates.The other is the Cu-vacancy scenario proposed by neutron and X-ray diffraction studies in Nd_2-xCe_xCuO_4 <cit.> and Pr_1-xLaCe_xCuO_4 <cit.>. The diffraction experiments found that (i) the occupancy at the Cu site is imperfect (perfect) in the as-grown (reduced) samples and (ii) the so-called secondary phase of RE_2O_3 appears (disappears) through the reduction (oxidation) annealing, both of which are reversible processes through the reduction (oxidation) annealing. Therefore, the role of the reduction annealing is to fill up Cu deficiencies in the as-grown samples where Cu deficiencies are regarded as causing the pair breaking. The secondary phase was precisely investigated in Nd_2-xCe_xCuO_4 <cit.>.The phase diagram of T'-cuprates was first obtained by μSR in Nd_2-xCe_xCuO_4 (Figure <ref>a) <cit.> and later in Pr_1-xLaCe_xCuO_4 <cit.>. The AF phase expands up to the optimally doped regime, and the optimal superconductivity suddenly sets in at the same time of the suppression of the AF order. In detail, both the superconductivity and AF order coexist around the boundary of two phases.With over-doping, T_ c monotonically decreases and disappears around the region of the solubility limit of Ce into the T'-structure. Neutron-scattering experiments in Nd_2-xCe_xCuO_4 around the optimally doped regime of x=0.15 revealed commensurate AF fluctuations <cit.>. This is in sharp contrast to incommensurate AF fluctuations <cit.> or stripe fluctuations <cit.> observed in the hole-doped cuprates. Following neutron-scattering experiments in Pr_1-xLaCe_xCuO_4 <cit.> uncovered that commensurate AF fluctuations existed also in the overdoped regime and that both the spin stiffness and relaxation rate of AF fluctuations decreased toward the end point of superconductivity in the heavily overdoped regime. These suggest the intimate relation between the superconductivity and AF fluctuations in the T'-cuprates, as well as in the hole-doped cuprates.In 1995, Brinkmann et al. reported pioneering results, which later led to the discovery of undoped (Ce-free) superconductivity <cit.>. They tried to expand the SC regime deeply into the AF underdoped regime in the phase diagram by improving the reduction process. A thin single crystal of Pr_2-xCe_xCuO_4 was sandwiched by polycrystalline pellets of the same component to protect the surface of the crystal and annealed at a higher temperature of 1080 ^∘C for a longer time than in the conventional annealing. They observed the superconductivity down to x=0.04 in the underdoped regime.Moreover, T_ c gradually increased with decreasing doping from the optimally doped regime. They suggested that the superconductivity in underdoped Pr_2-xCe_xCuO_4 was a product of the appropriate reduction of excess oxygen. Then, one would think of lower doping regime than x=0.04, namely whether Mott insulators like the hole-doped cuprates or superconductivity down to the parent compound. Ten years later, a surprising result was obtained by Tsukada et al. in T' thin films of Ce-free La_2-yRE_yCuO_4 (RE = Y, Lu, Sm, Eu, Gd, Tb) in which the superconductivity appears without Ce substitution (electron doping) <cit.>. Subsequent systematic investigation by the same group using Nd_2-xCe_xCuO_4 thin films uncovered that T_ c of ∼ 28 K in the parent compound of x=0 decreased monotonically with increasing x and disappeared at x ∼ 0.20 (Figure <ref>b) <cit.>. These findings were suggested to be due to the removal of excess oxygen from the as-grown thin films more effectively than that ever reported.Their unique procedure of the reduction annealing called two-step annealing was mentioned in the literature <cit.>. At the first step, the as-grown thin films were annealed at a high temperature in an intermediate oxygen pressure of ∼ 10^2 Pa. In this process, they insisted based on the increasing ab-plane electrical resistivity ρ_ ab and unchanged c-axis length that oxygen in the CuO_2 plane was mainly removed, and excess oxygen was not.At the second step, the thin films were further annealed at a low temperature in a low oxygen pressure of ∼ 10^-5 Pa. In this process, they insisted based on the decreasing ρ_ ab and c-axis length that excess oxygen moved to the oxygen-defect site in the CuO_2 plane, and the superconductivity was realized.In fact, optical studies of the SC thin film of Pr_2CuO_x with x ≃ 4 have revealed a Drude-like peak centered at zero frequency, suggesting a metallic state of the SC parent compound <cit.>. These results give us the following messages: (i) the newly-proposed phase diagram without the AF phase is seemingly different from the conventional one; and (ii) the AF Mott insulating state is not a starting point of the superconductivity in the T'-cuprates. The superconductivity in the parent compounds of the T'-cuprates was also confirmed using polycrystalline samples of Ce-free La_2-ySm_yCuO_4 <cit.> and Ce-free La_1.8Eu_0.2CuO_4 <cit.>, while it has not yet been confirmed in single crystals of the parent compound.From the theoretical viewpoint, the undoped (Ce-free) superconductivity in the parent compounds might be explained by the early local density energy band calculation in Nd_2-xCe_xCuO_4 <cit.> where the system was a simple band metal without electron correlation.On the assumption of the moderate electron correlation, the half-filled-band Hubbard model consisting of doublons and holons predicted the superconductivity in the parent compound <cit.>. Under the strong electron correlation generating the Mott–Hubbard gap between the upper Hubbard band (UHB) and the lower Hubbard band (LHB) of the Cu 3d_x^2-y^2 orbital, calculations based on the local density approximation (LDA) combined with the dynamical mean-field theory (DMFT) revealed the possibility of the closing of the so-called charge-transfer (CT) gap between UHB of the Cu 3d_x^2-y^2 orbital and the O 2p band observed in the hole-doped cuprates <cit.>. That is, both metallic and SC states were suggested to appear by eliminating the AF order in the parent compounds of the T'-cuprates.The recent calculation of a two-particle self-consistent analysis using the three-band model <cit.> revealed a monotonic decrease in T_ c with electron doping, which is consistent with the experimental observation shown in Figure <ref>b <cit.>.The undoped (Ce-free) superconductivity challenges the long-thought understanding that the parent compounds of the high-T_ c cuprates are AF Mott insulators.To understand the mechanism of the undoped superconductivity, it is extremely important to clarify the evolution of the electronic state through the reduction process, for which the detailed investigation using single crystals is necessary. One of critical questions is whether or not the undoped superconductivity in the T'-cuprates appears based on the strong electron correlation. To answer this issue, it is significant to investigate the Cu-spin correlation, because localized Cu spins in LHB of the Cu 3d_x^2-y^2 orbital are generated and expected to correlate with one another in the case of the strong electron correlation. To attain these aims, we used T' single crystals of Ce-underdoped Pr_1.3-xLa_0.7Ce_xCuO_4 + δ with x=0.10 and undoped (Ce-free) SC samples of La_1.8Eu_0.2CuO_4 + δ. In the next section, we start from the preparation of single crystals and the improved reduction annealing in Pr_1.3-xLa_0.7Ce_xCuO_4 + δ. § EVOLUTION OF THE ELECTRONIC STATE THROUGH THE REDUCTION ANNEALING Single crystals of Ce-underdoped Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 were grown by the traveling-solvent floating-zone method <cit.>. In order to reduce excess oxygen in the as-grown crystals, we grew single crystals in air atmosphere, which is at a lower partial pressure of oxygen than formerly reported <cit.>. The composition of as-grown crystals was determined by the inductively-coupled plasma (ICP) analysis to be Pr_1.17(1)La_0.73(1)Ce_0.10(1)Cu_1.00(1)O_4, which is close to the nominal composition.For the as-grown crystals, we performed the reduction annealing in vacuum of ∼ 10^-4 Pa for 24 h at various temperatures. In order to protect the surface of crystals during the annealing, the as-grown crystals were covered with polycrystalline powders having the same composition (protect annealing).This is an improved technique of the Brinkmann's <cit.>. From the iodometric titration, the oxygen content was confirmed to be reduced by 0.03(1) through the reduction annealing for Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10. Moreover, through the reduction annealing, the c-axis lengths estimated from the X-ray diffraction measurements were reduced by 0.028(1) Å for Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10, suggesting the removal of excess oxygen from the as-grown samples. The scanning electron microscope also revealed that, in the reduced SC crystal, neither Cu metals nor rare-earth oxides were observed.Therefore, both the absence of Cu deficiency within the error of ICP and the oxygen reduction through the annealing suggest that the Cu deficiency model is not applicable to the present samples. Polycrystalline samples of undoped (Ce-free) La_1.8Eu_0.2CuO_4+δ were prepared by a low-temperature technique using CaH_2 as a reductant and the subsequent annealing. The SC samples were obtained through the reduction annealing at 700 ^∘C for 24 h in vacuum. Details have been described in the literature <cit.>.Figure <ref> shows the temperature dependence of the magnetic susceptibility of Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 and La_1.8Eu_0.2CuO_4+δ. For Pr_1.3-xLa_0.7Ce_xCuO_4+δ, the Meissner diamagnetism appears in the 750 ^∘C-reduced crystal, and the relatively sharp SC transition with the onset of 27 K is observed in the 800 ^∘C-reduced crystal. Formerly, Sun et al. reported from transport properties of reduced single crystals of Pr_1.3-xLa_0.7Ce_xCuO_4+δ through the annealing in Ar that the crystals with x ≤ 0.10 are non-SC <cit.>. The present successful observation of bulk superconductivity in Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 suggests that the excess oxygen is effectively removed from the as-grown crystal through the protect annealing in vacuum at lower temperatures than in the conventional annealing. The La_1.8Eu_0.2CuO_4+δ sample also shows the bulk Meissner diamagnetism below ∼ 15 K, which is almost the same as that formerly reported <cit.>.The temperature dependence of ρ_ ab in Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 exhibits characteristic behaviors depending on the reduction condition, as shown in Figure <ref>a. The as-grown crystal shows semiconducting temperature dependence and no trace of superconductivity. For the 650 ^∘C-reduced crystal, ρ_ ab is semiconducting at high temperatures, while a drop of ρ_ ab is observed below ∼ 7 K due to the SC transition.The drastic decrease in ρ_ ab is observed in the 750 ^∘C- and 800 ^∘C-reduced crystals in which the bulk superconductivity appears. Both crystals show a metallic behavior at high temperatures and an upturn at low temperatures, followed by the SC transition. In the normal state below ∼ 50 K, ρ_ ab exhibits log T dependence. This behavior, as well as the saturation of ρ_ ab in the magnetic field and negative magnetoresistance <cit.> is most likely due to the Kondo effect <cit.>. The present evolution of ρ_ ab through the reduction annealing suggests that the strongly localized state of carriers in the as-grown crystal changes to a metallic state with the Kondo effect in Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10. Figure <ref>b shows the temperature dependence of the c-axis resistivity ρ_ c for Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10. For the as-grown and 650 ^∘C-reduced crystals, a hump is observed at ∼ 200 K, which is similar to that observed in Nd_2-xCe_xCuO_4 with x<0.14 <cit.> and Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x ≤ 0.03 <cit.> and due to the opening of an AF pseudogap. For the 750 ^∘C-reduced crystal, on the contrary, a simple metallic behavior without hump is observed, which resembles the behavior of ρ_ c in SC Nd_2-xCe_xCuO_4 with x=0.15 <cit.>.Therefore, it is inferred that the AF pseudogap disappears in the 750 ^∘C-reduced SC crystal. The electronic state of Pr_1.3-xLa_0.7Ce_xCuO_4+δ was further investigated by the angle-resolved photoemission spectroscopy (ARPES) by Horio et al. using our as-grown and reduced single crystals, as shown in Figure <ref> <cit.>. For the as-grown and 650 ^∘C-reduced crystals, the AF pseudogap is clearly observed around the hot spots, namely, crossing points of the Fermi surface with the AF Brillouin zone boundary. For the 800 ^∘C-reduced crystal, on the contrary, the sharp quasi-particle peaks are observed on the entire Fermi surface without the signature of the AF pseudogap unlike the previous works <cit.>. These results are consistent with the ρ_ c results in Figure <ref>b and indicate the dramatic reduction of the AF correlation length and/or of magnetic moments in the 800 ^∘C-reduced crystal. The quasi-particle peaks around the anti-nodal region are broader than those around the nodal region, suggesting the strong scattering of quasiparticles by the AF fluctuations and/or charge fluctuations in relation to the recent observation of the charge order in Nd_2-xCe_xCuO_4 <cit.>. The Hall effect of the electron-doped T'-cuprates is highly informative. Thin-film studies in Pr_2-xCe_xCuO_4 reduced in the conventional annealing revealed that the Hall coefficient R_ H was negative in the underdoped regime and increased with increasing x, followed by the sign change of R_ H at x ∼ 0.17 <cit.>. Moreover, the Hall resistivity ρ_xy exhibited a nonlinear behavior in the magnetic field in Pr_2-xCe_xCuO_4 thin films <cit.>. Our results of ρ_xy in Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 also revealed a nonlinear behavior in the magnetic field. These suggest that multiple carriers (most likely electrons and holes) reside in the T'-cuprates. The existence of multiple carriers of electrons and holes has also been suggested from NMR in single crystals of Nd_2-xCe_xCuO_4 and Pr_2-xCe_xCuO_4 <cit.>. Note that the undoped SC thin films exhibit positive R_ H in the ground state, suggesting the existence of predominant hole-like carriers <cit.>. § RELATIONSHIP BETWEEN CU-SPIN CORRELATION AND SUPERCONDUCTIVITY For the detailed investigation of the Cu-spin correlation, we used the μSR technique which is highly sensitive to the local magnetism in a sample and has an advantage to distinguish between a long-range and short-range magnetic order. Zero-field (ZF) and longitudinal-field (LF) μSR measurements of Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 and La_1.8Eu_0.2CuO_4+δ were performed using a pulsed positive muon beam at the Material and Life Science Experimental Facility at J-PARC in Japan and using a continuous positive muon beam at the Paul-Scherrer Institute (PSI) in Switzerland, respectively.Figure <ref>a,b show the ZF-μSR time spectra of as-grown non-SC and 700 ^∘C-reduced SC samples of La_1.8Eu_0.2CuO_4+δ, respectively <cit.>. For both samples, Gaussian-like slow depolarizations due to nuclear dipole fields are observed at high temperatures, indicating a paramagnetic state of Cu spins. With decreasing temperature, fast depolarizations of muon spins are observed, suggesting the development of the Cu-spin correlation. For the as-grown sample, clear muon-spin precessions appear at low temperatures, indicating the formation of a long-range magnetic order. For the reduced SC sample, the spectrum consists of a fast depolarization without precession and a following time-independent behavior above 1 μs, which is typical of a short-range magnetically ordered state. In the LF-μSR spectra at 1.6 K shown in Figure <ref>c <cit.>, a parallel shift up with LF indicates the formation of a static magnetically ordered state in ZF. In Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10, the as-grown non-SC crystal exhibits muon-spin precessions at low temperatures as shown in Figure <ref>d <cit.>.The spectra of the 800 ^∘C-reduced SC crystal in Figure <ref>e <cit.> show both fast and slow depolarizations without muon-spin precession at 3 K. The LF-μSR spectra at 10 K shown in Figure <ref>f <cit.> reveal a parallel shift as well as a slow depolarization with LF, suggesting the existence of both a short-range magnetic order and fluctuating Cu spins. These results indicate the coexisting state of the superconductivity and the short-range magnetic order in both La_1.8Eu_0.2CuO_4+δ and Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10. Note that μSR results of the undoped (Ce-free) thin film of La_2-xY_xCuO_4+δ have shown the development of the Cu-spin correlation at low temperatures <cit.>. The analysis of volume fractions of SC and short-range magnetically ordered regions leads to the detailed information on the electronic state in the T'-cuprates. The magnetic volume fractions were estimated from the μSR results <cit.>. For the reduced SC La_1.8Eu_0.2CuO_4+δ, the volume fraction of the short-range magnetically ordered region is almost 100% in the ground state. As the SC volume fraction is estimated from the Meissner fraction to be above 15%, both the SC and short-range magnetically ordered regions coexist in the sample. For the reduced SC Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10, the estimated magnetic volume fraction is 80% in the ground state. The SC volume fraction was estimated from the recovery of the electronic specific heat coefficient in the ground state by the application of magnetic field. It was at least 60%, suggesting the coexistence of the SC and short-range magnetically ordered regions in the sample. Numerically, a part of the short-range magnetically ordered regions appear to be spatially overlapped with the SC regions.However, it is plausible that the short-range magnetically ordered and SC regions are phase-separated, because the volume fraction of the short-range magnetically ordered region might be overestimated due to electronic dipole fields extending to the paramagnetic SC region. The Cu-spin dynamics was further investigated by NMR using our reduced SC T'-samples by Yamamoto et al. <cit.> and Fukazawa <cit.>. As shown in Figure <ref>, the temperature dependence of T_1 T, where 1/T_1 is the ^63Cu nuclear spin-lattice relaxation rate exhibited the Curie-Weiss behavior in a wide temperature range for Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 and 0.15, suggesting the existence of strong AF fluctuations in the underdoped and overdoped T'-cuprates. For the undoped (Ce-free) La_1.8Eu_0.2CuO_4+δ, the bulk superconductivity was evidenced by the significant decrease in the Knight shift below T_ c and AF fluctuations were observed in the normal state from the temperature dependence of 1/T_1. These results suggest that AF fluctuations are related to the formation of electron pairs in the T'-cuprates as well as in the hole-doped cuprates. § SUPERCONDUCTING PAIRING SYMMETRYWhile the SC pairing symmetry in the hole-doped cuprates is widely recognized to be d-wave, it is controversial in the electron-doped T'-cuprates. Early studies of impurity effects in Nd_2-xCe_xCu_1-y(Zn,Ni)_yO_4 with x=0.15 revealed that T_ c was depressed by the magnetic impurity Ni much more rapidly than by the nonmagnetic impurity Zn <cit.>. This is typical of conventional s-wave superconductors. On the contrary, ARPES <cit.> and penetration depth <cit.> experiments etc. insisted the occurrence of d-wave superconductivity in Nd_2-xCe_xCuO_4+δ with x=0.15 and Pr_2-xCe_xCuO_4+δ with x=0.15. To clarify the SC pairing symmetry of the undoped (Ce-free) superconductivity in the T'-cuprates, impurity effects on T_ c were precisely investigated in La_1.8Eu_0.2Cu_1-y(Zn,Ni)_yO_4+δ <cit.>. Figure <ref> displays the impurity-concentration dependence of T_ c normalized by T_ c of y=0 T_ c^y=0 in La_1.8Eu_0.2Cu_1-y(Zn,Ni)_yO_4+δ together with the preceding results of T'-cuprates <cit.> and hole-doped La_2-xSr_xCu_1-y(Zn,Ni)_yO_4 <cit.>. Obviously, the depression of T_ c is almost the same between the Zn and Ni substitution in contrast to the preceding results of Nd_2-xCe_xCu_1-y(Zn,Ni)_yO_4+δ with x=0.15 <cit.>. Moreover, the change of T_ c/T_ c^y=0 in La_1.8Eu_0.2Cu_1-y(Zn,Ni)_yO_4+δ follows the results of La_2-xSr_xCu_1-y(Zn,Ni)_yO_4 in the optimally doped x=0.15 and overdoped x=0.20.Therefore, in La_1.8Eu_0.2CuO_4+δ, the depression of T_ c by the Zn and Ni substitution is probably due to the pair-breaking effect and the SC pairing symmetry is d-wave mediated by spin fluctuations as in the case of the hole-doped cuprates. In fact, our μSR <cit.> and NMR <cit.> results also revealed strong spin fluctuations in La_1.8Eu_0.2CuO_4+δ. The characteristic temperature dependence of 1/T_1 observed in La_1.8Eu_0.2CuO_4+δ implies the existence of nodal lines in the SC gap, supporting the above statement of the d-wave superconductivity <cit.>. Moreover, optical studies in the SC thin film of Pr_2CuO_x with x ≃ 4 have suggested that the temperature dependence of the magnetic penetration depth λ exhibits the d-wave-like behavior <cit.>. In future, we will further investigate the SC pairing symmetry by the transverse-field μSR technique. Accumulating results of λ, inversely proportional to the SC carrier density, in electron-doped T'-cuprates <cit.> and infinite-layer cuprates <cit.> revealed that the T_ c vs. λ relation seemed not to intersect the origin, which is contrary to the results of hole-doped cuprates suggesting the Bose–Einstein-condensation-like pairing mechanism <cit.>. § ELECTRONIC STRUCTURE MODEL BASED ON THE STRONG ELECTRON CORRELATIONOur transport and μSR results of Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 and La_1.8Eu_0.2CuO_4+δ revealed the following evolution of the electronic state through the reduction annealing. The ρ_ ab showed that a strongly localized state of carriers in the as-grown sample changes to a metallic state with the Kondo effect in the reduced SC sample.The Hall resistivity of the reduced SC sample revealed the existence of multiple carriers.The μSR spectra revealed that, in the ground state, a long-range magnetic order of Cu spins in the as-grown sample changed to a short-range one coexisting with the superconductivity in the reduced SC samples.It would be a reasonable speculation that an ideal T'-cuprate in which the excess oxygen is completely removed exhibits no AF order. The formation of the short-range magnetic order due to a very small amount of excess oxygen in the reduced SC sample suggests that the Cu spins are correlated with one another in the absence of excess oxygen in the ideal T'-cuprates.Therefore, the T'-cuprates exhibiting the undoped (Ce-free) superconductivity are regarded as strongly correlated electron systems as well as the hole-doped SC cuprates.Not the simple band-metal state without the electron correlation <cit.> but both the doublon-holon model <cit.>, calculations using LDA with DMFT <cit.> and the two-particle self-consistent analysis <cit.> under the electron correlation would be able to explain the superconductivity in the undoped (Ce-free) and Ce-underdoped T'-cuprates. The present transport and μSR results are able to be understood in terms of the electronic structure model based on the strong electron correlation. The electronic structure models and schematic drawings of Cu spins and carriers in the CuO_2 plane are displayed in Figure <ref>a–c,e–g, . In the hole-doped high-T_ c cuprates, the parent compounds are CT insulators as shown in Figure <ref>a characterized by the CT gap between UHB of the Cu 3d_x^2-y^2 orbital and O 2p band. In the ideal undoped (Ce-free) T'-cuprates without excess oxygen shown in Figure <ref>c, on the other hand, the square-planar coordination of Cu and oxygen in the CuO_2 plane gives rise to a decrease in the Madelung energy of the Cu 3d orbitals compared with the octahedral coordination in the hole-doped cuprates.This brings about the lowering energy of UHB of the Cu 3d_x^2-y^2 orbital, leading to the mixing of UHB with the Zhang-Rice singlet band. That is, the CT gap is collapsed, so that both mobile electrons and holes simultaneously emerge at the Fermi level in the parent compounds of the T'-cuprates, leading to the appearance of superconductivity without nominal doping of electrons by the Ce substitution. This is plausible, because the calculation of the Madelung energy results in a decrease in the energy of the Cu 3d orbitals by ∼ 4 eV compared with the hole-doped cuprates with the CT gap ∼ 2 eV <cit.>.In the case of excess oxygen residing adjacent to the CuO_2 plane, assuming that two holes are produced in the CuO_2 plane by one excess oxygen, the doped holes in the CuO_2 plane tend to be localized near the excess oxygen due to the disorder of the electrostatic potential. The holes are doped into the Cu 3d and O 2p orbitals taking into account the strong electron correlation.This is because the energy of the Cu 3d orbitals is raised by the excess oxygen due to increasing coordination number, as shown in Figure <ref>d. The doped hole in the Cu 3d orbital resides not in LHB of the Cu 3d_x^2-y^2 orbital but in UHB of the Cu 3d_3z^2-r^2 orbital in order to gain the energy of the Zhang–Rice singlet of Cu 3d_x^2-y^2 (LHB) and O 2p spins. This is reasonable, because the splitting of Cu 3d_x^2-y^2 and Cu 3d_3z^2-r^2 in energy is smaller than the Mott–Hubbard gap of ∼ 8 eV. Accordingly, Cu 3d_3z^2-r^2 (LHB) free spins are probably induced at the Cu site adjacent to the excess oxygen. Both hole and electron carriers tend to be localized because of the disorder of the electrostatic potential near the excess oxygen, resulting in the recovery of the AF order of Cu 3d_x^2-y^2 (LHB) spins. This is able to explain the increase in ρ_ ab due to the strong localization and the hump of the temperature dependence of ρ_ c due to the opening of the AF pseudogap.In the CuO_2 plane in the as-grown non-SC samples of La_1.8Eu_0.2CuO_4+δ and Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10, because of the moderate number of excess oxygen, carriers are strongly localized, as confirmed by ρ_ ab <cit.>, and the AF order is long-ranged, as illustrated in Figure <ref>e.In reduced SC La_1.8Eu_0.2CuO_4+δ, as shown in Figure <ref>f, the CuO_2 plane should be mostly covered by short-range magnetically ordered regions, and superconductivity should appear around the boundary between the short-range magnetically ordered regions. In reduced SC Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10, because the amount of excess oxygen is smaller than that of reduced SC La_1.8Eu_0.2CuO_4+δ, the short-range magnetically ordered regions decrease in correspondence to the increase in the SC volume fraction, as shown in Figure <ref>g. It is noted that the introduction of excess oxygen gives rise to free Cu spins of LHB of the Cu 3d_3z^2-r^2 orbital in the CuO_2 plane just around itself, leading to the occurrence of the Kondo effect as confirmed by ρ_ ab of reduced Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 <cit.>. Accordingly, it is suggested that the short-range magnetically ordered regions are formed around excess oxygen and superconductivity appears far from the excess oxygen. As mentioned above, the ARPES experiment of Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 byrevealed that the AF pseudogap was closed on the whole Fermi surface, which is contrary to the former ARPES results in the T'-cuprates <cit.>. Considering both results of ARPES without the AF pseudogap and μSR with the short-range magnetic order, it is likely that Cu spins forming the short-range magnetic order have tiny magnetic moments.That is, the reduction annealing gives rise to not only the change of the long-range AF order to the short-range one but also the reduction of the magnetic moments of Cu spins, which is compatible with the decrease in the internal magnetic field at the muon site estimated from μSR results <cit.>. § FUTURE ISSUESThe present transport and μSR results of the undoped (Ce-free) and Ce-underdoped T'-cuprates uncovered that the superconductivity appeared under the strong electron correlation as in the case of the hole-doped cuprates. Here we discuss remaining unsolved issues of the T'-cuprates.One is that the oxygen dynamics through the reduction annealing must be directly clarified. To date, several candidates of effects of the reduction annealing have been proposed; the removal of excess oxygen at the apical site <cit.>, filling up Cu deficiencies <cit.>, etc. Our crystal of Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 seems to be the case of the reduction of excess oxygen. Intriguing in our results is that the introduction of Cu vacancies into SC La_1.8Eu_0.2Cu_1-yO_4+δ does not change T_ c so much up to y=0.015 <cit.>, which seems to be incompatible with the Cu-deficiency model. In any case, this issue would be clarified by the recent high resolution neutron and X-ray diffraction experiments.Another issue is about carriers in the undoped (Ce-free) SC T'-cuprates. Our proposed electronic structure model indicates the presence of both electron and hole carriers, which is compatible with the present and former <cit.> Hall resistivity and NMR <cit.> results suggesting the existence of multiple carriers. On the other hand, the ARPES results shown in Figure <ref> in Pr_1.3-xLa_0.7Ce_xCuO_4+δ with x=0.10 showed a hole-like Fermi surface without the AF pseudogap <cit.>. This is probably explained by considering that the overlapping UHB of the Cu 3d_x^2-y^2 orbital and Zhang–Rice band form a mixing band in which both hole-like and electron-like carriers reside.The actual carrier concentration is also under debate. It has been suggested from the Hall effect results <cit.>, etc. that the removal of excess oxygen does not simply correspond to the electron doping. Intriguing is the estimation of the electron concentration from the Fermi-surface volume observed in ARPES that T_ c is unchanged in a wide electron concentration range in Pr_1.3-xLa_0.7Ce_xCuO_4+δ <cit.>. On the contrary, very recent ARPES results in Pr_1-xLaCe_xCuO_4+δ revealed that T_ c exhibited a parabolic change against the electron concentration estimated from the Fermi-surface volume <cit.>. The hole doping into the parent compounds of the T'-cuprates is an interesting way to clarify the electronic structure. Based on the proposed electronic structure model, the parent compounds of the T'-cuprates are no longer reference materials and T_ c might continue to increase with hole doping into the parent compounds. Takamatsu et al. reported that the hole doping by the Sr/Ca substitution for La in La_1.8-xEu_0.2(Ca,Sr)_xCuO_4+δ results in the decrease in T_ c <cit.>. As Zn/Ni impurity effects on T_ c in La_1.8Eu_0.2Cu_1-y(Zn,Ni)_yO_4+δ suggest that the electronic state of the parent compounds is similar to the overdoped regime of the hole-doped cuprates <cit.>, it may be natural that further hole doping results in the decrease in T_ c.Our works were done in collaboration with Yosuke Mori, Akira Takahashi, Takuya Konno, Taro Ohgi, Tomohisa Takamatsu, Kensuke M. 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"authors": [
"T. Adachi",
"T. Kawamata",
"Y. Koike"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.str-el"
],
"primary_category": "cond-mat.supr-con",
"published": "20170726123921",
"title": "Novel Electronic State and Superconductivity in the Electron-Doped High-Tc T'-Superconductors"
} |
φ̌ ρ̌ G ℰ 𝒦 𝒪 ℂ ℙ ℝ ℚ ℤ Hom 𝒵Movable vs Monodromy Nilpotent Cones of Calabi–Yau ManifoldsMovable vs Monodromy Nilpotent Conesof Calabi–Yau Manifolds[This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.htmlhttp://www.emis.de/journals/SIGMA/modular-forms.html]Shinobu HOSONO and Hiromichi TAKAGIS. Hosono and H. TakagiDepartment of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan mailto:[email protected]@math.gakushuin.ac.jp, mailto:[email protected]@math.gakushuin.ac.jpReceived September 11, 2017, in final form April 23, 2018; Published online May 02, 2018We study mirror symmetry of complete intersection Calabi–Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.Calabi–Yau manifolds; mirror symmetry; birational geometry; Hodge theory14E05; 14E07; 14J33; 14N33 § INTRODUCTION A smooth projective variety X of dimension n is called a Calabi–Yau n-fold if the canonical bundle K_X is trivial and H^i(X,𝒪_X)=0, 1≤ i≤ n-1. In the 90's, the idea of mirror symmetry was discovered in theoretical physics and has long been a source of many mathematical ideas related to Calabi–Yau manifolds. After more than 20 years since its discovery, we have now several approaches <cit.> toward mathematical understanding of the symmetry.In this paper, we will focus on “classical” mirror symmetry of Calabi–Yau threefolds, i.e., we compare two different moduli spaces associated to Calabi–Yau threefolds, the Kähler moduli and the complex structure moduli spaces, considering Calabi–Yau threefolds which have several birational models. According to birational geometry of higher dimensional manifolds, if a Calabi–Yau threefold X has birational models, then the Kähler cone of X can be extended to the movable cone Mov(X) <cit.>. On the mirror side, corresponding to each birational model, there appears a special boundary point called large complex structure limit, which is characterized by unipotent monodromy <cit.>. Using this unipotent property, the so-called monodromy nilpotent cone is defined for each boundary point. We will find that, as a result of monodromy relations, the monodromy nilpotent cones glue together to define a larger cone which can be identified with the movable cone Mov(X) under mirror symmetry.Studying birational geometry in mirror symmetry (or string theory) goes back to papers by Morrison and his collaborators in the 90's <cit.>. The birational geometry discussed in the 90's was mostly for Calabi–Yau hypersurfaces in toric varieties, and it comes from the different resolutions of ambient toric varieties. In this paper, we will study two specific examples of complete intersection Calabi–Yau threefolds for which we have birational models in slightly different form, and also have birational automorphisms of infinite order.The construction of this paper is as follows: In Section <ref>, we will first recall some background material on mirror symmetry as formulated in the 90's. Restricting our attentions to three dimensional Calabi–Yau manifolds, we will summarize the basic properties of Calabi–Yau manifolds called A- and B-structures. In Section <ref>, we will introduce a specific Calabi–Yau threefold given by a complete intersection in ^4×^4, whose birational geometry and mirror symmetry were studied in detail in previous works <cit.>. We will describe its movable cone by studying the geometry of birational models. In Section <ref>, we will report some results of monodromy calculations, and describe the details of how the monodromy nilpotent cones glue together by monodromy relations. In Section <ref>, we will present another complete intersection given in ^3×^3. Although there do not appear other birational models to this Calabi–Yau threefold than itself, we will observe interesting gluing property of monodromy nilpotent cones which corresponds to the structure of the movable cone observed in <cit.>. Summary and discussions will be presented in Section <ref>. There we will also describe the corresponding calculations for a K3 surface in ^3×^3 which has a parallel description to the complete intersection in ^4×^4.§ CLASSICAL MIRROR SYMMETRY §.§ Mirror symmetry of Calabi–Yau threefolds Let us consider Calabi–Yau threefolds X and X^* which will be taken to be mirror to each other. For each of these, we have two different structures, called A-structure and B-structure.§.§.§ A-structure of XLet 𝒦_X be the Kähler cone of X and κ_1,…,κ_r∈ H^1,1(X,)=H^1,1(X)∩ H^2(X,) be generators of the Kähler cone, where for simplicity, we assume that the Kähler cone is a simplicial cone in H^2(X,ℝ). Let κ be the Kähler class which corresponds to the polarization of X and write κ byκ=t_1κ_1+⋯+t_rκ_r,with t_i>0. The Lefschetz operator L_κ(-):=κ∧(-) defines a nilpotent linear action on the even cohomology H^ even(X):=⊕_pH^p,p(X). In fact, this is a part of the Lefschetz 𝔰𝔩(2,ℂ) action, and defines the following decomposition:H^0,0 H^1,1 H^2,2 H^3,3=∙ ↓ ∙ ∙∙ ↓ ↓ ⋯ ↓ L_κ. ∙ ∙∙ ↓ ∙ From the viewpoint of homological mirror symmetry, it is natural to replace H^ even(X) with the Grothendieck group K(X) (modulo torsion) which is an abelian group equipped the symplectic formχ(-,-) K(X)× K(X)→ℤwith χ defined by χ(ℰ,ℱ):=∑(-1)^i H^i(X,ℰ^*⊗ℱ) for vector bundles. Based on this integral and symplectic structure on K(X), we can introduce the corresponding structure on H^ even(X,ℚ). A-structure of X is the nilpotent action L_κ on H^ even(X,ℚ) with this integral and symplectic structure.§.§.§ B-Structure of X^*Let X^*=X_b_0^* and consider a smooth deformation family π𝔛^*→ B of X_b_0^*(b_0∈ B) over some open parameter space B. We denote by X_b^*=π^-1(b) the fiber over b∈ B. Then we have Kodaira–Spencer map ρ_b T_bB→ H^1(X_b,𝒯X_b^*) which we assume to be an isomorphism. Associated to this family, we naturally have the local system R^3π_*ℂ_𝔛^* on B. In the 90's, mirror symmetry was recognized by finding some local family 𝔛|_Δ_r^*→Δ_r^* with special properties over the product of punctured disc Δ_r^*=(Δ^*)^r where Δ^*={ z∈ℂ | 0<|z|<1} and B=r. The required properties for the local family are described by the monodromy representation of the fundamental group π_1(Δ_r^*)≃ℤ^r for the local system R^3π_*ℂ_𝔛^* restricted to over Δ_r^*. Let T_i represent the monodromy matrix corresponding to the i-th generator of π_1(Δ_r^*) with fixing a base point b_0∈Δ_r^*. Assuming that all T_i are unipotent, we have nilpotent matrices N_i=log T_i=∑_k≥1(-1)^k-1/k(T_i- id)^k. The setΣ={∑λ_iN_i | λ_i∈ℝ_>0}is called monodromy nilpotent cone consisting of nilpotent matrices on H^3(X_b_0,ℚ). It is known that each element of Σ defines the same monodromy weight filtration on H^3(X_b_0,ℚ) (see <cit.>). The following definition is due to Morrison <cit.>. The degeneration of the local family 𝔛|_Δ_r^*→Δ_r^* at the origin is called a large complex structure limit (LCSL) if the following hold: =0pt(1) All T_i, i=1,…,r, are unipotent. (2) Let N_λ=∑_iλ_iN_i λ_i>0. This induces the monodromy weight filtration,W_0=W_1 ⊂W_2=W_3 ⊂W_4=W_5 ⊂W_6=H^3(X_b_0,ℚ) ∙ ← ∙ ← ∙ ← ∙ ∙ ← ∙ ⋮⋮ ∙ ← ∙with W_0=1 and W_2=1+r. (3) Let W_0=ℚw_0 and introduce a bi-linear form on W_0 by ⟨ w_0,w_0⟩=1. This defines m_jk:=⟨ w_0,N_jw_k⟩ for a ℚ-basis [w_1],…,[w_r] of W_2/W_0. Then the r× r matrix (m_jk)_1≤ j,k≤ r is an invertible ℚ-matrix.We note that there is a natural integral symplectic structure on H^3(X_b_0^*,ℤ), and the monodromy matrices T_i are given by integral and symplectic matrices if we fix a symplectic basis of H^3(X_b_0^*,ℤ). B-structure of X^* at LCSL is defined to be such an integral and symplectic basis of H^3(X_b_0^*,ℤ) with the monodromy matrices T_i which are compatible with the filtration (<ref>).§.§.§ Mirror symmetryIn classical mirror symmetry, X is called a mirror to X^* if the A-structure of X is isomorphic to the B-structure of X^*, i.e., the two nilpotent actions L_κ and N_λ are identified together with their integral and symplectic structures. To be more explicit, suppose we have a B-structure at a LCSL. Since N_λ^4=0 and N_λ^3W_6⊂ W_0, we haveN_iN_jN_k=C_ijk𝙽_0with a fixed rank one nilpotent matrix 𝙽_0 satisfying N_i𝙽_0=0. Corresponding to this products of nilpotent matrices, we have, in the A-structure, the cup-productκ_i∪κ_j∪κ_k=K_ijk𝚅_0,K_ijk∈ℤ,where 𝚅_0∈ H^3,3(X) normalized by ∫_X𝚅_0=1. After fixing a normalization of the matrix 𝙽_0, we have C_ijk=K_ijk if X and X^* are mirror to each other, in particular we have C_ijk∈ℤ_≥0. In fact, C_ijk is the leading coefficient of the so-called Griffiths–Yukawa coupling, and K_ijk is the leading term of the quantum product. Mirror symmetry implies the equality between the two in full orders under the so-called mirror map. §.§ Birational geometry and mirror symmetry Calabi–Yau threefolds often come with birational models. Mirror symmetry in such cases has been studied in <cit.> and is known as topology change in physics <cit.>. The purpose of this paper is to elaborate such cases in more details comparing the A-structure of X and the B-structure of X^*. In the 90's, Morrison considered the movable cone of X in the context of mirror symmetry and also the topology change. We will push this perspective further by finding the corresponding cone structure in terms of the monodromy nilpotent cones in the B-structure of X^*.§.§.§ Movable cones of XAs above, let us assume that Calabi–Yau threefold X=:X_1 comes with several other Calabi–Yau threefolds X_i, i=2,…,s, which are birational to each other. Let 𝒦_i⊂ H^2(X_i,ℝ) be the Kähler cone of X_i. Using the birational maps φ_i X X_i, these Kähler cones of X_i can be transformed to the corresponding cones in H^2(X,ℝ). The convex hull of the union of these cones is the movable cone Mov(X) of X. It is shown in <cit.> that the union of the transformed Kähler cones defines a chamber structure to the movable cone Mov(X) (see also <cit.>). To work with the classical mirror symmetry, in fact, we have to consider the movable cone in H^2(X,ℝ)⊗ℂ using the complexified Kähler cones 𝒦_i+√(-1)H^2(X_i,ℝ). However, in this paper, we will mostly focus on the structures in the real part of the complexified Kähler moduli.§.§.§ Compactification of the moduli space ℳ_X^*^ cpx Suppose that X^* is mirror to X, i.e., we have a mirror family 𝔛^*→ B:=_X^*^ cpx over a parameter space ℳ_X^*^ cpx on which we find a local (smooth) family 𝔛|_Δ_r^*→Δ_r^*⊂ℳ_X^*^ cpx to describe the B-structure which is mirror to the A-structure of X. In the classical mirror symmetry of Calabi–Yau complete intersections in toric varieties, there is a natural (toric) compactification ℳ_X^*^ cpx <cit.> of the moduli space ℳ_X^*^ cpx, and the geometry Δ_r^*⊂ℳ_X^*^ cpx is characterized by the corresponding normal crossing boundary divisors at the origin o∈Δ_r=ℂ^r.The following properties can be observed for an abundance of examples of complete intersection Calabi–Yau manifolds:Assume X and X^* are Calabi–Yau threefolds which are mirror to each other. If Calabi–Yau threefold X=:X_1 has birational models X_i, i=2,… ,s, then there appear the corresponding boundary points o=:o_1 and o_i, i=2,… ,s, given by normal crossing divisors in ℳ_X^*^ cpx such that =0pt(1) o_i are LCSLs, and(2) the A-structures of X_i are isomorphic to the B-structures arising from o_i.Let X and X^* be as above. Corresponding to the birational map φ_ji X_i X_j, there is a path connecting o_i to o_j and the connection matrix M_ji of the B-structures such that=0pt(1) it preserves the monodromy weight filtrations, and(2) it is integral and also compatible with the symplectic structures at each o_i, i.e., ^tM_jiΣ_jM_ji =Σ_i for the symplectic matrices Σ_i representing the symplectic forms on H^3(X_b_o_i^*,ℤ).Recently Calabi–Yau manifolds which are derived equivalent but not birational each other have been attracting attention (see, e.g., <cit.> and references therein). These are called Fourier–Mukai partners after the original work by Mukai for K3 surfaces <cit.>. As shown in examples <cit.>, if a Calabi–Yau threefold X has such Fourier–Mukai partners, then corresponding boundary points exist in ℳ_X^*^ cpx with the property (2) in Observation <ref>, but losing the property (1). If we have both the property (1) and (2), then we can see that the so-called prepotential for quantum cohomology is invariant up to quadratic terms under analytic continuations (see Proposition <ref> below), and hence the quantum cohomologies of birational Calabi–Yau threefolds are essentially the same (see <cit.> for example). However, as we see in <cit.>, quantum cohomologies of Fourier–Mukai partners are quite different to each other. In this paper we will focus on Calabi–Yau threefolds given by complete intersections in toric varieties. Showing two examples which exhibit interesting birational geometry, we will make Observation <ref> more explicit, e.g., we will give precise descriptions about the path connecting the boundary points. Also finding some monodromy relations, we will come to the following observation: Assume a Calabi–Yau threefold X and its mirror manifold X^* have the properties described in Observation <ref>. Then there are natural choices of path connecting o_i and o_j such that the monodromy nilpotent cones defined for each o_i in ℳ_X^*^ cpx are glued together. We identify the resulting structure as the mirror counter part of the movable cone obtained by gluing Kähler cones by birational maps. The gluing will be achieved by finding monodromy relations coming from boundary divisors which have multiple tangency with some component of the discriminant (see Section <ref>). When writing the monodromy relations, we find a certain monodromy action of a distinguished form, which we call “Picard–Lefschetz formula of flopping curves” based on the mirror correspondence (cf. the same forms are known in physics literatures, <cit.> for example, as strong coupling limits associated to certain contractions of curves).§ COMPLETE INTERSECTION CALABI–YAU SPACESFROM GORENSTEIN CONES In this section, we describe mirror symmetry of a Calabi–Yau complete intersection of the formX:=(ℙ^4|11111 ℙ^4|11111 )^2,52,i.e., a complete intersection of five general (1,1) divisors in ℙ^4×ℙ^4 which has Hodge numbers (h^1,1,h^2,1)=(2,52). In this section, we will study the A-structure of X. §.§ Cones for complete intersections and Calabi–Yau manifolds To describe the complete intersection X, let us note that we can write X=s^-1(0) with a generic choice of a section of the bundle 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4. We describe this starting with the affine cone over the generalized Segre embedding s_1,1,1(ℙ^4×ℙ^4×ℙ^4), which we write byU_0:=Specℂ[λ_iz_jw_k | 1≤ i,j,k≤5]with the homogeneous coordinates λ_i, z_j, w_k of ℙ^4's. Let U→ U_0 be the blow-up of the cone at the origin. It is easy to see that the exceptional divisor E is isomorphic to ℙ^4×ℙ^4×ℙ^4. In fact, U is isomorphic to the total space of the line bundle 𝒪(-1,-1,-1)→ℙ^4×ℙ^4×ℙ^4. Contracting one of the ℙ^4's (m-th factor of _λ^4×_z^4×_w^4), we have three possible contractions of U which fit in the following diagram:[dl]_π_1 U_1 @<–>[d]U_0[l]_π_2 U_2 @<–>[d][ul] [l] [dl] U. [ul]^π_3 U_3Again, it is easy to see that U_α→ U_0, α=1,2,3, are small resolutions, and the geometries of U_α are of the form 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4 that are birational to each other. It is worthwhile noting that if we start with the cone over s_1,1(ℙ^1×ℙ^1) in the above construction, the resulting geometry is the standard Atiyah flop for the small resolutions of the form 𝒪(-1)⊕𝒪(-1)→ℙ^1.Consider the potential function on U_0,W=∑_i,j,ka_ijkλ_iz_jw_kwith a_ijk∈ℂ being chosen generically. Let W_α:=π_α^#W be the potential functions on U_α. We denote the critical locus of W_α in each U_α byX_α:=Crit(W_α,U_α), α=1,2,3.The critical locus X_α is a Calabi–Yau complete intersection of the form (<ref>). By symmetry, we only consider the case X_1. To write the conditions for the criticality, it is helpful to use the homogeneous coordinate for the small resolution U_1, which is the total space (-1,-1)^⊕5→_z^4×_w^4. Let z_i,w_j denote the homogeneous coordinates of _z^4×_w^4 and λ_i be the fiber coordinate. Then the potential function is simply given by W_1=∑_i,j,ka_ijkλ_iz_jw_k, which gives the conditions for the criticality ∂ W_1/∂λ_i=∂ W_1/∂ z_j=∂ W_1/∂ w_k=0. If we denote ∂ W_1/∂λ_i=∑_j,ka_ijkz_jw_k=:f_i(z,w), the conditions ∂ W_1/∂ z_j=∂ W_1/∂ w_k=0 may be arranged into a matrix form(∇_zf_1 ⋯ ∇_zf_5 ∇_wf_1 ⋯ ∇_wf_5)([ λ_1; ⋮; λ_5 ])=0.The last equation gives the zero section {λ_1=⋯=λ_5=0}≃_z^4×_w^4 and the conditions f_1(z,w)=… =f_5(z,w)=0 give a smooth complete intersection in the zero section if we choose a_ijk sufficiently general. [<cit.>]X_α and X_β, α≠β, are birational. The birational maps φ_βα X_α X_β are given by the Atiyah flops associated to the contractions of 50 ℙ^1s, which we summarize in the following diagram:(-26,0)*++X_1="Xi", (0,0)*++X_2="Xii", (26,0)*++X_3="Xiii", (52,0)*++X_1,="XiR", (-13,-13)*+Z_2="Zii", (13,-13)*+Z_3="Ziii", (39,-13)*+Z_1="ZiR", _π_21 "Xi";"Zii" ^π_22 "Xii";"Zii" _π_32 "Xii";"Ziii" ^π_33 "Xiii";"Ziii" _π_13 "Xiii";"ZiR" ^π_11 "XiR";"ZiR" @<–> "Xi";"Xii" @<–> "Xii";"Xiii" @<–> "Xiii";"XiR"where Z_1⊂ℙ_z^4, Z_2⊂ℙ_w^4 and Z_3⊂ℙ_λ^4 are determinantal quintics defined by the 5×5 matrices (∑ z_ja_ijk), (∑ w_ka_ijk) and (∑λ_ia_ijk), respectively.We refer the references <cit.> for the proof of the above proposition. In the above proposition, we naturally come to birational Calabi–Yau complete intersections. Some remarks related to this are in order: =0pt* U_α's are birational to each other since they are all toric varieties with the same algebraic torus contained as a dense subset. In fact, they all have the form 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4. However, when defining X_α as the critical locus of the potential function, the zero section of 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4 is specified by the criticality condition. Hence, that U_α's are birational does not imply that X_α's are birational. The fact that X_α's are birational comes from different reasons as described in the above proposition. * The affine cone construction here is an example of more general method in toric geometry due to Batyrev and Borisov <cit.>. There, the affine cone is replaced by the so-called Gorenstein cones, and actually a pair of reflexive Gorenstein cone (C_∇,C_Δ) to describe mirror symmetry. The birational geometry we observed in the above proposition has been described by the property of so-called nef-partitions of ∇ by Batyrev and Nil <cit.>. They have found that the two different (but isomorphic) nef-partitions∇=∇_1+∇_2+⋯+∇_s=∇_1'+∇_2'+⋯+∇_s'sometimes results in dual nef-partitionsΔ=Δ_1+Δ_2+⋯+Δ_s,Δ'=Δ_1'+Δ_2'+⋯+Δ_k'with Δ and Δ' having completely different shapes to each other. We can describe our birational Calabi–Yau threefolds in this general setting. See references <cit.> for recent works which shed light on this general phenomenon from the derived categories of Calabi–Yau threefolds. §.§ Movable cone of X:=X_1Let us note that the Kähler cone of X(=X_1) is given by 𝒦_X=ℝ_>0H_1+ℝ_>0H_2 with the pull-backs H_1=π_11^*H_Z_1 and H_2=π_21^*H_Z_2 of the hyperplane classes H_Z_i of Z_i, where π_ji X_i→ Z_j is the projection in the diagram (<ref>). =0pt (1) Let 𝒦_X_2=ℝ_>0L_Z_2+ℝ_>0L_Z_3 be the Kähler cone with the generators L_Z_2=π_22^*H_Z_2 and L_Z_3=π_32^*H_Z_3. By the birational map φ_21 X_1 X_2, the Kähler cone is transformed toφ_21^*(𝒦_X_2)=_>0H_2+ℝ_>0(4H_2-H_1). (2) Similarly, let 𝒦_X_3=ℝ_>0M_Z_3+ℝ_>0M_Z_1 be the Kähler cone of X_3 generated by M_Z_3=π_33^*H_Z_3 and M_Z_1=π_13^*H_Z_1, then we haveφ_31^*(𝒦_X_3)=ℝ_>0(4H_1-H_2)+_>0H_1for the birational map φ_31 X_1 X_3.See Appendix <ref>. With the divisors L_Z_2, L_Z_3 and M_Z_3, M_Z_1 defined as above, we haveφ_32^*(M_Z_1)=4L_Z_3-L_Z_2,φ_32^*(M_Z_3)=L_Z_3for the birational map φ_32 X_2 X_3.The second relation holds by definition. For the first relation, see Appendix <ref>. Now, we define the following composite of the birational maps:ρ:=φ_13∘φ_32∘φ_21with the convention φ_ij=φ_ji^-1 X_j X_i (see the diagram (<ref>)).The birational map ρ is not an automorphism of X. It is of infinite order.We show thatρ^*H_1=-4H_1+15H_2,ρ^*H_2=-15H_1+56H_2for ρ^*=φ_21^*∘φ_32^*∘φ_13^*. Since φ_13^*=(φ_31^-1)^*=(φ_31)_* and using the relations φ_31^*(M_Z_3)=4H_1-H_2,φ_31^*(M_Z_1)=H_1 in Lemma <ref>(2), we haveM_Z_3=4M_Z_1-φ_13^*(H_2), M_Z_1=φ_13^*(H_1).Then, using Lemmas <ref> and <ref>, it is straightforward to evaluate ρ^*(H_i), e.g., ρ^*(H_1)=φ_21^*∘φ_32^*(M_Z_1)=φ_21^*(4L_Z_3-L_Z_2)=4(4H_2-H_1)-H_2. From these actions of ρ^*, we see that ρ^*(_X)≠_X and hence ρ∉Aut(X). Also, expressing the linear action (<ref>) by a matrix ([-4 -15;1556 ]), we see that ρ has an infinite order. Suppose X_i≄X_j, i≠j, then the groups of birational maps of X_i are given byBir(X_i)=Aut(X_i)·⟨φ_i1∘ρ∘φ_1i⟩.Since arguments are similar to <cit.>, here we only give a rough sketch. Also, we only describe the case i=1, φ_11=id_X. Take a birational map τ X X. We denote by 𝙴(τ) the locus where τ is not defined or non-isomorphic. Consider an ample divisor D and its transform D'=(τ^-1)_*D. Under this setting, we consider the two cases: (i) If D' is nef, then using <cit.> we have D|_𝙴(τ^-1)≡0, i.e., numerically equivalent to zero. Since D is ample, this implies 𝙴(τ^-1)=∅, i.e., τ∈Aut(X). (ii) If D' is not nef, the restriction D'|_𝙴(τ) is not nef, too. This is because if D'|_𝙴(τ) were nef, then D'=(τ^-1)_*D must be nef because D is ample. Therefore D'|_𝙴(τ) is not nef and there exists a curve C⊂𝙴(τ) such that D'· C<0. Now, since K_X|_𝙴(τ)≡0, we know that K_X+ε D', 0<ε≪1, is not nef and (X,ε D') is klt. From the theory of minimal models, we know that there exists an extremal ray of NE(X) and its associated contraction, which must be either X→ Z_1 or X→ Z_2 up to automorphisms. Now, corresponding to these two possibilities, we make the following diagrams:(-50,0)*++X="Xl", (-30,0)*++X_2="Xii", (-10,0)*++X="Xll", (-40,-13)*++Z_1="Zi",(10,0)*++X="Xr", (30, 0)*++X_3="Xiii", (50,0)*++X.="Xll", (20,-13)*++Z_2="Zii", (0,-5)*or @/^3mm/@–>^τ (-46,3);(-13,2) @–>^φ_21 "Xl";"Xii""Xl";"Zi""Xii";"Zi"@/^3mm/@–>^τ (14,3);(47,2) @–>^φ_31 "Xr";"Xiii""Xr";"Zii""Xiii";"Zii"Depending on the two cases, we set D”=(φ_21τ^-1)_*D or D”=(φ_31τ^-1)_*D and consider inductively the above two cases (i) and (ii) again. Due to <cit.>, this process terminates arriving at the case (i) in the end. We can deduce that there are only two possibilities under the assumption X≄X_i, i=2,3:(0,0)*+X=X_1_φ_21 X_2_φ_32 X_3_φ_13 X_1_φ_21 X_2⋯_φ_13 X_1 X_1=X,, (0,-12)*+X=X_1_φ_31 X_3_φ_23 X_2_φ_12 X_1_φ_31 X_3⋯_φ_12 X_1 X_1=X., @/_2.5mm/@–>^ρ (-35,-3);(-7,-3) @/_2.5mm/@–>^ρ^-1 (-35,-15);(-7,-15)Corresponding to these two, we have the decomposition τ=φ_Lρ^n or τ=φ_R(ρ^-1)^m with φ_L,R∈Aut(X). We use the assumption X_i≄X_j at the very end of the above proof. If X_1≃ X_i, then it is easy to deduce that we only have to include φ_i1 in the generators of Bir(X_1). Similar modification in Bir(X_i) is required if X_i≃ X_j, i≠j. These do not affect the form of the movable cone determined below. The assumption in the above proposition has been made just for simplicity. Let us denote by Mov(X_i) be the movable cones generated by movable divisors on X_i. Since the transforms of movable divisors by flops are movable, we haveMov(X)=Mov(X_1)=φ_21^*Mov(X_2)=φ_31Mov(X_3). The following result is known by <cit.>. For completeness of our arguments, we present it here with a general proof.The closure of the movable cone Mov(X) is given byMov(X)=ℝ_≥0(-H_1+(2+√(3))H_2) +ℝ_≥0(H_1+(-2+√(3))H_2). By Lemmas <ref> and <ref>, it is easy to see that the closure of the set φ_21^*(𝒦_X_2)∪𝒦_X_1∪φ_31^*(𝒦_X_3) is given byC_123:=ℝ_≥0(4H_2-H_1)+ℝ_≥0(4H_1-H_2).We defineM:=⋃_n∈ℤ(ρ^*)^nC_123:=⟨ρ^*⟩·C_123.Then, from a linear algebra, it is straightforward to see that the r.h.s. of (<ref>) coincides with the closure M. Since any automorphism of X_i preserves the generators of _X_i or exchanges them, using Proposition <ref>, we have ∪_iφ_i1^*(Bir(X_i)^*_X_i)=M. Hence we have M⊂Mov(X).To show the other inclusion, take a rational point d∈Mov(X). There exist m≫1 and an effective movable divisor D such that md=[D]. If D is nef, then d∈_X and hence d∈ M. If D is not nef, we do the same inductive process as in the proof of Proposition <ref> and find a birational map τ X X_i, τ∈⟨ρ,φ_21, φ_31⟩, such that D'=τ_*D is a nef divisor on X_i, i.e., D'∈_X_i. Namely, we have D=τ^*D'∈𝒦_X and 𝒦_X⊂ M, which imply Mov(X_i)()⊂ M. Hence we have Mov(X)⊂M.§.§ Mirror symmetry of XFor the complete intersection Calabi–Yau threefolds X (=X_1), the mirror family can be obtained by a straightforward application of the Batyrev–Borisov toric mirror construction. However, the construction involves complications in combinatorics for toric geometry. In our case, we can avoid these complications and find the mirror family of X by the so-called orbifold mirror construction starting with a special family <cit.>.Define the following special family of X_1:X_ sp:= { z_iw_i+a z_iw_i+1+b z_i+1w_i=0, i=1,…,5 }⊂ℙ_z^4×ℙ_w^4,where the indices of z_i, w_j should be considered modulo 5. For general values of a, b, we have the following properties: =0pt (1) X_ sp is singular along 20 lines of singularity of A_1 type. (2) There exists a crepant resolution X^*→ X_ sp with X^* being a Calabi–Yau threefold with h^1,1(X^*)=52, h^2,1(X^*)=2. (3) The resolution X^* parametrized by (a,b)∈ℂ^2 defines a family 𝔛^*→ℳ_X^*^ cpx∖ Dis with ℳ_X^*^ cpx=ℙ^2 and Dis=D_1∪ D_2∪ D_3∪ Dis_0 where D_i are the coordinate lines of ℙ^2 and Dis_0 is an irreducible (singular) curve of degree 5. The fiber over [a^5,b^5,1]∉ Dis is given by the resolution X^* with (a,b). Proofs of these properties are given in <cit.>. We can verify that all the properties in Observation <ref> hold for the family 𝔛^*→ℙ^2∖ Dis. We seto_1=D_1∩ D_2=[0,0,1], o_2=D_2∩ D_3=[1,0,0], o_3=D_3∩ D_1=[0,1,0].All these boundary points o_1, o_2, o_3 are LCSLs whose B-structures are identified with the A-structures of the birational models X_1, X_2 and X_3, respectively. See Fig. <ref> in the next section. The above proposition has been derived by introducing integral and symplectic structures at each o_i and calculating the monodromies around the divisors D_i, see <cit.> for details. Our focus in what follows will be gluing the monodromy cones (<ref>) which are defined for each boundary point o_i.§ GLUING MONODROMY NILPOTENT CONES IFor the example in the preceding section, we will find a path o_i→ o_j which we can identify with the birational map φ_ji X_i X_j as described in Observation <ref>. We will find that the monodromy nilpotent cones (<ref>) at each boundary point are naturally glued together by the monodromy relations coming from the path. Also, in the next section (Section <ref>) we will study another interesting example, which has no other birational models other than itself but has a birational automorphism of infinite order. §.§ B-structures of X^*Associated to the family π𝔛^*→ℳ_X^*^ cpx∖ Dis, we have the local system R^3π_*_𝔛^* which introduces the Gauss–Manin system on the moduli space, or equivalently the Picard–Fuchs differential equation for the period integrals of holomorphic three form. This Picard–Fuchs equation has been studied in our previous work <cit.>, where we have described the B-structure for the boundary points o_i, i.e., the integral and symplectic basis for the local solutions as well as integral monodromy matrices using the central charge formula given in <cit.> which goes back to the study of GKZ system <cit.> in the 90's (see <cit.> for details). Here we briefly recall the integral and symplectic basis referring to <cit.> for its explicit form, and define the monodromy nilpotent cones for each o_j from the monodromy matrices calculated there.§.§.§ B-structure at o_1 Let [-x,-y,1]∈ℙ^2 be the affine coordinate with the origin o_1 (where the minus signs are required to have the canonical integral and symplectic structure based on the central charge formula). The canonical, integral and symplectic structure appears from a unique power series solution w_0(x,y) of the Picard–Fuchs differential equation around the origin o_1. Including the logarithmic solutions, the result can be arranged as follows:Π(x,y) =^t(w_0(x,y),w_1^(1)(x,y), w_2^(1)(x,y),w_2^(2)(x,y),w_1^(2)(x,y),w^(3)(x,y))= ^t(∫_A_0Ω_x,∫_A_1Ω_x,∫_A_2Ω_x,∫_B_2Ω_x,∫_B_1Ω_x,∫_B_0Ω_x),where { A_0,A_1,A_2,B_2,B_1,B_0}⊂ H_3(X_b_o^*,ℤ) is a symplectic basis satisfying A_i∩ B_j=δ_ij, A_i∩ A_j=B_i∩ B_j=0 representing the integral and symplectic solutions of the Picard–Fuchs equation <cit.>. The monodromy matrix 𝚃_x of Π(x,y) for a small loop around x=0 and similarly 𝚃_y for y=0 have been determined as follows:𝚃_x=([100000;110000;001000;5 10 10100;25 10010; -5 -3 -50 -11 ]),𝚃_y=([100000;110000;101000;2 105100;5 10 10010; -5 -5 -3 -101 ]).We defineℬ_1:= {α_0,α_1,α_2,β_2,β_1,β_0}⊂ H^3(X_b_o^*,ℤ)to be the dual basis satisfying ∫_A_iα^j=δ_i^ j=∫_B_iβ^j and ∫_A_iβ^j=∫_B_iα^j=0. Since the monodromy actions on the period integrals, i.e., on H_3(X_b_o^*,ℤ), are translated into the dual space via the transpose and inverse, we define the linear action N_λ=∑λ_iN_i on H^3(X_b_o^*,ℤ) byN_1:=- ^t(log𝚃_x), N_2:=- ^t(log𝚃_y).Then we define the monodromy nilpotent cone at o_1 byΣ_o_1:={∑λ_iN_i | λ_i>0}⊂End(H^3(X_b_0,ℚ)).For general values of λ_i>0, it is easy to see that the nilpotent matrix N_λ induces the monodromy weight filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ) given byW_0=⟨α_0⟩, W_2=⟨α_0,α_1,α_2⟩, W_4=⟨α_0,α_1,α_2,β_2,β_1⟩, W_6=⟨α_0,α_1,α_2,β_2,β_1,β_0⟩. Using the matrices N_1, N_2, it is easy to see the following property: We haveN_iN_jN_k=C_ijk𝙽_0with totally symmetric C_ijk given by C_111=C_222=5, C_112=C_122=10 and C_ijk=0 for other cases, and 𝙽_0=([ 0 1; O_5 0 ]) where O_5 is the zero matrix of size 5×5.As we see above, the monodromy matrices of the period integrals act on H_3(X_b_o^*,ℤ) while the monodromy weight filtration is defined in the dual space H^3(X_b_o^*,ℤ). Hence, we translate any monodromy matrix 𝙰 obtained from the analytic continuations of the period integral Π(x,y) to the corresponding matrix A in the dual space by A= ^t𝙰^-1. §.§.§ B-structures at o_2, o_3 In a similar way to the last paragraph, we determine the B-structure from the boundary points o_2 and o_3, which are given by the origins of the affine charts [1,-y',-x']∈ℙ^2 and [-x”,1,-y”]∈ℙ^2. As described in detail in <cit.>, we have the canonical integral and symplectic basisΠ'(x',y')=x'Π(x',y')andΠ”(x”,y”)=y”Π(x”,y”)in terms of the same Π(x,y) as (<ref>) for o_2 and o_3, respectively. Since both of (<ref>) have essentially the same form as Π(x,y), we have𝚃_x'^'=𝚃_x”^''=𝚃_xand𝚃_y'^'=𝚃_y”^''=𝚃_yfor the monodromy matrices with the base points b_o' and b_o” near the origins. Hence for o_2 and o_3 we have isomorphic B-structures withÑ_1'=log T_x'',Ñ_2'=log T_y''andÑ_1”=log T_x””,Ñ_2”=log T_y””,where T_x''=( ^t𝚃_x'')^-1, T_y''=( ^t𝚃_y'')^-1 and similarly for T_x””, T_y””. These nilpotent matrices determine the respective monodromy weight filtrations in H^3(X_b_o'^*,ℚ) and H^3(X_b_o”^*,ℚ) with the basis{α_0',α_1',α_2',β_2',β_1',β_0'}and{α_0”,α_1”,α_2”,β_2”,β_1”,β_0”} ,as described above. We denote the monodromy nilpotent cones at o_2 and o_3 byΣ_o_2'={∑λ_iÑ_i' | λ_i>0}⊂End(H^3(X_b_o'^*,ℚ)), Σ_o_3”={∑λ_iÑ_i” | λ_i>0}⊂End(H^3(X_b_o”^*,ℚ)).These are the B-structures which we identify with the A-structures of the birational models X_2 and X_3, respectively, in <cit.>. §.§ Gluing the monodromy nilpotent conesThe monodromy matrices are transformed by conjugation when the base point is changed along a path. We can transform the monodromy nilpotent cones (<ref>) into H^3(X_b_o^*,ℚ) once we fix paths p_b_0'← b_o and p_b_0”← b_o. Let us denote by φ_b_o'b_o the resulting isomorphism φ_b_o'b_o H^3(X_b_o^*,ℚ)≃ H^3(X_b_o'^*,ℚ) and similarly for φ_b_o”b_o. We define the transforms of the nilpotent cones (<ref>) by these isomorphisms byΣ_o_2:=(φ_b_o'b_o)^-1Σ_o_2' φ_b_o'b_0 ,Σ_o_3:=(φ_b_o”b_o)^-1Σ_o_3”φ_b_o”b_o.Then the cones Σ_o_2 and Σ_o_3 are generated byN_i':=(φ_b_o'b_o)^-1Ñ_i' φ_b_o'b_0, N_i”:=(φ_b_o”b_o)^-1Ñ_i”φ_b_o”b_0,i=1,2,respectively. Note that Σ_o_1, Σ_o_2, Σ_o_3 are cones in End(H^3(X_b_o^*,ℚ)).§.§.§ Path p_o_2←o_1 The transform Σ_o_2 of the nilpotent cone obviously depends on the choice of the path. Looking the moduli space ℳ_X^*^ cpx closely, we find that there is a natural choice of the path by which the cone Σ_o_2 is glued with Σ_o_1 along a common face (boundary ray) of them.The moduli space ℳ_X^*^ cpx has been studied in detail in <cit.>. Here we recall the structure of the discriminant Dis= Dis_0∪ D_x∪ D_y∪ D_z. As we schematically reproduce the results in Fig. <ref>, the irreducible component Dis_0 of the discriminant touches the divisor D_y={ y=0} at (x,y)=(1,0) with fifth-order tangency as we can see in the expressionDis_0={ (1-x-y)^5-5^4xy(1-x-y)^2+5^5xy(xy-x-y)=0} .We introduce the affine chart ℂ_(1,0)^2 with the origin (1,0). After blowing-up at the origin five times, we can remove the tangential intersection of the proper transform Dis_0 of Dis_0 with the exceptional divisors (see Fig. <ref>). We denote the exceptional divisors by E_1,…,E_5.Let q_12 be a point near the intersection E_1∩ D_y, and b_o, b_o' be points near the origins o_1 and o_2, respectively. We define a path p_b_o'← b_0 to be the composite path p_b_o'← q_12∘ p_q_12← b_o of the following straight lines:p_q_12← b_o={(1-t)b_o+tq_12 | 0≤ t≤1},p_b_o'← q_12={(1-t)q_12+tb_o' | 0≤ t≤1}. §.§.§ The isomorphisms φ_b_o'b_0, φ_b_o”b_o' and φ_b_ob_o” We first calculate the connection matrix of the local solution Π(x,y) along the path p_b_o'← b_o.With respect to the basis (<ref>) and (<ref>), the isomorphism φ_b_o'b_o H^3(X_b_o^*,ℚ)≃ H^3(X_b_o'^*,ℚ) along the path p_b_o'← b_o is given byφ_b_o'b_o=(-1 0 0 0 0 00 1 -4 2 25 00 0 -1 0 -2 00 0 0 -1 -4 00 0 0 0 1 00 0 0 0 0 -1 ).This isomorphism preserves the monodromy weight filtrations and also the symplectic structures described in Section <ref>.To determine the matrix form of φ_b_o'b_o, we do first the analytic continuation of the period integral Π(x,y) along the path p_q_12← b_o by making local solutions around q_12=E_1∩ D_y in terms of the blow-up coordinates s_1=x-1, s_2=y/(1-x)^5 which represent q_12 by s_1=s_2=0. There are two local solutions which are given by regular powerseries, and others contain logarithmic singularities given by log s_1 and log s_2,…,(log s_2)^3. For a fixed value of y, |y|≪1, we analytically continue these solutions to Π(x,y) as functions of s_1=x-1. Note that, under the analytic continuation, the powers of log y are unchanged. Hence the connection matrix follows from the analytic continuation of the period integrals Π(x,0) where we set log y=0 and y=0. In our actural calculation, we set s_2=0 and log s_2=-5log(1-x) for the local solutions around (s1,s2)=(0,0), and relate these solutions numerically to Π(x,0) using powerseries expansions with sufficiently high degrees. In a similar way, we can calculate the connection matrix for the latter half p_b_o'← q_12 of the path p_b_o'← b_o. Actually, we can avoid the above numerical calculations finding an analytic formula for Π(x,0). However, since the details are technical, we will report them elsewhere. It is clear that the connection matrix φ_b_o'b_o preserves the filtrations since it is block diagonal with respect to the basis compatible with the filtrations W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ) and W_0'⊂ W_2'⊂ W_4'⊂ W_6'=H^3(X_b_o'^*,ℚ). Moreover, we can verify directly that it preserves the symplectic structure given by (<ref>). From the forms of period integrals given in (<ref>), it is easy to deduce that we have the isomorphismsφ_b_o”b_o' H^3(X_b_o'^*,ℤ)≃ H^3(X_b_o”^*,ℤ)andφ_b_ob_o” H^3(X_b_o”^*,ℤ)≃ H^3(X_b_o^*,ℤ)by simply exchanging the bases α_1↔α_2 and β_1↔β_2 suitably, i.e., φ_b_o”b_o'=φ_b_o'b_op_23p_45 and φ_b_ob_o”=p_23p_45φ_b_o'b_op_23p_45 with the permutation matrices p_ij for the transposition (i,j). Explicitly, they are given byφ_b_o”b_o'=([ -100000;0 -41 2520;0 -10 -200;000 -4 -10;000100;00000 -1 ]),φ_b_ob_o”=([-1 0 0 0 0 0; 0-1 0-2 0 0; 0-4 1 -25 2 0; 0 0 0 1 0 0; 0 0 0-4-1 0; 0 0 0 0 0-1 ]).Here we note that these isomorphisms preserve the monodromy weight filtrations and also the symplectic structures described in Sections <ref> and <ref>. Also it should be noted that we have verified Observation <ref> in Section <ref> in the present case.Let us introduce the following notation:_21:=φ_b_o'b_o,_32:=φ_b_o”b_o',_13:=φ_b_ob_o”and also set _ij:=_ji^-1. As this notation indicates, we expect certain correspondence of these _ij to the birational maps φ_ij X_j X_i under mirror symmetry. In order to make this more explicit, we note the groupoid structure associated to the isomorphisms _ij. We denote by G_{1,2,3} the groupoid generated by _21, _32, _13.Let G_ij be the subset of G_{1,2,3} consisting of elements _ii_1_i_1i_2⋯_i_kj, k≥0. It is easy to see thatG_11={^n |n∈ℤ} , G_21={_21^n |n∈ℤ} , G_31={_31^n |n∈ℤ} ,where set :=_13_32_21.§.§.§ Groupoid actions on the nilpotent cones We define the following conjugates of the nilpotent cones (<ref>) and (<ref>):Σ_o_1^(n):=(^-1)^nΣ_o_1^n, Σ_o_2^(n):=(^-1)^n_21^-1Σ_o_2'_21^n =(^-1)^nΣ_o_2^n, Σ_o_3^(n):=(^-1)^n_31^-1Σ_o_3”_31^n =(^-1)^nΣ_o_3^n.These are cones in End(H^3(X_b_o^*,ℝ)) and generalize the nilpotent cones Σ_o_k=Σ_σ_k^(0), k=1,2,3, introduced in the beginning of this subsection. It is easy to see that these cones are generated byN_i(n):=(^-1)^n N_i^n, N_i'(n):=(^-1)^n N_i' ^n, N_i”(n):=(^-1)^n N_i”^n,respectively, where we set N_i':=_21^-1Ñ_i'_21 and N_i”:=_31^-1Ñ_i”_31, i=1,2.§.§.§ Monodromy relationsTo see how the (closure of the) cone Σ_o_2=Σ_o_2^(0) is connected to (that of) Σ_o_1=Σ_o_1^(0), we calculate the generators N_i' in End(H^3(X_b_o^*,ℤ)). By the definition of N_i', it suffices to calculateT_x':=_21^-1 T_x'^'_21, T_y':=_21^-1 T_y'^'_21,since we can use T_x''=T_x, T_y''=T_y for the local monodromy matrices as we remarked in (<ref>). Similarly, using the connection matrix along the path p_q_12← b_o, we can express the local monodromy around the exceptional divisor E_1 as a linear (integral and symplectic) action on H^3(X_b_o^*,ℤ) which we denote by a matrix T_E_1 using the basis ℬ_1 in (<ref>). [`Picard–Lefschetz formula' for flopping curves] Using the basis ℬ_1 in (<ref>), we haveT_E_1=([1; 1 50;1; 1;1; 1 ]), i.e., α_1→α_1+50β_1, β_1→β_1, α_i=α_i,β_i=β_i, i≠1. As sketched briefly in the proof of Proposition <ref>, we make the local solutions of the Picard–Fuchs equation around the point of the blow-up q_12=E_1∩ D_y, and calculate the local monodromy around the divisor E_1. The claimed monodromy follows from the analytic continuation of the local solutions in the period integral Π(x,y) near the origin o_1. In our actual calculations, we only have powerseries expressions for the local solutions around q_12 and evaluate them numerically for the analytic continuation. However, as in Proposition <ref>, we can attain sufficient precision having an analytic formula for Π(x,0). The `Picard–Lefschetz formula' above is written using the symplectic basis {α_i,β_j} of H^3(X_b_o^*,ℤ). When we translate this into the dual basis { A_i,B_j} of H_3(X_b_o^*,ℤ), we haveA_1→ A_1, B_1→ B_1-50A_1with the rest of the basis left invariant. This should be contrasted to the genuine Picard–Lefschetz monodromyA_0→ A_0+B_0, B_0→ B_0,which we can see for the monodromy transformation around the proper transform Dis_0 of the discriminant. In the latter case, we see the topology of the cycles as A_0≈ T^3, B_0≈ S^3, where S^3 is a vanishing cycle and T^3 is its dual torus cycle. Recently, the construction of the A_k-cycles (k≠0) has been discussed in general in <cit.>. It is interesting to see how the dual B_k-cycles are constructed, and how the above `Picard–Lefschetz formula' are explained by the geometry of these cycles. We have the following monodromy relations:T_x'=T_E_1^-1T_x^-1T_y^4, T_y'=T_y. Recall that we have the relations T_x=( ^t𝚃_x)^-1, T_y=( ^t𝚃_y)^-1 (see Remark <ref>). Then both the relations can be verified directly using the explicit forms of T_x, T_y given in Section <ref> and T_x', T_y', T_E_1 above. The second relation also follows from the fact that the divisor { y=0} ={y'=0} intersects normally with the exceptional divisor E_1 of the blowing-up. We have arrived at (<ref>) by explicit monodromy calculations. It is natural to expect to have a conceptual derivation of (<ref>) by studying mirror symmetry of conifold transitions, but we have to this to future investigations. Instead, in the rest of this section, we will interpret the monodromy relation (<ref>). The following properties hold: =0pt(1) Generators N_i' are expressed asN_1'=4N_2-N_1+Δ_1,0', N_2'=N_2,where Δ_1,0' is a non-zero element of End(H^3(X_b_o^*,ℝ)) which annihilates the subspace W_2, i.e., Δ_1,0'|_W_2=0. (2) The monodromy nilpotent cones Σ_o_2=ℝ_>0N_1'+ℝ_>0N_2' and Σ_o_1 glue together along N_2'=N_2. They are not in a two dimensional plane in End(H^3(X_b_o^*,ℝ)). The properties in (1) are based on explicit calculations using (<ref>). The second relation N_2'=log T_y'=log T_y=N_2 is clear. For the first relation, by evaluating the matrix logarithms, we haveΔ_1,0'=N_1'-(4N_2-N_1)=([ 0 0 0 025 -25/3; 0 0 0 0 -5025; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0 ]).From this triangular form, we see the claimed property of Δ_1,0' (see also (<ref>)). The claims in (2) are clear from (1) and also from the fact that the cone Σ_o_1 is generated by N_1=log T_x and N_2=log T_y. (1) It should be observed that, under the identificationL_Z_3↔ N_1', L_Z_2↔ N_2'and H_1↔ N_1, H_2↔ N_2,Proposition <ref> above is the mirror counter part for the gluing of Kähler cones described in Lemma <ref>.(2) If the first monodromy relation of (<ref>) were T_x'=T_x^-1T_y^4, then we would haveN_1'=4N_2-N_1, N_2'=N_2,since T_x and T_y are commutative. These relations are exactly the same as those we have seen in Lemma <ref>. However the presence of T_E_1 prevents this exact correspondence. We will see that T_E_1 represents the first order quantum correction coming from the 50 flopping curves of the contraction X_1 Z_1. Thus the gluing relation found in Proposition <ref>(1) naturally encodes the first order quantum corrections. §.§.§ Gluing nilpotent conesBefore going into general descriptions, it will be helpful to see that the cone Σ_o_1 is glued with Σ_o_3 along N_1 in a similar way as above. Let us defineT_x”:=_31^-1T_x””_31, T_y”:=_31^-1T_y””_31,and also T_E_1” for the monodromy matrix around the exceptional divisor E_1”. Observing the symmetry in Fig. <ref> and (<ref>), it is easy to deduce the following monodromy relationsT_x”=T_x, T_y”=T_E_1”^-1T_y^-1T_x^4with T_E_1”=p_23p_45T_E_1p_23p_45 in End(H^3(X_b_o^*,ℤ)), where p_ij are the permutation matrices. Since the generators of the cone Σ_o_3 are given by N_1”=log T_y” and N_2”=log T_x”, we can evaluate these asN_1”=N_1, N_2”=4N_1-N_2+Δ_2,0”,where Δ_2,0” is given by Δ_2,0”=p_23p_45Δ_1,0'p_23p_45 with the vanishing property Δ_2,0”|_W_2=0. As before, Δ_2,0” is a non-vanishing element. Hence, the nilpotent cones Σ_o_1 and Σ_o_3 glue together along the common half line ℝ_≥0N_1 but do not lie on the same plane. Now we generalize these properties in the following proposition.=0pt(1) The matrixpreserves the monodromy weight filtrationW_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ). (2) The (closures of the) monodromy nilpotent cones Σ_o_1^(n), Σ_o_2^(n), Σ_o_3^(n) glue sequentially as in…,Σ_o_2^(1), Σ_o_1^(1), Σ_o_3^(1), Σ_o_2, Σ_o_1, Σ_o_3, Σ_o_2^(-1), Σ_o_1^(-1), Σ_o_3^(-1),… .(3) The generators of the cones satisfy(i) (N_1(n),N_2(n))=(N_1,N_2)(0 -11 4 )^3n+(Δ_1,n,Δ_2,n), (ii)(N_1'(n),N_2'(n))=(N_1',N_2')(0 -11 4 )^-3n+(Δ_1,n',Δ_2,n'),(iii) (N_1”(n),N_2”(n))=(N_1”,N_2”)(0 -11 4 )^-3n+(Δ_1,n”,Δ_2,n”), where Δ_i,n,Δ_i,n',Δ_i,n”∈End(H^3(X_b_o^*,ℚ)) and satisfy Δ_i,n|_W_2=Δ_i,n'|_W_2=Δ_i,n”|_W_2=0. (4) The following relations glue the nilpotent cones in (2) (see Fig. <ref>):N_1(n)=N_1”(n), N_2(n)=N_2'(n), N_2”(n)=N_1'(n-1). (1) Recall thatis defined by =_13_32_21. Each isomorphism _ij preserves the monodromy weight filtrations defined for each boundary point o_k (see Proposition <ref>). Hence, H^3(X_b_o,ℚ)→ H^3(X_b_o,ℚ) preserves the monodromy weight filtration as claimed.(2) We have introduced the generators of the nilpotent cones Σ_o_k^(n) by N_i(n),N_i'(n) and N_i”(n) for k=1,2,3, respectively, in Section <ref>. Then the claim follows from the properties (3) and (4) (see also Fig. <ref>).(3) By the definition of N_i(n), it is straightforward to calculate N_i(1) asN_i(1)=^-1N_i= -4N_1+15N_2+Δ_1,1, i=1,-15N_1+56N_2+Δ_2,1, i=2,where Δ_i,1 satisfy Δ_1,1|_W_2=Δ_2,1|_W_2=0 on the subspace W_2⊂ H^3(X_b_o^*,ℚ). We note the relation ([0 -1;14 ])^3=([-4 -15;1556 ]) and arrange the above relation into the claimed matrix form for n=1. Then we can obtain the claimed formula (i) for general n (in the first line) by evaluating (^-nN_1^n,^-nN_2^n) inductively. In the evaluation, we should note that ^-1Δ_i,n-1|_W_2=0 if Δ_i,n-1|_W_2=0 sincepreserves the monodromy weight filtration. For the second formula (ii), we note the relation(N_1',N_2')=(N_1,N_2)(-1 04 1 )+(Δ_1,0',0)obtained in Proposition <ref>(1). Taking the conjugations ^-n(-)^n on the both sides of this relation, and using the first formula (i) for ^-n(N_1,N_2)^n, we have the claimed formula. In the derivation, we use the relation(0 -11 4 )^3n(-1 04 1 )=(-1 04 1 )(0 -11 4 )^-3nand also the property ^-1Δ'_i,n-1|_W_2=0 if Δ'_i,n-1|_W_2=0. For the third relation (iii), calculations are similar but we need to use the relation ([0 -1;14 ])^3n([14;0 -1 ])=([14;0 -1 ])([0 -1;14 ])^-3n.(4) Since N_i(n), N_i'(n), N_i”(n) are defined by the conjugation of N_i(n-1), N_i'(n-1) and N_i”(n-1) by , it is sufficient to show the equalitiesN_1=N_1”, N_2=N_2', N_2”(1)=N_1'.The first two relations are verified already in Proposition <ref> and (<ref>). For the last relation, we evaluate N_2”(1)=^-1N_2” directly verifying its equality to N_1'.Consider the left ideal ℐ_2:={ X∈End(H^3(X_b_0^*,ℝ)) |X|_W_2=0} of End(H^3(X_b_0^*,ℝ)), and πEnd(H^3(X_b_0^*,ℝ))→End(H^3(X_b_0^*,ℝ))/ℐ_2 be the natural projection as a vector space. Then, taking the closure in End(H^3(X_b_0^*,ℝ))/ℐ_2, we have⋃_nπ(^-n(Σ_o_2∪Σ_o_1∪Σ_o_3)^n)=ℝ_>0c̅_1+ℝ_>0c̅_2,where c̅_1=-N̅_1+(2+√(3))N̅_2 and c̅_2=N̅_1-(2-√(3))N̅_2 with N̅_i=π(N_i), i=1,2. From Proposition <ref>, we haveπ(Σ_o_1∪Σ_o_2∪Σ_o_3) =ℝ_≥0π(N_1')+ℝ_≥0π(N_2”)=ℝ_≥0π(4N_2-N_1)+ℝ_≥0π(4N_1-N_2).Evaluating the matrix power ([0 -1;14 ])^3n, it is easy to see thatlim_n→∞ℝ_≥0π(N_1(n))=lim_n→∞ℝ_≥0π(N_2(n))=ℝ_≥0c̅_1andlim_n→-∞ℝ_≥0π(N_1(n))=lim_n→-∞ℝ_≥0π(N_2(n))=ℝ_≥0c̅_2.Then the claim follows from the gluing property (1) of Proposition <ref>.§.§ Flopping curves and T_E_1The matrix T_E_1 arises from the tangential intersection of the relevant components of the discriminant Dis in the moduli space ℳ_X^*^ cpx. As noted in the remark above, T_E_1 may be identified with the first order correction from the quantum cohomology of X_1. To see this, let us introduceN_1^𝚏:=log(T_x^-1T_y^4)=4N_2-N_1and N_2^𝚏=N_2'=N_2. Here, we should note the difference in N_1^𝚏 from the definition N_1'=log(T_E_1^-1T_x^-1T_y^4). Define C_ijk' and C_ijk^𝚏 by N_i'N_j'N_k'=C_ijk'𝙽_0 and N_i^𝚏N_j^𝚏N_k^𝚏=C_ijk^𝚏𝙽_0 with 𝙽_0 as given in Proposition <ref>. Non-vanishing (totally symmetric) C_ijk' and C_ijk^𝚏 are given by(C_111',C_112',C_122',C_222')=(5,10,10,5),(C_111^𝚏,C_112^𝚏,C_122^𝚏,C_222^𝚏)=(-45,10,10,5). We derive these numbers by direct calculations of matrix products. The nilpotent matrices N_1', N_2' follow from the B-structure at o_2, which has been identified with the A-structure of X_2. Hence the first equality in (<ref>) is a consequence from mirror symmetry. To see more details of the equality, let us recall the so-called mirror map which are defined byt_i'=∫_A_i'Ω_x'/∫_A_0'Ω_x', t_i=∫_A_iΩ_x/∫_A_0Ω_xfor each boundary point o_2 and o_1, respectively. If we relate these local definitions by the isomorphism ( ^tφ_b_o'b_o)^-1 H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) along the path p_b_o'← b_o (cf. Proposition <ref>), we havet_1'=-t_1, t_2'=4t_1+t_2. Let C_ijk be as defined in Proposition <ref>. Also set q_1':=e^t_1' and q_1=e^t_1. Then we have the following relationsC_ijk^𝚏=∑_l,m,nC_lmndt_l/dt_i'dt_m/dt_j'dt_n/dt_k'andC_111'+50q_1'/1-q_1'=C_111^𝚏+50q_1/1-q_1(dt_1/dt_1')^3. It is easy to verify these. For the second relation, we note that 50q_1/1-q_1(dt_1/dt_1')^3 =50+50q_1'/1-q_1' for q_1=1/q_1'. The equality (<ref>) is a consequence of the flop invariance of the quantum cohomology (see, e.g., <cit.>). As mentioned in Remark <ref>, the number 50 represents the flopping curves. Comparing this with Proposition <ref>, we see that the monodromy T_E_1 encodes the data of the flopping curves which is in the first order of the quantum cohomology of X_1. §.§ Prepotentials The flop invariance expressed in (<ref>) is known more precisely as the invariance of quantum cohomology under analytic continuations, where all higher order quantum corrections are taken into account. Here we rephrase this property as a property of the so-called prepotentials.For the B-structure at each boundary point, we can define the prepotential. For example for the B-structure at o_1 and o_2, respectively, they are given byℱ=1/2∑_i=0^3∫_A_iΩ_x∫_B_iΩ_x,ℱ'=1/2∑_i=0^3∫_A_i'Ω_x'∫_B_i'Ω_x'with the symplectic integral bases for period integrals in Π(x,y) and Π'(x',y'). By the isomorphism ( ^tφ_b_o'b_o)^-1 H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) along the path p_b_o'← b_o chosen as in Proposition <ref>, ℱ and ℱ' are related byℱ'=ℱ+1/2∑_i,j=1^2Q_ij∫_A_iΩ_x∫_A_jΩ_x,where (Q_ij)=([ -25-2; 2 0 ]). Using the basis { A_i,B_j}, { A_i',B_j'}, the connection matrix has the form(^tφ_b_o'b_0)^-1=([-1 0 0 0 0 0; 0 1 0 0 0 0; 0-4-1 0 0 0; 0 2 0-1 0 0; 0 -25-2-4 1 0; 0 0 0 0 0-1 ]),which gives the analytic continuation by Π'(x',y')=( ^tφ_b_o'b_0)^-1Π(x,y). From this, we read A_0'=-A_0, B_0'=-B_0 and([ A_1'; A_2' ])=R([ A_1; A_2 ]),([ B_1'; B_2' ])=( ^tR)^-1([ B_1; B_2 ])+([ -25-2; 2 0 ])([ A_1; A_2 ]),where we set R=([10; -4 -1 ]). The claimed formula is immediate from these. As we can deduce in the above proof, the prepotentials are invariant only up to quadratic terms of the A_i-periods under the analytic continuations even if they are symplectic and also preserve the monodromy weight filtrations. However the so-called Yukawa couplings are invariant since they are given by the third derivatives of the prepotentials with respect to the coordinates t_i (see (<ref>)).§ GLUING MONODROMY NILPOTENT CONES II We will study the following Calabi–Yau threefold of complete intersections:X=([ ℙ^3| 2 1 1; ℙ^3| 2 1 1 ])^2,66.We assume the defining equations of X are chosen general unless otherwise mentioned. For such X, there is no other birational model than X. However X has an interesting birational automorphism of infinite order <cit.>, and also has a non-trivial movable cone similar to the one in the preceding section. §.§ Birational automorphisms of infinite order Let π_i X→ℙ^3 be the projections to the first and second factor of ℙ^3×ℙ^3 for i=1 and 2, respectively. It is easy to see that the projection π_i is surjective and generically 2:1. We consider the Stein factorization X→ W_i→ℙ^3 of the morphism π_i X→ℙ^3 and denote the morphism by ϕ_i X→ W_i for i=1,2.For i=1,2, the morphism W_i→ℙ^3 is a double cover of ℙ^3 branched along an octic, and W_i is a (smooth) Calabi–Yau threefold. We omit proofs since they are standard (see, e.g., <cit.>). Let τ̃_i W_i≃ W_i^+ be the deck transformation of the covering W_i→ℙ^3. Then we have the map τ_i which covers τ̃_i as in the following diagram:(0,0)*++X="cX", (-15,0)*++X="lX", ( 15,0)*++X="rX", (-10,-20)*++ℙ^3="lP", ( 10,-20)*++ℙ^3="rP", (-10,-10)*+W_2="lW", (-26,-10)*+W_2^+="lWp", ( 10,-10)*+W_1="rW", ( 26,-10)*+W_1^+.="rWp", ( 17.5,-8)*++ ^τ̃_1, (-17.5,-8)*++ ^τ̃_2, ( 17.5,-10)*++≃, (-17.5,-10)*++≃, ( 7,1)*++∼, ( 7,3)*++ _τ_1, (-7,1)*++∼, (-7,3)*++ _τ_2, ( 7,-4)*++ _ϕ_1, (-7,-4)*++ _ϕ_2, @–>_ "cX";"lX" @–>^ "cX";"rX" ^π_2 "cX";"lP" _π_1 "cX";"rP""cX";"lW" _ "lW";"lP" "lX";"lWp" _2:1 "lWp";"lP""cX";"rW""rW";"rP" ^2:1 "rWp";"rP" "rX";"rWp" The following hold: =0pt(i) The map τ_i X X is birational but not bi-holomorphic.(ii)The morphism ϕ_i X→ W_i contracts 80 lines and 4 conics to points, and the birational map τ_i is an Atyah's flop of these curves. (i), (ii) See the reference <cit.>.(1) Bir(X)=Aut(X)·⟨τ_1,τ_2⟩. (2) τ_i^2= id for i=1,2. Also τ_1τ_2 has infinite order. See <cit.>.§.§ Mirror family of X We can describe the mirror family 𝔛^*→ℳ_X^*^ cpx of X by writing X in terms of a Gorenstein cone following Batyrev–Borisov. The parameter space of the defining equations up to isomorphisms naturally gives the moduli space ℳ_X^*^ cpx, which turns out to be compactified to ℙ^2 as before. Here we will not go into the details of the mirror family, but we only write the form of the Picard–Fuchs differential operator in the affine coordinate [1,x,y]∈ℳ_X^*^ cpx=ℙ^2.Picard–Fuchs equations of the family on the affine coordinate [1,x,y] are given by 𝒟_1w(x,y)=𝒟_2w(x,y)=0 with𝒟_1= (3θ_x^2-4θ_xθ_y+3θ_y^2)-(θ_x+θ_y)(2θ_x+2θ_y-1)(10x+6y)+4θ_x(2θ_x+2θ_y-1)(x-y), 𝒟_2= (θ_x^3-θ_x^2θ_y+θ_xθ_y^2-θ_y^3)-2(θ_x+θ_y)^2(2θ_x+2θ_y-1)(x-y),where θ_x=x∂ /∂ x,θ_y=y∂ /∂ y. The discriminant locus of this system is given by Dis=D_1∪ D_2∪ D_3∪ Dis_0 withDis_0={ (1-4x-4y)^4-128xy(17+56(x+y)+16(x^2+y^2))=0} ,and the coordinate lines D_i of ℙ^2.The differential operators 𝒟_1 and 𝒟_2 arise from the Gel'fand–Kapranov–Zelevinski system after finding suitable factorizations of differential operators. See <cit.> for more details. Once 𝒟_1 and 𝒟_2 are determined, it is straightforward to determine the discriminant locus (singular locus) of the system from the equations of the characteristic variety. From the forms of 𝒟_1and 𝒟_2, the origin x=y=0 is expected to be a LCSL. In fact, we can verify all the properties for the LCSL in Definition <ref>. We also verify that there is no other LCSL point in ℳ_X^*^ cpx=ℙ^2. In Fig. <ref>, we schematically describe the structure of the moduli space ℳ_X^*^ cpx. There, as in the preceding example, we see that the component Dis_0 intersects tangentially with the divisors D_1={ x=0} and D_2={ y=0}. This time, we blow-up at these two intersection points successively four times to make the intersections normal crossing (see Fig. <ref> right).As in the previous example, we should be able to arrive at the mirror family 𝔛^*→ℳ_X^*^ cpx starting with a special family { X_ sp} _a,b. But we leave this task for other occasions, since we have the mirror family in any case as above.§.§ B-structure of X at the origin oAs in Section <ref>, the canonical integral and symplectic structure can be introduced from the power series solutionw_0(x,y)=∑_n,mΓ(1+2n+2m)Γ(1+n+m)^2/Γ(1+n)^4Γ(1+m)^4x^ny^maround the origin o:=[1,0,0]. Simply replacing the necessary parameters in the general formula <cit.>, and fixing the so-called quadratic ambiguities there by C_kl=0, we obtain the canonical integral and symplectic structure in the form of period integrals Π(x,y) with the corresponding symplectic basis{ A_0,A_1,A_2,B_2,B_1,B_0}⊂ H_3(X_b_o^*,ℤ)with A_i∩ B_j=δ_ij,A_i∩ A_j=B_i∩ B_j=0,where a base point b_o is taken near the origin. We denote by 𝚃_x the matrix of the monodromy transformation of Π(x,y) for a small loop around the divisor D_1= { x=0 }, and similarly denote by 𝚃_y for a small loop around D_2= { y=0 }. Writing the local solutions explicitly, it is straightforward to have𝚃_x=([100000;110000;001000;366100;126010; -4 -1 -30 -11 ]),𝚃_y=([100000;010000;101000;162100;366010; -4 -3 -1 -101 ]).We introduce the dual basis ℬ= {α_i,β_i} of H^3(X_b_o^*,ℤ) and consider the dual actions T_x:=( ^t𝚃_x)^-1 and T_y:=( ^t𝚃_y)^-1 on H^3(X_b_o^*,ℤ) which are clearly unipotent.We define the monodromy nilpotent cone at o byΣ_o={∑λ_iN_i | λ_i≥0}⊂End(H^3(X_b_o^*,ℤ))with N_1=- ^t(log𝚃_x) and N_2=- ^t(log𝚃_y). Using the explicit forms of these matrices, we verify the following properties: =0pt(1) The nilpotent element N_λ:=∑_iλ_iN_i, λ_i>0, defines the weight monodromy filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_0^*,ℤ) with the same form W_2i as given in (<ref>). (2) We haveN_iN_jN_k=C_ijk𝙽_0with totally symmetric C_ijk defined by C_111=C_222=2, C_122=C_112=6, C_ijk=0 otherwise, and 𝙽_0=([ 0 1; O_5 0 ]) where O_5 is the zero matrix of size 5×5.From the above proposition, we see that the origin satisfies the conditions for LCSL. Also, looking other boundary points in the moduli space ℳ_X^*^ cpx, we see that no other LCSL exists in ℳ_X^*^ cpx. §.§ Gluing the monodromy nilpotent cone Σ_o Although there is only one LCSL in the mirror family 𝔛^*→ℳ_X^*^ cpx, we can find the monodromy transformations which correspond to the birational automorphisms τ_1 and τ_2 of X. We observe that the monodromy nilpotent cone Σ_o extends to a larger cone (or cone structure) using these monodromy transformations, and we will identify the resulting cone structure with the movable cone Mov(X) of X.§.§.§ Path p_o← E_i← o, i=1,2 As shown in Fig. <ref>, the discriminant locus Dis has non-normal crossing intersection at three points. To make the intersections normal, we blow-up successively four times at the two points near the origin o. We denote by E_1 and E_2, respectively, the exceptional divisors introduced by the blow-ups (see Fig. <ref> right). As we see in the form of the discriminant Dis, the family over ℳ_X^*^ cpx is symmetric under x↔ y. Because of this symmetry reason, it suffices to describe the divisor D_2= { y=0} which intersects with E_1 at [1,x,y]=[1,1/4,0]∈ℙ^2. Explicitly, we introduce the blow-up coordinate at the origin q_12:=E_1∩ D_y bys_1=4x-1, s_2=1/2^6y/(1-4x)^4.Let R_12={1/4+s_1/4e^iθ | 0≤θ≤2π} be a small loop around E_1 on D_2. We denote by p_q_12← b_o= { (1-t)b_o+tq_12 | 0≤ t≤1-ε} the straight line connecting the base point b_o near o and a point q_12 on the small loop R_12. Then we definep_b_o← E_1← b_0:=(p_q_12← b_o)^-1∘ R_12∘ p_q_12← b_oto be the composite path which encircles the divisor E_1 from the base point b_o. In a similar way, we define a closed path p_b_0← E_2← b_o which encircle the divisor E_2 from b_0 (see Fig. <ref>).§.§.§ Monodromy around E_iLet (x',y')=(1/x,y/x) be the affine coordinate with the origin [0,1,0]∈ℙ^2 and b_o' be a base point near the origin. We denote by 𝚃_x'' and 𝚃_y'' the local monodromy around x'=0 and y'=0, respectively. Conjugating 𝚃_x'', 𝚃_y'' by the connection matrix for the path p_b_o'← b_0=p_b_0'← q_12∘ p_q_12← b_o, we define the corresponding monodromy matrices 𝚃_x' and 𝚃_y' for loops with the base point b_o. We define T_x':=(^t𝚃_x')^-1 and T_y':=( ^t𝚃_y')^-1 to be the linear actions on the dual space H^3(X_b_o^*,ℤ).We haveT_x'=([-1-1 3 6 -10 2; 0 1-6 -12 8 2; 0 0-1 012-6; 0 0 0-1-6 3; 0 0 0 0 1-1; 0 0 0 0 0-1 ]), T_y'=([10 -1134;010 -6 -6 -3;001 -2 -6 -1;000101;000010;000001 ]).In particular we have T_y'=T_y.These are based on explicit calculations. Here we only sketch the calculations. We first make local solutions using the coordinate (s_1,s_2) centered at q_12. Then their domain of convergence have overlap both with the local solutions around (x,y)=(0,0) and (x',y')=(0,0). Then it is straightforward to obtain the connection matrices. The local monodromy matrices 𝚃_x'' and 𝚃_y'' are easily read off from the local solutions. Then by conjugating these local monodromy matrices by the connection matrix, we have the expressions for 𝚃_x', 𝚃_y' as the linear actions on H_3(X_b_0^*,ℤ). Translating these to H^3(X_b_o^*,ℤ), we obtain T_x' and T_y'. Similarly we define the monodromy matrix 𝚃_E_1 along the loop p_b_o← E_1← b_o and set T_E_1:=( ^t𝚃_E_1)^-1. Corresponding to Proposition <ref> we have [`Picard–Lefschetz formula' for the flopping curves]=0pt(1) The monodromy matrix is given byT_E_1=([ -100000;01 -60 480;00 -1000;000 -1 -60;000010;00000 -1 ]).In particular, this is quasi-unipotent. (2) For T_E_1^2 we haveT_E_1^2=([1; 1 96;1; 1;1; 1 ]),i.e.,α_1→α_1+96β_1, β_1→β_1, α_i=α_i, β_i=β_i,i≠1.(3) By symmetry, we have similar formula for T_E_2 and T_E_2^2. In particular, T_E_2^2 is given by α_2→α_2+96β_2, β_2→β_2, with α_i=α_i, β_i=β_i for i≠2. These results follow from making local solutions and the analytic continuations of them. Again, calculations are straightforward since local solutions around (x,y)=(0,0) and (s_1,s_2)=(0,0) have overlap in their domains of convergence.(1) As before, the monodromy action (2) in the above proposition is expressed in terms of the symplectic basis { A_i,B_j} of H_3(X_b_o^*,ℤ) asA_1→ A_1, B_1→ B_1-96 A_1. (2) We have seen in Proposition <ref> that each τ_i X X is an Atiyah's flop with respect to 80 lines and also 4 conics. We observe that 96=80+4×2^2 holds for the number in T_E_i^2. We can verify the corresponding relations also for other examples. Based on these, we conjecture the following general form:A_1→ A_1, B_1→ B_1-(n_0(1)+n_0(2)×2^2) A_1for the Atiyah's flops of n_0(1) lines and n_0(2) conics associated to the contractions to the double cover of ℙ^3. §.§.§ Monodromy relationsTake affine coordinates (x,y), (x',y') and (x”,y”) of ℙ^2 as shown in Fig. <ref>. Let T_x', T_y' be as defined in Proposition <ref>.The following monodromy relations holdsT_x'=T_E_1^-1T_x^-1T_y^3, T_y'=T_y, T_E_1T_y=T_yT_E_1.We have the second and the third relations since all the divisors are normal crossing after the blow-ups. We can verify the first relation directly by using T_x=( ^t𝚃_x)^-1, T_y=( ^t𝚃_y)^-1 given in Section <ref> and T_x', T_E_1 in Section <ref>.Define the following conjugations of T_x,T_y by T_E_1:T̃_x:=T_E_1^-1T_xT_E_1,T̃_y:=T_E_1^-1T_yT_E_1.Using these, we define the monodromy nilpotent cone byΣ̃_o:={∑λ_iÑ_i | λ_i>0}⊂End(H^3(X_b_o^*,ℝ)),where Ñ_1:=logT̃_x and Ñ_2:=logT̃_y. The (closures of the) monodromy nilpotent cones Σ_o and Σ̃_o glue along the ray ℝ_≥0N_2, but they are not on the same two dimensional plane.Using the monodromy relations in Proposition <ref>, we have T̃_y'=T_y. Hence the claim is immediate since we have Ñ_2=N_2 by definition. To see the second claim, we use again the monodromy relations to haveT̃_x=T_E_1^-1T_xT_E_1=T_E_1^-1T_y^3T_x'^-1=T_E_1^-1T_x'^-1T_y^3,which is reminiscent of the relation (<ref>). In fact, after some matrix calculations, we obtainÑ_1=6N_2-N_1+Δ_1,Δ_1=([ 0 0 0 048 -44/3; 0 0 0 0-11248; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0; 0 0 0 0 0 0 ]),where Δ_1 satisfies Δ_1|_W_2=0. Since the nilpotent cone Σ_o lies on the plane spanned by N_1 and N_2, and aN_1+bN_2|_W_2≠0 holds for any a, b, the basis element Ñ_1 does not lie on the same plane as Σ_o. §.§.§ Gluing nilpotent conesAs the example in the previous section, the structure of the moduli space ℳ_X^*^ cpx is symmetric under the exchange of x and y. Hence, corresponding to (<ref>), we haveT_x”=T_x, T_y”=T_E_2^-1T_y^-1T_x^3, T_E_2T_x=T_xT_E_2.When we define T̃_x':=T_E_2^-1T_xT_E_2, T̃_y':=T_E_2^-1T_yT_E_2, we have the following relationsÑ_1'=N_1,Ñ_2'=6N_1-N_2+Δ_1'for Ñ_1':=logT̃_x, Ñ_2':=logT̃_y with Δ'_1|_W_2=0. This entails the corresponding gluing property described in Proposition <ref>. We summarize these two actions into the following general form.We denote by τ_E_i the conjugations by T_E_i on End(H^3(X_b_o^*,ℚ)), which act on the nilpotent matrices N in general asτ_E_1(N)=T_E_1^-1NT_E_1,τ_E_2(N)=T_E_2^-1NT_E_2.We set G:=⟨τ_E_1,τ_E_2⟩, i.e., the group generated by τ_E_1 and τ_E_2. =0pt(1) The actions of τ_E_i^n∈ G on N_1=log T_x, N_2=log T_y are summarized as(τ_E_1^n(N_1),τ_E_1^n(N_2))=(N_1,N_2)(-1 06 1 )^n+(Δ_n,0), (τ_E_2^n(N_1),τ_E_2^n(N_2))=(N_1,N_2)(1 60 -1 )^n+(0,Δ_n'),where Δ_n, Δ_n' are elements in End(H^3(X_b_o^*,ℚ)) satisfying Δ_n|_W_2=Δ_n'|_W_2=0. In particular, we haveτ_E_1^n(N_2)=N_2,τ_E_2^n(N_1)=N_1.(2) The action of σ∈ G on Δ_n,Δ_n' preserves the vanishing properties of Δ_n, Δ_n' on W_2, i.e., σ(Δ_n)|_W_2=σ(Δ_n')|_W_2=0. (3) Δ_n, Δ_n' have the following forms:Δ_2m=( O_24 [ -96m0;0 -96m ]O_44O_42),Δ_2m-1=( O_24 [ 96(m-1/2) -44/3;-122 96(m-1/2) ]O_44O_42)and Δ_n'=p_23p_45Δ_np_23p_45, where O_ab is the a× b zero matrix and p_ij represents the permutation matrix for the transposition (i,j).These properties are verified by explicit calculations using the matrix representations T_x, T_y and T_E_i given previous sections. The vanishing properties follow inductively from Δ_1|_W_2=Δ_1'|_W_2=0 and the fact that both T_E_1 and T_E_2 preserve the monodromy weight filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ). As before, let ℐ_2:={ X∈End(H^3(X_b_0^*,ℝ)) |X|_W_2=0} be an left ideal of End(H^3(X_b_0^*,ℝ)), and πEnd(H^3(X_b_0^*,ℝ))→End(H^3(X_b_0^*,ℝ))/ℐ_2 be the natural projection. Since T_E_i preserve the monodromy weight filtration, and by the definition of τ_E_i, it is easy to see that σ(ℐ_2)⊂ℐ_2 for all σ∈ G. Hence we have the naturally induced G action on the quotient End(H^3(X_b_0^*,ℝ))/ℐ_2 by σ(X+ℐ_2):=σ(X)+ℐ_2. Note that, if we denote by σ̅ the action of σ∈ G on the quotient space, we have στ=τ̅σ̅ (i.e., anti-homomorphism by our convention for the adjoint action) for all σ,τ∈ G.Denote by N̅_i:=π(N_i) the basis of the cone π(Σ_o) in the quotient space. Then the following hold: =0pt(1) Define τ_12:=τ_1τ_2 with τ_i=τ_E_i. We have(τ̅_12^n(N̅_1),τ̅_12^n(N̅_2))=(N̅_1,N̅_2)(35 6-6 -1 )^n.(2) {σ∈ G | σ̅(N̅_i)=N̅_i, i=1,2 } =⟨τ_1^2,τ_2^2⟩. (3) Taking the closure in End(H^3(X_b_0^*,ℝ))/ℐ_2, we have⋃_σ∈ Gπ(σ(Σ_o))=ℝ_>0(-N̅_1+(3+2√(2))N̅_2)+ℝ_>0(N̅_1+(3-2√(2))N̅_̅2̅).The equality (1) follow from Proposition <ref>(1) and τ̅_12=τ̅_2τ̅_1. By definition, G is generated by τ_1, τ_2. Then the claim (2) follows from Proposition <ref>(1) and the above equality (1). To show the claim (3), we write by (N_1,N_2)_>0 the cone generated by N_1 and N_2. Then we first show that the following cones successively glue together to a large cone:(τ_12^n(N_1),τ_12^n(N_2))_>0,(τ_12^n(τ_1N_1),τ_12^n(τ_1N_2))_>0,n∈ℤ.Using the property τ_1(N_2)=N_2, τ_2(N_1)=N_1, we haveτ_12^n(τ_1N_1)=τ_12^n+1(N_1),τ_12^n(τ_1N_2)=τ_12^n(N_2),by which we can arrange a sequence of cones schematically as follows:[(τ_12^n+1(τ_1N_1), τ_12^n+1(τ_1N_2))_>0 (τ_12^n(τ_1N_1), τ_12^n(τ_1N_2))_>0⋯;; ⋯(τ_12^n+1(N_2),τ_12^n+1(N_1))_>0(τ_12^n(N_2),τ_12^n(N_1))_>0 ]Let us note that τ_2τ_1=τ_2^2τ_12^-1τ_1^2 and τ_2=τ_2^2τ_12^-1τ_1 hold. Then, using these relations, we can deduce the decompositionG=⟨τ_12,τ_1^2,τ_2^2⟩∪⟨τ_12,τ_1^2,τ_2^2⟩τ_1.Since τ_1^2, τ_2^2 have trivial actions on N̅_i, i=1,2, the above sequence of the cones explain the union ⋃_σ∈ Gπ(σ(Σ_o)). After some linear algebra of the matrix power ([ 356; -6 -1 ])^n, we can determine the infinite union in the claimed form.§.§ Flopping curves and T_E_1 The monodromy T_E_1 has appeared in the moduli space from the tangential intersection of the discriminants. This is quite parallel to Section <ref>. However, T_E_1 is not unipotent but only quasi-unipotent in the present case. This prevents a parallel definition to the second equation in (<ref>), but this time we setN_1^𝚏:=6N_2-N_1with Ñ_1=N_1^𝚏+Δ_1 (see (<ref>)) and also Ñ_2=N_2^𝚏=N_2. Then we haveLet Ñ_iÑ_jÑ_k=C̃_ijk𝙽_0 and N_i^𝚏N_j^𝚏N_k^𝚏=C_ijk^𝚏𝙽_0 with 𝙽_0 as given in Proposition <ref>. Non-vanishing (totally symmetric) C̃_ijk and C_ijk^𝚏 are given by(C̃_111,C̃_112,C̃_122,C̃_222)=(2,6,6,2),(C_111^𝚏,C_112^𝚏,C_122^𝚏,C_222^𝚏)=(-110,6,6,2).As before, the first equality of (<ref>) is explained by mirror symmetry, i.e., the isomorphism of the B-structure at o with the A-structure of X. To see this isomorphism more explicitly, we recall the mirror mapt_i=∫_A_iΩ_x/∫_A_0Ω_xdefined by the B-structure at the LCSL o. The monodromy matrix 𝚃_E_1=( ^tT_E_1)^-1 represents the isomorphism H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) which follows from the analytic continuation of the period integral Π(x,y) along the path p_b_0← E_1← b_o. After the continuation, the coordinate (t_1,t_2) transformed to (t_1',t_2') witht_1'=-t_1, t_2'=6t_1+t_2. Corresponding to Proposition <ref>, we now have Let C_ijk be as defined in Proposition <ref>. Also set q_1':=e^t_1' and q_1=e^t_1. Then we have the following relationsC_ijk^𝚏=∑_l,m,nC_lmndt_l/dt_i'dt_m/dt_j'dt_n/dt_k'andC̃_111+80q_1'/1-q_1'+42^3q_1'^2/1-q_1'^2=C_111^𝚏+(80q_1/1-q_1+42^3q_1^2/1-q_1^2)(dt_1/dt_1')^3.In the above equality, we see the invariance of the quantum cohomology of X under birational transformations. We note that the equality (<ref>) has a slightly more general form than the familiar form (<ref>) due to the existence of 4 conics in the flopping curves.§ SUMMARY AND DISCUSSIONS We have studied gluings of monodromy nilpotent cones through monodromy relations coming from boundary divisors. Under the mirror symmetry, we have identified them with the corresponding gluings along codimension-one walls of the Kähler cones in birational geometry. In this paper, we confined ourself to two specific examples by doing explicit monodromy calculations. However, it is naturally expected that the observed gluings of monodromy nilpotent cones and their interpretation in mirror symmetry hold in general.We present below some discussions and related subjects in order. In particular, we briefly report the gluing in the case of K3 surfaces whose moduli spaces have parallel structures to the Calabi–Yau threefolds X and X^* studied in Sections <ref> and <ref>.6.1. The gluing of monodromy nilpotent cones has been done naturally through the monodromy relations (<ref>), (<ref>) and also (<ref>), (<ref>). These relations came from boundary divisors which have tangential intersections with some component of discriminant and the blowing-ups at the intersection points. As remarked in Remarks <ref>, <ref>, these tangential singularities are related to the contractions in the birational geometry of the mirror Calabi–Yau manifolds. We expect some generality in the degenerations of the mirror families 𝔛^* when we approach to the exceptional divisors E_i of the blow-ups. We have to leave this for future investigations although we note that a categorical study of the mirror symmetry for conifold transitions has been put forward in a recent work <cit.>.6.2. In the homological mirror symmetry due to Kontsevich <cit.>, monodromy transformations in B-structures are interpreted as the corresponding transformations in the derived category of coherent sheaves D^b(X). From this viewpoint, the gluing of nilpotent cones in End(H^3(X^*,)) suggests the corresponding gluing of Kähler cones in End(K(X)) as a homological extension of the movable cones. The resulting wall structures of the gluing in End(K(X)) should be regarded as the wall structures in the stability space <cit.> of the objects in D^b(X).6.3. As addressed in Remark <ref>, one can expect non-trivial birational geometry also for other examples of complete intersections described by Gorenstein cones <cit.>. Among such examples, there are complete intersections whose projective geometry fits well to the so-called linear duality (see Appendix <ref>). We have for example the following complete intersection:X=(^4| 2 1 1 1 ^3| 1 1 1 1 )^2,56,which shares many properties with (<ref>) in Section <ref>, e.g., three birational models come together when we construct the complete intersection of the form X. Although we do not have birational automorphisms of infinite order in this example, these three birational models are explained nicely by “double linear duality”, a certain composite of two different linear dualities. We will report this elsewhere.6.4 (Cayley model of Reye congruences). Historically the Calabi–Yau complete intersection studied in Section <ref> is a generalization of the following K3 surface:X=(^3 |1 1 1 1 ^3 |1 1 1 1 ),which is called a Cayley model of Reye congruences. When we take the defining equations general, X is a smooth K3 surface of the Picard lattice isomorphic to M:=(^2,([ 4 6; 6 4 ])). This K3 surface has been studied in <cit.> as an example which has an automorphism ρ of infinite order and also positive entropy. Actually, we have the same diagram as (<ref>) with the parallel definitions of X_i (X_1:=X) and Z_i as well as ρ in Proposition <ref>. The difference is in that all X_i and Z_i are smooth K3 surfaces and hence isomorphic to each other under the morphisms, e.g., π_ij and φ_ij. For K3 surfaces, we have the so-called counting formula <cit.> for the number of Fourier–Mukai partners. Based on it, it is easy to see that the set FM(X) of Fourier–Mukai partners consists of only X itself.The construction of the mirror family of X is similar to Section <ref>, and there appear three LCSL o_i, i=1,2,3, on the compactified moduli space ℳ_X^*^ cpx=^2. As before, we determine the connecting matrices _ij by blowing-up at three points with (fourth) tangential intersections (cf. Fig. <ref>). Making similar canonical bases of period integrals as in (<ref>) at each point, which represents bases of the transcendental lattice T_X^*≃ U⊕ M of the mirror K3 surface X^*, we obtain_21=([ -1000;01 -30;00 -10;000 -1 ]),_32=([ -1000;0 -310;0 -100;000 -1 ]),_13=([ -1000;0 -100;0 -310;000 -1 ]), :=_13_32_21=([-1 0 0 0; 0 3-8 0; 0 8 -21 0; 0 0 0-1 ])as elements in O(U⊕ M,). Here we define U= e⊕ f to be the hyperbolic lattice (^2,([ 0 1; 1 0 ])) and order the bases of U⊕ M as e⊕ M⊕ f when writing the above matrix forms.The classical mirror symmetry summarized in Section <ref> applies to the so-called (families of) lattice polarized K3 surfaces replacing the Kähler cone with the ample cones <cit.>. In our case here, we consider a primitive lattice embedding M⊕ U⊕M̌⊂ L_K3 with a fixed decomposition M^⊥=U⊕M̌. Then X is a member of the M-polarized K3 surfaces, while the mirror X^* is a member of M̌-polarized K3 surfaces (whose transcendental lattice is M̌^⊥=U⊕ M). The classical mirror symmetry in this case may be summarized in the following isomorphism:V_M^++√(-1)M⊗≃Ω^+(U⊕ M)for the period domain Ω^+(U⊕ M)={ [ω]∈((U⊕ M)⊗) | ω.ω=0,ω.ω̅>0} ^+ where we take one of the connected components, and the corresponding component of the tube domain V_M^+={ v∈ M⊗ | (v,v)_M>0} ^+.Since there are no elements with (v,v)_M=-2 in M, the ample cones of general members of M-polarized K3 surfaces coincide with the positive cone, which is isomorphic to V_M^+. Similarly to what we described in Section <ref>, by gluing the cone _≥0H_1+_≥0H_2⊂ H^2(X,) by the morphisms φ_ij, we arrive at the positive cone V_M^+ which is an irrational cone (see <cit.> and <cit.>). This gluing exactly matches to the gluing the monodromy nilpotent cones at each boundary point o_i by the connection matrix φ̌_ij. The monodromy relations play the key roles for the gluing, and they follow from the parallel calculations to those in Section <ref>. For example, we haveT_x'=T_x^-1T_y^3, T_y'=T_y, T_x”=T_x, T_y”=T_y^-1T_x^3corresponding to (<ref>) and (<ref>), respectively, withT_x=([1 -10 -2;0104;0016;0001 ]), T_y=([10 -1 -2;0106;0014;0001 ]), T_x'=_21^-1T_x_21, T_y'=_21^-1T_y_21and T_x”=_31^-1T_x_31, T_y”=_31^-1T_y_31. As in Section <ref>, exceptional divisors E_1, E_1' and E_1” have to be introduced to determine the connection matrices _ij, but it turns out that their monodromies are trivial, i.e., T_E_1=T_E'=T_E_1”=id. Clearly, this is consistent to our interpretation of these monodromies in terms of the flopping curves (Proposition <ref>) for the case of Calabi–Yau threefolds.As this example shows, irrational ample cones indicate infinite gluings of the nilpotent cones in the mirror side. It is natural to expect that the corresponding property holds for the mirror symmetry of Calabi–Yau threefolds in general with ample cones replaced by movable cones and the morphisms by birational maps as known in the so-called movable cone conjecture <cit.>. We have shown in this paper that, in three dimensions, the gluings of monodromy nilpotent cones encode the non-trivial monodromies T_E_i which correspond to the flopping curves.§ PROOF OF LEMMAS <REF> AND <REF>§.§ Proof of Lemma <ref> Let us consider the projective spaces (V_i) with V_i≃^5, i=1,2. Here we will only present a proof of (1), but it should be clear how to modify the following setting to show (2).We start with our discussion with the following exact sequence, which we obtain by tensoring the Euler sequence of (V_2) with V_1:0→ V_1⊗_(V_2)(-1)→ V_1⊗ V_2⊗_(V_2)→ V_1⊗ T_(V_2)(-1)→0.In the following arguments, we denote this sequence by0→→ V_1⊗ V_2⊗_(V_2)→(^⊥)^*→0with defining :=V_1⊗_(V_2)(-1) and ^⊥:=V_1^*⊗Ω_(V_2)(1). We also have the following diagram of a linear duality (cf. <cit.>):(-25,0)*++X_1⊂="Xi", (-15,0)*++()="PE", (15,0)*++(^⊥)="PEp", (25,0)*++⊃ X_2, (-30,-13)*++(V_1⊗ V_2)="PVV", ( 0,-13)*++(V_2)="PV", ( 30,-13)*++(V_1^*⊗ V_2^*)="PVsVs", ( 45,-13)*+⊃Z_3. "PE";"PVV""PE";"PV""PEp";"PV""PEp";"PVsVs"Note that () is isomorphic to (V_1)×(V_2), and _()(1)≃_(V_1)×(V_2)(1,1) since it is the pull-back of _(V_1⊗ V_2)(1) by construction. Therefore X_1 is a codimension 5 complete intersection in () with respect to _()(1), and we have _()(1)|_X_1=H_1+H_2.We see that(^⊥)={(w,M) |Mw=0}⊂(V_2)×(V_1^*⊗ V_2^*),where we consider V_1^*⊗ V_2^*≃(V_2,V_1^*) and M is a 5×5 matrix. Therefore the image of the map (^⊥)→(V_1^*⊗ V_2^*) consists of 5×5 matrices of rank ≤4, thusis so-called the determinantal quintic. Note that we can write the determinantal quintic Z_3⊂ℙ_λ^4 in Proposition <ref> by Z_3=∩ P_4 for a 4-dimensional linear subspace P_4⊂(V_1^*⊗ V_2^*) with identifying P_4 with ℙ_λ^4. Moreover, the pull-back of Z_3 to (^⊥) is X_2.By a general fact on linear duality (<ref>) in Appendix <ref>, we have_()(1)|_X_1+_(^⊥)(1)|_X_1=^*=5H_2,where we denote by _(^⊥)(1)|_X_1 the strict transform of _(^⊥)(1)|_X_2 and abbreviate the notation for the pull-back for ^*. In this appendix, unless stated otherwise, we will write proper transforms of a divisor by the same symbol omitting the pull-backs by birational maps. Using this convention, we have _()(1)|_X_1=H_1+H_2 and also _(^⊥)(1)|_X_1=L_Z_3. Then we have(H_1+H_2)+L_Z_3=5H_2,which gives L_Z_3=4H_2-H_1. Therefore, restoring the pull-backs by birational maps, we haveφ_21^*L_Z_3=4H_2-H_1,φ_21^*L_Z_2=H_2in N^1(X), which determine φ_21^*(_X_2) as claimed. §.§ Proof of Lemma <ref> Basic idea is very similar to the linear duality in the previous section. We consider the following diagram:(-25,0)*++(V_1^*⊗Ω_(V_2)(1))="PVO", ( 0,-13)*++(V_1^*⊗ V_2^*).="PVsVs", ( 25,0)*++(Ω_(V_1)(1)⊗ V_2^*)="PVOt""PVO";"PVsVs""PVOt";"PVsVs" (V_1^*⊗Ω_(V_2)(1))→(V_1^*⊗ V_2^*) and (Ω_(V_1)(1)⊗ V_2^*)→(V_1^*⊗ V_2^*) are flopping contractions onto the common image . Moreover, it is of Atiyah type outside the locus inof corank ≥2. This is standard since (V_1^*⊗Ω_(V_2)(1))→(V_1^*⊗ V_2^*) and (Ω_(V_1)(1)⊗ V_2^*)→(V_1^*⊗ V_2^*) are the Springer type resolutions of the image 𝒵 (see (<ref>)). As we have seen in the proof of Lemma <ref>, X_2 is contained in (V_2^*⊗Ω_(V_1)(1)). Similarly, X_3 is contained in (Ω_(V_2)(1)⊗ V_1^*). Indeed, for the 4-dimensional linear subspace P_4⊂(V_1^*⊗ V_2^*) such that Z_3=∩ P_4, X_2 and X_3 are the pull-backs of P_4 to (V_1^*⊗Ω_(V_2)(1)) and (Ω_(V_1)(1)⊗ V_2^*), respectively.Now we take the fiber product:=(V_1^*⊗Ω_(V_2)(1))×_(V_1^*⊗ V_2^*)(Ω_(V_1)(1)⊗ V_2^*). It holds that =_(V_1)×(V_2)(Ω_(V_1)(1)⊠Ω_(V_2)(1)).Note that={(w,M,z) |Mw=0,^tzM=0}⊂(V_2)×(V_1^*⊗ V_2^*)×(V_1).Thus the fiber of →(V_1)×(V_2) over (w,z) is nothing but ((V_1/ w)^*⊗(V_2/ z)^*) and the assertion follows. Note that the tautological divisor _(1) of (Ω_(V_1)(1)⊠Ω_(V_2)(1)) defines a map to (V_1^*⊗ V_2^*) and it is the pull-back of _(V_1^*⊗ V_2^*)(1). We will denote it by L_(V_1^*⊗ V_2^*). By the canonical bundle formula of (Ω_(V_1)(1)⊠Ω_(V_2)(1)), we haveK_=-16L_(V_1^*⊗ V_2^*)+K_(V_1)×(V_2)+{Ω_(V_1)(1)⊠Ω_(V_2)(1)}^*,where we omit the notation of the pull-backs for K_(V_1)×(V_2) and {Ω_(V_1)(1)⊠Ω_(V_2)(1)}^*. Since K_(V_1)×(V_2)=-5L_(V_1)-5L_(V_2), where L_(V_1) and L_(V_2) are the pull-backs of _(V_i)(1)'s of (V_i) on the left and right factors of (V_1)×(V_2), respectively, and {Ω_(V_1)(1)⊠Ω_(V_2)(1)}^* =4L_(V_1)+4L_(V_2), we haveK_=-16L_(V_1^*⊗ V_2^*)-L_(V_1)-L_(V_2).By the canonical bundle formula of (V_1^*⊗Ω_(V_2)(1)), we have-K_(V_1^*⊗Ω_(V_2)(1))=20L_(V_1^*⊗ V_2^*).Pushing forwards (<ref>) to (V_1^*⊗Ω_(V_2)(1)), we obtain-K_(V_1^*⊗Ω_(V_2)(1))=16L_(V_1^*⊗ V_2^*)+L_(V_1)+L_(V_2).Therefore we haveL_(V_1)+L_(V_2)=4L_(V_1^*⊗ V_2^*). Now, restricting the above construction over the linear subspace P_4⊂(V_1^*⊗ V_2^*), we have(0,0)*++_|P_4="topP", (-15,-10)*++X_2="Xl", ( 15,-10)*++X_3,="Xr", ( 0,-20)*++P_4="botP""topP";"Xl""topP";"Xr""Xl";"botP""Xr";"botP"where we denote by _|P_4 the restriction ofover P_4. Restricting (<ref>) to X_2, we haveφ_32^*(M_Z_1)+L_Z_2=4L_Z_3.This is the claimed relation. (V_1^*⊗Ω_(V_2)(1))(Ω_(V_1)(1)⊗ V_2^*) is the flop. Similarly, X_2 X_3 is the flop. Note that L_(V_1) and L_(V_2) are relatively ample for (Ω_(V_1)(1)⊗ V_2^*)→(V_1) and (V_1^*⊗Ω_(V_2)(1))→(V_2), respectively. Since L_(V_1^*⊗ V_2^*) is the pull-backs of a divisor on (V_1^*⊗ V_2^*), we see that -L_(V_1) is relatively ample for (V_1^*⊗Ω_(V_2)(1))→(V_2) by (<ref>). Therefore (V_1^*⊗Ω_(V_2)(1))(Ω_(V_1)(1)⊗ V_2^*) is the flop. We can show the assertion for X_2 X_3 in the same way using (<ref>). § LINEAR DUALITY Having the case W=V_1⊗ V_2 and B=(V_2) in mind, we consider the exact sequence of sheaves (vector bundles) in the following general form with W=N:0→→ W⊗_B→(^⊥)^*→0,0→^⊥→ W^*⊗_B→^*→0.Under this general setting, we have the following natural morphisms:(-15,0)*++()="PE", ( 15,0)*++(^⊥)="PEp", (-30,-13)*++(W)="PW", ( 0,-13)*++B="B", ( 30,-13)*++(W^*).="PWs", _f "PE";"PW" "PE";"B" "PEp";"B" ^g "PEp";"PWs"Let _b and _b^⊥ be the fibers over b∈ B ofand ^⊥, respectively. Then it holds(_b∩ L_r^⊥)=(_b^⊥∩ L_r)for any r-dimensional linear subspace L_r⊂ W^* and the orthogonal linear subspace L_r^⊥ in W.We calculate the dimensions as follows: (_b∩ L_r^⊥)=_b+ L_r^⊥-(_b+L_r^⊥)=r+(N-r)-(_b+L_r^⊥)=(_b^⊥∩ L_r). The complete intersections X_1, X_2 in Appendix <ref> may be described, respectively, in general terms asX_L_r^⊥=f^-1(L_r^⊥)∩(), Y_L_r=g^-1(L_r)∩(^⊥)for a fixed subspace L_r⊂ W^*, which we call orthogonal linear sections. Consider the Grassmannian =Gr(r,N) of r-spaces in W^* and define the following family of orthogonal linear sections:𝒳_r:={ ([L_r],x)∈×() |f(x)∈(L_r^⊥)} , 𝒴_r:={ ([L_r],y)∈×(^⊥) |g(y)∈(L_r)} .Also we defineΣ_0:={ ([L_r],b)∈× B | _b∩ L_r^⊥≠0} ={ ([L_r],b)∈× B | _b^⊥∩ L_r≠0} ,where the second equality is valid due to Lemma <ref>. Then 𝒳_r, 𝒴_r are orthogonal linear sections fibered over Σ_0 and, with natural morphisms, they can be arranged in the following diagram:( 0,13)*++𝒳_r×_Σ_0𝒴_r="XrYr", (-15,0)*++𝒳_r="Xr", ( 15,0)*++𝒴_r="Yr", ( 0,-13)*++Σ_0="Sigma", (-30,0)*++()="PE", (30,0)*++(^⊥).="PEp", ( 12,-13)*⊂ × B, (-6,9.5);"Xr" (6,9.5);"Yr" "Xr";"Sigma" "Yr";"Sigma" "Xr";"PE" "Yr";"PEp"Let us introduce the following divisors related to the diagram:H_:=_()(1), H_^⊥:=_(^⊥)(1), H_:=_(1).Abbreviating the pull-back symbols by the morphisms in the diagram (<ref>), we haveK_𝒳_r×_Σ_0𝒴_r=-(N-2)H_-H_-H_^⊥+K_B+2 ^*andK_𝒳_r=-(N-1)H_+K_B+^*, K_𝒴_r=-(N-1)H_+K_B+(^⊥)^*. We leave the proofs for readers. It is easy to recognize that the proofs of the above proposition rely on the projective geometry behind the diagram (<ref>). We will report the proofs elsewhere with some additional properties which we can extract from the diagram (<ref>); for example, we can show that the morphisms 𝒳_r→Σ_0, 𝒴_r→Σ_0 are flopping contractions and the naturally induced birational map 𝒳_r𝒴_r in the diagram is the flop for these contractions. Pushing forward K_𝒳_r×_Σ_0𝒴_r to 𝒳_r, and equating toK_𝒳_r, we have a relationH_+H_^⊥=^*+H_on 𝒳_r. Similarly, we have a corresponding relation on 𝒴_r,H_+H_^⊥=(^⊥)^*+H_.Now restricting the relation (<ref>) on 𝒳_r to 𝒳_r|_[L_r]×()=X_L_r^⊥, we obtainH_|_X_L_r^⊥+H_^⊥|_X_L_r^⊥= ^*,which is the relation we used in (<ref>). §.§ AcknowledgementsThe results of this work have been reported by the first named author (S.H.) in several workshops; “Modular forms in string theory” at BIRS (2016), “Categorical and Analytic invariants IV” at Kavli IPMU (2016), “Workshop on mirror symmetry and related topics” at Kyoto University (2016) and “The 99th Encounter between Mathematicians and Theoretical Physicists” at IRMA, Strasbourg (2017). He would like to thank the organizers for the invitations for the workshops where he had valuable discussions with the participants. Writing this paper started when S.H. was staying at Brandeis University and Harvard University in March, 2017. He would like to thank B. Lian and S.-T. Yau for their kind hospitality and also valuable discussions during his stay. 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"authors": [
"Shinobu Hosono",
"Hiromichi Takagi"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170727072234",
"title": "Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds"
} |
Counting Pseudo Landau Levels in Spatially Modulated Dirac Systems Toshikaze Kariyado December 30, 2023 ================================================================== By the Gibbard–Satterthwaite theorem, every reasonable voting rule for three or more alternatives is susceptible to manipulation: there exist elections where one or more voters can change the election outcome in their favour by unilaterally modifying their vote. When a given election admits several such voters, strategic voting becomes a game among potential manipulators: a manipulative vote that leads to a better outcome when other voters are truthful may lead to disastrous results when other voters choose to manipulate as well. We consider this situation from the perspective of a boundedly rational voter,and use the cognitive hierarchy framework <cit.> to identify good strategies. We then investigate the associated algorithmic questions under the k-approval voting rule, k≥ 1. We obtain positive algorithmic results for k=1, 2 and NP- and coNP-hardness results for k≥ 4. § INTRODUCTIONImagine that you and your friends are choosing a restaurant to go to for dinner. Everybody is asked to name their two most preferred cuisines,and the cuisine named most frequently will be selected (this voting rule is known as 2-approval).Your favourite cuisine is Japanese and your second most preferred cuisine is Indian. Indian is quite popular among your friends and you know that if you name it among your favourite two cuisines, it will be selected. On the other hand, you also know that only a few of your friends like Chinese food. Will you vote for Japanese and Chinese to give Japanese cuisine a chance?This example illustrates that group decision-making is a complex process that represents an aggregation of individual preferences. Individual decision-makers would like to influence the final decision in a way that is beneficial to them, and hence they may be strategic in communicating their individual choices. Moreover, it is essentially impossible to eliminate strategic behavior by changing the voting rule: the groundbreaking result of <cit.> and <cit.> states that, under anyonto and non-dictatorial social choice rule, there exist situations where a voter can achieve a better outcome bycasting a strategic vote rather than the sincere one, provided that everyone else votes sincerely; in what follows, we will call such voters Gibbard–Satterthwaite (GS) manipulators. The Gibbard–Satterthwaite theorem alerts us that strategic behavior of voters cannot be ignored,but it does not tell us under which circumstances it actually happens. Of course, if there is just asingle GS-manipulator at a given profile, and he[We use `he' to refer to voters and `she' to refer to candidates.] is fully aware of other voters' preferences, it is rational for him to manipulate.However, even in this case this voter may prefer to vote truthfully, simply becausehe assigns a high value to communicating his true preferences; such voters are called ideological. Moreover, if there are two or more GS-manipulators, it is no longer easy for them to make up their mind in favour of manipulation: while the Gibbard–Satterthwaite theorem tells us that each of these voters would benefit from voting strategicallyassuming that all other voters remain truthful, it does not offer any predictions if several voters may be able to manipulate simultaneously.The issues faced by GS-manipulators in this case are illustrated by the following example. Suppose four people are to choose among three alternatives by means of 2-approval, with ties broken according to the order a>b>c. Let the profile of sincere preferences be as in Table <ref>. There are two voters who prefer b to c to a, one voter who prefers a to c to b, and one voter who prefers c to b to a. If everyone votes sincerely, then c gets 4 points, b gets 3 points and a gets 1 point, so c is elected.Voters 1 and 2 are Gibbard–Satterthwaite manipulators. Each of themcan make b the winner by voting bac, ceteris paribus.Let us consider this game from the first voter's perspective,assuming that he is strategic; let A_i denote the strategy set of voter i, i=1, 2, 3, 4.The strategy set of voter 1 can then be assumed to be A_1={bca, bac} (clearly, under 2-approval cba is indistinguishable from bca, abc is indistinguishable from bac,and the two votes that do not rank b first are less useful than either bca and bac). Voter 1 has a good reason to believe that voters 3 and 4 will vote sincerely,as voter 3 cannot achieve an outcome that he would prefer to the current outcome and voter 4 is fully satisfied.Case 1.If voter 1 believes that voter 2 is ideological, then he is analysing the game where A_2={bca}, A_3={acb}and A_4={cba}. In this case he just votes bac and expects b to become the winner. Case 2.Suppose now that voter 1 believes that voter 2 is also strategic. Now voter 1 has to analyse the game withA_1=A_2={bca, bac}, A_3={acb} and A_4={cba}. If either one of the strategic players—voter 1 or voter2—manipulates and another stays sincere, b will be the winner. However, if they both manipulate, their worstalternative a will become the winner. Thus, in this case voter 1's manipulative strategy does not dominate his sincere vote,and if voter 1 is risk-averse, he should refrain from manipulating. A popular approach (see Section <ref>) is to view voting as a strategic game among the voters,and use various game-theoretic solution concepts to predict the outcomes.The most common such concept is Nash equilibrium, whichis defined as a combination of strategies, one for each player,such that each player's strategy is a best response to other players' strategies.In these terms, the Gibbard–Satterthwaite theorem says that under every reasonable voting rulethere are situations where truthful voting is not a Nash equilibrium. As a further illustration, the game analysed in Example <ref> (Case 2) has two Nash equilibria: in the first one, voter 1 manipulates and voter 2 remains truthful, and in the second one the roles are switched. However, the principle that players can always be expected to choose equilibrium strategies isnot universally applicable. Specifically, if players have enough experiencewith the game in question(or with similar games), both theory and experimentalresults suggest that players are often able to learn equilibrium strategies <cit.>. However, it isalso well-known since the early work of <cit.> that learning dynamics may fail to converge to an equilibrium. Moreover, in many applications—and voting is one of them—players' interactions have only imperfect precedents,or none at all so no learning is possible. If equilibrium is justified in such applications, it must be viastrategic thinking of players rather than learning. However, in some games the required reasoning is too complex forsuch a justification of equilibrium to be behaviourally plausible<cit.>. This is fully applicable to voting, where such reasoning,beyond very simple profiles, is impossible because of the number of voters involved.In fact, a number of recent experimental and empirical studies suggest that players' responses in strategicsituations often deviate systematically from equilibrium strategies, and are better explained by the structural nonequilibriumlevel-k <cit.> or cognitive hierarchy (CH) models<cit.>; see also a survey by <cit.>.In a level-k model players anchor theirbeliefs in a non-strategic initial assessment of others' likely responses to the game.Non-strategic players are said to be level-0 players.Level 1 players believe that all other players are at level 0,and they give their best response on the basis of this belief.Level 2 players assume that all other players belong to level 1, and, more generally,players at level k give their best response assuming that all other players are at level k-1.The cognitive hierarchy model is similar, but with an essential difference:in this model players of level k respond to a mixture of types from level 0 to level k-1.It is frequently assumed that other players' levels are drawn from a Poisson distribution.Some further approaches based on similar ideas are surveyed by <cit.>. The aim of our work is to explore the applicability of these models to voting games. We believe that specifics of voting, and, in particular, the heterogeneity of types of voters in real electorates, make the cognitive hierarchy framework more appropriatefor our purposes. In more detail, an important feature that distinguishes voting from many other applicationsof both level-k and CH models is the role of level-0 players. Specifically, level-0 (non-strategic) players are typically assumed to choose their strategy at random,and this type practically does not appear in real games at all.In contrast, in applications to voting it is natural to associate level-0 playerswith ideological voters, who have a significant presence in real elections.For instance, in the famous Florida vote (2000), where Bush won over Gore by just 537 votes, 97,488 Nader supporters voted for Nader—even though in such a close election every strategic votershould have voted either for Gore or for Bush (and an overwhelming majority of Nader supporterspreferred Gore to Bush). However, in the level-k analysis voters of level 2 assumethat all other voters have level 1, i.e., level-k models cannot be used to accommodate ideological voters. We therefore focus on the cognitive hierarchy approach. Moreover, we limit ourselves to considering the first three levels of the hierarchy (i.e., level-0, level-1, and level-2 players), as it seems plausible that very few voters are capable of higher-level reasoning (see the survey by <cit.> for some evidence in support of this assumption). Adapting the cognitive hierarchy framework to voting games is not a trivial task. First, it does not make sense to assume that voters' levels follow a specific distribution. Second, in the standard model of social choice voters' preferences over alternatives are ordinal rather than cardinal. The combination of these two factors means that,in general, for voters at level 2 or higher their best response may not be well-defined. We therefore choose to focus on strategies that are not weakly dominated according to the voter's beliefs. We present our formal definitions and the reasoning that justifies them in Section <ref>.To develop a better understanding of the resulting model, we instantiate it for a specific family of voting rules, namely, k-approval with k≥ 1. We develop a classification of strategies under k-approval and clarify the relationship between level-1 reasoning and the predictions of theGibbard–Satterthwaite theorem (Section <ref>). We then switch our attention to level-2 strategies,and, in particular, to the complexity of computing such strategies. For k-approval with k=1 (i.e., the classic Plurality rule) we describe an efficient algorithmthat decides whether a given strategy weakly dominates another strategy; as a corollary of this result,we conclude that under the Plurality rule level-2 strategies can be efficiently computedand efficiently recognised (Section <ref>). We obtain a similar result for 2-approval underan additional assumption of minimality (Section <ref>). Briefly, this assumption means that the level-2 player expects all level-1 players to manipulate by making as few changes to their votes as possible. However, for larger values of k finding level-2 strategies becomes computationally challenging: we show that this problem is NP-hard for k-approval with k≥ 4 (Section <ref>). As the problem of finding a level-1 strategy under k-approval is computationally easy for any value of k≥ 1 (this follows immediately by combining our characterization of level-1 strategies with the classic results of <cit.>), this demonstrates that higher levels of voters' sophistication come with a price tag in terms of algorithmic complexity. §.§ Related workThere is a substantial body of research in social choice theory and in political sciencethat models non-truthful voting as a strategic interaction, with a strong focus on Plurality voting; this line of work dates back to <cit.> and includes important contributionsby <cit.>, <cit.> and <cit.>, to name a few. More recently, voting games and their equilibria have also receiveda considerable amount of attention from computer science researchers, with a variety of approaches used to eliminate counterintuitive Nash equilibria. For instance, some authors assume that voters have a slight preference for abstaining or for voting truthfully when they are not pivotal <cit.>. Other works consider refinements of Nash equilibrium, such as subgame-perfect Nash equilibrium<cit.>, strong equilibrium <cit.>or trembling-hand equilibrium <cit.>, or model the reasoning of voters who have incomplete or imperfect information about each others' preferences <cit.>. Dominance-based solution concepts have been investigated as well<cit.>,albeit from a non-computational perspective. All the aforementioned papers do not impose any restrictions on the voters' reasoning ability,de facto assuming that they are fully rational. Boundedly rational voters are considered by <cit.>; however, their work focuses on strategic interactions amongGibbard–Satterthwaite manipulators, and studies conditions that ensure existence ofpure strategy Nash equilibria in the resulting games. In contrast, in this paper we go further and formally define the degree of voters' rationality by using the cognitive hierarchy approach. Level-k models and the cognitive hierarchy framework have been long used to modela variety of strategic interactions; we refer the reader to the survey of <cit.>. Nevertheless, to the best of our knowledge, ours is the first attempt to apply these ideas in the context of voting.Atopic closely related to voting games is voting dynamics, where players change their votes one by one in response to the current outcome <cit.>;see also a survey by <cit.>. However, this line of work assumes the voters to be myopic.Our work can also be seen as an extension of the model of safe strategic voting proposed by <cit.>.However, unlike us, Slinko and White focus on a subset of GS-manipulators who (a) all have identical preferences and (b) choose between truthtelling and using a specific manipulative vote, and on the existence of a weakly dominant strategic vote in this setting (such votes are called safe strategic votes). In contrast, our decision-maker takes into account that manipulators may have diverse preferences and have strategy sets that contain more than one strategic vote. It is therefore not surprising that computing safe strategic votes is easier than finding level-2 strategies: <cit.> show that safe strategic votes with respect to k-approval can be computed efficiently for every k≥ 1, whereas we obtain hardness results for k≥ 4. One of our contributions is a classification of manipulative votes under k-approval withlexicographic tie-breaking. <cit.> propose a similar classification for severalapproval-based voting rules. However, they view k-approval as a non-resolute voting rule, and thereforetheir results do not apply in our setting. Paper outline. The paper is organised as follows. We introduce the basic terminology and definitions in Section <ref>. Section <ref> presents the adaptation of the cognitive hierarchy framework to the setting of voting games. We then focus on the study of k-approval. Section <ref>describes the structure of level-1 strategies under k-approval. In Section <ref> we provide an efficient algorithm for identifying level-2 strategies with respect to the Plurality rule. Section <ref> contains our results for 2-approval, and in Section <ref> we present our hardness results for k-approval with k≥ 4. Section <ref> summarises our results and suggests directions for future work.§ PRELIMINARIESIn this section we introduce the relevant notation and terminologyconcerning preference aggregation and normal-form games. §.§ Preferences and Voting RulesWe consider n-voter elections over a candidate set C={c_1, …, c_m}; in what follows we use the terms candidates and alternatives interchangeably. Let (C) denote the set of all linear orders over C. An election is defined by a preference profile V=(v_1, …, v_n), where each v_i, i∈ [n], is a linear order over C; we refer to v_i as the sincere vote, or preferences, of voter i. For two candidates c_1, c_2∈ C we write c_1≻_i c_2, if voter i ranks c_1 above c_2, and say that voter i prefers c_1 to c_2. For brevity we will sometimes write ab… z to represent a vote v_i such that a≻_i b≻_i⋯≻_i z. We denote the top candidate in v_i by (v_i).Also, we denote the set of top k candidates in v_i by _k(v_i); note that _1(v_i)={(v_i)} and a≻_i b for all a∈_k(v_i) and b∈ C∖{_k(v_i)}.A (resolute) voting ruleis a mapping that, given a profile V, outputs a candidate (V)∈ C, which we call the winner of the election defined by V, or simply the winner at V.In this paper we focus on the family of voting rules known as k-approval. Under k-approval, k∈ [m-1], each candidate receives one point from each voter who ranks her in top k positions; the k-approval score of a candidate c, denoted by _k(c, V), is the total number of points that she receives. The winner is chosen among the candidates with the highest score according to a fixed tie-breakinglinear order > on the set of candidates C: specifically, the winner is the highest-ranked candidate with respect to this order among the candidates with the highest score. The 1-approval voting rule is widely used and known as Plurality. We will denote the k-approval rule (with tie-breaking based on a fixed linear order >) by _k. We say that a candidate x beats candidate y at V with respect to _k and the tie-breaking order > if _k(x, V) > _k(y, V) or _k(x, V)=_k(y, V) and x>y.§.§ Strategic Voting Given a preference profile V=(v_1, …, v_n), and a linear order v'_i∈(C), we denote by (V_-i, v'_i) the preference profile obtained from V by replacing v_i with v'_i; for readability, we will sometimes omit the parentheses around (V_-i, v'_i) and write V_-i, v'_i. We will often use this notation when voter i submits a strategic vote v'_i instead of his sincere vote v_i. Consider a profile V=(v_1, …, v_n), a voter i, and a voting rule .We say that a linear order v'_i is a manipulative vote of voter i at V with respect toif (V_-i, v'_i)≻_i (V). We say that i manipulates in favour of candidate c by submitting a vote v'_i if c is the winner at (V_-i, v'_i).A voter i is a Gibbard–Satterthwaite manipulator, or a GS-manipulator,at V with respect toif the set of his manipulative votes at V with respect tois not empty. We denote the set of all GS-manipulators at V by N(V, ). Note that a voter may be able to manipulate in favour of several different candidates.Let F_i={c∈ C|(V_-i, v'_i)=c for some v'_i∈(C)}; we say that the candidates in F_i are feasible for i at V with respect to .Note that F_i≠∅ for all i∈ [n], as this set contains the -winner at V under truthful voting. We say that two votes v and v' over the same candidate set C are equivalent with respect to a voting ruleif (V_-i, v)=(V_-i, v') for every voter i∈[n] and every profile V_-i of other voters' preferences.It is easy to see that v and v' are equivalent with respect to k-approval if and only if _k(v)=_k(v').§.§ Normal-form GamesA normal-form game (N, (A_i)_i∈ N, (≽_i)_i∈ N)is defined by a set of players N, and, for each i∈ N,a set of strategies A_i and a preference relation ≽_i defined on the space of strategy profiles, i.e., tuples of the form s⃗=(s_1, …, s_n), where s_i∈ A_i for all i∈ N[While one usually defines normal-form games in terms of utility functions,defining them in terms of preference relations is more appropriate for our setting, as preference profiles only provide ordinal information about the voters' preferences.]. For each pair of strategy profiles s⃗, t⃗ and a player i∈ N, we write s⃗≻_i t⃗ if s⃗≽_i t⃗ and t⃗⋡_i s⃗. A normal-form game is viewed as a game of complete and perfect information, which means that all players are fully aware of the structure of the game they are playing.Given a strategy profile s⃗=(s_1, …, s_n) and a strategy s'_i∈ A_i, we denote by (s⃗_-i,s'_i) the strategy profile (s_1,…, s'_i,…,s_n), which is obtained from s⃗ by replacing s_i with s'_i. We say that a strategy s_i∈ A_i weakly dominates another strategy s'_i∈ A_i iffor every strategy profile s⃗_-i of other players we have(s⃗_-i, s_i)≽_i (s⃗_-i,s'_i) and there exists a profile s⃗_-i of other players' strategies such that (s⃗_-i, s_i)≻_i (s⃗_-i,s'_i).§ THE MODELAs suggested in Section <ref>, our goal is to analyse voting as a strategic game and consider it from the perspective of the cognitive hierarchy model. As we reason about voters' strategic behavior, we consider games where players are voters, their strategiesare ballots they can submit, and their preferences over strategy profilesare determined by election outcomes under a given voting rule. We then use the cognitive hierarchy framework to narrow down the players' strategy sets. §.§ Cognitive Hierarchy Framework for Voting GamesRecall that, in the general framework of cognitive hierarchy, players at level 0 aretypically assumed to choose their action at random. This is because in general normal-form games a player who is unable to deliberate about other players' actionsusually has no reason to prefer one strategy over another. In contrast, in the context of voting, there is an obvious focal strategy, namely, truthful voting. Thus, in our model we associate level-0 voters with ideological voters,i.e., voters who always vote according to their true preferences.At the next level of hierarchy are level-1 voters. These voters assume that all other voters are ideological (i.e., are at level 0),and choose their vote so as to get the best outcome they consider possible under this assumption. That is, voter i votes so as to make his most preferred candidate in F_i the election winner (in particular, if F_i is a singleton, voter i votes truthfully). We say that a vote v'_i of a voter iis a level-1 strategy at profile V with respect to if (V_-i, v'_i)≻_i c for all c∈ F_i∖{(V_-i, v'_i)}. Note that a level-1 voter that is not a Gibbard–Satterthwaite manipulatorhas no reason to vote non-truthfully, as he does not expect to be able to change the election outcome according to his tastes; hence we assume that such voters are truthful.We are now ready to discuss level-2 voters. These voters believe that all other voters are at levels 0 or 1 of the cognitive hierarchy. We will further assume that level-2 voters are agnostic about other voters' levels; thus, from their perspective every other voter may turn out to be a level-0 voter (which in our setting is equivalentto being sincere) or a level-1 voter. Thus, from the point of view of a level-2 voter,a voter who is not a GS-manipulator will stick to his truthful vote, whereas a GS-manipulator will either choose his action among level-1 strategies or(in case he is actually a level-0 voter) vote truthfully. Thus, when selecting his vote strategically, a level-2 voter takes into account the possibility that other voters—namely, the GS-manipulators—may be strategic as well.We further enrich the model by assuming that a level-2 voter may be able to identify, for each other voter i, a subset of level-1 strategies such that i always chooses his vote from that subset, i.e., a level-2 voter may be able to rule out some of the level-1 strategies of other voters. There are several reasons to allow for this possibility. First, the set of all level-1 strategies for a given voter can be very large, and a voter may be unable or unwilling to identify allsuch votes.For example, our level-2 voter may know or believe that other voters use a specific algorithm (e.g., that of <cit.>) to find their level-1 strategies;in this case, his set of strategies for each voter i would consist of the truthful vote v_iand the output of the respective algorithm. Also, voters may be known not to choosemanipulations that are (weakly) dominated by other manipulations. Finally, voters may prefer not to change their vote beyond what is necessary to make their targetcandidate the election winner, either because they want their vote to be as close to the true preference order aspossible (see the work of <cit.>),or for fear of unintended consequences of such changes in the complex environment of the game. Thus, a preference profile together with a voting rule define not just a single game, but a family of games, which differ in sets of actions available to GS-manipulators. §.§ Gibbard–Satterthwaite GamesWe will now describe a formal model that will enable us to reason about the decisionsfaced by a level-2 voter. For convenience, we assume that voter 1 is a level-2 voter and describe a normal-form game that captures his perspective of strategic interaction, i.e., his beliefs about the game he is playing.Fix a voting rule , let V be a profile over a set of candidates C,let N=N(V,) be the set of GS-manipulators at V with respect to , and set N_1=N∪{1}. We consider a family of normal-form games defined as follows.In each game the set of players is N_1, i.e., voter 1 is a player irrespective of whether he is actually a GS-manipulator.For each player i∈ N_1∖{1}, i's strategy set A_i consists of his truthful vote and a (possibly empty) subset of his level-1 strategies; for voter 1 we have A_1=(C), i.e., 1 can submit an arbitrary ballot. It remains to describe the voters' preferences over strategy profiles. For a strategy profile V^* = (v^*_i)_i∈ N_1, where v^*_i∈ A_i for i∈ N_1, let V[V^*]=(v'_1, …, v'_n) be the preference profile such that v'_i=v_i for i∉N_1 and v'_i=v^*_i for i∈ N_1. Then, given two strategy profiles V^* and V^** and a voter i∈ N_1, we write V^*≽_i V^** if and only ifi prefers (V[V^*]) to (V[V^**]) or (V[V^*])=(V[V^**]). We refer to any such game as a GS-game.We denote the set of all GS-games for V andby (V, ). Note that an individual game in (V, ) is fully determined by theGS-manipulators' sets of strategies, i.e., (A_i)_i∈ N(V, ) (player 1's set of strategies is always the same, namely, (C)).Thus, in what follows, we write G=(V, , (A_i)_i∈ N(V, )); when V andare clear from the context, we simply write G = (A_i)_i∈ N. We refer to a strategy profile in a GS-game as a GS-profile, and we will sometimes identify the GS-profile V^* = (v^*_i)_i∈ N_1 with the preference profile V[V^*]. We denote the set of all GS-profiles in a game G by (G). We will now argue that games in (V, ) reflect the perspective of voter 1 when he is at the second level of the cognitive hierarchy. Fix a game G∈(V, ). Note first that, since voter 1 believes that all other voters belong to levels 0 and 1 of the cognitive hierarchy,he expects all voters who are not GS-manipulators to vote truthfully, i.e., he does not need to reason about their strategies at all. This justifies having N_1 = N∪{1} as our set of players. On the other hand, consider a voter i∈ N∖{1}. Voter 1 considers it possible that i is a level-0 voter, who votes truthfully. Voter 1 also entertains the possibility that i is a level-1 voter, in which case i's vote has to be a level-1 strategy; as argued above, voter 1 may also be able to rule out some of i's level-1 strategies. Consequently, the set A_i, which, by definition, contains v_i, consists of all strategies that voter 1 considers possible for i. Thus, voter 1's view of other voters' actions is captured by G.We are now ready to discuss level-2 strategies. In game-theoretic literature, it is typical to assume that a level-2 player is endowed with probabilistic beliefs about other players' types as well as a utility function describing his payoffs under all possible strategy profiles. Under these conditions, it makes sense to define level-2 strategies as those that maximise player 2's expected payoff. However, in the absence of numerical information,as in the case of voting games, we cannot reason about expected payoffs. We can, however, compare different strategies pointwise, and remove strategies that are weakly dominated by other strategies.On the other hand, if a strategy v is not weakly dominated, a level-2 player may hold beliefs that make him favour v, so no such strategy can be removed from consideration without making additional assumptions about the behavior of players in N(V, ). This reasoning motivates the following definition of a level-2 strategy. Given a GS-game G=(V, , (A_i)_i∈ N(V, )), we say that a strategy v∈ A_1 of player 1 is a level-2 strategy if no other strategy of player 1 weakly dominates v. We note that being weakly undominated is not a very demanding property: a strategycan be weakly undominated even if it fares badly in many scenarios. This is illustratedby the following example. Consider the 4-voter profile over {a, b, c, d} given in Table <ref>.Suppose that the voting rule is Plurality and the tie-breaking rule is a>b>c>d. As always, we assume that voter 1is the level-2 voter. Voters 2, 3, and 4 are GS-manipulators; their most preferred feasible candidates are, respectively, d, b, and c. Consider the GS-game where A_2={bdac, dbac}, A_3={cbad, bcad}, A_4={dcab, cdab}. In this game every vote that does not rank d first is a level-2 strategy for the first player. Indeed,a vote that ranks a first is optimal when all other players submit their sincere votes; a vote that ranks b first is optimal when players 2 and 3 stay sincere, but player 4 votes for c; and a vote that ranks c first is optimal when player 2 votes for d, but players 3 and 4 stay sincere. Note, in particular, that, by changing his vote from abcd (his sincere vote) to cabd,player 1 changes the outcome from a (his top choice)to c (his third choice) when other players vote truthfully; however, this behavior is rational if player 1 expects players 3 and 4 (but not player 2) to vote sincerely.Example <ref> illustrates that level-2 strategies are not `safe': there can be circumstanceswhere a level-2 strategy results in a worse outcome than sincere voting. Now, a cautious level-2 player may prefer to stick to his sincere vote unless he can find a manipulative vote which leads to an outcome that is at least as desirable as the outcome under truthful voting, for any combination of actions of other players that our level-2 player considers possible. The following definition, which is motivated by the concept of safe strategic voting <cit.>, describes the setof strategies that even a very cautious level-2 player would prefer to sincere voting. Given a GS-game G=(V, , (A_i)_i∈ N(V, )), we say that a strategy v∈ A_1 of player 1 is an improving strategy if v weakly dominates player 1's sincere strategy v_1. We note that a level-2 strategy may fail to be an improving strategy,and conversely, an improving strategy is not necessarily a level-2 strategy. For instance, in Example <ref> the strategy bac is a level-2 strategy, but not an improving strategy, and none of the level-2 strategies in Example <ref> is improving. However, it is easy to see that if a player has an improving strategy,he also has an improving strategy that is a level-2 strategy. Moreover, an improving strategy exists if and only if sincere voting is not a level-2 strategy.One can also ask if a given strategy weakly dominates all other (non-equivalent) strategies.However, while strategies with this property are highly desirable, from the perspective of a strategic voter it is more important to find out whether his truthful strategy is weakly dominated. Indeed, the main issue faced by a strategic voter is whether to manipulate at all, and if a certain vote can always ensure an outcome that is at least as good, and sometimes better,as that guaranteed by his truthful vote,this is a very strong incentive to use it, even if another non-truthful vote may be better in some situations.This issue is illustrated by Example <ref> below, whichdescribes a profile where a player has two incomparable improving strategies. Let the profile of sincere preferences be as in Table <ref>, and assume thatthe voting rule is Plurality and the tie-breaking order is given by w>d>c>b>a.The winner at the sincere profile is w. All level-1 strategies of voter 2 are equivalent to cbdwa, whereas all level-1 strategies of voter 3 are equivalent to dcbwa; voters 4 and 5 are not GS-manipulators. Consider the GS-game where for i∈{2, 3} the set of strategies of player i consists of his truthful vote and all of his level-1 strategies. Voter 1, who is our level-2 player,can manipulate either in favour of b or in favour of d, by ranking the respective candidate first.Indeed, for player 1 both badwc and dabwc weakly dominate truthtelling. However, neither of these strategies weakly dominates the other: badwc is preferable if no other player uses a level-1 strategy, whereas dabcw is preferable if player 2 uses his level-1 strategy, but player 3 votes sincerely.We note that a level-2 voter may find it useful to act as a counter-manipulator <cit.>, i.e.,to submit a vote that is not manipulative with respect to the truthful profile, but minimises the damage from someone else's manipulation. Let the profile of sincere preferences be as in Table <ref>, and assume thatthe voting rule is Plurality and the tie-breaking order is given by a>b>c.Under truthful voting a wins, so voter 6 is the only GS-manipulator: if he changes his vote to bca then b wins, and b≻_6 a. Therefore, for voter 1 voting acb is preferable to voting truthfully: this insincere vote has no impact if voter 6 votes truthfully, but prevents b from becoming a winner when voter 6 submits a manipulative vote. Thus, in this example acb is an improving strategy,and truthful voting is not a level-2 strategy as it is weakly dominated by voting acb. In contrast, acb is a level-2 strategy, as no other strategy weakly dominates it. §.§ Algorithmic QuestionsFrom an algorithmic perspective, perhaps the most natural questionssuggested by our framework are how to decide whether a given strategy is a level-2 strategy, or how to compute a level-2 strategy. A related question is whether a given strategy is an improving strategy and whether an improving strategy can be efficiently computed. These questions offer an interesting challenge from an algorithmic perspective: the straightforward algorithm for deciding whether a given strategy is a level-2 strategy or an improving strategy relies on considering all combinations of other players' strategies, and hence has exponential running time. It is therefore natural to askwhether for some voting rules exhaustive choice can be avoided. We explore this question in Sections <ref>–<ref>; for concreteness, we focus on k-approval, for various values of k.§ LEVEL-1 STRATEGIES UNDER K-APPROVALThe goal of this section is to understand and classify level-1 strategies under the k-approval voting rule; this will help us reason about level-2 strategies in subsequent sections. In what follows, we fix a linear order > used for tie-breaking. We start with a simple, but useful lemma.Fix k≥ 1. Consider a profile V over C, let w be the k-approval winner at V,and let x be an alternative in C∖{w}. Then any manipulative vote by a voter i in favour of x at V falls under one of the following two categories: Type 1 Voter i increases the score of x by 1 without decreasing the score ofw. In this case w, x∉_k(v_i), x≻_i w, and the manipulative vote v'_i satisfies x∈_k(v'_i), w∉_k(v'_i). In such cases voter i will be referred to as a promoter of x.Type 2Voter i decreases the score of w (and possibly that of some other alternatives) by 1 without increasing the score of x. In this case w, x∈_k(v_i), x≻_i w, and the manipulative vote v'_i satisfies x∈_k(v'_i), w∉_k(v'_i). In such cases voter i will be referred to as a demoter of w. Manipulations of type 2 only exist for k≥ 2.Suppose that voter i manipulates in favour of x. If i can increase the score of x, then x∉_k(v_i). However, i must rank x higher than w (otherwise, this would not be a manipulation). Thus, w∉_k(v_i) and therefore voter i cannot decrease w's score. Moreover, if w∈_k(v'_i), then w would beat x under k-approval in (V_-i, v'_i); thus, w∉_k(v'_i). On the other hand, suppose that i cannot increase the score of x. This means that x∈_k(v_i) and hence i is left with reducing the scores of some of x's competitors including the current winner w. For this to be possible, it has to be the case that w∈_k(v_i) and x≻_i w. Also, we have w∉_k(v'_i), as otherwise w would beat x under k-approval in (V_-i, v'_i). Finally, as w≠ x, we can only have x, w∈_k(v_i) if k≥ 2. The classification in Lemma <ref> justifies our terminology: a promoter promotes a new winner and a demoter demotes the old one. Under Plurality, i.e., when k=1, we only have promoters. Let X = {x_1, …, x_ℓ} and Y = {y_1, …, y_ℓ} be two disjoint sets of candidates. Given a linear order v over C, we denote by v[X; Y] the vote obtained by swapping x_j with y_j for j∈[ℓ]. If the sets X and Y are singletons, i.e., X={x}, Y={y}, we omit the curly braces, and simply write v[x;y]. Clearly, under k-approval any manipulative vote of voter i is equivalentto a vote of the form v_i[X; Y], where X⊆_k(v_i), Y⊆ C∖_k(v_i).We can now state a corollary of Lemma <ref>, which characterises the possible effects of a manipulative vote underk-approval. Let w be the k-approval winner at a profile V, let v^*_i=v_i[X; Y],where X⊆_k(v_i) and Y⊆ C∖_k(v_i). Let V'=(V_-i,v^*_i), and let w' w be the k-approval winner at V'.Then eitherw∈ X or w'∈ Y but not both.Lemma <ref> implies that either the new winner is promoted or the old winner is demoted, but not both. Consider a manipulative vote v_i[X;Y] of voter i at V under k-approval; we say that v_i[X;Y] is minimal if for every other manipulative vote v_i' of voter i there is a vote v_i[X';Y'] that is equivalent to v'_i and satisfies |X'|≥ |X|. That is, amanipulative vote is minimal if it performs as few swaps as possible.Arguably, minimal manipulative votes are the main tool that a rational voter would use, as they achieve the desired result in the most straightforward way possible.We now introduce some useful notation.Fix a profile V. Let w be the k-approval winner at V,and let t = _k(w, V). SetS_1(V, k)= {c∈ C|_k(c)=t, w>c}, S_2(V, k)= {c∈ C|_k(c)=t-1, c>w},and set S(V, k)=S_1(V, k)∪ S_2(V, k). We say that a candidate c is k-competitive at V if c∈ S(V, k). The following proposition explains our choice of the term: only k-competitive candidates can become k-approval winners as a result of a manipulation.Suppose that some voter can manipulate in favour of a candidate p∈ C at a profile V with respect to k-approval.Then p∈ S(V, k). Let w be the k-approval winner at V; clearly, w≠ p. Suppose that voter i can manipulate in favour of p at V by submitting a vote v'_i; let V'=(V_-i, v'_i). Set t=_k(w, V); then _k(p, V)≤ t. Note that if _k(p, V)=t, it has to be the case that w > p, since otherwise p would beat w at V. Thus, in this case p∈ S_1(V, k). Now, suppose that _k(p, V)=t-1.By Corollary <ref> we have either _k(w, V')=_k(p, V')=t (if p was promoted) or _k(w, V')=_k(p, V')=t-1 (if w was demoted). In both cases we have to have p > w,as otherwise w would beat p at V'. Therefore, in this case p∈ S_2(V, k). Finally, note that it cannot be the case that_k(p, V)≤ t-2,since in this case by Corollary <ref> we would have either _k(w, V')≥ t-1, _k(p, V')≤ t-2 or _k(w, V')=t, _k(p, V')≤ t-1, i.e., w would beat p at V'. Suppose that S(V, k)≠∅. If S_1(V,k)∅, then by p^*(V, k) we denote the top-ranked candidate in S_1(V,k) with respect to >; otherwise, we denote by p^*(V, k) the top-ranked candidate in S_2(V,k) with respect to >. Thus, p^*(V, k) beats all candidates other than w at V, and would become a winner if it were to gain one point or if w were to lose one point. We omit V and k from the notation when they are clear from the context.We are now ready to embark on the computational complexity analysis of level-2 strategiesunder k-approval, for various values of k. § PLURALITYPlurality voting rule is k-approval with k=1.For this rule we only have manipulators of type 1, and all manipulative votes of voter i in favour of candidate c are equivalent: in all such votes c is placed in the top position. The main result of this section is that the problem of deciding whether a given strategyof voter 1 weakly dominates another strategy of that voter admits a polynomial-time algorithm. Note that,since under Plurality there are only m votes that are pairwise non-equivalent,this means that we can check if a given strategy is a level-2 strategy or an improving strategy, or find a level-2 strategy or an improving strategy (if it exists) in polynomial time; we formalise this intuition inCorollary <ref> at the end of this section.Fix a preference profile V over a candidate set C and consider a GS-game(V, _1, (A_i)_i∈ N), where N=N(V, _1). Let w be the Plurality winner at V.As argued above, for each i∈ N∖{1} the set A_i consists of v_i and possibly a number of pairwise equivalent manipulative votes; without loss of generality, we can remove all but one manipulative vote, so that |A_i|≤ 2 for all i∈ N∖{1}. We will now explain how, given two votes v_1' and v_1”, voter 1can efficiently decide if one of these votes weakly dominates the other.We will first describe a subroutine that will be used by our algorithm.There is a polynomial-time procedure =(G, r, r', x, y, C^[1], C^[0], C^[-1], C^[-2]) that,given a GS-game G=(V, _1, (A_i)_i∈ N(V, _1)) with |V|=n,two integers r, r'∈{0, …, n}, two distinct candidates x, y∈ C, and a partition of candidates in C∖{x, y} into C^[1], C^[0], C^[-1] and C^[-2], decides whether there is a strategy profile V^* in G such that* _1(x, V[V^*]_-1)=r,* _1(y, V[V^*]_-1)=r', and* for each c∈ C∖{x, y} and each ℓ∈{1, 0, -1, -2} if c∈ C^[ℓ] then_1(c, V[V^*]_-1)≤ r+ℓ.We proceed by reducing our problem to an instance of network flow with capacities and lower bounds, as follows. We construct a source, a sink, a node for each voter i∈[n]∖{1} and a nodefor each candidate in C. There is an arc from the source to each voter node; the capacity and the lower bound of this arc are set to 1, i.e., it is required to carry one unit of flow. Also, there is an arc with capacity 1 and lower bound 0 from voter i to candidate c ifi∈ N(V, _1)∖{1} and c=(v) for some v∈ A_i or if i∈ [n]∖(N(V, _1)∪{1})and c=(v_i). Finally, there is an arc from each candidate c to the sink. The capacity of this arc is set to r+ℓ if c∈ C^[ℓ] for some ℓ∈{1, 0, -1, -2}; the lower bounds for these arcs are 0. For x,both the capacity and the lower bound of the arc to the sink are set to r, and for y they are both set to r'.We note that some of the capacities may be negative, in which case there is no valid flow. It is immediate that an integer flow that satisfies all constraints corresponds to a strategy profile in G where all candidates have the required scores; it remains to observe that the existence of a valid integer flow can be decided in polynomial time. We are now ready to describe our algorithm. Given a GS-game G=(V, _1, (A_i)_i∈ N(V, _1)) and two strategies v_1', v_1”∈(C) of player 1 we can decide in polynomial time whether v_1' weakly dominates v_1”. We will design a polynomial-time procedure that, given two strategies u, v of player 1, decides if there exists a profile V^*_-1 of other players' strategies such that _1(V[V^*_-1, u])≻_1 _1(V[V^*_-1, v]); by definition, v'_1 weakly dominates v”_1 if this procedure returns `yes' for u=v_1', v=v_1” and `no' for u=v_1”, v=v_1'. Let a=(u), b=(v). We can assume without loss of generality that a≠ b,since otherwise u and v are equivalent with respect to Plurality.Consider an arbitrary profile V^*_-1 of other players' strategies, and let V^u=V[V^*_-1, u], V^v=V[V^*_-1, v], w^u=_1(V^u), w^v=_1(V^v). We note that w^u≠ a implies w^v≠ a: if w^u beats a at V^u, this is also the case at V^v. Similarly,if w^v≠ b then also w^u≠ b. Now, suppose thatw^u≠ a and w^v≠ b. We claim that in this case w^u=w^v. Indeed, suppose for the sake of contradiction that w^u≠ w^v. As w^u≠ a, w^v≠ b, the argument above shows that {w^u, w^v}∩{a, b}=∅. Thus, both w^u and w^v have the same Plurality score at V^u and V^v; as w^u beats w^v at V^u, this must also be the case at V^v, a contradiction.Note that _1(V[V^*_-1, u])≻_1 _1(V[V^*_-1, v])if and only if w^u≻_1 w^v. By the argument in the previous paragraph,this can happen in one of the following three cases: (i) w^u=a, w^v=b and a≻_1 b; (ii)w^u=a, w^v=w for some w≠ b, a≻_1 w; (iii) w^u=w, w^v=b for some w≠ a, w≻_1 b. (We note that we can merge case (i) into case (ii) or case (iii); we choose not to do so for the sake of clarity of presentation.) We will now explain how to check if there exists a profile V^*_-1 that corresponds to any of these three situations.* Case (i): w^u=a, w^v=b. Suppose first that a > b. Then a desired profile V^*_-1 exists if and only if there is some value t∈[n] such that _1(a, V^u)=t and(a)_1(b, V^u)=t, _1(c, V^u)≤ t for all c∈ C∖{a, b}with a>c, (c, V^u)≤ t-1 for all c∈ C∖{a, b} with c>a, or(b)_1(b, V^u)=t-1, _1(c, V^u)≤ t for all c∈ C∖{a, b} with b>c, and (c, V^u)≤ t-1 for all c∈ C∖{a, b} with c>b.Note that _1(a, V^u_-1)=_1(a, V^u) - 1 and _1(c, V^u_-1)=_1(c, V^u) for c∈ C∖{a}. Thus, to check if condition (a) is satisfied for some t∈ [n],we set C^[1] ={c∈ C∖{a, b}| a > c}, C^[0] ={c∈ C∖{a, b}| c > a}, C^[-1]=C^[-2]=∅ and call (G, t-1, t, a, b, C^[1], C^[0], C^[-1], C^[-2]).Similarly, to determine whether condition (b) is satisfied for some t∈ [n],we set C^[1] ={c∈ C∖{a, b}| b>c}, C^[0] ={c∈ C∖{a, b}| c > b}, C^[-1]=C^[-2]=∅ and call(G, t-1, t-1, a, b, C^[1], C^[0], C^[-1], C^[-2]).The answer is `yes' if one of these calls returns `yes' for some t∈ [n].For the case b>a the analysis is similar. In this case, we need to decide whether there exists a value oft∈ [n] such that _1(a, V^u)=t and(a)_1(b, V^u)=t-1, _1(c, V^u)≤ t for all c∈ C∖{a, b} with a>c, and (c, V^u)≤ t-1 for all c∈ C∖{a, b} with c>a, or(b) _1(b, V^u)=t-2, _1(c, V^u)≤ t-1 for all c∈ C∖{a, b} with b>c, and (c, V^u)≤ t-2 for all c∈ C∖{a, b} with c>b.Again, this can be decided by calling the procedurewith appropriate parameters; we omit the details. * Case (ii): w^u=a, w^v=w for some w with a≻_1 w. In this case, we go over all candidates w∈ C∖{a, b} with a≻_1 w and all values of t∈ [n] and callwith appropriate parameters.Specifically, if a>w, we start by setting r =t-1, r'=t, andC^[1] ={c∈ C∖{a, w, b}| w>c},C^[0] = {c∈ C∖{a, w, b}| c>w},C^[-1] = C^[-2] = ∅.We then place b in C^[0] if w>b and in C^[-1] otherwise; our treatment of b reflects the fact that she gets an extra point at V^v.If w>a we start by setting r=t-1, r'=t-1, andC^[1] =∅, C^[0] = {c∈ C∖{a, w, b}| w>c}, C^[-1] = {c∈ C∖{a, w, b}| c>w}, C^[-2] = ∅.We then place b in C^[-1] if w>b and in C^[-2] otherwise.Finally, we call(G, r, r', a, w, C^[1], C^[0], C^[-1], C^[-2]).The answer is `yes' if one of these calls returns `yes' for some t∈ [n] and some w with a≻_1 w. * Case (iii): w^u=w, w^v=b for some w with w≻_1 b. The analysis is similar to the previous case; we omit the details. Theorem <ref> immediately implies that natural questions concerning level-2 strategiesand improving strategies are computationally easy. Given a GS-game G=(V, _1, (A_i)_i∈ N(V, _1)) and a strategy v_1'∈(C) of player 1 we can decide in polynomial time whether v_1' is a level-2 strategy or an improving strategy. Moreover, we can decide in polynomial time whether player 1 has a level-2 strategy or an improving strategy in G. Let a=(v'_1). To decide whether v'_1 is an improving strategy, we use the algorithmdescribed in the proof of Theorem <ref> to check whether v'_1 weakly dominates v_1. Similarly, to decide whether v'_1 is a level-2 strategy, for each c∈ C∖{a} we construct a vote v^c with (v^c)=c and check whether v^c weakly dominates v'_1 using the algorithm from the proof of Theorem <ref>.As every strategy of player 1 is equivalent either to v'_1 or to one of the votes we constructed,v'_1 is a level-2 strategy if and only if it is not weakly dominated by any of the votes v^c, c∈ C∖{a}.Similarly, to decide whether v_1 has a level-2 strategy (respectively, an improving strategy),we consider all of his m pairwise non-equivalent strategies, and check if any of them is a level-2 strategy (respectively, an improving strategy), as described in the previous paragraph. § 2-APPROVALIn this section, we study the computational complexity of identifying level-2 strategiesand improving strategies in GS-games under 2-approval. We show that if the level-2 player believes that level-1 players can only contemplate minimal manipulations,he can efficiently compute his level-2 strategies as well as his improving strategies.As argued in Section <ref>,minimality is a reasonable assumption, as level-1 players have no reason to use complex strategies when simple strategies can do the job.Specifically, we prove that, under the minimality assumption, given two strategies v' and v”,the level-2 player can decide in polynomial time whether one of these strategies weakly dominates the other; just as in the case of Plurality, this implies that hecan check in polynomial time whether a given strategy is a level-2 (respectively, improving) strategy or identify all of his level-2(respectively, improving) strategies.The following observations play a crucial role in our analysis. Consider a GS-game G=(V, _2, (A_i)_i∈ N(V, _2)). Let w be the 2-approval winner at V.Then for each player i∈ N(V, _2)∖{1} such that w∈_2(v_i) it holds that (v_i)≠ w andthe candidate (v_i) is ranked in top two positions in every vote v∈ A_i.Since i is a GS-manipulator, he does not rank w first; therefore, he ranks w second.Then the only way for him to improve the outcomeis to vote so that his top candidate p = (v_i) wins.For this i must demote w and promote a candidate c which will not overtake p,i.e., i is necessarily a demoter. Moreover, it must be the case that p beats every candidate in C∖{w} at V. Thus, i can manipulate in favour ofp = (v_i)only if S(V, 2)≠∅ and p=p^*(V,2), andevery non-truthful strategy in A_i is of the form v_i[w;c]for some c∉_2(v). Thus,p∈∩_v∈ A_i_2(v), which proves our claim. Proposition <ref>concerns voters who are demoters, and follows immediately from Lemma <ref>; note also that it does not depend on the minimality assumption.Consider a GS-game G=(V, _2, (A_i)_i∈ N(V, _2)). Let w be the 2-approval winner at V.Consider a player i∈ N(V, _2)∖{1}such that w∉_2(v_i) and the set A_i consists of i's truthful voteand a subset of i's minimal manipulations. Let _2(v) = {a, a'}. Then there is a candidate c∈ C∖{a, a'} such that for each v∈ A_i we have _2(v)∈{{a, a'}, {a, c}, {a',c}}. Player i cannot lower the score of w by changing his vote. However, he can raise the scores of some candidates in C∖_2(v_i) by moving these candidates into top two positions. In general, i can do that for two candidates simultaneously; however, the minimality assumption implies that i only moves one candidate into the top two positions.Thus, i is a promoter (see Section <ref>). For a vote v'_1 to be a level-1 strategythe promoted candidate has to be i's most preferred candidate in S(V,2)∖_2(v_i)(let us denote this candidate by p).Thus, in this case voter i has 3 options: (1) to vote truthfully,(2) to swap p with the candidate that he ranks first or(3) to swap p with the the candidate he ranks second. This completes the proof Propositions <ref> and <ref> enable us to establish an analogue of Lemma <ref> for 2-approval under the minimality assumption. There is a polynomial-time procedure '='(G, r, r', x, y, C^[1], C^[0], C^[-1], C^[-2]) that,given a GS-game G=(V, _2, (A_i)_i∈ N(V, _2)) with |V|=n, where for each i∈ N∖{1} the set A_i consists of i's truthful voteand a subset of i's minimal manipulations,two integers r, r'∈{0, …, n}, two distinct candidates x, y∈ C, and a partition of candidates in C∖{x, y} into C^[1], C^[0], C^[-1] and C^[-2], decides whether there is a strategy profile V^* in G such that* _2(x, V[V^*]_-1)=r,* _2(y, V[V^*]_-1)=r', and* for each ℓ∈{1, 0, -1, -2} and each c∈ C^[ℓ] it holds that_2(c, V[V^*]_-1)≤ r+ℓ.Let w be the 2-approval winner at V. If S(V, 2)≠∅, set p^*=p^*(V, 2). We use essentially the same construction as in the proof of Lemma <ref>. Specifically, the set of nodes consists of a sink, a source, one node for each voter in [n]∖{1}, and one node for each candidate c∈ C. For each i∈ N∖{1}, the capacity and the lower boundof the arc from the source to node i are equal to 2, and the capacitiesand lower bounds of the arcs from candidatesto the source are defined as in the proof of Lemma <ref>.It remains to describe the arcs connecting voters and candidates.If i∉N, we add an arc from i to c for each c∈_2(v_i); the capacity and the lower bound of these arcs are 1, encoding the fact that i has to vote for his top 2 candidates.Now, consider a voter i∈ N∖{1} who is a demoter; if such a voter exists, we have S(V, 2)≠∅ and hence p^* is well-defined.By Proposition <ref> we have _2(v_i)={p^*, w} and A_i={v_i[w;c]| c∈ C_i} for some C_i⊂ C∖{p^*, w}.Then we introduce an arc from i to p^* whose capacity and lower bound are both set to 1,and arcs with capacity 1 and lower bound 0 from i to each c∈ C_i∪{w}.Finally, consider a voter i∈ N∖{1} who is a promoter; let _2(v_i)={a, a'} and let p be i's most preferred candidate in S(2, V)∖{a, a'}. If A_i contains both a vote v' with _2(v')={a, p} and a vote v” with _2(v”)={a', p}, then by Proposition <ref> it suffices to add arcs with capacity 1 and lower bound 0 that go from i to a, a', and p. If we have _2(v)∈{{a, a'}, {a, p}} for each v∈ A_i, we add an arc with capacity 1 and lower bound 1 from i to a and arcs with capacity 1 and lower bound 0 from i to a' and p. Similarly, if we have _2(v)∈{{a, a'}, {a', p}} for each v∈ A_i, we add an arc with capacity 1 and lower bound 1 from i to a'and arcs with capacity 1 and lower bound 0 from i to a and p.It is clear from the construction that a valid integer flow in this networkcorresponds to a strategy profile V^* with the desired properties. We are now ready to prove the main result of this section. Given a GS-game G=(V, _2, (A_i)_i∈ N(V, _2)), where for each i∈ N∖{1} the set A_i consists of i's truthful vote and a subset of i's minimal manipulations, and two strategies v_1', v_1”∈(C) of player 1,we can decide in polynomial time whether v_1' weakly dominates v_1”.Just as in the proof of Theorem <ref>, it suffices to design a polynomial-time procedure that,given two strategies u, v of player 1, decides if there exists a profile V^*_-1 of other players' strategies such that _2(V[V^*_-1, u])≻_1 _2(V[V^*_-1, v]). Let _2(u)={a, a'}, _2(v) = {b, b'}. We can assume that {a, a'}≠{b, b'},and we will focus on the case where {a, a'}∩{b, b'}=∅; the casewhere {a, a'}∩{b, b'} is a singleton is similar (and simpler).We use the same notation as in the proof of Theorem <ref>: given a profile V^*_-1 of other players' strategies,we let V^u=V[V^*_-1, u], V^v=V[V^*_-1, v], w^u=_2(V^u), w^v=_2(V^v). Our goal then is to decide if there exists a profile V^*_-1 such that w^u≻_1 w^v. To this end, we go over all values of t∈[n-1] and all candidates w, w'∈ C with w≻_1 w', and ask if there is a profile V^*_-1 such that w wins at V[V^*_-1, u] with t points,whereas w' wins at V[V^*_-1, v]. As in the proof of Theorem <ref>,for each triple (t, w, w')we have to consider a number of possibilities, depending on whether w∈{a, a'},w'∈{b, b'} as well as on the relative positions of w, w', a, a', b, and b' with respect to the tie-breaking order. The analysis is as tedious as it is straightforward; to illustrate the main points, we consider two representative cases. w=a, w'=b, a>a'>b>b'In this case,a wins with t points at V^uif and only if _2(a', V^u_-1)≤ t-1 and for each c∈ C∖{a, a'} we have _2(c, V^u_-1)≤ t if a>c and _2(c, V^u_-1)≤ t-1 if c>a. Suppose that these conditions are satisfied. Then b can win at V^v with t+1 or t points. The former case is possible if and only if _2(b, V^u_-1)= t.The latter case is possible if and only if _2(b, V^u_-1)= t-1, _2(b', V^u_-1)≤ t-1,and for each c∈ C∖{a, b, b'} we have _2(c, V^u_-1)≤ t if b>c and _2(c, V^u_-1)≤ t-1 if c>b.Thus, to decide whether this situation is possible, we have to call ' twice. For our first call, we setC^[1]={c∈ C∖{a', b}| a > c},C^[0]={c∈ C∖{a'}| c > a}∪{a'},C^[-1]=C^[-2]=∅ and call'(G, a, b, t-1, t, C^[1], C^[0], C^[-1], C^[-2]).For our second call, we set C^[1]={c∈ C∖{b'}| b > c},C^[0]= {c∈ C∖{a}| c > b}∪{b'},C^[-1]=C^[-2]=∅ and call'(G, a, b, t-1, t-1, C^[1], C^[0], C^[-1], C^[-2]).w∉{a, a', b, b'}, w'=b, b' > b > a' > w > aIf w wins at V^u with t points, this means that_2(w, V^u_-1)=t, _2(a', V^u_-1)≤ t-2, _2(a, V^u_-1)≤ t-1, _2(c, V^u_-1)≤ t for all c∈ C∖{w, a, a'} with w > c, and _2(c, V^u_-1)≤ t-1 for all c∈ C∖{w, a, a'} with c > w. Suppose that these conditions are satisfied. As w still receives t points at V^v,this means that b wins at V^v if and only if _2(b, V^u_-1)=t-1, _2(b', V^u_-1)≤ t-2. Thus, we set C^[1]=∅, C^[0]={c∈ C∖{a}| w > c},C^[-1]={c∈ C∖{a', b, b'}| c > w}, C^[-2]={a', b'} and call'(G, w, b, t, t-1, C^[1], C^[0], C^[-1], C^[-2]). Just as for Plurality, we obtain the following corollary that describes the complexityof finding and testing level-2 strategies and improving strategies under 2-approval. Given a GS-game G=(V, _2, (A_i)_i∈ N(V, _2)), where for each i∈ N the set A_i consists of i's truthful vote and a subset of his minimal manipulations,and a strategy v_1'∈(C) of player 1 we can decide in polynomial time whether v_1' is a level-2 strategy and whether v'_1 is an improving strategy.Moreover, we can decide in polynomial time whether player 1 has a level-2 strategyor an improving strategy in G. We remark that the minimality assumption plays an important role in our analysis. Indeed, in the absence of this assumption a promoter i may manipulate by swapping two different candidates (one of which is his most preferred 2-competitive candidate p) into the top two positions. Let v_i[_2(v_i);{p, c}] be some such manipulation. If we try to model this possibility via a network flow construction, we would have to add edges from i to both p and c; the lower bounds on these edges would have to be set to 0,to allow i to vote truthfully. However, there may then be a flow that uses the edge (i, c), but not (i, p), which corresponds toa vote that promotes c, but not p; such a vote is not a level-1 strategy.Interestingly, a level-2 player may want to swap two candidates into the top two positions,even if he assumes that all level-1 players use minimal strategies. In fact, the following example showsthat a strategy of this form may weakly dominate all other non-equivalent strategies. Let the profile of sincere preferences be as in Table <ref>, and assume thatthe voting rule is 2-approval and the tie-breaking order is given bya>b>c>d>…. Assume that player 1 is the level-2 player. The winner under 2-approval is a with two points; candidates b, c, and d also have two points each. Voters 1, 4 and 5 are GS-manipulators; voter 4 may manipulate by swapping c into top two positions, and voter 5 may manipulate by swapping d into top two positions.Consider the GS-game where N={1, 4, 5}, A_4={v_4, v_4[b;c]},and A_5={v_5, v_5[b;d]} (note that both A_4 and A_5 only contain a proper subset of the respective player's minimal manipulations; for instance, v_4[u_1;c]∉A_4).We claim that v_1[{u_5, u_6};{b, c}] is a weakly dominant strategy for player 1. Indeed, consider the four possible scenarios: * Players 4 and 5 are truthful. Then the best outcome that voter 1 can ensure is that b wins.* Player 4 is truthful, but player 5 manipulates. Then the best outcome that voter 1 can ensure is that c wins.* Player 4 manipulates, but player 5 is truthful. Then the best outcome that voter 1 can ensure is that c wins.* Players 4 and 5 both manipulate. Then the best outcome that voter 1 can ensure is that c wins.Now, it is clear that only votes that rank b and c in top two positions achieveall of these objectives simultaneously. § K-APPROVAL FOR K≥ 3Regrettably, our analysis of k-approval under the minimality assumption does not extendfrom k=2 to k=3. Specifically, the argument breaks down when we consider a potential demoter under 3-approval who can only help his top candidate by swapping his second and third candidateout of the top three positions. If he chooses to manipulate, he has to perform bothof these swaps at once; he can also remain truthful and not perform any swaps.It is not clear how to capture this all-or-nothing behavior via network flows. We conjecture that finding a level-2 strategy under 3-approval is computationally hard, even under minimality assumption. We will now prove a weaker result, showing that this problem is NP-hard for k-approval with k≥ 4 (and without the minimality assumption). Moreover, we will also show that it is coNP-hard to decide whether a given strategy is improving. For every fixed k≥ 4,given a GS-game G=(V, _k, (A_i)_i∈ N) and a strategy v of voter 1, it is NP-hard to decide whether v is a level-2 strategy,and it is coNP-hard to decide whether v is an improving strategy. Our hardness proof proceeds by a reduction from the classic NP-complete problem Exact Cover by 3-Sets (X3C). An instance of this problem is given by a ground set Γ = {g_1, …,g_3ν} and a collection Σ = {σ_1, …, σ_μ} of 3-element subsets of Γ. It is a `yes'-instance if there is a subcollection Σ'⊆Σ with |Σ'|=ν such that ∪_σ∈Σ'σ = Γ, and a `no'-instance otherwise.We will first establish that our problems are hard for k=4; towards the end of the proof, we will show how to extend our argument to k>4.Consider an instance I^0 = (Γ^0, Σ^0) of X3C with |Γ|=3ν'.We will first modify this instance as follows. We add three new elements to Γ^0 and a set containing them to Σ^0. We then add ν'+2 triples x_i, y_i, z_i,i∈[ν'+2], of new elements to Γ^0 and for each such triple we addthe set S_i= {x_i, y_i, z_i} to Σ^0. Finally, we add sets S'_i = {y_i, z_i, x_i+1},i∈[ν'+1], and S'_ν'+2 = {y_ν'+2, z_ν'+2, x_1} to Σ^0. We then renumber the elements of the ground set so that the elements added at the first step are numbered g_1, g_2, g_3. We denote the resulting instance by (Γ, Σ), and let ν=|Γ|/3, μ=|Σ|. Clearly, I = (Γ, Σ) is a `yes'-instance of X3C if and only if I = (Γ^0, Σ^0) is. We let Σ={S_i, S'_i| i∈[ν'+2]};we have ν = 2ν'+3, |Σ|=2(ν'+2).In what follows, when writing X ≻ Y in the description of an order ≻, we mean that all elements of X are ranked above all elements of Y, but the order of elements within X and within Y is not specified and can be arbitrary.We construct a GS-game as follows. We introduce a set of candidates C'={c_1, …, c_3ν} that correspond to elements of Γ,three special candidates w, p, c, and, finally, a set of dummy candidatesD = ⋃_i=0^μ D_i ∪ D_c ∪⋃_j=1^ν+1 E_j ∪⋃_i=1^3ν⋃_j=1^ν+1 F_i, j,where |D_i|=4 for i=0, …, μ, |D_c|=3, and |E_j|=2, |F_i, j|=3 for i∈ [3ν], j∈ [ν+1].Thus, the set of candidates is C={w,p,c}∪ C'∪ D. We define the tie-breakingorder > on C by setting w>c>p>c_1>…>c_3ν>D. For each j∈[μ], we let C_j={c_i| g_i∈σ_j}. The profile V consists of 2+μ+(3ν+1)(ν+1) votes defined as follows:z_0= D_0≻ p≻ c_1≻ c≻ C'∖{c_1}≻…,z_i= D_i≻ C_i ≻ c ≻…,i∈ [μ], u= c≻ D_c ≻ w ≻…, u_j = w≻ p ≻ E_j ≻…,j∈ [ν+1], u_i, j = c_i ≻ F_i,j≻ w ≻… i ∈ [3ν], j∈ [ν+1].We have _4(w, V)=_4(p, V)=_4(c_i, V)=ν+1 for all i∈ [3ν], _4(c, V)=1, and _4(d, V)≤ 1 for each d∈ D. Thus, w wins under 4-approval because of the tie-breaking rule.We have S(V,4)=C'∪{p}. The set of GS-manipulators in this profile consists of the first μ+1 voters; we assume that the first voter (i.e., voter 0) is the level-2 voter.We now define a GS-game for this profile by constructing the players' sets of strategies as follows:z'_i = z_i[D_i; C_i∪{c}],A_i = {z_i, z'_i} for all i∈[μ]. Observe that for each i∈ [μ] the vote z'_i is a level-1 strategy for voter i, which makes i's top candidate in C_i the winner with ν+2 points (note that voter i orders C_i in the same way as > does,so tie-breaking favours i's most preferred candidate in C_i). This completes the description of our game G.Fix some d,d'∈ D_0 and let z'_0 = z_0[{d, d'}; {p, c}], z”_0=z_0[d;p].Note that both z'_0 and z”_0 are level-1 strategies for voter 0,which make p the winner with ν+2 points.Clearly, we can construct the profile V and the players' sets of strategiesin polynomial time given I.We will now argue that z'_0 is an improving strategy if and only if I=(Γ, Σ) is a `no'-instance of X3C,and that z”_0 is a level-2 strategy if and only if I=(Γ, Σ) is a `yes'-instance of X3C.As a preliminary observation, consider some strategy z of voter 0 such that _4(z) consists of pand three dummy candidates. By construction,for every profile of other players' strategies, z and z_0” result in the same outcome. Moreover, if everyone except voter 0 votes truthfully, voter 0 strictly prefers z”_0 to every strategy ẑ with _4(ẑ)⊆ D. Thus, z”_0 can only be weakly dominated by a strategy that places at least one candidate in C'∪{c, w}in top 4 positions.Suppose first that I is a `yes'-instance of X3C. Fix a subcollection Σ' witnessing this, and consider a profile V' where the GS-manipulators that correspond to sets in Σ'vote strategically, whereas everyone else votes truthfully.We have _4(p, V')=_4(c, V')=_4(w, V') =ν+1, _4(c_i, V')=ν+2 for all c_i∈ C',so c_1 wins.However, if voter 0 changes his vote to z'_0, the winner would be c, and voter 0 prefers c_1 to c, so voter 0 strictly prefers voting z_0 over voting z'_0 in this case,i.e., z'_0 is not an improving strategy.Now, if voter 0 changes her vote to z”_0 instead, p becomes the election winner,which is the best feasible outcome from voter 0's perspective. The only way for voter 0 to achieve this outcome is to rank p and some dummy candidates in the top 4 positions; any vote z with _4(z)∩ (C'∪{c, w})≠∅ is strictly worse for voter 0, and hence cannot weakly dominate z”_0. As we have already observed that no strategy zwith _4(z)∩ (C'∪{c, w})=∅ can weakly dominate z”_0, it follows that if I is a `yes'-instance of X3C then z”_0 is a level-2 strategy.On the other hand, suppose that I is a `no'-instance of X3C. Consider a strategy profile V^* in G, and letΣ” be a subcollection of Σ that corresponds to players in [μ]who vote non-truthfully in V^*; we know that Σ” is not an exact cover of Γ. We will argue that voter 0 weakly prefers z'_0 to both z_0 and z”_0 for every choice of Σ”, and there are choices of Σ” for which this preference is strict.If Σ”=∅, i.e., all voters in [μ] are truthful, then voter 0 benefits from changing his vote from z_0 to z'_0, as this vote makes p the winner.Similarly, suppose that all sets in Σ” are pairwise disjoint (and hence |Σ”| ≤ν-1). Then candidate c gets at most ν points and the winner in V[V^*_-0, z_0] is one of the candidates from C' (with ν+2 points). On the other hand, the winner in V[V^*_-0, z'_0]is p (with ν+2 points), so voter 0 benefits from changing his vote to z'_0. In both of these cases, z”_0 has the same effect as z'_0.Now, suppose that the sets in Σ” are not pairwise disjoint.Let X be the set of elements that appear in the largest number of sets in Σ”, and let g_ℓ be the element of X with the smallest index. Note that g_ℓ≠ g_1, since we modified our instance of X3C so thatg_1 only occurs in one set. The winner in V[V^*_-0, z_0] is either c_ℓ or c, and the winner's score is at least ν+3. Suppose that voter 0 changes his vote from z_0 to z'_0. If the winner in V[V^*_-0, z_0] was c, this remains to be the case,and if the winner was c_ℓ then either c_ℓ remains the winner or c becomes the winner,and voter 0 prefers c to c_ℓ. Thus, in this case voting z'_0 is at least as good as voting z_0, and voting z”_0 has the same effect as voting z_0.We conclude that whenever Σ” is not an exact cover of Γ, voting z'_0 is at least as good as voting z_0 or z”_0.It remains to establish that z'_0 is sometimes strictly better than either of these strategies. To this end, suppose that Σ”=Σ.If voter 0 votes z'_0, then the scores of the candidates covered by sets in Σ are ν+3, the score of c is 1+2(ν'+2) +1 =2ν'+6=ν+3, and all other candidates have lower scores, so c wins. However, if voter 0 votes z_0 or z”_0, c's score is ν+2, and therefore the winner is a candidate in C'. Thus, in those circumstances, voter 0 strictly prefers z'_0 to both z_0 and z”_0. Hence, if I is a `no'-instance of X3C, z'_0 weakly dominates z_0 and z”_0,and hence z”_0 is not a level-2 strategy. This completes the proof for k=4. For k>4, we modify the construction by introducing |V| additional groups of dummy candidates H_1, …, H_|V| of size k-4 each. We renumber the voters from 1 to |V| and modifythe preferences of the i-th voter, i∈ [μ], by inserting thegroup H_i in positions 5, …, k, and adding all other new dummy candidates at the bottom of his ranking. Then the k-approval scores of all candidates in C remain the same as in the original construction, and the k-approval score of each new dummy candidate is 1. The rest of the proof then goes through without change.We note that the strategies of level-1 players in our hardness proof are not minimal; determining whether our hardness result remains true under the minimality assumptionis an interesting research challenge. Our complexity lower bounds are not tight: we do not know whether the computational problems we consider are in, respectively, NP and coNP. The following argument provides upper boundson their complexity.For every n-player game G=(N, (A_i)_i∈ N, (≽_i)_i∈ N),where each relation ≽_i is represented by a polynomial-time computable function of its arguments, andfor every pair of strategies u, v of player 1, the problem of deciding whether u weakly dominates vbelongs to the complexity class DP (difference polynomial-time) <cit.>.Indeed, u weakly dominates v if and only if(a)for every profile P_-1 of other players' strategies we have (P_-1, u)≽_1 (P_-1, v) (which can be checked in coNP), and(b)for some profile P_-1 of other players' strategies we have (P_-1, u)≻_1 (P_-1, v) (which can be checked in NP), i.e., the language associated with our problem is an intersection of an NP-language and a coNP-language.Thus, for every GS-game based on a polynomial-time voting rule (including k-approval) the problem of checking whether a given strategy is improving is in DP. This also means that for k-approval with a fixed value of k the problem of checking whether a given strategy is a level-2 strategy belongs to the boolean hierarchy <cit.>,as there are only m k≤ m^k pairwise non-equivalent votes, andit suffices to check that none of these votes weakly dominates the given strategy.§ CONCLUSIONS AND FURTHER RESEARCHWe have initiated the analysis of voting games from the perspective of the cognitive hierarchy. We have adopted a distribution-free approach that uses the concept of weak dominance in order to reason about players' actions. The resulting framework is mathematically rich, captures some interesting behaviours,and presents a number of algorithmic challenges, even for simple voting rules. To illustrate this, we focused on a well-known family of voting rules, namely, k-approval with k≥ 1, and investigated the complexity of finding level-2 strategies and improving strategies under various rules in this family. For Plurality, i.e., for k=1,level-2 strategies and improving strategies are easy to find, and for k≥ 4 these problems are computationally hard,but for k=2, 3 we do not have a full understanding of the complexity of these problems.We identify a natural assumption (namely, the minimality assumption), which is sufficient to obtain an efficient algorithm for k=2; however, it isnot clear if it remains useful for larger values of k.We list a few specific algorithmic questions that remain open: * Is there a polynomial-time algorithm for computing level-2 strategies and improving strategies under 2-approvalwithout the minimality assumption?* Does Theorem <ref> remain true under the minimality assumption?* What can be said about 3-approval, with or without the minimality assumption?* What can be said about other prominent voting rules, most importantly the Borda rule? In our analysis, we have focused on level-1 and level-2 voters. It would also be interesting to extend our formal definitions to level-ℓ players for ℓ≥ 3 andto investigate the associated algorithmic issues.While it is intuitively clear that the view of the game for these players will be more complex,it appears that for Plurality our algorithm can be extended in a straightforward manner;however, it is not clear if this is also the case for 2-approval. Another interesting question, which can be analysed empirically, is whether truthful voting is likelyto be a level-2 strategy, or, more broadly, how many votes in (C) are level-2 strategies; again,this question can also be asked for level-ℓ strategies with ℓ≥ 2.Note that the analysis in our paper contributes algorithmic tools tools to tackle this issue.A yet broader question, which can only be answered by combining empirical data and theoretical analysis, is whether the cognitive hierarchy approach provides a plausible description of strategic behavior in voting. While our paper makes the first steps towards answering it,there is more to be done to obtain a full picture.abbrvnat | http://arxiv.org/abs/1707.08598v1 | {
"authors": [
"Edith Elkind",
"Umberto Grandi",
"Francesca Rossi",
"Arkadii Slinko"
],
"categories": [
"cs.GT"
],
"primary_category": "cs.GT",
"published": "20170726182212",
"title": "Cognitive Hierarchy and Voting Manipulation"
} |
Serious Games Application for Memory Training Using Egocentric Images Gabriel Oliveira-Barra1 Marc Bolaños1 Estefania Talavera1,2 Adrián Dueñas1 Olga Gelonch3 Maite Garolera3 December 30, 2023 ============================================================================================================= We investigate the nearest common ancestor (NCA) function in rooted trees. As the main conceptual contribution, the paper introduces universal trees for the NCA function: For a given family of rooted trees, an NCA-universal tree S is a rooted tree such that any tree T of the family can be embedded into S such that the embedding of the NCA in T of two nodes of T is equal to the NCA in S of the embeddings of the two nodes.As the main technical result we give explicit constructions of NCA-universal trees of size n^2.318 for the family of rooted n-vertex trees and of size n^1.894 for the family of rooted binary n-vertex trees. A direct consequence is the explicit construction of NCA-labeling schemes with labels of size 2.318log_2 n and 1.894log_2 n for the two families of rooted trees. This improves on the best known such labeling schemes established by Alstrup, Halvorsen and Larsen [SODA 2014]. Keywords: Rooted Trees, NCA, Nearest Common Ancestor, Lowest Common Ancestor, Universal Trees, Labeling Schemes, Embedding Schemes§ INTRODUCTIONThe nearest common ancestor[In the literature, the nearest common ancestor of two vertices in a rooted tree is sometimes also referred to as the lowest or least common ancestor (LCA).] (NCA) of two vertices u and v of a rooted tree T is the first common vertex of the paths connecting u and v to the root of T. Finding the nearest common ancestor appears as an essential operation in many algorithms and applications (see for example the survey by Alstrup, Gavoille, Kaplan, and Rauhe <cit.>). NCA-Universal Trees. The present paper introduces the notion of NCA-universal trees as a novel tool to study and algorithmically deal with the NCA function in rooted trees. We define an NCA-universal tree S for a family of rooted trees 𝒯 as such that every tree T∈𝒯 can be embedded into S such that the NCA function is preserved by the embedding. More formally, an embedding of T into S is an injective mapping φ_T of V(T) into V(S) such that the embedding function φ_T and the NCA function commute. NCA-Labeling Schemes. As an immediate application of an NCA-universal tree S for a family 𝒯 of rooted trees, S directly implies an NCA-labeling scheme <cit.> for the family 𝒯. Generally, a labeling scheme is a way to preprocess the structure of a graph to later allow simple and fast queries. A labeling scheme consists of an encoder and a decoder, where the encoder must be able to label a family of graphs such that the decoder can answer queries, given just the labels and no additional information about the underlying graph. More specifically, an NCA-labeling scheme assigns a unique label to each node of a rooted tree T such that given the labels of two vertices u and v of T, it is possible to compute the label of the NCA of u and v in T. If an NCA-univeral tree S for a family 𝒯 of rooted trees is given, we can get an NCA-labeling scheme for 𝒯 as follows. Let |S| be the number of vertices of S and assume that the vertices of S are labeled from 0 to |S|-1 in a arbitrary fixed way. Given an embedding of a tree T∈𝒯 into S, we then get the labeling of a vertex v of T by using the label of the vertex x of S to which v is embedded. The size of the labels (in bits) of the labeling scheme is therefore exactly ⌈log|S|⌉. We remark that throughout the paper, all logarithms are to base 2. Contribution. We show that the family of all rooted trees with at most n vertices has an NCA-universal tree of size O(n^2.318) and that the family of all binary rooted trees with at most n vertices has an NCA-univeral tree of size O(n^1.894). This implies that the families of rooted n-vertex trees and of rooted n-vertex binary trees have labeling schemes with labels of size 2.318log n and 1.894log n, respectively. This improves on the best previous NCA-labeling schemes that were developed by Alstrup, Halvorsen and Larsen <cit.> and which require labels of size 2.772log n for general rooted trees and of size 2.585log n for binary rooted trees. In <cit.>, it is also shown that any NCA-labeling scheme for general n-vertex rooted trees requires labels of size at least 1.008log n.As we show how to explicitly construct the NCA-universal trees, our labeling schemes are constructive. Note that the best NCA-labeling schemes of <cit.> are not constructive and that the best previous constructive NCA-labeling scheme for n-vertex rooted trees requires labels of size 3log n. Further, our NCA-labeling schemes are efficient, the embedding of a rooted tree into the constructed NCA-universal tree can be computed efficiently and a single query can be answered in time O(log^2n) (O(log n) for binary trees). We believe that our new NCA-labeling schemes are not only interesting because they improve upon the best existing schemes, but also because our approach leads to more intuitive and significantly simpler constructions. Related Work. Graph labeling schemes are an elegant way to store structural information about a graph. As every vertex is only assigned a small label, the information is stored in a completely distributed way and graph labelings therefore are particularly interesting in a distributed context, where labeling schemes are used for various kinds of graph queries <cit.>. In addition, labeling schemes can be used in a context where extremely large graphs are processed and where accessing the data is expensive. To answer a pair-wise query, only the two labels of the corresponding vertices need to be accessed. The first labeling schemes that appear in the literature are adjacency labeling schemes (given the labels of two vertices, determine whether the vertices are adjacent). They were introduced among others by Breuer <cit.> and Folkman <cit.>. In the context of adjacency labeling schemes, it is well known that they are tightly connected to induced universal graphs. A graph G is an induced universal graph for a graph family ℋ if G contains every graph H∈ℋ as an induced subgraph. Induced universal graphs were first described by Rado <cit.> and Kannan et al.<cit.> noted the equivalence between adjacency labeling schemes and induced universal graphs.When considering the family of rooted trees we are interested indifferent queries such as whether a vertex is an ancestor of the other. Ancestry labeling has been studied and labeling schemes of size log(n) + Θ(log(log(n))) are known to be tight <cit.>. If the tree has low depth, then a scheme of size log(n) +2log(d) +O(1) is known, where d is the depth of the tree <cit.>.For NCA-labeling schemes, a linear-size labeling scheme that answers queries in constant time was introduced by Harel and Tarjan in <cit.>. In the following, there was a series of significant improvements in <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and most recently in <cit.>. In particular in <cit.>, a lower bound for NCA labeling schemes of 1.008 log n is shown, which separates NCA-labelings, that need labels of size log n + Ω(log n), from ancestry labeling schemes, where labels of size log n + O(loglog n) are sufficient. There is concurrent work by Gawrychowski and Łopuszański <cit.> who reach the exact same bounds for labeling schemes and construct almost identical universal trees as we do. The proof method is a bit different and in their paper they also have lowerbounds for the size of a universal tree for the NCA function. Outline. The rest of the paper is organized as follows. In the remainder of this section, we first formally define the problems and state our results in <Ref>. In <Ref>, we first prove a simpler upper bound of O(n^2) on the size of NCA-universal trees for the family of binary rooted trees. We extend the construction of <Ref> to obtain the stronger and more general results stated above in <Ref>. Finally, in <Ref>, we sketch how to efficiently implement our labeling scheme and in <Ref>, we conclude the paper and discuss some open issues.§.§ DefinitionsWe next define the necessary graph-theoretic concepts and notation and we in particular formally introduce the notion of universal trees for the NCA function. In the following let 𝒯_n be the family of unlabeled rooted trees with at n vertices and ℬ_n the family of unlabeled rooted binary trees with n vertices. In a rooted tree T=(V,E) a vertex u is an ancestor of a vertex v if u is contained in the (unique) path from v to the root of T. Note that according to the above definition, a node u is an ancestor of itself. Let T=(V,E) be a rooted tree. For a pair of vertices u and v their nearest common ancestor (NCA) _T(u,v) is the unique common ancestor that is furthest from the root of T. With this notation at hand we can define the notion of a universal tree for NCA.A rooted tree S is called an NCA-universal tree for a family of rooted trees 𝒯, if for every tree T∈𝒯 there is an embedding function φ_T:V(T)↦ V(S) such that φ_T commutes with the NCA function, i.e., for all u,v∈ V(T), φ_T(_T(u,v)) =_S(φ_T(u),φ_T(v)). Hence, the embedding has the property that the NCA of two nodes u and v of T is mapped to the NCA of φ_T(u) and φ_T(v) in S. Note that we do not require the root of T to be embedded to the root of S. In the following, a rooted tree that is universal for the NCA function is also called an NCA-universal tree.An NCA-labeling scheme for a family of rooted trees 𝒯 is a pair of functions called the encoder (f) and decoder (g) with f : { v|v∈ T ∈𝒯}↦ [m] and g:[m]× [m] ↦ [m] satisfying the following properties. * for every T∈𝒯 and every u,v∈ V(T), f(u)≠ f(v) and* for every T∈𝒯 and every u,v∈ V(T), g(f(u),f(v))=f(_T(u,v)).For a node v∈ T∈𝒯, f(v) is called the label of v. The size of the labeling scheme defined by f and g is ⌈log m⌉, i.e., the number of bits required to store the largest label. Given an NCA-universal tree S for a family of rooted trees 𝒯, we directly obtain an NCA-labeling scheme for 𝒯.Let S be an N-vertex rooted tree that is universal for the NCA function and the family 𝒯 of rooted trees. Then, there exists an NCA-labeling scheme of size ⌈log N⌉ for 𝒯. We assign unique names from 0 to N-1 to the N vertices of S. Consider a tree T∈𝒯 and let f be an embedding of T into S. The label of a vertex v of T is the name assigned to vertex f(v) of S. Given the labels x_u∈0,…,N-1 and x_v∈0,…,N-1 of two nodes u and v of T, the decoder outputs the name x_w∈0,…,N-1 of the NCA of the vertices u' and v' with names x_u and x_v in S.§.§ Main Results Our main result is an explicit construction of universal trees for the families of all trees and binary trees with n vertices. The same bounds can be found in the concurrent work of Gawrychowski and Łopuszański <cit.>. Let n∈ℕ. Then: * There is a rooted tree S_n of size less than n^ 2.318 that is universal for the NCA function and the set 𝒯_n of rooted trees of size n. * There is a rooted tree S_n^bin of size less than n^1.894 which is universal for the NCA function and the set ℬ_n of rooted binary trees of size n. The proof can be found in Section 3. The direct implication of this for labeling schemes is summarized in the following statement. For any n∈ℕ there exists an NCA-labeling of size less than 2.318log n and an NCA-labeling for binary trees of size less than 1.894log n.This is an improvement of the current best known bound of 2.772log n from <cit.>. Further for the specific case of binary trees of particular interest is that the constant is now below 2 and therefore will likely not be an integer. This is exactly what we have shown in <Ref> and therefore follows from <Ref>. Although this is just an existential proof, from the construction we will see later, it is clear that a reasonably fast algorithmic implementation is possible and we will give a sketch in <Ref>.§ BASIC UNIVERSAL TREE CONSTRUCTIONThe NCA-universal trees of <Ref> are constructed recursively.Before proving the general statements of <Ref>, we describe a simpler, slightly weaker construction that provides an NCA-universal tree of size O(n^2) for n-vertex binary trees. The full constructions required to prove <Ref> appears in <Ref>. For any n∈ℕ there exists a rooted tree S_n of size less than n^2 which is universal for the NCA function and the rooted binary trees of size at most n. Our recursive universal tree construction requires two kinds of NCA-universal trees. In addition to ordinary unlabeled rooted trees, we also need to define NCA-universal trees for the family of rooted binary trees where one leaf node is distinct (marked). Recall that ℬ_n denotes the set of all n-vertex unlabeled rooted binary trees, so let ℬ_n' denote the family of unlabeled rooted binary trees on at most n vertices and with one marked leaf. A rooted tree S' with one marked leaf vertex w is called a NCA-universal tree for a family of rooted trees 𝒯' with one marked leaf if for every tree T∈𝒯', there exists an embedding function φ_T : V(T) ↦ V(S') that maps the marked leaf of T to the marked leaf w of S' and where φ_T commutes with the NCA function, i.e.,φ_T(_T(u,v)) =_S(φ_T(v),φ_T(u)) for all u,v∈ T.As in <Ref>, we do not require that the root of T is mapped to the root of S'. Overview of the Construction. The construction of the NCA-universal tree for binary rooted trees is done recursively as illustrated in <Ref>. The universal tree S_n for binary trees of size at most n consists of three NCA-universal trees for binary trees of size at most n/2, where one of these three universal trees needs to work for the more general family trees with one marked leaf vertex. Universal trees for the family of n-vertex trees with a marked leaf are constructed recursively in a similar way. They consist of two NCA-universal trees for n/2-vertex binary trees with a marked leaf and of a single NCA-univeral tree for ordinary n-1-vertex binary trees (see <Ref>).In order to show that the recursive construction of <Ref> results in an NCA-universal tree we need to argue that any n-vertex binary tree T can be embedded. To achieve this, we show that any rooted tree T has a vertex v such that v splits T into three subtrees of size at most n/2. Vertex v is then embedded to the vertex marked in red in the left part of <Ref>. The three subtrees of T induced by v are then embedded recursively into the three parts of the universal tree construction. Further, we need to show that any rooted tree T with a marked leaf can be partitioned in a similar way to be consistent with the recursive structure in the right part of <Ref>. In the following, we first give the basic technical lemmas required to partition n-vertex trees T into the required smaller subtrees. Based on these partitioning results, we the analyze the recursive NCA-universal tree construction in more detail and prove <Ref>. As the same partitioning lemmas will also be needed in the general NCA-universal tree constructions in <Ref>, they are stated more generally than what we require for the simple construction of the present section. For every rooted n-vertex tree T and for every parameter λ∈ (0,1], there exists a vertex v∈ V(T) such that removing the edges from v to its children splits the tree into components such that each component rooted at a child of v has size at most ⌊ (1 - λ) · n ⌋ and such the remaining component containing v and the root of T has size at most ⌈λ· n ⌉. We determine v using the following simple iterative procedure. We initialize v to be the root of T. We stop the procedure as soon as v satisfies the conditions of the lemma. For some vertex u of T, let 𝑠𝑖𝑧𝑒(u) be the number of vertices in the subtree rooted at u. If v does not split the tree as required, we let w be the child vertex of v that maximizes 𝑠𝑖𝑧𝑒(w) and we set v:=w. Since v goes from being the root of T to being a leaf of T during this process, the component containing the root goes from being of size 1 to a set of size n. We claim that for the last vertex v where the connected component of the root is still of size at most ⌈λ· n ⌉, the lemma holds.Clearly, the component with the root is of size at most ⌈λ· n ⌉ and it thus suffices to show that all the subtrees of v are of size at most ⌊ (1-λ)n⌋. Assume for contradiction that v has a child w such that 𝑠𝑖𝑧𝑒(w)≥ 1+ ⌊ (1-λ)n⌋. Then, removing all subtrees of w from T would result in a component of size at most n-⌊ (1-λ)n⌋=⌈λ n⌉ and thus v would not be the last vertex for which the connected component of the root is of size at most ⌈λ n⌉.For trees with a marked leaf we can get a similar tree splitting lemma.Given a rooted n-vertex tree T with one marked leaf vertex w and a parameter λ∈(0,1]. If n≥1/1-λ, there exists a vertex v∈ V(T) such that when removing the edges connecting v to its children, T is split into components satisfying the following properties. The component containing the root of T and vertex v has size at most ⌈λ n⌉, the component containing the marked leaf w has size at most ⌊ (1-λ)n⌋, and all other components have size at most n-1.Let r bet the root vertex of T. We choose v to be last vertex on the path from r to w such that when removing the subtrees of v, the remaining component has size at most ⌈λ n ⌉. ⌈λ n ⌉<n, so v cannot be a leaf and thus v≠ w. Let v' be the root of the subtree of v containing w. To prove the lemma, it suffices to show that the subtree rooted at v' has size 𝑠𝑖𝑧𝑒(v')≤⌊(1-λ)n⌋. For the sake of contradiction, assume that 𝑠𝑖𝑧𝑒(v')≥⌊(1-λ)n⌋+1. In this case, the total size of all subtrees of v' is at least ⌊(1-λ)n⌋ and thus removing all subtrees of v' would leave a component of size at most ⌈λ n⌉. This contradicts the assumption that v is the last vertex on the path from r to w for which this is true.For every integer n≥ 1, we show how to construct an NCA-universal tree S_n for the family ℬ_n of n-vertex rooted binary trees and an NCA-universal tree S_n' for the family ℬ'_n of n-vertex binary rooted trees with one marked leaf. We will prove by induction on n that |S_n|≤ n^2 and that |S_n'|≤ 2n^2-1 for all n≥ 1.For the induction base, note that S_1 and S_1' clearly need to only consist of a single vertex and we thus have |S_1|=|S_1'|=1. Thus, the bounds on |S_n| and |S_n'| hold for n=1. For the induction step, assume that n≥ 2 and that |S_k|≤ k^2 and |S_k'|≤ 2k^2-1 for all 1≤ k<n. We build the two NCA-universal trees S_n and S_n' by using smaller NCA-universal trees as given in <Ref>. That is, S_n is composed of one copy of S_⌈ n/2⌉'and two copies of S_⌊ n/2⌋and S_n' is composed of one copy of S_⌈ n/2⌉', S_⌊ n/2⌋', and S_n-1. We need to show that the constructed trees S_n and S_n' are in fact NCA-universal trees and that they satisfy the required size bounds. We first show that the trees are or the right size. Using the induction hypothesis, we have|S_n| =|S_⌈n/2⌉'| + 2·|S_⌊n/2⌋|≤ 2 ·⌈n/2⌉^2-1 + 2·⌊n/2⌋^2≤ n^2,and|S_n'| = |S_⌈n/2⌉'| +|S_⌊n/2⌋'| +|S_n-1| ≤ 2·⌈n/2⌉^2 - 1+2·⌊n/2⌋^2 -1 +(n-1)^2 ≤ 2 n^2-1.It thus remains to prove that the recursive construction of S_n and S_n' allows to find a proper embedding for every T ∈ℬ_n into S_n and every T' ∈ℬ_n into S_n', respectively.To show this, we use <Ref> and <Ref>. We first show how to construct an embedding φ_T of a binary n-vertex tree T∈ℬ_n into S_n. For this purpose, we apply <Ref> with parameter λ=1/2 to tree T. Let v∈ V(T) be the vertex of T that splits the tree such that the part containing the root of T and v has size at most ⌈ n/2⌉ and such that all other components have size at most ⌊ n/2⌋. For the embedding take the splitting vertex v∈ T given by <Ref> which will be embedded to the marked vertex of the copy of S_⌈n/2⌉ ' (cf.<Ref>). Consider the three components of T after splitting. By the induction hypothesis there is an embedding function to embed the child components (components below the splitting vertex v) into the two copies of S_⌊n/2⌋. Then we take the component with the root and the marked vertex to get a tree with a single marked leaf of size at most ⌈n/2⌉ and the induction hypothesis again provides with an embedding function of this component into S_⌈n/2⌉ '. The embedding is depicted in <Ref>.To see that the NCA-function for any two vetices u_1 and u_2 and the described embedding function φ_T commute, we need that _S_n(φ_T(u_1), φ_T(u_2)) = φ_T(_T(u_1,u_2)). This can be verified through the following case analysis considering the three components after the splitting process.If u_1 and u_2 are in the same component, then we have embedded them into the same subtree and by the induction hypothesis, we have φ_T(_T(u_1,u_2))=_S_n(φ_T(u_1),φ_T(u_2)). If u_1 and u_2 are in different child components, we have _T(u_1,u_2)=v and the embedding is therefore also correct because is embedded to the vertex marked in red in <Ref>. Finally, if u_1 is from the component containing the root and vertex v and u_2 is a vertex from a child component, we have _T(u_1,u_2)=_T(u_1,v) and similarly_S_n(φ_T(u_1),φ_T(u_2))=_S_n(φ_T(u_1),φ_T(v)) and the embedding is therefore again correct by the induction hypothesis (applied to the partial embedding into the subtree S_⌈ n/2⌉').For the family of n-vertex binary trees with a marked leaf, the embedding into the recursively constructed tree S_n' (cf.<Ref>) works in similar way. Let T' be a binary tree of size at most n and with a marked leaf. We apply <Ref> with parameter λ=1/2 to T' to obtain a vertex v∈ V(T') that splits T' into a) a subtree of size at most ⌈ n/2⌉ that contains the root of T' and that contains v as a leaf vertex, b) a subtree size at most ⌊ n/2⌋ that is rooted at a child of v and contains the marked leaf of T', and c) a (possibly empty) subtree of size at most n-1 rooted at a child of v. The tree T' is embedded into S_n' by embedding vertex v to the center red node separating the three recursive subtrees in <Ref>. The three subtrees resulting after splitting T' are embedded into the three recursively constructed subtrees S_⌈ n/2⌉', S_⌊ n/2⌋', and S_n-1 in the natural way. The proof that the embedding is correct is done in the same way as for the embedding of T into S_n. The details of the embedding are illustrated in <Ref>. § GENERAL UNIVERSAL TREE CONSTRUCTIONThe proof of the basic construction was handled in detail. We now adjust the construction to improve on the exponent in the size of S_n and to deal with general rooted trees. Throughout the section, we omit floor and ceiling functions. They do not change the calculations significantly, but hinder the readability of the proof. As suggested in <Ref> and <Ref>, the adjustment of the basic construction can be made by choosing λ≠ 1/2 for the size of the splitted components.We start with the binary tree case. We apply the same induction as in the proof of <Ref>. In the general case, we prove that |S_k| ≤ k^β and |S_k'| ≤ c · k^β ∀ 1≤ k < n and some constants c and β that will be determined later. For the construction of S_n we take S_λ n ' and attach copies of S_ (1-λ )n and S_n/2 to the marked vertex for some λ∈ (0,1/2]. The construction of S_n' remains the same as in <Ref>. For an illustration, see <Ref>. Note that although in <Ref> the child components can be of size (1- λ) · n, the two components together can have size at most n, so the smaller of the components is always of size at most n/2. We obtain[ |S_n|≤ |S_λ n '| + |S_ (1-λ )n| + |S_n/2| ≤ c(λ n)^β + ((1-λ ) n)^β + ( n/2) ^β!≤ n^β;|S_n'| ≤ |S_n/2'| + |S_n/2'| +|S_n |≤ c( n/2) ^β + c ( n/2) ^β + n^β !≤ c n^β. ]We would like to choose β as small as possible. Note that with the second inequality, we obtainc( n/2) ^β + c ( n/2) ^β + n^β≤ c n^βand we thus get that c ≥1/1-2^β-1.In our construction we are allowed to freely choose λ∈ (0,1/2]. We can thus choose β and λ such that β is minimized and following inequality is still satisfied:(1-λ )n^β + cλ n^β +·n/2^β≤n^β.To achieve this, we choose λ = 0.296149... and the corresponding β≤ 1.89311.... This proves the claim of <Ref> about binary rooted trees. In a general tree, a vertex can have many children. Therefore we adjust the construction to deal with this fact as shown in <Ref>. Take any λ∈ (0,1/2]. Let S_n be composed of a copy of S_λ n' and attached to the marked vertex of that tree copies of S_(1-λ)n, S_n/2, S_n/3, S_n/4, etc., up to S_1. Similarly, let S_n' be composed of a copy of S_n/2' and attached to the marked vertex of that tree copies of S_n/2', S_n, S_n/2, S_n/3, S_n/4, etc., up to S_1.For the embedding, note that we can sort the child components by size.<Ref> states that any child component is of size at most (1-λ) · n. In addition, the total size of all components cannot add up to more than the entire tree of size n. This implies that after ordering the components by size, the i^𝑡ℎ child component without a marked vertex is of size at most n/i.With the induction hypothesis that |S_k| ≤ k^β and |S_k| ≤ c · k^β, the recursion gives[ |S_n| ≤ |S_ (1-λ )n| + |S_λ n '| +∑_i=2^n |S_n/i| ≤( (1-λ )n)^β + c (λ n) ^β +( ζ(β) -1)n^β !≤n^β;|S_n'| ≤ |S_n/2'| + |S_n/2'| + ∑_i=1^n |S_n/i| ≤ c( n/2) ^β + c ( n/2) ^β + ζ(β) n^β !≤ c n^β, ], where ζ(β) is the Riemann zeta function (ζ(β) =∑_i≥1i^-β ). Again we can deduce from the second inequality that c ≥ζ(β) /1-2^β-1. By using λ = 0.341395..., we get that β≤ 2.31757... .In both constructions, the fact that the NCA function and the embedding function commute follows in the same way as in the proof of <Ref>.This concludes the proof of <Ref>. § IMPLEMENTATION OF THE NCA-LABELING SCHEME The labeling schemes described in <Ref> can be constructed efficiently. Further, given two labels, the label of the nearest common ancestor can be determined in O(log^2n) time(in O(log n) time in the binary tree case). For R ∈ℝ we write [R] := {x: 1 ≤ x ≤ R}. In order to assign labels to the vertices of S_n we proceed as follows. Set s(n) =n^β, andrecall that |S_n| ≤ s(n) and |S'_n| ≤ c s(n). Moreover, as depicted in <Ref> the tree S_n is composed out of the n+1 treesT_0 = S'_λ nT_1 = S_(1-λ)nT_ℓ= S_n/ℓ,where2 ≤ℓ≤ n.Define the corresponding counting sequencet_-1= 0 t_0 = cs(λ n) t_1 = t_0 + s((1-λ)n) t_ℓ= t_ℓ-1 + s(n/ℓ),where2 ≤ℓ≤ n.For S'_n we proceed similarly. As depicted in <Ref> S'_n is composed out of the n+2 treesT'_-1= S'_n/2T'_0 = S'_n/2T'_ℓ= S_n/ℓ,where1 ≤ℓ≤ n.and the corresponding counting sequence is given byt'_-2= 0 t'_-1= cs(n/2) t'_0 = t_-1 + cs(n/2) t'_ℓ= t_ℓ-1 + s(n/ℓ),where1 ≤ℓ≤ n.Given these sequences, in order to assign labels to the vertices in S_n we assign to the vertices of T_i, 0 ≤ i ≤ n the labels in [t_i] ∖ [t_i-1]. The assignment is performed recursively, in the sense that as soon the labels in T_i, 0 ≤ i ≤ nare assigned, they are translated by an additive ⌈ t_i-1⌉, so that they all lie (with room to spare) in the required set [t_i] ∖ [t_i-1]. The assignment is performed analogously for S'_n, where we use the corresponding counting sequence instead.Given the label of a vertex in S_n, its location in the tree can be found with this preprocessing in O(log^2 n) time. Indeed, in every step we have to decide in which of the at most n+2 subtrees we have to branch to; however, this can be decided with binary search on the sequences (t_i)_0 ≤ i ≤ n or (t'_i)_-1 ≤ i ≤ n. As the depth of the recursive construction of S_n is O(log n), the claim follows.§ CONCLUSIONWe introduced NCA-universal trees and gave simple recursive constructions of such trees that in particular lead to improved NCA-labeling schemes for rooted trees. The paper leaves several interesting open questions. The current upper bound of 2.318log n bits per label is still quite far from the 1.008log n-bit lower bound proven in <cit.> and it remains an intriguing open problem to close this gap. In addition, given that NCA-universal trees provide an intuitive way to argue about NCA-labeling schemes, it is natural to ask whether the approach can lead to optimal NCA-labeling schemes or whether every NCA-labeling scheme for a given tree family can be turned into an equivalent one that can be characterized by an NCA-universal tree for the tree family. The following observation shows that NCA-universal trees are equivalent to a certain well-structured class of NCA-labeling schemes.We call an NCA-labeling scheme consistent if any three labels can occur together in some tree. More formally, an NCA-labeling scheme is called consistent if it satisfies the following three properties for any 3 possible labels x, y, and z. In the following, g is the decoder function. * If g(x,y)=z, then g(x,z)=z and g(y,z)=z(i.e.,if z is the NCA of x and y, then z is an ancestor of x and y)* If g(x,y)=y and g(y,z)=z, then g(x,z)=z(i.e.,if y is an ancestor of x and zan ancestor of y, then also z is an ancestor of x)* If g(x,y)=y and g(x,z)=z, then g(y,z)∈y,z(i.e., if y and z are ancestors of x, then z is an ancestor of y or y is an ancestor of z)Every NCA-universal tree S for a given family 𝒯 of trees leads to a consistent NCA-labeling scheme for 𝒯 with labels of size ⌈log|S|⌉. Conversely, every consistent NCA-labeling scheme for 𝒯 and with ℓ-bit labels induces an NCA-universal tree of size 2^ℓ for 𝒯. The first claim of the lemma is immediate because S is a tree and therefore any three vertices of S (i.e., any three labels) are consistent.For the second claim, define a directed graph G=(V,E) as follows. The vertex set V of G is the set of labels of the given NCA-labeling scheme. Assume that g is the decoder function of the labeling scheme.We add a directed edge from u∈ V to v∈ V if g(u,v)=v and there is no vertex w such that g(u,w)=w and g(w,v)=v (i.e., if v is the parent of u). We claim that G is a rooted tree.First observe that G is acyclic. Otherwise, by using Property (II) several times, we can find three vertices u, v, and w such that g(u,v)=v, g(v,w)=w, and g(w,u)=u. However from Property (II) we then also have g(u,w)=w, a contradiction.Second, we show that the out-degree of each vertex of G is at most 1. For contradiction, assume that there exists a vertex u that has out-going edges to v and w. We then have g(u,v)=v and g(u,w)=w and by Property (III) of consistent labeling schemes, we thus also have g(v,w)=w or g(w,v)=v. Thus, one of the two edges (u,v) and (u,w) cannot be in G.Finally, we show that there can be at most one vertex with out-degree 0. For the sake of contradiction assume that u and v both have out-degree 0 and let g(u,v)=w. By Property (I), we then also have g(u,w)=w and g(v,w)=w. If w≠ u, this implies that u has out-degree at least 1 and if w≠ v, it implies that v has out-degree at least 1.Hence, G is a rooted tree on the set of labels of the labeling scheme. Because the ancestry relationship of G is consistent with the labeling scheme, G is an NCA-universal tree for the family 𝒯.abbrv § APPENDIX §.§ Induction Basis For n equals one through four it is simple to find a universal tree of size at most n^2. For example:S_n' shall have size at most 2n^2. We want for any n a large tree S_n' with a marked leaf such that we can embed any tree on n vertices with a marked leaf with the nca-function commuting. Again as example and induction basis: It is easy to check that ∀ T ∈ℬ_l and ∀ T' ∈ℬ_l' the NCA-query is equivalent to the NCA-query in S_l or S_l' ∀ l ∈{1,.., 4} resp. | http://arxiv.org/abs/1707.08807v1 | {
"authors": [
"Fabian Kuhn",
"Konstantinos Panagiotou",
"Pascal Su"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170727100533",
"title": "Nearest Common Ancestors: Universal Trees and Improved Labeling Schemes"
} |
Leibniz Universität Hannover, Institut für Theoretische Physik, Appelstr. 2, 30167 Hannover, GermanyCorrelation functions and low-energy excitations are investigated in the asymmetric two-leg ladder consisting of a Hubbard chain and a noninteracting tight-binding (Fermi) chain using the density matrix renormalization group method.The behavior of charge, spin and pairing correlations is discussed for the four phases found at half filling, namely, Luttinger liquid, Kondo-Mott insulator, spin-gapped Mott insulator and correlated band insulator. Quasi-long-range antiferromagnetic spin correlations are found in the Hubbard leg in the Luttinger liquid phase only. Pair-density-wave correlations are studied to understand the structure of bound pairs found in the Fermi leg of the spin-gapped Mott phase at half filling and at light doping but we find no enhanced pairing correlations. Low-energy excitations cause variations of spin and charge densities on both legs that demonstrate the confinement of the lowest charge excitations on the Fermi leg while the lowest spin excitations are localized on the Hubbard leg in the three insulating phases. The velocities of charge, spin, and single-particle excitations are investigatedto clarify the confinement of elementary excitations in the Luttinger liquid phase. The observed spatial separation of elementary spin and charge excitations could facilitate the coexistence of different (quasi-)long-range orders in higher-dimensional extensions of the asymmetric Hubbard ladder. 71.10.FdLattice fermion models (Hubbard model, etc.) 71.10.PmFermions in reduced dimensions (anyons, composite fermions, Luttinger liquid, etc.) 71.27.+aStrongly correlated electron systems; heavy fermions Correlations and confinement of excitations in an asymmetric Hubbard ladder Anas Abdelwahab Eric Jeckelmann Received: date / Revised version: date =========================================================================== § INTRODUCTIONAsymmetric ladders with two inequivalent legs have attracted significant attention in recent years. The one-dimensional (1D) Kondo-Heisenberg model was used to study exotic correlations in stripe-ordered high-temperature superconductors <cit.> and quantum phase transitions in heavy fermions <cit.>.A study of pairing mechanisms in repulsive fermion systems was also based on a two-band Hubbard ladder model <cit.>. Additionally, the effect of asymmetric couplings on exotic spin orders was investigated in a frustrated Heisenberg model <cit.>. Finally, the stability of a Luttinger liquid coupled to an environment was examined using an asymmetric two-chain model <cit.>.The asymmetric Hubbard ladder with one Hubbard leg and one noninteracting (Fermi) leg was first proposed to study proximity effects on antiferromagnetic spin correlations <cit.>. This study was motivated by the coexistence of antiferromagnetism and superconducting correlations in multi-layered high-temperaturesuperconductors. Later, this model was the subject of a more systematic investigation <cit.> that uncovered a rich phase diagram at half filling, although the model was found to be inappropriate for the primary motivation of that work (atomic wires deposited on semiconducting substrates, see <cit.> for recent progress). In particular, some features of the asymmetric Hubbard ladder resemble those of the Kondo-Heisenberg model <cit.> andthe symmetric two-leg Hubbard model <cit.>.Our first investigation <cit.>focused on the analysis of limiting cases as well as the calculation of physical properties such as excitation gaps, density profiles and spectral functions. In the present paper, we discuss correlation functions corresponding to various types of symmetry-breaking orders such as spin density waves (SDW) or pair density waves (PDW). These correlation functions were calculated numerically using the density-matrix renormalization group (DMRG) method <cit.>. Naturally, (spontaneous) long-range order is not possible in the 1D model discussed here. Nevertheless, the coexistence and competition between quasi-long range orders or enhanced fluctuations in two-leg asymmetric Hubbard ladders may yield useful knowledge about the long-range orders induced byproximity effects that play a role in two-dimensional layered systems <cit.>.In addition, we will investigate the distributions of charge and spin on the Hubbard and Fermi legs for low-energy excitations by varying the number of electrons of each spin. Finally, we will discuss the spin and charge velocities of elementary excitations in the Luttinger liquid. These data allow us to understand the spatial separation of elementary charge and spin excitations in the asymmetric Hubbard ladder, in particular in the Luttinger liquid. This spatial separation could facilitate the coexistence of various (quasi-)long-range orders for spin and chargein the Hubbard and Fermi subsystem of (quasi-)two-dimensional extensions of the asymmetric Hubbard ladder.§ MODEL AND METHOD §.§ ModelThe asymmetric Hubbard ladder model consists in one Hubbard leg (y=H) described by a Hubbard chain <cit.>and one Fermi leg (y=F) described by a tight-binding chain. The two legs are connected by a single-particle hopping between adjacent sites. The model is sketched in Fig. <ref>. Its Hamiltonian is H=- t_∥∑_x,y,σ ( c_x+1,y,σ^†c_x,y,σ^†+ c_x,y,σ^†c_x+1,y,σ^† )- t_⊥∑_x,σ ( c_x,F,σ^† c_x,H,σ^†+ c_x,H,σ^†c_x,F,σ^† ) + U ∑_x ( n_x,H,↑-1/2 )( n_x,H,↓-1/2 ) .The parameters t_∥ and t_⊥ describe the nearest-neighbor intra-leg and inter-leg hoppings, respectively. The strength of the electron-electron repulsion is denoted U. The operatorsc_x,y,σ(c^†_x,y,σ) annihilate (create) an electron with spin σ on site (x,y) while n_x,y,σ = c_x,y,σ^† c_x,y,σ^† denote the electron number operators. The rung index x runs from 1 to the ladder length L.The Hamiltonian is particle-hole symmetric, i.e. invariant under the transformation c_x,y,σ→ (-1)^x c^†_x,y,σ. Therefore, a half-filled ladder corresponds to N=2L electrons and its Fermi energy is always equal to 0. Moreover, it is sufficient to consider doping with additional electrons only, i.e. N≥ 2L. In the singlet ground state, the numbers electrons with up and down spins are given by N_↑=N_↓=N/2. We set the energy unit using t_∥=1.In Ref.<cit.> four different phases of the half-filled Hubbard ladder were found for varying electron-electron coupling U and inter-leg hopping t_⊥.The schematic phase diagram is shown in Fig. <ref>. These phases are distinguished by the wave number of their low-energy excitations, see <cit.> for details. Here we just summarize their main features.The first phase (starting from the right-hand side of Fig. <ref>) is a correlated band insulator for large inter-leg hopping t_⊥.(The boundary is t_⊥ = 2t_∥ for U→ 0.) This phase is characterized by charge and spin gaps approaching the same values and increasing linearly with t_⊥.The second phase is a spin-gapped Mott insulator characterized byfinite but different charge, spin and single particle gaps at intermediate values of U and t_⊥.The single-particle gap is larger than the charge gap resulting in a finite pair-binding energy of the order of the spin gap.The gaps are non-monotonic functions of the inter-leg hopping t_⊥ and the lowest excitations have incommensurate wave number in this phase only.These first two phases exhibit some similarities with those observed in the symmetric Hubbard ladder <cit.>. The third phase, called a Kondo-Mott insulator, is found for large repulsive interaction U and weak to intermediate inter-leg hopping t_⊥. It is similar (but not equivalent) to the ground state of the Kondo-Heisenberg model with charge and spin gaps induced by the effective exchange coupling J ∼ t_⊥^2/U on a rung. The last phase is a Luttinger liquid at weak to intermediate electron-electron repulsion U and weak inter-leg hoping t_⊥. It exhibits gapless charge and spin excitationswith different velocities, a characteristic feature of the dynamical separation between charge and spin in Luttinger liquids <cit.>. §.§ MethodThe Hamiltonian (<ref>) is not exactly solvable and field-theoretical methods have not yield much information about asymmetric ladders so far <cit.>. However, two-leg ladders have been studied for more than two decades with great success using DMRG methods <cit.>. DMRG is the most powerful numerical method for 1D dimensional correlated electron systems with short interactions.In this work, the finite-system DMRG was used to calculate the ground-state properties of Hamiltonian (<ref>) as described in <cit.>. The calculations were performed on ladders with open boundary conditions and up to L=200 rungs. We kept up to m=3072 density-matrix eigenstates to reach discarded weights smaller than 10^-6. Moreover, we extrapolated the ground-states energies to the limit of vanishing discarded weights by varying the number of density-matrix eigenstates. We investigated various correlation functions of the asymmetric Hubbard ladder model (<ref>) using DMRG. The DMRG method has often been used to investigate static correlation functions of ladder systems <cit.>. Typically, we can obtain accurate correlation functions for finite system lengths L or for short distances x in infinite systems. Consequently, the asymptotic behavior of correlations must be inferred from the short-range data using a priori knowledge or hypotheses about the system properties. Despite the lower accuracy of DMRG for correlation functions and local densities than for energies, truncation errors are negligible for the results presented here unless otherwise mentioned. Uncertainties are mostly due to finite size and open boundary effects. § CORRELATION FUNCTIONSIn this section we discuss the ground-state correlation functions calculated with DMRG for the four phases found in our analysis of low-energy excitation properties <cit.>. The charge density operatorN(x,y) = n_x,y,↑+n_x,y,↓is used to define the density-density correlation functionC_c^α(x) =⟨ N(x_0,y)N(x_0+x,y^') ⟩ -⟨ N(x_0,y)⟩⟨ N(x_0+x,y^')⟩ .Intra-leg correlations corresponds to α=y=y^'=F for the Fermi leg and α=y=y^'=H for the Hubbard leg while inter-leg correlations (α = ⊥) are given by setting y ≠ y^'.Here, we will discuss intra-leg correlations only because inter-leg correlations are always weaker. The correlation functions are calculated from the middle of the ladder, x_0= L/2, so that open boundary effects affect the results for large distances x only. Similarly, the spin density operator S(x,y) = n_x,y,↑-n_x,y,↓is used to define the spin-spin correlation functionC_s^α(x)= ⟨ S(x_0,y)S(x_0+x,y^') ⟩ .Figure <ref> illustrates the density-density correlations for U=5 and U=8. Note that we use a double logarithmic scale in Fig. <ref>(a) but Fig. <ref>(b) is a semilogarithmic plot. For U=5 and t_⊥=0.1 the system is in the Luttinger liquid phase. Figure <ref>(a) shows a power-law decay in the Fermi leg but an exponential decay in the Hubbard leg.The power-law behavior is expected for Luttinger liquid with gapless charge excitations while an exponential decay is expected for the 1D half-filled Hubbard model.This result confirms that low-energy charge fluctuations are localized on the Fermi leg in this phase. For the parameter sets (U=5, t_⊥=0.5) [in Fig. <ref>(a)] and(U=8, t_⊥=0.5) [in Fig. <ref>(b)], the ladder is in the Kondo-Mott phase and charge correlations decrease exponentially in both legs. Similarly to the findings for the half-filled Kondo-Heisenberg model <cit.>,a charge gap is induced in the Fermi leg by the effective exchange coupling J ∼ t_⊥^2/U between both legs. For stronger U the decay becomes faster in the Hubbard leg but slower in the Fermi leg because the Mott gapincreases with U but the effective exchange coupling decreases. Density-density correlations decay exponentially in the spin-gaped Mott phase with slightly weaker amplitudes in the Hubbard leg as seen in Fig. <ref>(a)forU=5 and t_⊥=1 and in Fig. <ref>(b) for U=8 and t_⊥=1.5. Finally, in the correlated band insulator (not shown) these correlations decay exponentially and similarly fast in both legs, as expected. DMRG truncation and convergence errors are responsible for the saturation(i.e., the apparent long-range correlations) observed in some cases in Fig. <ref> for large distances x whenC_c^α(x)≈ 10^-6 - 10^-8.The strong antiferromagnetic correlations of the Hubbard chain induce antiferromagnetic correlationsin the Fermi leg for t_⊥≠ 0 <cit.>. Spin correlation functions are depicted in Fig. <ref> for the same model parameters as used for the charge correlations in Fig. <ref>. For the Luttinger liquid phase (U=5, t_⊥=0.1), Fig. <ref>(a) shows that the spin correlation function decays with a power-law with exponent -1 in the Hubbard leg.Thus the behavior of the charge and spin correlations in the Hubbard leg resembles that of the 1D Hubbard model <cit.>. The spin correlations in the Fermi leg follow a faster power law than in the Hubbard leg, quite close to the one found for the density-density correlations. This similarity between spin and charge fluctuations suggests that the Luttinger liquid in the Fermi legis only weakly correlated.The spin-spin correlations in the Kondo-Mottphase [shown for (U=5, t_⊥=0.5) in Fig. <ref>(a) and (U=8, t_⊥=0.5) in Fig. <ref>(b)]are weaker in the Fermi legthan in the Hubbard leg but decay at the same rate with an apparent power law in both legs.Actually, they seem to be as strong as in the gapless Luttinger liquid phase. This contradicts the observation of a finite spin gap in our previous study <cit.>.The existence of this gap agrees with previous findings in the half-filled Kondo-Heisenberg model <cit.>, however,and results from the effective exchange coupling J ∼ t_⊥^2/U between both legs. Thus this apparent power-low behavior can only be explained by the small value of the spin gap, resulting in correlation lengths larger than the ladder size that we can simulate with DMRG.The spin correlations of the half-filled asymmetric Hubbard ladder were studied previously in <cit.> for inter-leg hoppings t_⊥ corresponding to the Kondo-Mott phase.An apparent power-law decay was also observed (for smaller ladder sizes than in the present study) leading to the erroneous conclusion that the system must be gapless. The main finding in Ref. <cit.> was a non-monotonic behavior of the induced antiferromagnetic correlations in the Fermi leg with increasing U.Our work confirms this finding and explains it as the result of the competition between the increasing antiferromagnetic correlations in the Hubbard leg and the decrease of the effective rung exchange coupling J ∼ t_⊥^2/U in the Kondo-Mott insulator.A similar problem with apparent power-law SDW correlations occur in the spin-gapped Mott phase for (U=5, t_⊥=1), see Fig. <ref>(a), but for (U=8, t_⊥=1.5) Fig. <ref>(b) shows clearly that the spin-spin correlations decay exponentially. Note that the spin gap is much smaller in this phase of the asymmetric two-leg Hubbard ladder <cit.> than in the symmetric one <cit.> for similar parameters U and t_⊥. Consequently, it is more difficult to examine the asymptotic behavior of spin correlations.In the correlated band insulator (not shown), spin-spin correlation functions always decayexponentially fast.The spin-gapped Mott phase is characterized by a finite pair-binding energy, which is comparable in sizeto the spin gap <cit.>. Furthermore, the spin and charge density profiles show that added electrons (or holes) tendstay close together like a bound pair on the Fermi leg. Both features persist if the ladder is lightly doped, e.g. for four added electrons in a 2×128-site ladder. In contrast, the three other phases do not exhibit any sign of pairing. In particular, the density profiles show that added particles tend to stay away from one another as expected for identical fermions.We have calculated various correlation functions to investigate the nature of this pairing. The features observed in the (lightly doped) spin-gapped Mott phase are reminiscent of the pairingtendency observed in the symmetric Hubbard ladder <cit.>. There, the pairing is related to the so-called d-wave correlations <cit.>.This notion of d-wave order parameter is not meaningful on an asymmetric Hubbard ladder, however.Actually, we have not found any enhanced pairing correlation in this model and we do not understand the nature of the observed pair binding. Among all the pairing correlations that we have examined (singlet and triplet PDW, on-site pairs, doublon-doublon, …), singlet PDW correlations decrease most slowly.The singlet PDW order parameter is defined asΔ^†(x,y)=1/2( c_x,y,↑^†c_x+1,y,↓^†- c_x,y,↓^†c_x+1,y,↑^†).Thus, the (intra-leg) PDW correlation function takes the formC_PDW^α(x)= ⟨Δ^†(x_0,y)Δ^†(x_0+x,y^') ⟩where the notation is similar to (<ref>).It was reported that the 1D Kondo-Heisenberg model away from half filling exhibits a spin-gapped phase with dominant PDW correlations <cit.>. This quasi-long-range order could be the 1D precursor to striped-order in high-temperature superconductors and has attracted much attention in recent years <cit.>.Because of thesimilarity between the Kondo-Heisenberg model and the asymmetric Hubbard ladder in the strong coupling regime <cit.>, we expected to find enhanced PDW correlations in the (lightly) doped ladder (<ref>). In the Kondo-Mott phase, we have not find any sign of pairing, however. Actually, the pair binding energy vanishes in this phase. We think that the major reason for this discrepancyis that the strong rung exchange coupling (i.e., of the order of t_∥), for which enhanced PDW correlations were found in the Kondo-Heisenberg model <cit.>, cannot be realized in the Kondo-Mott phase of the asymmetric Hubbard ladder, where typically J ∼ t_⊥^2/U ≪ t_∥.Figure <ref> shows the PDW correlations for the same model parameters as in Figs. <ref>and <ref>. We found that all PDW correlations decay exponentially at half filling.The strongest PDW correlations occur in the Fermi leg for the Luttinger liquid phase, as shown for (U=5, t_⊥=0.1) in Fig. <ref>(a). In the Kondo-Mott phase PDW correlations are also weaker in theHubbard leg than in the Fermi leg [see (U=5, t_⊥=0.5) in Fig. <ref>(a) and(U=8, t_⊥=0.5) in Fig. <ref>(b)] but decrease at the same rate for large distances x.In contrast, in the spin-gapped Mott insulator [(U=5, t_⊥=1) in Fig. <ref>(a) and (U=8, t_⊥=1.5) in Fig. <ref>(b)] and in the correlated band insulator (not shown) PDW correlations are almost equal in both legs.PDW correlations in the lightly doped spin-gapped Mott phase exhibit a power law with an exponent close to -2 as depicted in Fig. <ref>. Thus they are not (or barely) enhanced in comparison to a noninteracting ladder (U=0) and cannot explain the pair binding observed in the excitation energies and local densities of that phase <cit.>. Density-density correlations (not shown) decrease as fast as the PDW correlations. The dominant correlations seem to be pow-law SDW correlations with exponents close to -1, which are also shown in Fig. <ref>. Here, we cannot decide whether this power-law behavior is real (as in the Luttinger liquid phase) or a finite-size effect (as in the Kondo-Mott insulator) because we could not perform an accurate finite-size scaling of the spin gap with the accessible ladder lengths (see Sec. <ref>). Enhanced PDW correlations were found in doped Kondo-Heisenberg ladders with a substantial spin gap <cit.>. These results suggest a competition between the SDW and PDW fluctuations in asymmetric ladders. Note that the analysis of correlation functions away from half filling is a delicate problem because of the inhomogeneous distribution of charge and spin along the ladder <cit.> and between both legs (see the next section). In summary, this investigation of correlation functions is compatible with the phase diagram deduced in our previous work <cit.>. The only discrepancy is the apparent power-law behavior of spin correlations in the spin-gapped Kondo-Mott phase, which we canunderstand as a finite-size effect but should be checked using longer ladder lengths in a future study. In the Luttinger liquid phase, charge and spin fluctuations appear to be spatially separated with the stronger spin fluctuations in the Hubbard leg and the stronger charge fluctuations in the Fermi leg. Despite the strong pair binding in the Fermi leg of the (doped) spin-gapped Mott phase, which we deduced from the energy and density observables,we could not identify any enhanced pairing correlations and the dominant correlations seem to be SDW in that regime. Therefore, we think that the coexistence of (quasi-)long range orders is likely in systems of coupled asymmetric Hubbard ladders or in two-dimensional systems made of a Hubbard layer and a Fermi layer. We expect antiferromagnetic spin order in the interacting Hubbard subsystem while various pairing or charge orders could dominate the noninteracting Fermi subsystem. § LEG DENSITIES We can obtain interesting information about the low-energy excitations of the asymmetric ladder (<ref>)using the changes in the total charge and spin densities on both legs for variable numbers of electrons in the system. The deviations from the ground-state charge and spin distributions at half filling are given byN_m(y)= ∑_x⟨ N(x,y) ⟩ -L_xandS_m(y)= ∑_x⟨ S(x,y) ⟩ .Here, we will consider the deviations caused by one (m=1p) or two electrons (m=2p) added and by a spin triplet (m=1s). This corresponds to the lowest single-particle, charge and spin excitations, respectively (see the discussion of gaps below).Figure <ref>(a) shows N_2p(y) and S_1s(y) for the lowest charge and spin excitation as a function of t_⊥.Clearly, most of the excess charge is concentrated on the Fermi leg while most of the excess spin is localized on the Hubbard leg.The confinement of additional charges in the Fermi leg is expected because of the repulsive interaction on the Hubbard leg. The confinement of the excess spin in the Hubbard leg is more surprising because both legs have gapless spin excitations when decoupled (t_⊥=0).This uneven distribution is probably related to the fact that spin excitations have a lower velocity in the 1D Hubbard model for U>0 than in the tight-biding chain <cit.>. Thus the lowest charge and spin excitations are separated in real space in this model. In the Luttinger liquid phase, we see that this separation is almost perfect with most of the density variations concentrated in opposite legs. We will showd in the next section that the lowest excitations are also dynamically separated, i.e. have different velocities, in that phase. For increasing coupling t_⊥, the distributions of charge and spin become progressively more even and converge to the same values for both legs in the dimer limit (t_⊥≫ t_∥, U). However, anon-monotonic behavior of the spin distribution S_1s(y) is observed for hopping terms t_⊥ corresponding to the Kondo-Mott and spin-gapped Mott phases. The picture is somewhat different for the single-particle excitation in Fig. <ref>(b). Both the additional charge N_1p(y) and spin S_1p(y)are localized in the Fermi leg in the Luttinger liquid phase.However, the excess spin moves onto the Hubbard leg upon entering the Kondo-Mott phase with increasing t_⊥. Then the density deviations for the single electron behave similarly to those for charge and spin excitations in Fig. <ref>(a): overall convergence toward equal values in the dimer limit and non-monotonic behavior inthe Kondo-Mott and spin-gapped Mott phases.In summary, the low-energy excitations are mostly confined to one leg when the rung hopping t_⊥ is not too strong. This explains the different behavior of correlation functions on Hubbard and Fermi legs. In most cases charge excitations tend to stay on the Fermi leg, while spin excitations prefer the Hubbard leg. Thus low-energy spin and charge excitations are spatially separated in the 1D correlated electron model (<ref>). The single-particle excitations in the Luttinger liquid phase constitute the only exception. The difference between pure spin excitations and the spin associated to single-particle excitations in that phase is intriguing and we have investigate the velocities of these excitations to gain more information.§ VELOCITIES Similarly to the analysis of the leg density distributions for excited states (<ref>) and (<ref>), we calculate the gaps for single-particle, charge, spin and excitations from the change in ground-state energies for one added particle (electron or hole), two added particles (electrons or holes) and a spin triplet. The charge gap is defined asE_c=1/2 [ E_0(N_↑+1,N_↓+1)+E_0 (N_↑-1,N_↓-1).. -2E_0(N_↑,N_↓))]where E_0(N_↑,N_↓) refers to the ground-state energy of the Hamiltonian (<ref>) for N_σ electrons of spin σ. This gap is the lowest excitation energy seen in the dynamical structure factor, which can be measured in experiments such as electron energy loss spectroscopy. The spin gap is defined asE_s=E_0(N_↑+1,N_↓-1)-E_0(N_↑,N_↓) .This gap is the lowest excitation energy seen in the the dynamical spin structure factor, which can be measured in experiments such as inelastic neutron-scattering. Finally, the single-particle gap is defined asE_p=E_0(N_↑+1,N_↓)+ E_0(N_↑-1,N_↓)-2E_0(N_↑,N_↓).This gap is the lowest excitation energy seen in the single-particle spectral functions (Green's functions), which can be probed in experiments such asangle resolved photoemission spectroscopy.In a ladder (<ref>) of finite length L, these gaps are always finite (excluding accidental degeneracies). To determine the true gaps of an infinite ladder, one has to analyze the scalingof the finite-size gaps with the ladder length. The finite-size scaling is performed by calculating these gaps for several system sizes up to L=200 using DMRG andextrapolating the values to L →∞ numerically.For most of the parameter space (U, t_⊥) of the half-filled Hamiltonian (<ref>) we found that the gaps remain finite in the thermodynamic limit. This corresponds to the three insulating phases in the phase diagram in Fig. <ref>. These results were presented in detail in our previous work <cit.>. Here, we focus on the gapless Luttinger liquid phase. Figure <ref> shows that the extrapolation of the finite-size gapsindicate gapless excitations in the thermodynamic limit for U=8 and t_⊥=0.3, which corresponds to the Luttinger liquid phase. Moreover, the gaps tend to vanish linearly with the inverse system length as expected for 1D correlated conductors <cit.>. The slope corresponds to the excitation velocity up to a constant prefactor π (assuming ħ=1 and a lattice constant a=1). The deviations from the linear behavior for small 1/L are due to rapidly increasing relative errors because the absolute DMRG errors for the energies E_0(N_↑,N_↓)scale as L while the energy differences (<ref>), (<ref>), and (<ref>) are of the order of 1/L.The finite-size charge gap scales as E_c≈ 5.8/L for large ladder length L, in agreement with the finite-size scaling in the half-filled tight-binding chain with open boundary conditions,i.e. E_c≈ 2π/L. This confirms that the added charges are concentrated mostly on the Fermi leg. Similarly, the finite-size spin gap scales as E_s≈ 1.49/L in agreement with the Bethe Ansatz solution for the one-dimensional Hubbard model with U=8 and open boundary conditions, which yields E_s≈ 1.51/L <cit.>.This also confirms the localization of the lowest triplet excitation in the Hubbard leg.The finite-size single-particle gap is very close to the charge gap as seen in Fig. <ref>.In a Luttinger liquid the single-particle velocity is the average of the velocities for the elementary charge and spin excitations that contribute to the lowest single-particle excitations. Consequently, the elementary spin excitation contributing to the single-particle excitation in Fig. <ref> must have almost the same velocity than the elementary charge excitation and thus is not responsible for the finite-size spin gap seen in that figure.Therefore, the nature of elementary excitations in the Luttinger liquid phase is relatively simple. In an electronic two-leg ladder, two charge modes and two spin modes can exist <cit.>. We have seen in the previous section that, in the Luttinger liquid phase of the asymmetric Hubbard ladder,each mode is concentrated in one leg. The charge mode in the Hubbard leg is gapped and thus not relevant for the Luttinger liquid properties. The spin mode in the Hubbard leg is gapless, determines the finite-size spin gap seen in Fig. <ref> for triplet excitations, and is responsible for the critical antiferromagnetic correlations that can be seen in Fig. <ref>(a) and the spin density deviations caused by the triplet excitation in Fig. <ref>(a). The low-energy single-particle excitations are essentially made of charge and spin excitations localized in the Fermi leg. The spin mode in the Fermi leg is gapless but has a higher velocity and thus larger finite-size gaps thanthe spin mode in the Hubbard leg. It is responsible for the weak antiferromagnetic correlations in the Fermi leg that can be seen in Fig. <ref>(a) and the spin density deviations caused by the single-particle excitation in Fig. <ref>(b). Finally, the charge mode in the Fermi leg is gapless, has approximately the same velocity than the spin mode and is responsible for the charge density deviations seen in Fig. <ref>(a) and (b), as well as the power-lawdensity correlations in Fig. <ref>(a). The near equality of the spin and charge correlations and velocities in the Fermi leg suggest that the effective Luttinger liquid induced in this leg is only weakly correlated.§ CONCLUSIONS We have investigated the four ground-state phases found previously <cit.>in the half-filled asymmetric Hubbard ladder using the DMRG method. The correlation functions studied in Sec. <ref> are fully compatible with our previous findings, besides the problem of apparent power-law correlations in phases with very small spin gaps. Quasi-long-range antiferromagnetic order is found only in the Hubbard leg of the Luttinger liquid phase and, possibly, upon doping of the spin-gapped Mott phase. An open issue is the absence of enhanced pairing correlations despite the strong pair binding observed in the excitation energies and local densities in the Fermi leg of the (lightly doped) spin-gapped Mott phase.The leg density distributions discussed in Sec. <ref> confirm the existence and the (rough) locationof the four ground-state phases.So far our investigations have not yield precise phase boundaries (and consequently no information on the nature of the phase transitions), except in limiting cases. Entanglement measurements <cit.> based on the DMRG method,such as the block entanglement entropy, are the most promising approachto determine these phase boundaries and are in progress.Correlation functions, leg densities, and excitation velocities show that low-energy spin and charge degreesof freedom can be spatially separated in the asymmetric Hubbard ladder. This confinement results in different correlations and low-energy excitations in both legs in some cases. Thus itcould facilitate the coexistence of various (quasi-)long-range orders in the Hubbard and Fermi subsystem of generalizations of the asymmetric Hubbard ladder. In particular, the Hamiltonian (<ref>) can be generalized to allow for different intra-leg hoping terms in both legs and thusto reach the regime ofstrong spin-exchange coupling on the rung (compared to the hopping in the Fermi leg),where enhanced PDW correlations were found in the Kondo-Heisenberg model <cit.>. Therefore, we think that the coexistence of (quasi-)long range orders could be possible in two-dimensional models made of coupled (generalized) asymmetric Hubbard laddersor of a Hubbard layer and a Fermi layer. Antiferromagnetic spin order should occur in the interacting Hubbard subsystem whilepairing or other charge orders could dominate the noninteracting Fermi subsystem. These models could describe real quasi-two-dimensional materials such as thelayered high-temperature superconductors or arrays of linear atomic chains deposited on semiconducting substrates <cit.>.This work was done as part of the Research Units Metallic nanowires on the atomic scale: Electronic and vibrational coupling in real world systems (FOR1700) of the German Research Foundation (DFG) and was supported by grant JE 261/1-2. The DMRG calculations were carried out on the cluster system at the Leibniz University of Hannover and at the Sudan Center for HPC and Grid Computing.§ AUTHOR CONTRIBUTION STATEMENTEric Jeckelmann initiated and guided the project. Anas Abdelwahab adapted the DMRG code and performed the DMRG simulations. Both authors analyzed the data and contributed to the writing of the manuscript. sik97 A. E. Sikkema, I. Affleck, S. R. White, Phys. Rev. Lett. 79, 929 (1997) zac01b O. Zachar, A. M. Tsvelik, Phys. Rev. B 64, 033103 (2001) ber10 E. Berg, E. Fradkin, S. A. Kivelson, Phys. Rev. Lett.105, 146403 (2010) dob13 A. Dobry, A. Jaefari, E. Fradkin, Phys. Rev. B 87, 245102 (2013) eid11 E. Eidelstein, S. Moukouri, A. Schiller, Phys. Rev. B 84, 014413 (2011) alh09 K. A. Al-Hassanieh, C. D. Batista, P. Sengupta, A. E. Feiguin, Phys. Rev. B 80, 115116 (2009) pan17 L. Pan, D. Zhang, H.-H. Hung, Y.-J. Liu, Eur. Phys. J. B 90 105 (2017) das01 I. K. Dash, A. J. Fisher, J. Phys.: Condens. 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Settelein, R. Claessen, Phys. Rev. Lett. 111, 137203 (2013) | http://arxiv.org/abs/1707.08780v1 | {
"authors": [
"Anas Abdelwahab",
"Eric Jeckelmann"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727085142",
"title": "Correlations and confinement of excitations in an asymmetric Hubbard ladder"
} |
Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, SwedenWe investigate the emergence and consequences of odd-frequency spin-triplet s-wave pairing in superconducting hybrid junctions at the edge of a two-dimensional topological insulator without any magnetism. More specifically, we consider several different normal-superconductor hybrid systems at the topological insulator edge, where spin-singlet s-wave superconducting pairing is proximity-induced from an external conventional superconductor. We perform fully analytical calculations and show that odd-frequency mixed spin-triplet s-wave pairing arises due to the unique spin-momentum locking in the topological insulator edge state and the naturally non-constant pairing potential profile in hybrid systems.Importantly, we establish a one-to-one correspondence between the local density of states (LDOS) at low energies and the odd-frequency spin-triplet pairing in NS, NSN and SNS junctions along the topological insulator edge; at interfaces the enhancement in the LDOS can directly be attributed to the contribution of odd-frequency pairing. Furthermore, in SNS junctions we show that the emergence of the zero-energy LDOS peak at the superconducting phase ϕ=π is associated purely with odd-frequency pairing in the middle of the junction. Odd-frequency superconducting pairing and subgap density of states at the edge of a two-dimensional topological insulator without magnetism Jorge Cayao and Annica M. Black-Schaffer December 30, 2023 ===========================================================================================================================================§ INTRODUCTION Superconductivity is strongly characterized by the Cooper pair wave function, or the pairing amplitude, obeying the antisymmetry condition imposed by Fermi-Dirac statistics, which restricts its spin and spatial symmetries. The antisymmetry condition also allows for the emergence of odd-frequency superconductivity, where the pairing amplitude is odd in the time, or equivalently frequency, parameter. The existence of odd-frequency superconductivity was first postulated by Berezinskii<cit.> in 1974, when he pointed out that there is no symmetry restriction for the existence of odd-frequency spin triplet s-wave pairing in ^3He.Subsequent works first took this idea to analyze spin-triplet s-wave superconductivity in disordered systems<cit.> and later extended this suggestion to proposed odd-frequency superconductivity in bulk systems with spin-singlet p-wave pairing.<cit.> More recently, bulk odd-frequency superconductivity has also been found in multiband superconductors<cit.> and in superconductors subjected to a time-dependent drive.<cit.>Following another route for non-bulk materials, it has been shown that odd-frequency superconductivity also emerges as an induced effect in hybrid systems where superconductivity is induced by proximity effect. For ferromagnet-superconductor (FS) junctions it is now well established, both theoretically and experimentally, that it is odd-frequency spin triplet s-wave pairing that explains the long-range proximity effect into the ferromagnet even when using conventional spin-singlet s-wave superconductors.<cit.> In these systems the spin-rotation symmetry is broken by the ferromagnet, which allows for the formation of the spin-triplet state.Afterwards, it was also shown that normal metal-superconductor (NS) junctions also exhibit odd-frequency pairing.In this case, however, a conventional superconductor can only induce odd-frequency spin-singlet p-wave pairing,since only translational symmetry is broken.<cit.>However, according to Anderson's theorem<cit.> only s-wave pairing is intrinsically stable againstever present non-magnetic disorder, and therefore this latter odd-frequency p-wave is much less stable than theodd-frequency s-wave state present in FS junctions. Moreover, normal metals are also fragile to disorder due toAnderson localization, namely they allow for finite elastic backscattering processes from non-magnetic impurities,causing dissipation of electric current.Odd-frequency spin-triplet s-wave pairs can however appear at interfaces of diffusive normal metals if the external superconductor is unconventional with spin-triplet p-wave symmetry.<cit.>More recently, the appearance of odd-frequency pairing was also found in systems withRashba spin-orbit coupling.<cit.>On the other hand, topological insulators,<cit.> characterized by having an insulating bulk but metallic surface states, have strongly suppressed backscattering in their surface states. In fact, for two-dimensional topological insulators (2DTI) backscattering is completely absent due to the perfect helicity of the edge states. Induced s-wave superconductivity in a 2DTI edge therefore represents a promising and very disorder-insensitive platform for the search of robust odd-frequency pairing. Indeed, it has recently been demonstrated that the combination of surface state helicity and a finite in-surface gradient in a proximity-induced spin-singlet s-wave superconducting state gives rise to odd-frequency spin-triplet s-wave pairing.<cit.> It is the helicity of the TI surface state that allows a spatial symmetry breaking (finite in-surface gradient) to be effectively converted into a change of the spin symmetry, such that spin-triplet s-wave pairing is generated. Notably this mechanism does not need any presence of ferromagnetic regions or even a magnetic field, and thus does not destroy the topological protection of the TI surface states. There exits now a growing body of work focusing in detail on behavior of odd-frequency pairing in superconducting hybrid systems at both the surface of 3DTIs<cit.> and edge of 2DTIs.<cit.> However, in these cases finite ferromagnetic regions has been assumed, which destroys the topological protection of the TI surface states. For the situation of odd-frequency pairing without magnetism in 2DTIs there exists no detailed analysis of the pairing amplitudes and especially not of their relationship to the local density of states (LDOS). In this work we fill this gap and investigate in detail NS, NSN, and SNS hybrid junctions at the edge of a 2DTI without any magnetism present. Our study is based on retarded Green's functions extracted from scattering states and allow us to analytically extract all pairing amplitudes as well as the LDOS in the junctions.Very generally, we show that breaking translation symmetry at the NS interface(s) gives rise to four different symmetry classes atthe NS interfaces: Even-frequency, spin-Singlet, Even in space (ESE), i.e. the conventional order, Odd-frequency, spin-Singlet, Odd in space (OSO), Even-frequency spin-Triplet, Odd in space (ETO), and Odd-frequency, spin-Triplet, Even in space (OTE). In the absence of magnetism in the system we only generate mixed spin-triplet pairing, such that m_z = 0 for the Cooper pairs. Disorder stability is moreover preferential for the local s-wave states present in the ESE and OTE classes and we therefore focus primarily on these. These local pairing terms are also naturally the contributions most directly connected with the LDOS which is also a local quantity.A strong relationship between LDOS andodd-frequency pairing has been previously already established in systems formed out of normal metals with unconventional superconductors<cit.> or when magnetism is present<cit.>. With this work we extend this relationship also to 2DTI superconducting hybrid junctions without any magnetism. In both NS and NSN junction we find strongly dominating OTE pairing at very low energies in the superconducting interface region, with an exponential decay into the bulk of S. In fact, at zero energy the ESE contribution is completely suppressed and only OTE pairing is present. We also find that the LDOS in the S region experience the same frequency dependence and exponential decay into the bulk of S as the OTE pairing. This shows that the very low energy contribution to the LDOS arises entirely from the OTE pairing. Moreover, we also find that the conductance in NS junctions follows the same behavior as the OTE component at low energies, suggesting that the main contribution to the conductance is also of OTE nature. Our results here are strongly connected to the fact that the Andreev reflection magnitude at the NS interface reaches its maximum for energies within the gap, a unique characteristic of the helical edges of 2DTIs.<cit.>In SNS junctions the LDOS reveals the formation of Andreev bound states (ABSs) for energies within the energy gap ofthe superconductor. We find that the ESE and OTE pairing magnitudes exactly capture their emergence, but they behavevery differently for different values of the superconducting phase ϕ across the junction. Indeed, for very shortjunctions we obtain that at ϕ=0 the ESE pairing dominates over a completely suppressed OTE contribution.At ϕ=π the ESE is instead zero while the OTE becomes dominant and even exhibits a zero-energy peak just asthe LDOS thanks to the topologically protected zero-energy crossing of the ABSs. Moreover, thesupercurrent across short junctions exhibits its maximum value at ϕ=π, as a result of theresonant zero-energy peak which possesses purely OTE pairing. In longer junctions, the increasing number of ABSs within the gap is also exactly captured in the ESE and OTE amplitudes. In fact, in the middle part of the N region the situation is very similar to the short junction case with only OTE pairing present at ϕ=π. Here the ABSs again have protected crossings generating resonant peaks in the LDOS, which exactly correspond to resonant peaks in the OTE pairing. Thus the LDOS signatures of the ABSs at ϕ = π is entirely a consequence of OTE pairing. On the experimental side, HgCd/HgTe<cit.> and InAs/GaSb<cit.> heterostructures represent two of the most promising 2DTI. In both cases, induced superconductivity has already been demonstrated.<cit.> Moreover, the superconducting junction geometries we consider comprise of a realistic platform for both LDOS and conductance measurements, as has also been demonstrated in experiments. <cit.> Therefore, all experimental prerequisites already exist for the systems we study. This paper is organized as follows. In Sec. <ref> we present the model and the method based on retarded Green's functions calculated from scattering states. In Sec. <ref> we perform a detailed analysis of the pairing amplitudes and investigate their strong relationship with the LDOS for subgap energies for NS, NSN, as well as SNS junctions along the 2DTI edge. We present our conclusions in Sec. <ref>. For completeness, in Appendices <ref>, <ref>, <ref>, and <ref> we provide a detailed account of the method and analytical calculations reported in this work.§ MODEL AND METHOD We consider a 2D topological insulator (2DTI), with its 1D metallic edge in proximity to a conventional spin-singlet s-wave superconductor. This system is modeled by the Bogoliubov-de Gennes (BdG) HamiltonianH_BdG = v_Fp_xτ_zσ_z-μτ_zσ_0+Δ(x)τ_xσ_0 ,in the basis Ψ(x)= (ψ_↑,ψ_↓,ψ_↓^†,-ψ_↑^†)^T, where T denotes the transpose operation, ψ^†_σ(x) adds an electron with spin σ=↑,↓ at position x along the edge. The first term represents the 1D metallic edge of a 2DTI,<cit.> where the spin quantization direction is along the z-axis, p_x=-i∂_x, v_F is the velocity of the edge state, σ_i is the ith Pauli spin matrix and τ_i is the ith Pauli matrix in Nambu space. The chemical potential is represented by μ and determines the filling.In the last term Δ(x)=Δ(x)e^iϕ is the induced superconducting pairing potential at the edge. A good interface enables this pairing potential to be induced into the region(s) of the metallic edge in contact with an external superconductor as a result of proximity effect.<cit.> Notice that the pairing potential Δ(x) generally depends on the position at the edge in hybrid junctions. In the normal state (N) the edge states exhibit a linear energy versus momentum dispersion with Δ = 0, while in superconducting (S) regions a finite Δ mixes the electron and hole branches and opens a gap at the Fermi momenta as shown in Fig. <ref>(a,b). We will consider NS, NSN and SNS junctions formed at the edge of the 2DTI, such that the profile of the induced pairing potential is approximated by the step-like functions indicated in green in Fig. <ref>(c). The NS junction captures the physics of the N-S interface, while the NSN and SNS configurations represent finite S regions and Josephson junctions, respectively.Here the length, L_ N and L_ S, respectively, of the middle regions become important parameters. Also, in the SNS junction it is possible to attain a finite phase difference between the two S regions, with pairing potentials Δ_ L=Δ and Δ_ R=Δe^iϕ, respectively. §.§ Retarded Green's function In this work we primarily investigate the pairing amplitudes and LDOS in different hybrid junctions.For this we follow the formalism based on retarded and advanced Green's functions, corresponding to outgoing and incoming waves, respectively.In general, the retarded Green's function can be calculated using<cit.> G^r(x,x',ω)=α_1Ψ_1(x)Ψ̃_3^T(x')+α_2Ψ_1(x)Ψ̃_4^T(x') + α_3Ψ_2(x)Ψ̃_3^T(x')+α_4Ψ_2(x)Ψ̃_4^T(x'), x>x',β_1Ψ_3(x)Ψ̃_1^T(x')+β_2Ψ_4(x)Ψ̃_1^T(x') + β_3Ψ_3(x)Ψ̃_2^T(x')+β_4Ψ_4(x)Ψ̃_2^T(x'), x<x',where Ψ_i, i=1,2,3,4 are the wave functions representing the four scattering processes at the NS interface at x = x' indicated by black arrows in Fig. <ref>(b), i.e. right-moving electron, right-moving hole, left-moving quasi-electron, and left-moving quasi-hole. Ψ̃_i corresponds to the conjugated processes obtained using H̃_BdG(k)=H_BdG^*(-k)=H_BdG^T(-k) instead of Eq. (<ref>). The detailed form of these wave functions is provided in the appendix for NS, NSN, and SNS junctions.The retarded Green's function satisfies the equation of motion[ω-H_BdG(x)]G^r(x,x',ω)=δ(x-x') ,where its integration around x=x',[G^r(x>x')]_x=x'-[G^r(x<x')]_x=x'=σ_zτ_z/iv_f ,provides a system of equations that allows us to find the coefficients α_i and β_i in Eq. (<ref>). Once the retarded Green's functions are found, the advanced Green's function can be calculated using G^a(x,x',ω)=[G^r(x',x,ω)]^†. The Green's functions G^r(a) are 4×4 matrices in Nambu space, G^r(x,x',ω)= [ G^r_ee(x,x',ω) G^r_eh(x,x',ω); G^r_he(x,x',ω) G^r_hh(x,x',ω) ] ,where the diagonal elements are the regular electron-electron and hole-hole Green's functions, while the off-diagonal corresponds to the anomalous electron-hole part. Here each element is a 2 × 2 matrix in spin-space. The electron-electron part allows us to calculate the LDOSρ_(x,ω)=-lim_x→ x'1/π Im Tr[G^r_ee(x,x',ω)],while the electron-hole part provides the pairing amplitudes. The LDOS is generally one of the most experimentally accessible quantities in a system and can be measured directly using, for example, different scanning tunneling probes.<cit.> Notice that the LDOS is proportional to the trace of the imaginary local (x=x') Green's function. Therefore, any connection between LDOS and pairing amplitudes is most naturally occurring for local pair amplitudes, which have s-wave symmetry.Once the retarded and advanced Green's functions are calculated, we can decompose the spin symmetry of the anomalous electron-hole part as G^r(a)_eh(x,x',ω)=f^r(a)_0(x,x',ω)σ_0+∑_j=1^3f^r(a)_j(x,x',ω)σ_j ,where f_0^r(a) corresponds to spin-singlet, f_1,2^r(a) to equal spin-triplet and f_3^r(a) to mixed spin-triplet retarded (advanced) pairing amplitudes.These components all necessarily obey Fermi-Dirac statistics and are therefore antisymmetric under the simultaneous exchange of positions and frequency, f^r_0(x,x',ω) =f_0^a(x',x,-ω) ,f^r_j(x,x',ω) =-f_j^a(x',x,-ω) ,with j=1,2,3 labeling the spin-triplet pairing amplitudes. Note that when analyzing the symmetry with respect to frequency, the change ω→-ω requires a pass from the retarded Green's function into the advanced function since each are only defined on parts of the time (frequency) axis, see Appendix <ref> for more details. From this we can isolate the even- and odd-frequency components asf_i^r,E(x,x',ω) =f_i^r(x,x',ω)+f_i^a(x,x',-ω)/2 ,f_i^r, O(x,x',ω) =f_i^r(x,x',ω)-f_i^a(x,x',-ω)/2for all spin components (i = 0,1,2,3).Taking the full antisymmetry condition of the anomalous Green's functions into account, which includes spin, spatial, and frequency dependence, we are left with a total of four allowed symmetry classes.<cit.> These symmetries are Even-frequency, spin-Singlet, Even in space (ESE), Odd-frequency, spin-Singlet, Odd in space (OSO), Even-frequency, spin-Triplet, Odd in space (ETO), and Odd-frequency, spin-Triplet Even in space (OTE).As we will see, NS structures along 2DTI edges generally host all of these four components even though the superconductor itself only has conventional spin-singlet s-wave symmetry (ESE).However, considering the stability against disorder and also favoring a direct comparison with theLDOS which is a fully local property (extracted at x' = x), we will here primarily be concerned with states with local or equivalently uniform s-wave symmetry with f_i(x,x,ω)≡ f_i(x,ω), which are the ESE and OTE components. § PAIRING AMPLITUDES AND LDOS We now turn to the main objective in this work which is to analyze the pair amplitudes in NS, NSN, and SNS superconducting hybrid junctions at the edge of a 2DTI and its consequences for the LDOS. In normal NS junctions, i.e. not in TIs, it has previously been shown that when a superposition of spin-singlet and singlet-triplet superconductivity is present, the state with larger pairing magnitude dominates the induced superconductivity<cit.> and with remarkable consequences for the LDOS.<cit.> Before going into details, we stress that in this work both the retarded and advanced equal spin-triplet pairing amplitudes are zero since we do not have any magnetic order in our hybrid junctions, i.e. f^r(a)_1,2(x,x',ω)=0. We will therefore refer to the mixed spin-triplet component f_3^r simply as the spin-triplet component. Moreover, we refer to f^r_i as the pairing amplitude, while |f^r_i|=√(f_i^rf_i^r,*) is the pairing magnitude, unless otherwise specified. §.§ NS junctionWe first focus on the simplest situation, which consists of a semi-infinite NS junction, where the N region occupies x<0 and S x>0. Before proceeding further we point out that NS junctions at the metallic edge have previously been found, both theoretically and experimentally, to allow perfect (local) Andreev reflection for energies within the superconducting gap with totally suppressed normal reflection processes (backscattering).<cit.> Extracting the Green's function from the interface scattering processes we find in the normal region N for x<0 (see Appendix <ref> for details) the even and odd-frequency pairing amplitudesf^r, E_0,N(x,x',ω) =a_2(ω)/2iv_F C_xx'e^-i(x+x')/ξ_ω,f^r, O_0, N(x,x',ω) =-a_2(ω)/2v_F S_xx'e^-i(x+x')/ξ_ω,f^r, E_3,N(x,x',ω) =a_2(ω)/2v_F S_xx'e^-i(x+x')/ξ_ω,f^r, O_3, N(x,x',ω) =-a_2(ω)/2iv_F C_xx'e^-i(x+x')/ξ_ω,where C_xx'= cos[k_μ(x-x')],S_xx'= sin[k_μ(x-x')], and k_μ=μ/v_F, ξ_ω=v_F/ω. Here the superscripts indicate the retarded Green's function (r), the even (E) and odd (O) frequency components, while subscripts stand for spin-component (0 or 3) and normal region (N). Moreover, a_2 represents the Andreev reflection amplitude for an incident hole from N at the NS interface. At the metallic edge of a 2DTI a_2 it is the same as the Andreev reflection for an incident electron from Nat the NS interface, a_1. For energies within the superconducting gap, a_1,2=e^-iη(ω), where η(ω)= arccos(ω/Δ).Despite the non-obvious frequency dependence ofEqs. (<ref>) it is straightforward to verify their symmetry with respect to frequency.For instance, the local ESE amplitude f^r,E_0(x,ω) ≡ f^r,E_0(x,x,ω)=a_2(ω) e^-2ix/ξ_ω/(2iv_F) and thus reversing the sign of ω leads to f^r,E_0(x,-ω)≡ f^a,E_0(x,-ω)=-a^*_2(-ω) e^-2ix/ξ_ω/(2iv_F)=f^r,E_0(x,ω), since a_2^*(-ω) = -a_2(ω). This confirms the even frequency dependence of the ESE pairing amplitude. The amplitudes in Eqs. (<ref>) correspond to ESE, OSO, ETO, OTE amplitudes, respectively, and they are all finite for x≠ x' andfinite Andreev reflection amplitude a_2.The emergence of all symmetry classes at interfaces of junctions, as the ones discussed here, is not surprising. Indeed, the junction breaks translational symmetry which allows for mixing between different spatial symmetries. Moreover, the 2DTI also allows for mixing between different spin states, without the presence of a magnetic field. Thus all symmetry classes preserving the antisymmetric Fermi-Dirac statistic will appear at the interface. Moreover, we observe that the even-frequency (odd-freqeuncy) spin-singlet and odd-frequency (even-frequency) spin-triplet pairing amplitudes are equal to each other in magnitude, namely |f^r, E(O)_0(x,x',ω)|=|f^r, O(E)_3(x,x',ω)|. Considering only s-wave pairing, this means that ESE and OTE contribute equally to the superconducting state in N. This situation with very strong odd-frequency components is due to the unique nature of the 1D metallic edge. We also note that Eqs. (<ref>) suggest that the pairing amplitudes in N do not decay into N, but rather remain finite as a pure consequence of the NS interface. This has to be understood to happen only very close to the NS interface, where Andreev reflection is manifested.Performing the same analysis as above for the superconducting region S, x>0, leads us to naturally distinguish between two different terms in the pairing amplitudes, attributed to the bulk (B) and to the interface (I), with the total pairing amplitude f = f_ B + f_ I. For energies within the superconducting gap, |ω|<Δ, we findf^r, E_0, B(x,x',ω) =Z(ω) e^-κ(ω)|x-x'| C_xx' ,f^r, E_3, B(x,x',ω) =iZ(ω) e^-κ(ω)|x-x'| S_xx' ,f^r, O_i, B(x,x',ω) =0andf^r, E_0, I(x,x',ω) =a_3(ω)/2iv_F e^-κ(ω)(x+x') B(ω) C_xx' ,f^r, O_0, I(x,x',ω) =a_3(ω)/2v_F e^-κ(ω)(x+x') S_xx' ,f^r, E_3, I(x,x',ω) =a_3(ω)/2v_F e^-κ(ω)(x+x') B(ω) S_xx' ,f^r, O_3, I(x,x',ω) =a_3(ω)/2iv_F e^-κ(ω)(x+x') C_xx' ,with Z(ω)=(1/iv_F)/[ e^iη(ω)- e^-iη(ω)], B(ω)=-i cot[η(ω)], κ(ω)=√(Δ^2-ω^2), and a_3 = -a_1. For details on the derivation we refer to Appendix <ref>. The division into bulk and interface properties is evident for energies within the superconducting gap. Here the interface components exhibit an exponential decay<cit.> proportional to e^-κ(ω)(x+x'), while the bulk components have an exponential decay proportional to e^-κ(ω)|x-x'|. Thus, locally at x=x', the bulk components are independent of spatial coordinates, confirming their bulk nature, while the interface components display an exponential decay into the bulk of the S region. Notice also that the interface amplitudes are all proportional to the Andreev reflection amplitude a_3 = -a_1 and thus purely a consequence of the NS interface. Analyzing Eqs. (<ref>-<ref>) we find that only the even-frequency ESE and ETO components are finite in the bulk. At the interface, however, all symmetry classes (ESE, OSO, ETO, OTE) are present. Even limiting the interest to the disorder robust s-wave states leaves both ESE and OTE amplitudes in the interface region. Note that an OTE component does not appear in normal metal NS junctions, but there instead a magnetic field is necessary to generate the OTE component. In Fig. <ref> we display the frequency dependence of the pairing amplitudes at the interface,x=x'=0. This confirms the odd frequency dependence of the OTE component, while the ESE is fully even in frequency. While the figure only shows the x=x'=0, we have verified the frequency dependence also for x ≠ x'. Notice that the real part of OTE exhibits an abrupt transition across zero, which arises due to the discontinuity between f^a and f^r at zero frequency. Thus, the magnitude of the OTE component is finite even for ω=0, as is shown in Fig. <ref>(b). One could argue that at ω=0 the retarded and advanced Green's functions are not well defined and therefore this discussion might lead to a wrong interpretation of the emergence of OTE. We can rule out such a possibility by pointing out that away from zero frequency, but well below Δ, the OTE component is still finite and even clearly dominant over the ESE component.In fact, we find that the OTE amplitude dominates over ESE at the interface for |ω|<Δ/√(2).In order to establish a connection with the LDOS, we compare the LDOS with the local, s-wave, pairing amplitudes at x=x'. In Fig. <ref> we present the spatial dependence of the LDOS and the local pairing magnitudes in the normal and interface regions with the interface placed at x=0. In the normal region, x<0, the LDOS acquires a finite value ρ_N(x,ω) =1/(π v_F) independent of both position and energy, as shown in Fig. <ref>(a). This naturally arises because, at the metallic edge of a 2DTI, an incident electron is purely reflected back as a hole without interference.<cit.> The equal and finite values of ESE f^r, E_0,N(x,ω) and OTE f^r, O_0,N(x,ω) in N, first and last equations in Eqs. (<ref>), suggest that ρ_N arises due to their simultaneous contribution. In the superconducting region, x>0, the situation is more interesting.Within the superconducting gap, |ω|<Δ, the LDOS is ρ_ S(x,|ω|<Δ)=1/π v_F e^-2κ(ω)x .At the interface, x=0, the LDOS is thus finite for energies within the superconducting gap Δ, acquiring its maximum value and then decaying into the bulk of S with adecay length given by 1/[2κ(ω)], and also seen in Fig. <ref>(a).For |ω|>Δ the inset in Fig. <ref>(a) shows the energy dependence and spatial dependence in the S region. Well within the S region the LDOS is depleted below the gap and shows an oscillatory pattern for |ω|>Δ, which arises due to interference between the incident and reflected quasiparticles,<cit.> unlike in the N region. For the local pairing amplitudes in S we simplify Eqs. (<ref>) by setting x = x' and obtainf^r, E_0, I(x,ω) =a_3(ω)/2iv_F e^-2κ(ω)x B(ω) ,f^r, O_3, I(x,ω) =a_3(ω)/2iv_F e^-2κ(ω)x ,for the ESE and OTE interface amplitudes. Notice that the ESEinterface component is multiplied by the factor B(ω), which leads to important consequences at low energies.Remarkably, at the interface and zero energy the ESE interface component is totally suppressed because B(ω=0)=0, whilethe OTE amplitude is still non-zero and even acquires its maximum value, as seen in Fig. <ref>(b). We can therefore directly associate the enhancement of the LDOS close to the interface in S, Fig. <ref>(a), with the OTE pairing and not with the conventional ESE pairing.[A similar conclusion was achieved in junctions between normal metals and unconventional superconductors,<cit.> which are sensitive to backscattering and disorder unlike our 2DTI system.] Interestingly, this behavior is preserved for energies away from (but close to) zero. For energies close to Δ, however, there is an equal contribution of the ESE component. We have verified that these results do not change by introducing an insulating barrier at the NS interface, since the Andreev reflection coefficient a_1 remains unchanged. Thus the 1D metallic edges of 2DTI offers more robust predictions unlike its 2D surface counterparts, which are sensitive to an interface barrier.<cit.> We have also only considered a step-like pairing potential Δ. Preliminary results (not shown) using a smoother Δ(x) across the NS interfaces, present due to inverse proximity effect, show that there is a reduction of the pairing magnitudes at the NS interfaces, characterized by the sharpness of the Δ(x) profile. However, the classification of the pairing symmetries as well as the main conclusions remain unchanged. This is consistent with the OTE pairing amplitude being generated by an in-surface gradient,<cit.> which naturally is maximized for step-edge profiles.Based on our results we can also relate the conductance across the NS junction to the OTE pairing. The zero-temperature single mode conductance for an incident electron from N is given by G_ NS(ω)=(e^2/h)(1+|a_1|^2-|b_1|^2), where a_1 and b_1 are the Andreev and normal reflection amplitudes, respectively. Since backscattering is forbidden in the 2DTI edge b_1=0. Therefore, the conductance is fully describedby the Andreev reflection coefficient a_1. At the same time, the interface ESE and OTE pairing amplitudes are both proportional to the Andreev reflection coefficient a_3 = -a_1. At low energies the B(ω) factor, however, dramatically suppresses the ESE pairing and left is therefore only the OTE contribution. Thus, any change in the Andreev reflection process is directly manifested in both the conductance across the NS junction and OTE pairing magnitude at low energies. This allows us to conclude that the main contribution to G_ NS is given by the OTE component only. Notice that the conductance across NS junctions at the edge of a 2DTI is independent of the chemical potentials in the two regions, unlike the situation with 3DTIs,<cit.> and also becomes constant within the gap while exhibiting a decay outside,<cit.> as is evident from the behavior of a_1 in Figs. <ref>(b) and also <ref>(a). To conclude this part we especially stress that the contribution of the OTE pairing to the enhancement of LDOS and conductance across the NS junction is a direct consequence of the metallic edge of 2DTI, where the Andreev reflection is perfect for energies below the gap<cit.> and also independent of barrier imperfections at the NS interface.<cit.> Moreover, we point out that Andreev reflection at the edge of a 2DTI has already been experimentally demonstrated<cit.> and significant effort has also been devoted to investigate induced superconductivity at the edge of 2DTIs. <cit.> Our findings can thereforehelp to elucidate the nature of the superconducting pairing in both LDOS and conductance measurements in NS junctions.§.§ NSN junctionHaving investigated the simplest NS junction we now turn to the situation with a finite S region.We consider a NSN junction at the edge of a 2DTI, where the S region is placed within 0<x<L_ S, surrounded by N regions on each side. We arrive at the pairing amplitudes and the LDOS following the same procedure as above, constructing the retarded Green's function from the allowed scattering processes, see Appendix <ref> for details.In the two normal regions we find the pairing amplitudes proportional to the Andreev reflection amplitude a_2(ω,L_ S)= sin[iκ(ω)L_ S]/ sin[iκ(ω)L_ S-η(ω)] but otherwise exactly following the same form as for the pairing amplitudes in the N region of NS junctions given in Eqs. (<ref>). Thus the only change for the N pairing amplitudes compared to the NS junction is a_2(ω)→ a_2(ω,L_ S). This is fully consistent with the results in N being purely due to the interfaces.In particular, notice that the dependence on L_ Sis only present through a_2(ω,L_ S). The magnitude |a_2(ω,L_ S)| as a function of ω for different L_ S is plotted in Fig. <ref>(a). For |ω|<Δ the magnitude increases as the length of the S region increases and saturates to the value found in NS junctions |a_2(ω)|=|a_2(ω,L_ S→∞)| for L_ S> ξ, where ξ=v_ F/Δ is the superconducting coherence length. For energies above Δ, |a_2(ω,L_ S)| develops an oscillatory decay whose amplitude for ξ<L_ S≪∞, is higher than that in NS junctions. Turning to the middle superconducting region and concentrating on energies below the gap Δ, we obtain the following even and odd-frequency pairing amplitudes divided into bulk contributionsf_0, B^r, E(x,x',ω) =2C_xx'[β_2 e^-κ(ω)|x-x'|+β_3 e^κ(ω)|x-x'|] ,f_3,B^r, E(x,x',ω) =2iS_xx'[β_2 e^-κ(ω)|x-x'|+β_3 e^κ(ω)|x-x'|] ,f_0,3, B^r, O(x,x',ω) =0and interface contributionsf_0, I^r, E(x,x',ω) =C_xx'[β_42^- e^-κ(ω)(x+x')+β_13^- e^κ(ω)(x+x')] ,f_0, I^r, O(x,x',ω) =-iS_xx'[β_42^+ e^-κ(ω)(x+x')+β_13^+ e^κ(ω)(x+x')] ,f_3, I^r, E(x,x',ω) =iS_xx'[β_42^- e^-κ(ω)(x+x')+β_13^- e^κ(ω)(x+x')] ,f_3, I^r, O(x,x',ω) =-C_xx'[β_42^+ e^-κ(ω)(x+x')+β_13^+ e^κ(ω)(x+x')] ,where β_42(13)^±=β_4(1)±β_2(3), β_4,3=- e^-2iη(ω)β_2,1, β_1=-iZ(ω) e^iη(ω)-κ(ω)L_ S/4 sin[iκ(ω)L_ S-η(ω)] and β_2=-β_1 e^2κ(ω)L_ S. First of all, we stress that these results are in agreement with the expressions for the NS junction. Indeed, we find that in the bulk only even-frequency pairing amplitudes exist, while all pairing symmetries have finite contributions at the interface.Observe also that the interface components develop an exponential decay from both interfaces proportional to e^±κ(ω)(x+x'), while the bulk components are proportional to e^±κ(ω)|x-x'| and thus become independent of space for x=x'. Next we proceed to analyze how the local pairing amplitudes for x=x' are related to the LDOS. The regular Green's functions in the N region are exactly the same as in the NS junction case and the LDOS in this N region is subsequently also the same, ρ_N = 1/(π v_F). On the other hand, in the S region we find a much more complex expression for the LDOS (see Appendix <ref> for details). However, it acquires a simple expression at zero energy ρ_ S(x,ω=0)=1/π v_F e^-2L_ S/ξ e^2x/ξ+ e^-2x/ξ/1+ e^-2L_ S/ξ .In Fig. <ref>(a) we plot the spatial and frequency dependence of the LDOS. At the interfaces, x=0, 10ξ, the LDOS is finite and exhibits its maximum value 1/(π v_F). It then exponentially decays from both interfaces into the S region with a decay length for ω=0 given by ξ/2. This behavior is preserved even for energies away from zero as can be observed in Fig. <ref>(a). For ω≠0 the decay length is given by 1/[2κ(ω)].We compare the LDOS with the local pairing amplitudes at x = x', derived from Eqs. (<ref>) and (<ref>), giving the finite componentsf_0, B^r, E(x,ω) =β_2-β_1 e^-2iη(ω) ,f_0, I^r, E(x,ω) = B_1(ω) [β_1 e^2κ(ω)x -β_2 e^-2κ(ω)x] ,f_3, I^r, O(x,ω) =B_2(ω) [β_1 e^2κ(ω)x +β_2 e^-2κ(ω)x] ,which correspond to the ESE class in the bulk and the ESE and OTE classes at the interface, where B_1,2(ω)=±(1± e^-2iη(ω))/2. Figure <ref>(b) shows the spatial dependence of the interface and normal region pairing magnitudes for different values of ω. The pairing amplitudes decay from the interfaces at x=0 and x=L_ S into the bulk of S with a decay length given by 1/[2κ(ω)]. Remarkably, this spatial decay is also observed in the LDOS in Fig. <ref>(a), directly supporting their relationship. Interestingly, at zero energy the coefficients of the interface pairing amplitudes acquire different prefactors, B_1(ω=0)=0 and B_2(ω=0)=2, which gives rise to purely OTE pairing at the interface. A strongly dominating OTE component is preserved even for energies away from ω=0 but well below Δ as clearly seen in Fig. <ref>(b).The previous discussion can be clarified by analyzing the frequency dependence of |f_ I^r|, plotted in Fig. <ref>(b-d) for different positions in S.At the left interface, x=0, Fig. <ref>(b) shows the OTE pairing being finite and remaining constant as ω increases within the gap. On the other hand, the ESE pairing is zero at ω=0 and increases only following a V-dependenceas ω increases.To derive the condition for the parameter space where OTE pairing dominates over the ESE pairing, we solve |f_0, I^r, E(ω,x)|=|f_3, I^r, O(ω,x)| for ω. As for NS junctions, we find a simple expression at x=0, where the OTE component is larger for |ω|<Δ/√(2). We also point out that the Andreev reflection amplitude a_1(ω,L_ S) = a_2(ω,L_ S) is directly connected to the pairing amplitudes. To visualize this statement we compare the Andreev reflection and pairing magnitudes at the the left interface x=0 in Figs. <ref>(a) and (b). For energies within Δ the Andreev reflection magnitude is finite and constant for finite L_ S≳ξ. In these junctions it is only the OTE amplitude that is finite and constant, while the ESE amplitude is zero at zero energy, increases linearly with ω, and is subdominant all the way until ω = Δ/√(2).As we go inside the S region, at x=ξ, the OTE pairing magnitude is reduced and the ESE pairing enhanced, both taking a U-shape form, see Fig. <ref>(c). Finally in the middle of the S regionat x = 5ξ, the OTE magnitude is totally suppressed, while a BCS-like gap is fully developed for the ESE component, see Fig. <ref>(d). This is also what we expect due to the ESE nature of the induced superconducting pairing in the bulk. Notice that in the inset of Fig. <ref>(a) we see how the LDOS at low energies is finite at the interface, x=0, exhibits the same U-shape observed in the pairing magnitudes at x=ξ, while deep in the S bulk at x=5ξ the expected BCS-like induced gap is totally developed.Thus the LDOS directly follows the magnitude of the OTEcomponent |f_3, I^r, O| and we conclude that the enhancement of the LDOS in S close to the interfaces arises mainly due to the OTE contribution. This relationship is exact at ω=0, as the pairing there is purely OTE. For 0<ω<Δ/√(2), when the ESE pairing magnitude is finite, the OTE magnitude is still dominating close to the interface and thus has the largest influence on the LDOS. This confirms that the emergence of the exotic odd-frequency superconductivity is intrinsically contained in the LDOS at the interface of NSN junctions at the edge of 2DTI, which supports our findings for NS junctions. §.§ SNS junctionWe finally turn to a SNS junction along the edge of a 2DTI, where the finite normal region with Δ=0 is restricted to 0<x<L_ N. We further set the order parameters inthe left and right S region to Δ_ L=Δ and Δ_ R=Δe^iϕ, respectively. We arrive at the expressions for the LDOS and all pairing amplitudes following the same formalism as used for NS and NSN junctions, see Appendix <ref> for a detailed derivation.As for NS and NSN junctions, all pairing amplitude symmetry classes are also present in SNS junctions. Here we however also obtain a dependence on the superconducting phase across the junction. As we will see, this phase dependence gives rise to a very strong connection between the LDOS and the local s-wave pairing amplitudes.Let us first focus on the normal N region. We find the even and odd-frequency components of the pairing amplitudes given byf_0, N^r, E(x,x',ω) =W_+(ω) cos[k_μ(x-x')] ,f_0, N^r, O(x,x',ω) =W_-(ω)i sin[k_μ(x-x')] ,f_3, N^r, E(x,x',ω) =W_+(ω)i sin[k_μ(x-x')] ,f_3, N^r, O(x,x',ω) =W_-(ω) cos[k_μ(x-x')]where W_±(ω)=[m_5 e^i(x+x')/ξ_ω± m_6 e^-i(x+x')/ξ_ω ]/2.The coefficients m_5=p_1r̃_4α_1 andm_6=q_2s̃_3α_4 contain important information about the scattering processes acrossthe junction.In fact, p_1 is the amplitude for electron transmission from left S into N,r̃_4 gives the Andreev reflection at the left SN interface of a left-moving electron into a hole,q_2 gives the Andreev reflection at the right SN interface of a right-moving hole into an electron, and s̃_3 is the hole transmission from right S into N. Moreover, we findm_6(ω,L_ N,ϕ)=m_5(ω,L_ N,-ϕ) e^i(2L_ N/ξ_ω+ϕ)and for energies within the gap, |ω|<Δ,m_5(ω,L_ N,ϕ) =-1/2v_F e^i(ϕ/2- L_ N/ξ_ω)/ sin[η(ω)-L_ N/ξ_ω+ϕ/2] .The first general observations are that the pairing amplitudes, given by Eqs. (<ref>), become dependent on the superconducting phase difference ϕ through the coefficients m_5,6. Note also that all symmetry classes have finite amplitudes. By simple inspection we notice that they do not exhibit any decay but, for frequencies away from zero, rather develop an oscillatory behavior which extends over the whole N region. For the two S regions we obtain similar expressions for the pairing amplitudes as for S in the NS junction case. In fact, they include two components that represent the contributions from the interface, proportional to e^±κ(ω)(x+x'),and two components associated with the bulk and thus proportional to e^-κ(ω)|x-x'|. Here the ± signs indicate that the exponential decay occurs from both interfaces and into the bulk of the left and right S regions, respectively. Exact expressions are given in Appendix <ref>, but they are not needed here for the future discussion.Let us now focus on the local pairing amplitudes ESE and OTE present at x = x' and theirrelation to the LDOS. We have verified that the results found in N and in the left and right S regions provide the same information and thus for clarity we only discuss the results in the N region.The N region components, given by Eqs. (<ref>), for energies within the gap |ω|<Δ at x =x' reduce tof_0, N^r, E(x,ω) =1/2[m_5 e^2i x/ξ_ω+m_6 e^-2ix/ξ_ω] ,f_3, N^r, O(x,ω) =1/2[m_5 e^2ix/ξ_ω-m_6 e^-2ix/ξ_ω] .These expressions for the pairing amplitudes can be shown to be directly connected to the LDOS and especially its phase dependence. By analyzing m_5,6 we notice that the zeros of their denominators appear at2η(ω)-2L_ N/ξ_ω±ϕ=2π n . But this equation is also the exact quantization condition of the Andreev bound states (ABSs) in SNS junctions at the edge of a 2DTI with a normal region of length L_ N,<cit.> see also Appendix <ref>.These ABSs naturally give very strong peaks in the subgap LDOS in the N region, but here we, quite remarkably, instead obtained them from the pairing amplitudes given by Eqs. (<ref>). This immediately establish a direct relationship between local pairing amplitudes and the LDOS.A short SNS junction, L_ N≪ξ, only hosts one pair of ABSs at energies ω_±(ϕ)=±Δ cos(ϕ/2), which are 4π-periodic and develop zero-energy crossings at ϕ=π(2n-1), n=1,2,… protected by time-reversal symmetry.<cit.>These zero-energy protected crossings give rise to a bound state with Majorana-like properties.<cit.> In a long SNS junction, L_ N≫ξ, on the other hand, many more ABSs fit within the superconducting energy gap. Here the ABSs appear atω_±,n(ϕ)=(v_F/2L)[2π(n+1/2)±ϕ], which gives a linear dispersion with respect to the superconducting phase difference ϕ.§.§.§ Short junctionsTo further analyze the connection between pairing amplitudes and ABSs energies we first investigate the situation of a short junction, L_ N≪ξ, corresponding to the limit L_ N→ 0.From Eq. (<ref>) we obtainm_5(ω,L_ N=0,ϕ)=-(1/2v_ F) e^iϕ/2/ sin[η(ω)+ϕ/2] and m_6(ω,L_ N=0,ϕ)=m_5(ω,L_ N=0,-ϕ) e^iϕ. By simple inspection we notice that this simplified dependence on ϕ has profound consequences for the pairing amplitudes in Eqs. (<ref>). At ϕ=2π n, for n=0,1,2,…, the coefficients are the same, i.e., m_5(ω,L_ N=0,2π n)=m_6(ω,L_ N=0,2π n), and thereforef_0, N^r, E(x=0,ω) =-1/2v_F1/ sin[η(ω)] ,f_3, N^r, O(x=0,ω) =0 . Thus only ESE pairing survives, while the OTE term is completely vanishing.This can be understood from the meaning of coefficients m_5,6. Although they correspond to different (electron and hole) scattering processes, at zero phase difference, they are equal and destructively interfere, which gives rise to zero OTEbut finite ESE pairing. To visualize these results, we plot in Fig. <ref>(a) the ESE and OTE pairing magnitudes as a function of ω at ϕ=0. Observe that the OTE magnitude is completely suppressed, while the ESE contribution exhibits resonant peaks at the gap edges ±Δ. These ESE peaks are in full agreement with the ABSs energies, ω_±(2π n)=±Δ forn=0,1,2,…, which at ϕ = 0 merges with the continuum (indicated by blue arrows). At ϕ=π(2n-1) for n=1,2,…, the coefficients instead acquire opposite values, m_5(ω,L_ N=0,π(2n-1))=-m_6(ω,L_ N=0,π(2n-1)), which again affects the pairing amplitudes in a crucial wayf_0, N^r, E(x=0,ω) =0 ,f_0, N^r, O(x=0,ω) =-i/2v_F1/ cos[η(ω)] .Here the ESE contribution is instead completely suppressed, while the OTE term acquires a finite value, as is also shown in Fig. <ref>(b). At ϕ=π(2n-1) the ABSs cross zero energy, ω_±(π(2n-1))=0 for n=1,2,…, developing a protected crossing and also introducing a resonant peak at ω=0in the LDOS. Other work has found similar phase dependence for a combination of spin-orbit coupling and magnetism.<cit.> Very importantly, in the 2DTI we obtain that no magnetism is necessary. This LDOS peak (indicated by red arrow) at zero energy is directly reflected in the OTE pairing amplitude also peaking at zero energy.[A zero energy peak in the odd-frequency pairing was also reported in junctions with unconventional superconductors,<cit.> where the pairing function is peaked due to the sign change of the pairing potential. Moreover, a zero energy peak in the OTE amplitude was also predicted in Rashba double wires,<cit.> where Rashba coupling and the additional wire index allows for OTE pairing.] The mechanism is here the same as mentioned before: two scattering processes, Andreev reflection and normal transmission for electrons and holes, become exactly opposite at ϕ=π, unlikeat ϕ=0 where they are the same. Therefore it exists a clear interference pattern with distinguishablesignatures for each process. We stress that both results, the total suppression of OTE and ESE at ϕ=0 and ϕ=π, respectively, arise due to the metallic nature of the 2DTI edge. In normal metals finite backscattering processes usually induce finite normal reflection amplitudes.Before concluding we would like to also point out the connection between OTE and supercurrents I(ϕ). In both short and long junctions the supercurrentcan be calculated from the energy levels ω_nas<cit.> I(ϕ)=-(2e/ħ) ∑_n>0dω_n(ϕ)/dϕ. In short junctions only the pair of ABSs within the gap contribute to I(ϕ). Thus, for short junctions with the ABSs given by ω_±(ϕ)=±Δ cos(ϕ/2), the individual current contribution becomes I_±(ϕ)= ± I_ c sin(ϕ/2), where I_ c=eΔ/ħ is the maximum supercurrent across the junction. Provided fermion parity conservation I(ϕ)notably exhibits the well-known 4π-periodic fractional Josephson effect<cit.> in ϕ. Notice that I_±(ϕ) iszero at ϕ=0 while at ϕ=π it acquiresits maximum value. The former arises because there are no ABSs in the junction, while in the latter case the ABSs develop a protected crossing at zero-energy, giving rise to the resonant peak in the LDOS. This resonant peak is, as shown above, purely due to OTE pairing. Thus, the maximum supercurrent, i.e. the critical current I_ c, exhibits contributions from purely OTE pairing, since at ϕ=π only OTE pairing is present.We can thus conclude that the pairing amplitudes entirely capture the LDOS through the emergence of ABSs, and, remarkably, the zero energy peak in the LDOS is purely a consequence of the OTE pairing as shown in Fig. <ref>(b). These findings are in agreement with the odd-frequency nature previously found for a single Majorana state in time-reversal symmetry breaking systems.<cit.> However, note that in our setup time-reversal symmetry is preserved and the Majorana mode is therefore necessarily two-fold degenerate.§.§.§ Long junctionsNext we proceed by increasing the normal region length L_ N. Then the number of energy levels, bound in the junction within the energy gap Δ, increases and is proportional to L_ N/ξ. In this situation the ESE and OTE pairing amplitudes, given by Eqs. (<ref>), strongly depends on x. In what follows we primarily only need to consider twosituations of this spatial dependence, at the left interface, x=0, and in the middle of N, x=L_ N/2.At the left interface, x=0, the pairing amplitudes for ϕ=0 read f_0, N^r, E(x=0,ω) =-1/2v_F cos(ω L_ N)/ sin[η(ω)-L_ Nω] ,f_3, N^r, O(x=0,ω) =i/2v_F sin(ω L_ N)/ sin[η(ω)-L_ Nω]and for ϕ=πf_0, N^r, E(x=0,ω) =-1/2v_F sin(ω L_ N)/ cos[η(ω)-L_ Nω] ,f_3, N^r, O(x=0,ω) =1/2iv_F cos(ω L_ N)/ cos[η(ω)-L_ Nω] . The magnitude of these ESE and OTE pairing terms is plotted in Fig. <ref>(a,b) at ϕ=0 and ϕ=π, respectively.At ϕ=0 the ESE and OTE magnitudes reach zero value when ω L_ N=(2n+1)π/2 and ω L_ N=π n for n=0,1,2,…, respectively. In fact, the zeros of ESE correspond to the maxima of OTE and vice-versa. Specifically, for ω=0 the OTE pairing magnitude is vanishing, while the ESE magnitude remains finite and proportional to 1/(2v_F).On the other hand, at ϕ=π the position of the zeros of the ESE and OTE magnitudes is reversed.Remarkably, the OTE magnitude now has a resonant peak at zero energy, while the ESE magnitude remains finite at, again, approximately 1/(2v_F).Thus, although the OTE pairing magnitude is dominant around zero energy, the finite value of ESE obscures the pure OTE contribution to the LDOS we found in the short junction limit. The coexistence of ESE and OTE pairing in long junctions at the interface x=0 can be avoided by probing the pairing amplitudes in the middle of the N region at x=L/2, as presented in Fig. <ref>(c,d). In this case, Eqs. (<ref>) reduce tof_0, N^r, E(x=L_ N/2,ω) = m̅_5+m̅_6 ,f_3, N^r, O(x=L_ N/2,ω) = m̅_5-m̅_6 ,where m̅_5(ω,L_ N,ϕ)=(-1/2v_F) e^iϕ/2/ sin[η(ω)-L_ N/ξ_ω+ϕ/2] and m̅_6(ω,L_ N,ϕ)=m̅_5(ω,L_ N,-ϕ) e^iϕ. Thus, in the middle of the N region the coefficients m̅_5,6 lose their length dependence in the numerator, unlike m_5,6. Moreover, Eqs. (<ref>) very much resemble the behavior of the pairing amplitudes in short junctions discussed before. From Eqs. (<ref>) it is simple to conclude that at ϕ=0, m̅_5(ω,L_ N,0)=m̅_6(ω,L_ N,0) and thusf_0, N^r, E(x=L_ N/2,ω) = -1/2v_F1/ sin[η(ω)-L_ N/ξ_ω] ,f_3, N^r, O(x=L_ N/2,ω) =0 .Thus the OTE pairing is now completely suppressed for all energies within the energy gap. The ESE component, on the other hand, is still finite with multiple resonant peaks within the energy gap. These resonant peaks in the ESE magnitude directly correspond to the energies of the ABSs ω_±,n(ϕ=0)=(v_F/2L)[2π(n+1/2)] bound within Δ, as indicated by blue arrows in Fig. <ref>(c).Remarkably, the situation at ϕ=π implies m̅_5(ω,L_ N,π)=-m̅_5(ω,L_ N,π), leading to insteadf_0, N^r, E(x=L_ N/2,ω) = 0 ,f_3, N^r, O(x=L_ N/2,ω) =1/2iv_F1/ cos[η(ω)-L_ N/ξ_ω] .Thus, at ϕ=π and in the middle of the N region, the OTE component is the only local pairing amplitude, since ESE is completely vanishing. Furthermore, the OTE resonant peaks reveal the emergence of the ABSs with energies ω_±,n(ϕ=π)=(v_F/2L)[2π(n+1/2)±π], as indicated by red arrows in Fig. <ref>(d).Thus the peaks appearing in the subgap LDOS from the ABSs in the middle of the N regioncan be directly associated with different pairing amplitudes. At ϕ=0 the LDOS peaks are associated with pure ESE pairing, while at ϕ = π they are solely emerging due to the OTE pairing. Also, importantly, the zero energy bias peak at ϕ signaling the presence of Majorana fermions in the SNS junction is entirely associated with OTE pairing. This extends the results for a single pair of ABS in the short junction regime to include all subgap ABSs in longer junctions. Above we found that for both short and long junctions it is only the OTE pairing that is intimately connected to the energy resonance at zero energy. However, the dominance of the OTE pairing is not restricted to zero energy, ϕ=π phase, or the middle of the junction region. This can clearly be appreciated in Fig. <ref>, where we plot the spatial dependence of the ESE and OTE pairing magnitudes over the whole N region for energies away from zero and phase π. Although at the interface x=0, L_N, both ESE and OTE pairing magnitudes usually coexist at all phases, it is in the middle of the N region at x=L_ N/2, where OTE dominates over the completely suppressed ESE component for ϕ=π. The dominating OTE pairing extends over a notably finite region in the middle of the junction. It also survives to quite large subgap energies, ω≳ 0.4Δ, and also for phase values.Despite ESE often exhibiting a finite value in these regimes, the dominant behavior of OTE over ESE is clearly observed. Also notice how the spatial and energy dependences give rise to an oscillatory behavior of the pairing magnitudes given by Eqs. (<ref>) as can be seen in Fig. <ref>.Overall this allows us to conclude that OTE pairing and its influence on the LDOS is not restricted neither to zero energy nor ϕ=π.§ CONCLUSIONS In this work we have analytically studied the emergence of odd-frequency superconductivity in NS, NSN, and SNS junctions at the edge of a 2DTI without magnetism or any other time-reversal symmetry breaking perturbations, where the S regions have spin-singlet s-wave superconducting pairing induced by proximity to an external conventional superconductor. We have shown that odd-frequency mixed spin-triplet s-wave superconductivity does not require the presence of magnetic order but naturally arises at any NS interface as a result of breaking translation invariance in combination with the helicity of the 2DTI edge state. Moreover, we have clearly extended previous studies<cit.> andestablished a one-to-one correspondence between the subgap LDOS and odd-frequency pairing in NS, NSN, and SNS junctions at the edge of 2DTI without any magnetism. These geometries are all suitable for LDOS as well as conductance measurements as has recently been demonstrated in experiments. <cit.>For NS and NSN junctions, we have derived analytical expressions for both the pairing amplitudes and LDOS. Focusing on local, s-wave, pairing we have shown that both even-frequency, spin-singlet, even-parity (ESE) and odd-frequency, spin-triplet, even-parity (OTE) pairing is generally present in these systems.In the normal N region(s) the ESE and OTE magnitudes are equal and proportional to the Andreev reflection magnitude.In the superconducting S region the pairing amplitudes have two components arising from either the bulk or the interface(s). In the bulk of the S region, only the ESE pairing magnitude survives. However, there are both ESE and OTE interface components that develop an exponential decay into the bulk of the S region with the decay length set by the superconducting coherence length for small energies.Very close to the interfaces in the S region the OTE pairing becomes very dominant over an extremely suppressed ESE amplitude for energies well below the superconducting gap. Moreover, the behavior of the low-energy LDOS in the S region close to the interface has the same dependence on theenergy and distance from the interface as that of the OTE pairing. This allows us to associate the induced low-energyLDOS in the S region purely with OTE pairing.In SNS junctions we have demonstrated an even stronger functional relationship between superconducting pairing andlow-energy LDOS inside the N junction region. In this case the pairing amplitudes become dependent on thesuperconducting phase difference ϕ and on the length of the N region.In fact, we find that the condition giving the Andreev bound states (ABSs) energies in the junction is exactly thesame as the condition generating resonant peaks in either the ESE and OTE pairing amplitudes, independent of junction length. For short junctions, at ϕ=0 the ESE pairing dominates over a completely suppressed OTE pairing, while atϕ=π the ESE is instead zero and the OTE amplitude large. Notably, the pair of ABSs crosses zero atϕ = π, which is directly reflected in the OTE amplitude which has a resonant peak at zero energy.We have also shown that the supercurrent across such short junctions acquires its maximum value at ϕ=π as a result of the resonant peak at zero energy which is purely due to OTE pairing. In long junctions more ABSs fit within the N region, but still all ABS energies correspond to resonant peaks inthe pairing amplitudes. In this case we find the clearest distinction between ESE and OTE pairing in the middle ofthe N region. The ESE is completely suppressed at ϕ = π, while the OTE is finite and has resonant peaks ateach ABS energy. At ϕ=0 the relation is reversed with ESE having resonances at the ABS energies. Dominant OTEpairing exists also at distances away from the middle of the junction and at phase differences away from ϕ=π.In summary, our findings show that the finite LDOS, as well as local conductance, at the interfaces of NS or NSN junctions at the edge of a 2DTI is entirely a consequence of pure OTE pairing. This results from the unique nature ofthe helical edges of 2DTIs, where the Andreev reflection magnitude, strongly connected to LDOS and conductance, at the NS interface reaches its maximum for energies within the gap.<cit.> In SNS junctions the relationship is even stronger with the ABSs directly corresponding to peaks in the pairing amplitudes. In particular, the zero-energy ABS at ϕ = π, protected by topology, is of complete OTE nature.§ ACKNOWLEDGEMENTSWe thank P. Burset, D. Kuzmanovski and C. Triola for motivating and helpful discussions. This work was supported by the Swedish Research Council (Vetenskapsrådet), the Göran Gustafsson Foundation, the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation through the Wallenberg Academy Fellows program.§ RETARDED AND ADVANCED GREEN'S FUNCTIONS The structure of the Green's function G^r is given by Eq. (<ref>) in the main text. The elements G_ee,hh^r we refer to as the regular parts of G^r, while G_eh,he^r are the anomalous electron-hole parts.Electron-hole symmetry imposes for the BdG Hamiltonian PH_BdG^*P^†=-H_BdG, while for the Green's function P[G^r(x,x',ω)]^*P^†=-G^r(x,x',-ω), where P=σ_yτ_y.<cit.> This therefore connects the two diagonal (off-diagonal) elements of G^r.The advanced Green's function, G^a, has the same matrix structure as the retarded function given by Eq. (<ref>) and can be calculated from incoming boundary conditions. Alternatively, we can use the relation between retarded and advanced Green's functions: G^a(x,x',ω)=[G^r(x',x,ω)]^†.We are interested in the pairing amplitudes, that is, the anomalous (electron-hole) part of the retarded Green's function, G^r_eh(x,x',ω)= [[G^r_eh(x,x',ω)]_↑↓ -[G^r_eh(x,x',ω)]_↑↑;[G^r_eh(x,x',ω)]_↓↓ -[G^r_eh(x,x',ω)]_↓↑ ] ,where the minus signs arise due to the specific choice of basis. In order to decompose the spin symmetry, we write the anomalous Green's function according to Eq. (<ref>) in the main text. Then we arrive at the pairing amplitudes f^r_i:f_0^r(x,x',ω) =[G^r_eh(x,x',ω)]_↑↓-[G^r_eh(x,x',ω)]_↓↑/2 ,f_3^r(x,x',ω) =[G^r_eh(x,x',ω)]_↑↓+[G^r_eh(x,x',ω)]_↓↑/2 ,f_1^r(x,x',ω) =-[G^r_eh(x,x',ω)]_↑↑+[G^r_eh(x,x',ω)]_↓↓/2 ,f_2^r(x,x',ω) =i[G^r_eh(x,x',ω)]_↑↑+[G^r_eh(x,x',ω)]_↓↓/2 .From these equations we observe that f_0^r acquires a minus sign under the exchange of spins and is thus an odd function under spin exchange, while f^r_1,2,3 acquire a plus sign and are even under spin exchange. f^r_0 is therefore referred to as the spin-singlet component, while f^R_1,2,3 are the spin-triplet components.§.§ Antisymmetry The anomalous Green's function represents the wave function of a two-electron system which must obey antisymmetry upon the simultaneous exchange of spins (↑↔↓), spatial coordinates (x↔ x') and time (or energy/frequency) coordinates (t↔ t' or ω→-ω). In this work, we use retarded and advanced Green's functions which donot respect symmetry under frequency. These Green's functions are only partially defined on the time axis and therefore, when the sign of the frequency changes (or time), we should pass from one to the other,<cit.>G^r_s_1s_2(x,x',ω)=-G^a_s_2s_1(x',x,-ω) .By using Eqs. (<ref>) and (<ref>) we get[G^r_eh(x,x',ω)]_↑↓ =[G^a_eh(x',x,-ω)]_↓↑ ,[G^r_eh(x,x',ω)]_↑↑ =-[G^a_eh(x',x,-ω)]_↑↑ ,[G^r_eh(x,x',ω)]_↓↓ =-[G^a_eh(x',x,-ω)]_↓↓ ,[G^r_eh(x,x',ω)]_↓↑ =[G^a_eh(x',x,-ω)]_↑↓ .Similarly, we arrive at the conditions of antisymmetry for the spin-singlet and triplet amplitudesf^r_0(x,x',ω) =f_0^a(x',x,-ω) ,f^r_1(x,x',ω) =-f_1^a(x',x,-ω) ,f^r_2(x,x',ω) =-f_2^a(x',x,-ω) ,f^r_3(x,x',ω) =-f_3^a(x',x,-ω) .These equations represent the full antisymmetry conditions under the exchange of spin, spatial coordinates, and frequency imposed by Fermi-Dirac statistics on the pairing amplitudes and they are presented in the main text as Eqs. (<ref>). § NS JUNCTION We model a NS junction at the metallic edge of a 2DTI by considering a step-like profile of the induced pairing potential, with the normal region at x<0 and the superconducting region atx>0, Δ(x)=θ(x)Δ=0,x<0,Δ , x>0 ,where we can set the overall superconducting phase to zero.§.§ Scattering processesIn general, there are four different scattering processes at the NS interface which readΨ_1(x) = ϕ_1^Ne^ik_ex+a_1ϕ_3^Ne^ik_hx +b_1ϕ_2^Ne^-ik_ex, x<0 c_1ϕ_1^Se^ik_e^Sx +d_1ϕ_4^Se^-ik_h^Sx, x>0 Ψ_2(x) = ϕ_4^Ne^-ik_hx+a_2ϕ_2^Ne^-ik_ex +b_2ϕ_3^Ne^ik_hx, x<0 c_2ϕ_4^Se^-ik_h^Sx +d_2ϕ_1^Se^ik_e^Sx,x>0 Ψ_3(x) =c_3ϕ_2^Ne^-ik_ex +d_3ϕ_3^Ne^ik_hx,x<0ϕ_2^Se^-ik_e^Sx+a_3ϕ_4^Se^-ik_h^Sx+b_3ϕ_1^Se^ik_e^Sx, x>0Ψ_4(x) =c_4ϕ_3^Ne^ik_hx +d_4ϕ_2^Ne^-ik_ex, x<0ϕ_3^Se^ik_h^Sx+a_4ϕ_1^Se^ik_e^Sx+b_4ϕ_4^Se^-ik_h^Sx, x>0while the conjugated processes areΨ̃_1(x') = ϕ̃_1^Ne^ik_ex'+ã_1ϕ̃_3^N e^ik_hx' +b̃_1ϕ̃_2^N e^-ik_ex',x<0c̃_1ϕ̃_1^S e^ik_e^Sx' +d̃_1ϕ̃_4^Se^-ik_h^Sx', x>0Ψ̃_2(x') = ϕ̃_4^N e^-ik_hx'+ã_2ϕ̃_2^N e^-ik_ex' +b̃_2ϕ̃_3^N e^ik_hx', x<0c̃_2ϕ̃_4^S e^-ik_h^Sx' + d̃_2ϕ̃_1^S e^ik_e^Sx', x>0 Ψ̃_3(x') = c̃_3ϕ̃_2^N e^-ik_ex' +d̃_3ϕ̃_3^N e^ik_hx',x<0 ϕ̃_2^S e^-ik_e^Sx'+ã_3ϕ̃_4^S e^-ik_h^Sx'+b̃_3ϕ̃_1^S e^ik_e^Sx', x>0Ψ̃_4(x') = c̃_4ϕ̃_3^N e^ik_hx' +d̃_4ϕ̃_2^N e^-ik_ex',x<0ϕ̃_3^S e^ik_h^Sx'+ã_4ϕ̃_1^Se^ik_e^Sx'+b̃_4ϕ̃_4^S e^-ik_h^Sx',x>0 ,where ϕ_1^N =[ 1, 0, 0, 0 ]^T, ϕ_2^N =[ 0, 1, 0, 0 ]^T, ϕ_3^N =[ 0, 0, 1, 0 ]^T, ϕ_4^N =[ 0, 0, 0, 1 ]^T, ϕ_1^S =[ u, 0, v, 0 ]^T, ϕ_2^S =[ 0, u, 0, v ]^T, ϕ_3^S =[ v, 0, u, 0 ]^T, ϕ_4^S =[ 0, v, 0, u ]^T,are wave functions of H_BdG(k), while ϕ̃_1^N =[ 0, 1, 0, 0 ]^T, ϕ̃_2^N =[ 1, 0, 0, 0 ]^T, ϕ̃_3^N =[ 0, 0, 0, 1 ]^T, ϕ̃_4^N =[ 0, 0, 1, 0 ]^T,ϕ̃_1^S = [ 0, u, 0, v ]^T, ϕ̃_2^S = [ u, 0, v, 0 ]^T, ϕ̃_3^S = [ 0, v, 0, u ]^T, ϕ̃_4^S = [ v, 0, u, 0 ]^T,are the wave functions of the conjugated Hamiltonian H̃_BdG(k). Notice that the conjugated scattering processes are constructed after solving the eigenvalue problem for H̃_BdG(k)=H_BdG^*(-k)=H_BdG^T(-k) instead of Eq. (<ref>) in the main text. In these equations we have used the following relations:k_e,h =(μ±ω)/v_F ,k_e,h^S =(μ±√(ω^2-Δ^2))/v_F=k_μ± k(ω) , u,v =√(1/2[1±√(ω^2-Δ^2)/ω]) .In this work we mainly focus on energies within Δ and then k_e,h^S=k_μ± iκ(ω), with κ(ω)=√(Δ^2-ω^2)/v_F and the coherence factors can be written as u=√(Δ/2ω)e^iη(ω)/2 and v=√(Δ/2ω)e^-iη(ω)/2, where η(ω)= arccos(ω/Δ). At the NS interface, Ψ_1 represents the following process:an incoming electron (right-moving with spin up) from the N region with wave function ϕ_1^Ne^ik_exexperiences reflection and transmission at the NS interface with certain probabilities. It can be reflected into a left-moving electron with spin-down with wave functionϕ_2^Ne^-ik_ex and amplitude b_1 or Andreev reflected into a left-moving hole with spin down with wave function ϕ_3^Ne^ik_hx and amplitude a_1, or transmitted to into the S region in the form of a right-moving quasielectron with wave functionϕ_1^Se^ik_e^Sx and amplitude c_1 or a right-moving quasihole with wave function ϕ_4^Se^-ik_h^Sx and amplitude d_1.Thus, a_1, b_1, c_1 and d_1 are the amplitudes of reflection into a hole (Andreev reflection), reflection into an electron (normal reflection), transmission into an electron and transmission into a hole.Likewise, Ψ_2,3,4 correspond to scattering processes for an incoming hole from the N region and incoming electron or incoming hole from the S region, respectively.The processes Ψ_i and Ψ̃_i are fully determined after finding the coefficients a_i, b_i, c_i and d_i. These in turn are calculated by matching the functions Ψ_i at the NS interface x=0,[Ψ_i(x>0)]_x=0 =[Ψ_i(x<0)]_x=0 .Each Ψ_i is a four column vector and therefore provides four equations. At the end we have a system of 16 equations for the 16 unknown coefficients a_i, b_i, c_i and d_i, which is generally solvable and completely determines thescattering states Ψ_i. The same holds for the conjugated processes Ψ̃_i.Notice that we here have written the scattering wave functions, Eqs. (<ref>-<ref>),in a general form. However, the processes with amplitudes b_i and d_i are forbidden by helicityof the 2DTI edge; an incident electron can be only reflected as a hole by a superconducting barrier ortransmitted as an electron through it.Thus, normal reflection and non-local Andreev transmission are forbidden by helicity conservation<cit.> and b_i=d_i=0.For the Andreev reflection amplitudes we generally also obtain a_1,2=v/u=-a_3,4. §.§ Green's functions and pairing amplitudes§.§.§ Normal region After finding Ψ_i and Ψ̃_i, given by Eqs. (<ref>) and (<ref>), we construct the retarded Green's functions using Eq. (<ref>) in the main text. In the normal (N) region, we obtain the following expressions for the regular and anomalous partsG_ee^r(x,x',ω)= 1/iv_f[ G_ee,↑↑^r 0; 0 G_ee,↓↓^r ],G_hh^r(x,x',ω)=1/iv_f[ G_hh,↓↓^r 0; 0 G_hh,↑↑^r; ],G_eh^r(x,x',ω)= 1/iv_f[ 0 0; 0 a_1(ω) e^-i(k_ex-k_hx'); ],G_he^r(x,x',ω)= 1/iv_f[ a_1(ω) e^i(k_hx-k_ex')0;00 ],with G_ee,↑↑^r=θ(x-x') e^ik_e(x-x'),G_ee,↓↓^r=θ(x'-x) e^-ik_e(x-x'),G_hh,↓↓^r=θ(x'-x) e^ik_h(x-x'), G_hh,↑↑^r=θ(x-x') e^-ik_h(x-x') anda_1=v/u the Andreev reflection amplitude for an incident electron from N. For energies within the gap a_1(ω)= e^-iη(ω).Now, by using G^r_ee, we calculate the LDOS in the N region following Eq. (<ref>) in the main text and getρ_ N(x,ω)=1/π v_F ,where we have used lim_x→ x'θ(x-x')=1/2.Notice how the LDOS is independent of both energy and position, as is expected for the helical edge state in a 2DTI.By decomposing the spin symmetry of the anomalous part of the retarded and advanced Green's functions, employing Eq. (<ref>), we get the pairing amplitudesf_0, N^r(x,x',ω) =a_1(ω)/2i v_F e^-iμ(x-x')+ω(x+x')/v_F ,f_3, N^r(x,x',ω) =-f_0^r(x,x',ω),and f_1,2, N^r=0. As discussed in Sec. <ref>, the pairing amplitudes must obey Fermi-Dirac statistics. We use Eqs. (<ref>) and check the antisymmetry of previous pairing functions. For example, for the singlet component we havef^a_0, N(x',x,-ω) =-a_1^*(-ω)/2iv_F e^iμ(x'-x)-ω(x+x')/v_F ,=a_1(ω)/2iv_F e^-iμ(x-x')+ω(x+x')/v_F ,=f_0^r(x,x',ω) ,where we have used a_1^*(-ω)= e^iη(-ω)=- e^-iη(ω)=-a_1(ω). Thus, we conclude that f_0^r is antisymmetric.Likewise, the triplet component is fully antisymmetric and obeys the relation given by Eq. (<ref>). Notice that in order to check antisymmetry we have to use the advanced pairing functions as discussed in Appendix <ref>. We now use Eqs. (<ref>) in order to decompose into the even and odd-frequency componentsf^r, E_0, N(x,x',ω) =a_2(ω)/2iv_F C_xx'e^-iω(x+x')/v_F,f^r, O_0, N(x,x',ω) =-a_2(ω)/2v_F S_xx'e^-iω(x+x')/v_F,f^r, E_3, N(x,x',ω) =a_2(ω)/2v_F S_xx'e^-iω(x+x')/v_F,f^r, O_3, N(x,x',ω) =-a_2(ω)/2iv_F C_xx'e^-iω(x+x')/v_F,where C_xx'= cos[k_μ(x-x')], S_xx'= sin[k_μ(x-x')], and a_2=a_1 the Andreev reflection amplitude. These are Eqs. (<ref>) given in the main text.Before going further it is worth to point out the following. To check the antisymmetry of the retarded even- and odd-frequency pairing amplitudes is not trivial and it is important to write down their respective advances functions. We have explicitly verified that the given expressionsfollow the antisymmetry relations. For a quick check, however, we can focus on the parity in space which follows directly from cos[k_μ(x-x')] and sin[k_μ(x-x')]. Since the spin is already explicit this means the only symmetry left is that of the frequency. For example, for OSO we see that sin[k_μ(x-x')] makes the amplitude odd in space and since it is a spin singlet, the only possibility is for an odd-frequency dependence.§.§.§ Superconducting region In the superconducting region (S) we proceed similarly as in N. We obtain the retarded Green's function, which for energies below the gap Δ, reads G^r_ee,↑↑(x,x',ω) =Z(ω) e^ik_μ(x-x') [ ã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|N(x,x',ω)] ,G^r_ee,↓↓(x,x',ω) = Z(ω) e^-ik_μ(x-x')[ã_3(ω)e^-κ(ω)(x+x') + e^-κ(ω)|x-x'|N(x',x,ω)],G^r_hh,↓↓(x,x',ω) = Z(ω) e^ik_μ(x-x')[ã_3(ω)e^-κ(ω)(x+x') + e^-κ(ω)|x-x'|N(x',x,ω)],G^r_hh,↑↑(x,x',ω) =Z(ω) e^-ik_μ(x-x') [ ã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|N(x,x',ω)] ,G^r_eh,↑↓(x,x',ω) = Z(ω) e^ik_μ(x-x') [ u/vã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|] ,G^r_eh,↓↑(x,x',ω) = Z(ω) e^-ik_μ(x-x') [ v/uã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|] ,G^r_he,↓↑(x,x',ω) = Z(ω) e^ik_μ(x-x') [ v/uã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|] ,G^r_he,↑↓(x,x',ω) = Z(ω) e^-ik_μ(x-x') [ u/vã_3(ω)e^-κ(ω)(x+x') +e^-κ(ω)|x-x'|] ,G^r_eh(he),↑↑,↓↓(x,x',ω) =0 , G^r_ee(hh),↑↓,↓↑(x,x',ω)=0 , where N(x,x',ω)=θ(x-x')(u/v)+θ(x'-x)(v/u), Z(ω)=(1/iv_F)/[ e^iη(ω)- e^-iη(ω)], and ã_3=a_3=-a_2.Notice that, while in the anomalous parts (eh, he) only mixed spin components are finite, in the regular parts (ee, hh) we also obtain finite equal spin components. If we were to consider finite magnetic order in the system the situation would dramatically change with additional off-diagonal (mixed spin) terms in the regular part and equal spin terms in the anomalous components.<cit.>By using the regular electron-electron part of G^r, given by the first two expressions of Eqs. (<ref>), we obtain the LDOS in the superconducting region,ρ_ S(x,ω)= Im{i/π v_F[ ρ̅+(1- ρ̅) e^ik(ω)2x]} ,where ρ̅(ω)=ω/√(ω^2-Δ^2), and k(ω)=iκ(ω). Within the gap, ω^2<Δ^2, this reduces toρ_ S(x,|ω|<|Δ|)=1/π v_F e^-κ(ω)2x ,which is given as Eq. (<ref>) in the main text.The pairing amplitudes are calculated by decomposing the spin components according to Eq. (<ref>) and we arrive at f_0,S^r(x,x',ω) = Z(ω) e^-κ(ω)|x-x'| C_xx' + Z(ω)ã_3(ω)/2 e^-κ(ω)(x+x')[e^iη(ω) e^ik_μ(x-x')+ e^-iη(ω) e^-ik_μ(x-x')] ,f_3,S^r(x,x',ω) = Z(ω) e^-κ(ω)|x-x'| iS_xx' + Z(ω)ã_3(ω)/2 e^-κ(ω)(x+x')[e^iη(ω) e^ik_μ(x-x')- e^-iη(ω) e^-ik_μ(x-x')] , with f_1,2,S^r=0.It is important to notice that the pairing amplitudes given by Eqs. (<ref>) contain two terms, which arise from different parts in the S region. The first term is proportional to e^-κ(ω)|x-x'| and is associated with the bulk deep inside the S region. Indeed, locally (x = x'), such a term becomes independent of the space coordinate. The second term is proportional to the Andreev reflection amplitude a_3 and therefore arises due the presence of the NS interface. This term also keeps a spatial dependence even locally that gives the decay into the S region. Thus, in the S region, the pairing amplitudes can be written with as bulk (B)and interface (I) components f_0, B^r(x,x',ω) = Z(ω) e^-κ(ω)|x-x'| C_xx' ,f_3, B^r(x,x',ω) = Z(ω) e^-κ(ω)|x-x'| iS_xx' ,f_0, I^r(x,x',ω) = Z(ω)ã_3(ω)/2 e^-κ(ω)(x+x') K_+ ,f_3, I^r(x,x',ω) = Z(ω)ã_3(ω)/2 e^-κ(ω)(x+x') K_- ,where we have used K_±(ω,x,x')=e^iη(ω) e^ik_μ(x-x')± e^-iη(ω) e^-ik_μ(x-x'). We have explicitly checked that all these pairing amplitudes are fully antisymmetric, obeying Eqs. (<ref>). For example, for f_3, I^r we findf_3, I^a(x',x,-ω) = Z^∗(-ω)ã^*_3(-ω)/2 e^-κ(-ω)(x+x')× K_-(-ω,x',x) ,=-Z(ω)ã_3(ω)/2 e^-κ(ω)(x+x')× K_-(ω,x,x') ,=-f_3, I^r(x,x',ω)where we have used ã_3^*(-ω)=-a_3(ω), Z^∗(-ω)=Z(ω) and K_-(-ω,x',x)=K_-(ω,x,x'). We have also used the advanced pairing function f^a_3, I which iscalculated from the relation between retarded and advanced Green's functions G^a(x,x',ω)=[G^r(x',x,ω)]^†.We now use Eqs. (<ref>) in the main text to isolate the even- and odd-frequency components, which gives in the bulk of the S regionf^r, O_0,3, B(x,x',ω) =0 ,f^r, E_0, B(x,x',ω) =Z(ω) e^-κ(ω)|x-x'| C_xx' ,f^r, E_3, B(x,x',ω) =iZ(ω) e^-κ(ω)|x-x'| S_xx' .and close to the NS interface on the S sidef^r, E_0, I(x,x',ω) =a_3(ω)/2iv_F e^-κ(ω)(x+x') B(ω) C_xx' ,f^r, O_0, I(x,x',ω) =a_3(ω)/2v_F e^-κ(ω)(x+x') S_xx' ,f^r, E_3, I(x,x',ω) =a_3(ω)/2v_F e^-κ(ω)(x+x') B(ω) S_xx' ,f^r, O_3, I(x,x',ω) =a_3(ω)/2iv_F e^-κ(ω)(x+x') C_xx' ,where B(ω)=[ e^iη(ω)+ e^-iη(ω)]/[ e^iη(ω)- e^-iη(ω)].Notice that in the bulk we only obtain trivial even-frequency spin-singlet and -triplet components, namely the ESE and ETO pairing amplitudes. At the interface, however, hosts all symmetry classes (ESE, OSO, ETO, and OTE). Previous two sets of equations, for the bulk and interface, correspond to Eqs. (<ref>-<ref>) in the main text. § NSN Here we consider a NSN junction, where the S region has a finite length L_ S restricted to 0<x<L_ S, i.e., Δ(x)=0,x<0 ,Δ , 0<x<L_ S , 0,x>0 .§.§ Scattering processesThe four scattering processes in a NSN junction readΨ_1(x) = ϕ_1^N e^ik_ex+a_1ϕ_3^N e^ik_hx+b_1ϕ_2^N e^-ik_ex,x<0∑_ip_iϕ_i^S e^ik_i^Sx, 0<x<L_ Sc_1ϕ_1^N e^ik_ex +d_1ϕ_4^N e^-ik_hx, x>L_ SΨ_2(x) = ϕ_4^N e^-ik_hx+a_2ϕ_2^N e^-ik_ex+b_2ϕ_3^N e^ik_hx,x<0∑_iq_iϕ_i^S e^ik_i^Sx, 0<x<L_ Sc_2ϕ_4^N e^-ik_hx +d_2ϕ_1^N e^ik_ex, x>L_ SΨ_3(x) = c_3ϕ_2^N e^-ik_ex +d_3ϕ_3^N e^ik_hx,x<0∑_ir_iϕ_i^S e^ik_i^Sx, 0<x<L_ Sϕ_2^N e^-ik_ex+a_3ϕ_4^N e^-ik_hx+b_3ϕ_1^N e^ik_ex, x>L_ SΨ_4(x) = c_4ϕ_3^N e^ik_hx +d_4ϕ_2^N e^-ik_ex,x<0∑_is_iϕ_i^Se^ik_i^Sx, 0<x<L_ Sϕ_3^N e^ik_hx+a_4ϕ_1^N e^ik_ex+b_4ϕ_4^N e^-ik_hx, x>L_ S ,while the conjugated processes areΨ̃_1(x) = ϕ̃_1^N e^ik_ex+ã_1ϕ̃_3^N e^ik_hx+b̃_1ϕ̃_2^N e^-ik_ex, x<0∑_ip̃_iϕ̃_i^S e^ik_i^Sx, 0<x<L_ Sc̃_1ϕ̃_1^N e^ik_ex +d̃_1ϕ̃_4^N e^-ik_hx, x>L_ S ,Ψ̃_2(x) = ϕ̃_4^N e^-ik_hx+ã_2ϕ̃_2^N e^-ik_ex+b̃_2ϕ̃_3^N e^ik_hx,x<0∑_iq̃_iϕ̃_i^S e^ik_i^Sx, 0<x<L_ Sc̃_2ϕ̃_4^N e^-ik_hx +d̃_2ϕ̃_1^N e^ik_ex, x>L_ SΨ̃_3(x) = c̃_3ϕ̃_2^N e^-ik_ex +d̃_3ϕ̃_3^N e^ik_hx,x<0∑_ir̃_iϕ̃_i^S e^ik_i^Sx, 0<x<L_ Sϕ̃_2^N e^-ik_ex+ã_3ϕ̃_4^N e^-ik_hx+b̃_3ϕ̃_1^N e^ik_ex, x>L_ SΨ̃_4(x) = c̃_4ϕ̃_3^N e^ik_hx +d̃_4ϕ̃_2^N e^-ik_ex,x<0 ∑_is̃_iϕ̃_i^S e^ik_i^Sx, 0<x<L_ Sϕ̃_3^N e^ik_hx+ã_4ϕ̃_1^N e^ik_ex+b̃_4ϕ̃_4^N e^-ik_hx, x>L_ Swhere i=1,2,3,4, k_1,2^S=± k_e^S, k_3,4^S=± k_h^Sand the spinors ϕ_i^N,S and ϕ̃_i^N,S are given by Eqs. (<ref>)and (<ref>), respectively. Notice that, as for NS junctions, in this part we have written the general form of the scattering wave functions in the N and S regions.The S region is in general formed by a linear combination of four elements given in terms of the amplitudes p_i, q_i, r_i and s_i. Their meaning is as follows: p_1 represent electron transmission from left N to S, p_2 the normal reflection at the right SN interface, p_3 the Andreev reflection at the SN interface, and p_4 the Andreev reflection at the left NS interface. Similar ideas apply to the amplitudes q_i, r_i and s_i. The amplitudes of all these processes are calculated after matching the wave-functions at the left NS and right SN interfaces. Due tohelicity conservation in the 2DTI edge states we can again directly obtainb_i=d_i=b̃_i=d̃_i=0 and p_2,4=s_2,4=r_1,3=q_1,3=0. Similar relations hold for the conjugated processes. §.§ Green's function and pairing amplitudes§.§.§ Normal regions Finding Ψ_i and Ψ̃_i, given by Eqs. (<ref>) and (<ref>), allows us to construct the retarded Green's functions using Eq. (<ref>) in the main text. In the two normal regions the Green's functions and pairing amplitudes acquire the same form as in the NS junction, with the only difference that the Andreev reflection amplitude becomes dependent on the length L_ S of the S region and readsa_1(ω,L_ S)= a_2(ω,L_ S)= sin[iκ(ω)L]/ sin[iκ(ω)L-η(ω)]. Notice that a_2(ω,L_ S)→ a_2(ω)=v/u for L_ S→∞, as expected.§.§.§ Superconducting region Due to the finite length of the S region in NSN junctions we find quite different results here compared to the NS junction. In the S region the retarded Green's function contains the componentsG^r_ee,↑↑(x,x',ω) = e^i(k_h^Sx-k_e^Sx')γ_2+e^i(k_e^Sx-k_h^Sx')γ_3+θ(x-x')[ e^ik_e^S(x-x')γ_1+ e^ik_h^S(x-x')γ_4]-θ(x'-x)[ e^ik_e^S(x-x')γ_2+ e^ik_h^S(x-x')γ_3] ,G^r_ee,↓↓(x,x',ω) = e^i(k_e^Sx'-k_h^Sx)γ_3+e^i(k_h^Sx'-k_e^Sx)γ_2-θ(x-x')[ e^-ik_e^S(x-x')γ_2+ e^-ik_h^S(x-x')γ_3]+θ(x'-x)[ e^-ik_e^S(x-x')γ_1+ e^-ik_h^S(x-x')γ_4] ,G^r_hh,↓↓(x,x',ω) = e^i(k_h^Sx-k_e^Sx')γ_2+e^i(k_e^Sx-k_h^Sx')γ_3-θ(x-x')[ e^ik_h^S(x-x')γ_2+ e^ik_e^S(x-x')γ_3]+θ(x'-x)[ e^ik_h^S(x-x')γ_1+ e^ik_e^S(x-x')γ_4] ,G^r_hh,↑↑(x,x',ω) = e^i(k_e^Sx'-k_h^Sx)γ_3+e^i(k_h^Sx'-k_e^Sx)γ_2+θ(x-x')[ e^-ik_h^S(x-x')γ_1+ e^-ik_e^S(x-x')γ_4]-θ(x'-x)[ e^-ik_h^S(x-x')γ_2+ e^-ik_e^S(x-x')γ_3] ,G_eh,↑↓^r(x,x',ω) =- e^i(k_e^Sx-k_h^Sx')β_2-e^i(k_h^Sx-k_e^Sx')β_3+ e^i k_e^S(x-x')[θ(x-x')β_2+θ(x'-x)β_3]+ e^i k_h^S(x-x')[θ(x-x')β_3+θ(x'-x)β_2] ,G_eh,↓↑^r(x,x',ω) = e^i(k_h^Sx'-k_e^Sx)β_1+e^i(k_e^Sx'-k_h^Sx)β_4+ e^-i k_e^S(x-x')[θ(x-x')β_3+θ(x'-x)β_2]+ e^-i k_h^S(x-x')[θ(x-x')β_2+θ(x'-x)β_3] ,G_he,↓↑^r(x,x',ω) = e^i(k_h^Sx-k_e^Sx')β_1+e^i(k_e^Sx-k_h^Sx')β_4+ e^i k_e^S(x-x')[θ(x-x')β_2+θ(x'-x)β_3]+ e^i k_h^S(x-x')[θ(x-x')β_3+θ(x'-x)β_2] ,G_he,↑↓^r(x,x',ω) =- e^i(k_e^Sx'-k_h^Sx)β_2-e^i(k_h^Sx'-k_e^Sx)β_3+ e^-i k_e^S(x-x')[θ(x-x')β_3+θ(x'-x)β_2]+ e^-i k_h^S(x-x')[θ(x-x')β_2+θ(x'-x)β_3] ,where we have definedγ_1 =1/iv_fu^4/(u^2-v^2)[u^2- e^i(k_e^S-k_h^S)Lv^2] ,γ_2=1/iv_fu^2v^2/(u^2-v^2)[-u^2 e^i(k_h^S-k_e^S)L+v^2] , γ_3 =1/iv_fu^2v^2/(u^2-v^2)[-u^2+ e^i(k_e^S-k_h^S)Lv^2] ,γ_4=1/iv_fv^4/(u^2-v^2)[u^2 e^i(k_h^S-k_e^S)L-v^2] , β_1 = 1/iv_fu^3v/(u^2-v^2)[-u^2 e^i(k_h^S-k_e^S)L+v^2] ,β_2=1/iv_fu^3v/(u^2-v^2)[u^2- e^i(k_e^S-k_h^S)Lv^2] , β_3 =1/iv_fu v^3/(u^2-v^2)[u^2 e^i(k_h^S-k_e^S)L-v^2] ,β_4=1/iv_fu v^3/(u^2-v^2)[-u^2+ e^i(k_e^S-k_h^S)Lv^2] .The regular electron-electron part of G^r gives the LDOS in the S region which readsρ_S(x,ω)= -1/π v_F Im{γ_1/i[v^2/u^2 e^2ik(ω)L(1+v^2/u^2-2 e^-2ik(ω)x)+(1+v^2/u^2-2v^2/u^2 e^2ik(ω)x)]} .For energies within Δ this expression reduces to ρ_ S(x,|ω|<Δ)=ω^2(1- e^-2κ(ω)L)^2+Δ^2(1+ e^-2κ(ω)L)[ e^-2κ(ω)L( e^2κ(ω)x-ω^2/Δ^2) +( e^-2κ(ω)x-ω^2/Δ^2)]/π v_F[Δ^2(1+ e^-2κ(ω)L)^2-4ω^2 e^-2κ(ω)L] .At ω=0, this expression can be simplified even further and we obtain ρ_ S(x,0)=1/π v_F e^-2L/ξ e^2x/ξ+ e^-2x/ξ/1+ e^-2L/ξ ,where we have used ξ = v_F/Δ. This is the expression for the LDOS given in Eq. (<ref>) in the main text.For the pairing amplitudes, limiting ourselves to energies within the gap, we arrive atf_0, S^r(x,x',ω) = -β_2 e^-κ(ω)(x+x')[ e^-ik_μ(x-x')-2iη(ω) + e^ik_μ(x-x')]+β_1 e^κ(ω)(x+x')[ e^-ik_μ(x-x') + e^ik_μ(x-x')-2iη(ω)]+2 cos[k_μ(x-x')][β_2 e^-κ(ω)|x-x'|-β_1 e^-2iη(ω) e^κ(ω)|x-x'|] ,f_3, S^r(x,x',ω) = -β_2 e^-κ(ω)(x+x')[ e^ik_μ(x-x')- e^-ik_μ(x-x')-2iη(ω)]+ β_1 e^κ(ω)(x+x')[ e^ik_μ(x-x')-2iη(ω) - e^-ik_μ(x-x')]+2i sin[k_μ(x-x')][β_2 e^-κ(ω)|x-x'|-β_1 e^-2iη(ω) e^κ(ω)|x-x'|] ,and f^r_1, S = f^r_2, S = 0, where we have used γ_4,2=- e^-2iη(ω)γ_2,1 and β_4,3=- e^-2iη(ω)β_2,1,with β_1=-iZ(ω) e^iη(ω)-κ(ω)L_ S/4 sin[iκ(ω)L_ S+η(ω)] ,β_2=-β_1 e^2κ(ω)L_ S, γ_1= e^iη(ω)β_2,and γ_2= e^-iη(ω)β_1.From these and the related advanced expressions we can extract the even- and odd-frequency components which readsf_0, S^r, E(x,x',ω) =C_xx'[ e^-κ(ω)(x+x')β_42^-+ e^κ(ω)(x+x')β_13^-+2β_2 e^-κ(ω)|x-x'|+2β_3 e^κ(ω)|x-x'|] ,f_0, S^r, O(x,x',ω) =-iS_xx'[ e^-κ(ω)(x+x')β_42^++ e^κ(ω)(x+x')β_13^+ ,f_3, S^r, E(x,x',ω) =iS_xx'[ e^-κ(ω)(x+x')β_42^-+ e^κ(ω)(x+x')β_13^-+2β_2 e^-κ(ω)|x-x'|+2β_3 e^κ(ω)|x-x'|] ,f_3, S^r, O(x,x',ω) =-C_xx'[ e^-κ(ω)(x+x')β_42^++ e^κ(ω)(x+x')β_13^+] ,where β_42(13)^±=β_4(1)±β_2(3).These pairing amplitudes in S can be composed into bulk (B) and interface (I) contributions. Bulk contributions we designate terms which are independent of the average distance from either interface, i.e. with an overall |x-x'| spatial dependence. The remaining terms we label interface contributions, as they all have a decay length 1/[2κ(ω)] from the interface. With this division we directly arrive at the results given in Eqs. (<ref>)-(<ref>) in the main text.§ SNS JUNCTION Finally we consider SNS junctions, where the normal region has a finite length L_ N, while the S regions are semi-infinite such thatΔ(x)=Δ ,x<0, 0 , x<0<L_ N, Δe^iϕ ,x>0,where ϕ is the superconducting phase difference across the junction. §.§ Scattering processesIn this case the processes readΨ_1(x) = ϕ_1^S_Le^ik^S_ex+a_1 ϕ_3^S_Le^ik^S_hx+b_1 ϕ_2^S_Le^-ik_e^Sx,x<0∑_ip_i ϕ_i^Ne^ik_ix,0<x<L_ N c_1 ϕ_1^S_Re^ik_e^Sx +d_1 ϕ_4^S_Re^-ik_h^Sx, x>L_ NΨ_2(x) = ϕ_4^S_L e^-ik_h^Sx+a_2ϕ_2^S_L e^-ik^S_ex+b_2ϕ_3^S_L e^ik_h^Sx,x<0∑_iq_iϕ_i^N e^ik_ix,0<x<L_ Nc_2ϕ_4^S_R e^-ik^S_hx +d_2ϕ_1^S_R e^ik^S_ex, x>L_ NΨ_3(x) = c_3ϕ_2^S_L e^-ik_e^Sx +d_3ϕ_3^S_L e^ik^S_hx,x<0∑_ir_iϕ_i^N e^ik_ix,0<x<L_ Nϕ_2^S_R e^-ik^S_ex+a_3ϕ_4^S_R e^-ik^S_hx+b_3ϕ_1^S_R e^ik^S_ex, x>L_ NΨ_4(x) = c_4ϕ_3^S_L e^ik^S_hx +d_4ϕ_2^S_L e^-ik^S_ex,x<0∑_is_iϕ_i^N e^ik_ix,0<x<L_ Nϕ_3^S_R e^ik^S_hx+a_4ϕ_1^S_R e^ik_e^Sx+b_4ϕ_4^S_R e^-ik^S_hx, x>L_ NΨ̃_1(x) = ϕ̃_1^S_L e^ik^S_ex+ã_1ϕ̃_3^S_Le^ik^S_hx+b̃_1ϕ̃_2^S_L e^-ik^S_ex,x<0∑_ip̃_iϕ̃_i^N e^ik_ix,0<x<L_ Nc̃_1ϕ̃_1^S_R e^ik_e^Sx +d̃_1ϕ̃_4^S_R e^-ik^S_hx, x>L_ NΨ̃_2(x) = ϕ̃_4^S_L e^-ik^S_hx+ã_2ϕ̃_2^S_L e^-ik_e^Sx+b̃_2ϕ̃_3^S_L e^ik^S_hx,x<0∑_iq̃_iϕ̃_i^N e^ik_ix,0<x<L_ Nc̃_2ϕ̃_4^S_R e^-ik^S_hx +d̃_2ϕ̃_1^S_R e^ik^S_ex, x>L_ NΨ̃_3(x) = c̃_3ϕ̃_2^S_L e^-ik^S_ex +d̃_3ϕ̃_3^S_L e^ik^S_hx,x<0∑_ir̃_iϕ̃_i^N e^ik_ix,0<x<L_ Nϕ̃_2^S_R e^-ik^S_ex+ã_3ϕ̃_4^S_R e^-ik^S_hx+b̃_3ϕ̃_1^S_R e^ik^S_ex, x>L_ NΨ̃_4(x) = c̃_4ϕ̃_3^S_L e^ik_h^Sx +d̃_4ϕ̃_2^S_L e^-ik_e^Sx,x<0∑_is̃_iϕ̃_i^N e^ik_ix,0<x<L_ Nϕ̃_3^S_R e^ik_h^Sx+ã_4ϕ̃_1^S_R e^ik_e^Sx+b̃_4ϕ̃_4^S_R e^-ik^S_hx , x>L_ Nwhere i=1,2,3,4 and the spinors ϕ_i^N, ϕ̃_i^N,ϕ_i^S_L, ϕ̃_i^S_L acquire the same form as in the NS junction,given by Eqs. (<ref>) and (<ref>). In the right superconductor,the wave functions of H_ BdG readϕ_1^S_R =[ ue^iϕ/2, 0, ve^-iϕ/2, 0 ]^T ϕ_2^S_R =[ 0, ue^iϕ/2, 0, ve^-iϕ/2 ]^T ϕ_3^S_R =[ ve^iϕ/2, 0, ue^-iϕ/2, 0 ]^T ϕ_4^S_R =[ 0, ve^iϕ/2, 0, ue^-iϕ/2 ]^T,while the corresponding conjugated wave functions areϕ̃_1^S_R = [ 0, ue^-iϕ/2, 0, ve^iϕ/2 ]^T ϕ̃_2^S_R = [ ue^-iϕ/2, 0, ve^iϕ/2, 0 ]^T ϕ̃_3^S_R = [ 0, ve^-iϕ/2, 0, ue^iϕ/2 ]^T ϕ̃_4^S_R = [ ve^-iϕ/2, 0, ue^iϕ/2, 0 ]^T . As before,helicity conservation directly imposes b_i=d_i=b̃_i=d̃_i=0 and also q_1,3=r_1,3=p_2,4=s_2,4=q̃_1,3=r̃_1,3=p̃_2,4=s̃_2,4=0. §.§ Green's functions and pairing amplitudes§.§.§ Normal regionWe construct the retarded Green's function from the scattering states in the same was as for SNand NSN junctions. In the normal region N the regular and anomalous Green's functionsbecomeG_ee^r(x,x',ω) = [ e^ik_e(x-x') M_1(x,x')0;0 e^-ik_e(x-x')M_2(x,x') ] ,G_hh^r(x,x',ω) = [e^ik_h(x-x')M_1(x',x)0;0 e^-ik_h(x-x')M_2(x',x) ] ,G_eh^r(x,x',ω) = [e^i(k_ex-k_hx') m_50;0 e^-i(k_ex-k_hx') m_6 ] ,G_he^r(x,x',ω) = [e^i(k_hx-k_ex') m_70;0 e^-i(k_hx-k_ex') m_8 ] ,where M_1(x,x')=θ(x-x')m_1+θ(x'-x)m_3, M_2(x,x')=θ(x-x')m_2+θ(x'-x)m_4,m_1=u/vm_5, m_2=v/um_6, m_3=v/um_7, m_4=u/vm_8, andm_5 ≡ p_1r̃_4α_1=1/iv_fuv/u^2-v^2e^i(k_e-k_h)L_ N-iϕ ,m_6 ≡ q_2s̃_3α_4=1/iv_fuv/u^2e^i(k_h-k_e)L_ N-iϕ-v^2 ,m_7 ≡ r_1r̃_2α_1=1/iv_fuv/u^2e^i(k_h-k_e)L_ N+iϕ-v^2 ,m_8 ≡ q_4s̃_1α_4=1/iv_fuv/u^2-v^2e^i(k_e-k_h)L_ N+iϕ .From these equations, we observe thatit is enough to specify the form of m_5.Thus, for energies within the superconducting gap previous expressions reduce to m_5(ω,L_ N,ϕ) =-1/2v_F e^i(ϕ/2- L_ N/ξ_ω)/ sin[η(ω)-L_ N/ξ_ω+ϕ/2] m_6(ω,L_ N,ϕ) =m_5(ω,L_ N,-ϕ) e^i(2L_ N/ξ_ω+ϕ) ,m^*_7,8(-ω) =m_6,5(ω) ,which correspond to Eq. (<ref>) presented in the main text. Instead of deriving the full expression for the LDOS from the regular part of G^r, we here focus on the main peaks as that is much easier and, for this discussion, as enlightening. In the N region the low-energy peaks in the LDOS are from the discrete ABSs, whose number depend on the ratio L_ N/ξ. There exist different ways to calculate these energy levels. One efficient way is to locate the poles (zeros of the denominator) of the Andreev reflection coefficients, a_1,2, which area_1(ω,L_ N,ϕ)=uv[ e^i(k_eL_ N-ϕ/2)- e^i(k_hL_ N +ϕ/2)]/u^2 e^i(k_hL_ N+ϕ/2)-v^2 e^i(k_eL_ N-ϕ/2) .For energies within Δ, and using u/v= e^iη(ω), this expression can be written asa_1(ω,L_ N,ϕ)= sin[L_ N/ξ_ω-ϕ/2]/ sin[η(ω)-L_ N/ξ_ω+ϕ/2]and also a_2(ω,L_ N,ϕ)=a_1(ω,L_ N,-ϕ). With these expressions, the poles of a_1,2 can be shown to give rise to the following condition for the ABSs, which is also given in the main text:2η(ω)-2L_ N/ξ_ω±ϕ=2π n , n=0,1,….For a short junction, L_ N≪ξ, the condition for ABSs reduces to 2η(ω)±ϕ=2π n, with two Andreev levels at energiesω_±(ϕ)=±Δ cos(ϕ/2) ,while for a long junction, L_ N≫ξ we obtainω_±^n(ϕ)=v_F/2L_ N[2π(n+1/2)±ϕ] .Notice that the Andreev bound states given by Eqs. (<ref>) and(<ref>),and plotted in Fig. <ref>(a-b), develop crossings (indicated by red arrows) around zero energyω=0 at π(2n-1) for n=1,2,3,…,which are protected by time-reversal symmetry. The Andreev reflection magnitude |a_1(ω,L_ N,ϕ)| at ϕ=0,π is also plotted in Fig. <ref>(c-d) for short and long junctions. Observe that it develops resonances at the energies of the protected crossings. In particular, when ϕ=π a single zero-energy peak emerges in short junctions as shown in Fig. <ref>(c). In longer junctions the number of such resonances (that correspond to ABSs) increases, being proportional to L_ N/ξ, and the zero-energy peak coexists with additional peaks as we observe in Fig. <ref>(d).Remarkably, the same discussion was performed in the main text but there instead the ABSs energies was derived from the pairing amplitudes.In terms of the pairing amplitudes in the N region we obtain after some algebraf_0, N^r(x,x',ω) = e^i(k_ex-k_hx') m_5+ e^-i(k_ex-k_hx') m_6/2 ,f_3, N^r(x,x',ω) = e^i(k_ex-k_hx') m_5- e^-i(k_ex-k_hx') m_6/2 ,and f^r_1,N = f^r_2,N = 0, where m_5,6 are given by Eqs. (<ref>).We can explicitly check the antisymmetry condition of these expressions. For example, for the singlet component f_0, N^r(x,x',ω) we havef_0, N^a(x',x,-ω) = e^-i(k_ex-k_hx') m_7^*(-ω)+ e^i(k_ex-k_hx') m_8^*(-ω)/2 ,= e^i(k_ex-k_hx') m_5+ e^-i(k_ex-k_hx') m_6/2 ,=f_0, N^r(x,x',ω) ,and therefore we conclude that f_0, N^r(x,x',ω) is antisymmetric according to the relations in Eqs. (<ref>). Likewise, we have verified the antisymmetry condition f^r_3,N(x,x',ω)=-f_3,N^a(x',x,-ω). By using Eq. (<ref>) we write down the even- and odd-frequency componentsf_0, N^r, E(x,x',ω) =W_+(ω) cos[k_μ(x-x')] ,f_0, N^r, O(x,x',ω) =W_-(ω)i sin[k_μ(x-x')] ,f_3, N^r, E(x,x',ω) =W_+(ω)i sin[k_μ(x-x')] ,f_3, N^r, O(x,x',ω) =W_-(ω) cos[k_μ(x-x')]where W_±(ω)=[m_5 e^i(x+x')/ξ_ω± m_6 e^-i(x+x')/ξ_ω ]/2.These equations correspond to Eqs. (<ref>) presented in the main text. We have checked that Eqs. (<ref>) fulfill the antisymmetry conditions given by Eqs. (<ref>).§.§.§ Superconducting regionsIn the left superconducting region we obtainG^r_ee,↑↑(x,x',ω) =Z [ a_1(ω,L_ N,ϕ) e^i(k_h^Sx-k_e^Sx')+P(x,x',ω)] ,G^r_ee,↓↓(x,x',ω) =Z [a_2(ω,L_ N,ϕ) e^i(k_h^Sx'-k_e^Sx)+P(x',x,ω)] ,G^r_hh,↓↓(x,x',ω) =Z [ a_1(ω,L_ N,ϕ) e^i(k_h^Sx-k_e^Sx')+Q(x,x',ω)]G^r_hh,↑↑(x,x',ω) =Z [a_2(ω,L_ N,ϕ) e^i(k_h^Sx'-k_e^Sx)+Q(x',x,ω)] ,G^r_eh,↑↓(x,x',ω) =Z [ a_1(ω,L_ N,ϕ)v/u e^i(k_h^Sx-k_e^Sx')+P̅(x,x',ω)]G^r_eh,↓↑(x,x',ω) =Z [a_2(ω,L_ N,ϕ)u/v e^i(k_h^Sx'-k_e^Sx)+P̅(x',x,ω)] ,G^r_he,↓↑(x,x',ω) =Z [ a_1(ω,L_ N,ϕ)u/v e^i(k_h^Sx-k_e^Sx')+Q̅(x,x',ω)] ,G^r_he,↑↓(x,x',ω) =Z [a_2(ω,L_ N,ϕ)v/u e^i(k_h^Sx'-k_e^Sx)+Q̅(x',x,ω)] ,where P(x,x',ω) =θ(x-x')u/ve^ik_e^S(x-x')+θ(x'-x)v/ue^ik_h^S(x-x') , Q(x,x',ω) =θ(x-x')v/ue^ik_e^S(x-x') +θ(x'-x)u/ve^ik_h^S(x-x') , P̅(x,x',ω) =[θ(x-x')e^ik_e^S(x-x') +θ(x'-x)e^ik_h^S(x-x')] , Q̅(x,x',ω) =[θ(x-x')e^ik_e^S(x-x') +θ(x'-x)e^ik_h^S(x-x')] ,with Z=1/iv_f1/(u/v)-(v/u)and Z̅=1/-iv_f1/(u/v)^*-(v/u)^*. In this case the Andreev reflection amplitudes obeya_2(ω,L_ N,ϕ)=a_1(ω,L_ N,-ϕ)with a_1(ω,L_ N,ϕ) given by Eqs. (<ref>). This results in the pairing amplitudesf^r_0, S_L(x,x',ω) = Z/2 e^-ik(ω)(x+x')[ a_1(ω,L_ N,ϕ)v/u e^ik_μ(x-x')+a_2(ω,L_ N,ϕ)u/v e^-ik_μ(x-x')] +Z e^ik(ω)|x-x'|C_xx' ,f^r_3, S_L(x,x',ω) = Z/2 e^-ik(ω)(x+x')[ a_1(ω,L_ N,ϕ)v/u e^ik_μ(x-x')-a_2(ω,L_ N,ϕ)u/v e^-ik_μ(x-x')] +Z e^ik(ω)|x-x'|iS_xx' ,and f^r_1, S_L = f^r_2, S_L = 0. In a similar way we proceed for the right superconducting region to calculate the pairing amplitudes.We can then extract the odd- and even-frequency components. For energies within the superconducting gap we obtainf_0,±^r, E(x,x',ω) =Z e^iϕ(1∓1)/2{ e^±κ(ω)(x+x')[a_1(4)(ω,L_ N,ϕ)a_1(ω)+a_2(3)(ω,L_ N,ϕ)a_1^*(ω)]+2 e^-κ(ω)|x-x'|}C_xx' ,f_0,±^r, O(x,x',ω) =Z e^iϕ(1∓1)/2{ e^±κ(ω)(x+x')[a_1(4)(ω,L_ N,ϕ)a_1(ω)-a_2(3)(ω,L_ N,ϕ)a_1^*(ω)]}iS_xx' ,f_3,±^r, E(x,x',ω) =Z e^iϕ(1∓1)/2{ e^±κ(ω)(x+x')[a_1(4)(ω,L_ N,ϕ)a_1(ω)+a_2(3)(ω,L_ N,ϕ)a_1^*(ω)]+2 e^-κ(ω)|x-x'|}iS_xx' ,f_3,±^r, O(x,x',ω) =Z e^iϕ(1∓1)/2{ e^±κ(ω)(x+x')[a_1(4)(ω,L_ N,ϕ)a_1(ω)-a_2(3)(ω,L_ N,ϕ)a_1^*(ω)]}C_xx' ,where ± subscripts correspond to results for the left and right superconducting regions, respectively. Here we have used the notation a_1(ω)=v/u= e^-iη(ω) and a_1(ω,L_ N,φ) given by Eq. (<ref>). Moreover, we find that a_3,4(ω,L_ N,φ)=a_2,1(ω,L_ N,φ) e^2κ(ω). As with NS junctions, we associate bulk behavior to elements in the pairing amplitudes that do not depend on the Andreev reflections and have an exponential form e^-κ(ω)|x-x'|. We therefore conclude that the second term in the curly brackets in the ESE and ETO pairing amplitudes emerge in the bulk of S.However, at the interface, we observe all different allowed pairing amplitudes.Notice that all the pairing amplitudes in the right superconductor acquire a phase factor e^iϕ. | http://arxiv.org/abs/1707.08530v2 | {
"authors": [
"Jorge Cayao",
"Annica M. Black-Schaffer"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726164016",
"title": "Odd-frequency superconducting pairing and subgap density of states at the edge of a two-dimensional topological insulator without magnetism"
} |
^1Physics Department, College of Science, University of Sulaimani, Kurdistan Region, Iraq^2 Department of Mechanical Engineering, National United University, 1, Lienda, Miaoli 36003, Taiwan^3 Reykjavik University, School of Science and Engineering, Menntavegur 1, IS-101 Reykjavik, Iceland^4 Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, [email protected], [email protected] Spin-dependent heat and thermoelectric currents in a quantum ring with Rashba spin-orbit interaction placed in a photon cavity are theoretically calculated. The quantum ring is coupled to two external leads withdifferent temperatures. In a resonant regime, with the ring structure in resonance with the photon field, the heat and the thermoelectric currents can be controlled bythe Rashba spin-orbit interaction. The heat current is suppressed in the presence of thephoton field due to contribution of the two-electron and photon replica states to the transport whilethe thermoelectric current is not sensitive to changes in parameters of the photon field.Our study opens a possibility to use the proposed interferometric device as a tunable heat current generatorin the cavity photon field.78.20.N-,73.23.-b, 42.50.Pq, 78.20.JqSpin-dependent heat and thermoelectric currents in a Rashba ring coupled to a photon cavity Nzar Rauf Abdullah^1, Chi-Shung Tang^2, Andrei Manolescu^3 and Vidar Gudmundsson^4 December 30, 2023 ===========================================================================================§ INTRODUCTION Thermal properties of nanoscale systems have attracted much interest due to their highefficiency of converting heat into electricity <cit.> which has been studied by bothexperimental <cit.> and theoretical <cit.> groups. This growing interest in thermal properties on the nanoscale is mainly caused by the peculiar thermal transport behaviors of the systems, which followfrom their very special electronic structure.Traditionally, thermal transport can be obtained by a temperature gradient across a system that contains mobile charge, which in turn create a thermoelectric current (TEC) <cit.>.Detailed experimental and theoretical tests have provided new insight into the thermoelectrics oflow dimensional structures such as quantum dots <cit.>, double quantum dots <cit.>, quantum point contacts <cit.>, quantum wires <cit.>, and quantum rings <cit.>.These nano-structures show that high thermoelectric efficiency may be achieved by using the quantum properties of the systems <cit.>, such as quantized energy <cit.>, and interference effects <cit.>. On the other hand, it has been shown that the spin polarization induced by an electric field in a two-dimensionalelectron gas with a Rashba spin-orbit interaction influences the thermal transport <cit.>. This phenomenon has been investigated in various systems exhibiting Rashba spin-orbit interaction <cit.>. In this system, the temperature gradient is utilized as a possibility to generate spin-dependent thermoelectricand heat currents, in an analogy to the generation of a charge current in conventional thermoelectrics. Until now, earlier work focused mostly on thermal transport without the influences of a cavity photon field onthe electronic structure.In a previous paper, we studied the influences of a quantized photon field on TEC <cit.>. We assumed a quantum wire coupled to a photon cavity and found thatthe TEC strongly depends on the photon energy and the number of photons initially in the cavity. In addition,the current is inverted for the off-resonant regime and a reduction in the current is found for a photon field in resonance to electronic systems,a direct consequence of the Rabi-splitting.In the present work, we study the thermoelectric effect in a quantum ring taking into accountthe electron-electron and electron-photon interactions in the presence of a Rashba spin-orbit coupling.The spin-dependent heat and thermoelectric currents are calculatedusing the generalized non-Markovian master equation when the bias voltage difference betweenthe two leads tends to zero. Moreover, the influences of the photon field on thermal transport of the system is presented. We investigate these effects in the late transient time regime before thephoton leak of the cavity influences the results.§ THEORY We model the thermal properties of the quantum device based on a quantum ring coupled toa cavity photon field. The quantum ring is assumed to be realized in a two dimensional electron gas of an GaAs/AlGaAs hetero-structure in thexy-plane and the photon field is confined to a three-dimensional cavity.The quantum ring is embedded in a cavity that is much larger than the ring. In addition, the quantum ring is diametrically coupled to two semi-infinite leads. §.§ Quantum ring coupled to a cavity photon field The quantum ring embedded in the central system with length L_x = 300 nm is schematically shown in fig01.The ring is parabolically confined with characteristic energy ħΩ_0 = 1.0 meV along the y-direction and hard-wallconfined in the transport direction (x-direction). The potential of the ring is expressed asV_r(𝐫) = ∑_i=1^6V_iexp[-(γ_xi(x-x_0i))^2 - (γ_yiy)^2] +1/2m^* Ω_0^2y^2, where V_i, γ_xi, and γ_yi are constants presented in Table <ref>. x_03=ϵ is a small numerical symmetry breaking parameter with |ϵ|=10^-5 nm to guarantee a numerical stability.The second term of V_r indicates the characteristic energy of the electron confinementof the short quantum wire the quantum ring is embedded in.The central system is coupled to a photon cavity much larger than the central system. The total momentum operator of the quantum ring coupled to thephoton field is defined as 𝐩̂(𝐫) = ħ/i∇ +e/c[𝐀̂(𝐫) +𝐀̂γ(𝐫)], where 𝐀̂(𝐫) = -Byx̂is the vector potential of the static classical external magnetic field with𝐁 = B ẑ, and 𝐀̂_γ(𝐫) is the vector potential of the quantized photon field in the cavity that is introduced in terms of the photon creation (â^†) and annihilation (â) operators 𝐀̂_γ=A(𝐞â+𝐞^*â^†) with 𝐞= 𝐞_x for the longitudinal photon polarization (x-polarization) and 𝐞= 𝐞_y forthe transverse photon polarization (y-polarization) <cit.>. The Hamiltonian for two-dimensional electrons in the quantum ring coupled to a photon cavity is Ĥ_S = ∫ d^2 r Ψ̂^†(𝐫)[(𝐩̂^2/2m^* +V_r(𝐫)) + H_Z.+. Ĥ_R(𝐫)]Ψ̂(𝐫)+Ĥ_ee+ħωâ^†â, with the electron spinor field operatorsΨ̂(𝐫)= ( [ Ψ̂(↑,𝐫); Ψ̂(↓,𝐫) ]), Ψ̂^†(𝐫)= ([ Ψ̂^†(↑,𝐫),Ψ̂^†(↓,𝐫) ]), where Ψ̂(x)=∑_aψ_a^S(x)Ĉ_a is the field operator withx≡ (𝐫,σ), σ∈{↑,↓} and the annihilation operator, Ĉ_a, for the single-electron state (SES) ψ_a^S(x) in the central system.The second term of H^S is the Hamiltonian that givesthe Zeeman interaction of the static magnetic field with spin of the electron. It can be described byH_Z = 1/2 (μ_B g_S B σ_z), where μ_B is the Bohr magnetron and g_S refers tothe electron spin g-factor.The third terms of H^S is the Rashba-spin orbit coupling that describesthe interaction between the orbital motion and the spin of an electron Ĥ_R(𝐫)=α/ħ( σ_xp̂_y(𝐫) -σ_yp̂_x(𝐫) ) , where α is a coupling constant that can be tuned by an external electric field, and σ_x and σ_y are the Pauli matrices. In addition, Ĥ_ee stands for the electron-electron interaction <cit.>,and ħω_γâ^†â is the free photon Hamiltonianin the cavity with ħω_γ as the photon energy.A time-convolutionless generalized master equation (TCL-GME) is utilized to investigate the transport properties of thesystem <cit.>. The TCL-GME is local in time and satisfies the positivity for the many-body state occupation described the reduced density operator (RDO).Before the central system is coupled to the leads, the total density matrix isthe product of the density matrices of the system and the leads ρ̂_T.The RDO of the system after the coupling is defined asThe reduced density operator is calculated using a TCL non-Markovian generalized master equation (GME) valid for a weak coupling of the leads and system. The GME is derived according to a Nakajima-Zwanzigprojection approach with the coupling Hamiltonian entering the dissipation kernel of the integro-differential equation up to second order. The coupling of the central system and the leads is expressed by a many-body coupling tensor derived from the geometry of the single-electrons states in the contact area of the leads and system <cit.>.ρ̂_S(t) =Tr_l (ρ̂_T) where l ∈{L,R} refers to the two electron reservoirs, the left (L) and the right (R) leads, respectively. The time needed to reach the steady state depends on the chemical potentials in each lead, the bias window, and their relation to the energy spectrum of the system. In our calculations we integrate the GME to t = 220 ps, a point in time late in the transient regime when the system is approaching the steady state.The heat current (I^H) can be calculated from the reduced density operator.It is the rate at which heat is transferred through the system over time. Therefore, the heat current in our system can be introduced as I^H_l = c_l d/dt[ρ̂_S,l(t) (Ĥ_S-μN̂_ e)]= c_l∑_αβ (α̂ | ρ̇̂̇_S,l | β̂)(E_α - μN̂_ e) δ_αβ , where ρ̂_S,l is the reduced density operator in terms of the l lead, c_L=+1, but c_R=-1, andĤ_S is the Hamiltonian of the central system coupled to a cavity,μ = μ_ L = μ_ R, and N̂_ e is the number operator of the electrons in the ring system. The thermoelectric current (I^ TH) in terms of the reduced density operator can be defined as I^ TH_l = c_lTr[ρ̇̂̇_S,lQ̂] , where the charge operator is Q̂ = e∫ d^2r Ψ̂^†(𝐫)Ψ̂(𝐫) <cit.>. In the next section, we present our main results of the thermal transport of a quantum ring coupled to aphoton field.§ RESULTS We assume a single cavity mode with photon energy ħω_γ = 0.55 meV. The applied perpendicular magnetic field is B = 10^-5 T to lift the spin degeneracy. The value of the magnetic field is out of the Aharonov-Bohm (AB) regime becausethe area of the ring structure is A = π a^2 ≈ 2 × 10^4 nm^2 leading toa magnetic field B_0 = ϕ_0/A ≈ 0.2 T corresponding to one flux quantum ϕ_0 = hc/e <cit.>. The applied magnetic field is B = 10^-5 T is much smaller than B_0, orders of magnitudes outside the AB regime.A temperature difference is applied betweenthe left and the right leads, which induces a current to flow in the central system.A temperature gradient emerges as the leads are coupledwith different thermal baths. Therefore, a thermal current is driven to flowthrough the ring due to the Seebeck effect.We begin our description by showing the energy spectrum of the ring versus the Rashba coupling constantin fig02, where the states 0ES (green rectangles) are zero-electron states,1ES (red circles) are one-electron states, and 2ES (blue circles) are two-electron states.Figure <ref>(a) displays the many-electron energy of the quantum ring without a cavity photon field. The energy of the states decreases with increasing Rashba coupling constant. As a result,crossing of the one-electron states at α≈ 12 meV (green arrow) are formed corresponding to the AC destructive phase interference <cit.>. Figure <ref>(b) displays an energy spectrum of photon-dressed many-body states of the quantum ring in the presenceof the photon field with energy ħω_γ = 0.55 meV andcoupling g_γ = 0.05 meV. Comparing to the energy spectrum in fig02(a),where the photon field is neglected, photon replica states are formed. The energy spacing between the photon replicas is approximately equal to thephoton energy at low electron-photon coupling strength.Generally, the perturbational idea of a simple replica with an integer photon number only applies for a weakly coupled electrons and photons out of resonance, but we use here the terminology to indicate the more general concept of cavity-photon dressed electron states. For instance at the Rashba coupling constant α = 0.0 meV nm, the state at E_μ≃ 1.5 meV, thefirst replica of the ground state, is formed near the second-excited statethat can not be seen in the absence of the photon field (fig02(a)).The ring system here under this condition is in a resonance with the photon field. These photon replicas have a important role in the electron transport through the system that will be shown later.In addition, the energy spectrum of the leads has a subband structure (not shown)since the leads contain semi-infinite quasi-one-dimensional non-interacting electron systems <cit.>. §.§ Heat current Heat current is the rate of change in the thermal energy as it is presented in Heat_Current. In nanoscale systems coupled to electron reservoirs with zero bias window,the heat current takes on zero or positive values depending on the location of the chemical potential of the leads with respect tothe energy states of the system. If the chemical potential is equal to the value of the energy of an isolated state of the system,the quantum system is resonant with the leads, and the heat current is close to zero as is seenin quantum dots <cit.>.Otherwise, the heat current has a positive value. We observe that for our rather large ring structure the heat current has always a nonvanishing positive value. This has to do with the high density of states or the near degeneracy of the states of the system, that the tiny Zeeman spin term orthe Rashba spin-orbit coupling in the ring does not drastically change. Important here is the thermal energy due to the higher temperature in the left lead and the photon energy with respect to the energy scale of the rings. They are all of a similar order. Figure <ref> indicates theheat current versus the chemical potential of the leads for three different values of theRashba coupling constant: α = 0.0 (blue squares), α = 6.0 (red circles), andα = 12.0 meV nm (green diamonds). As we see, the heat current is zero below μ = 1.0 meV becausethis region is below the lowest subband energy of the leads. In the absence of the Rashba spin-orbit interaction (α = 0.0 meV nm),for the selected range of the chemical potentialsfrom μ = 1.0 to 1.75 meV,we observe 6 photon dressed electron states, the lowest of which is the ground state as is shown in fig02(a).Therefore, a change of the chemical potential μ brings these ring states into resonance with the leads. For the low magnetic field the two spin components of these states are almost degenerate and the ring structure supplies a further near orbital degeneracygiving the heat current a nonzero value at the above mentioned resonant states.Two current dips (blue arrows) are formed at μ = 1.2 and 1.35 meV corresponding to thesecond- and third-exited states, respectively.In the presence of the Rashba spin-orbit interaction when α = 6.0 meV nm (red circles)two current dips are observed at μ = 1.25 and1.70 meV corresponding to the third- and fifth-excited states, respectively. Tuning the Rashba coupling constant to α = 12.0 meV nm (green diamonds) the two dips are formed at μ = 1.112 and1.473 meV corresponding to an additional degeneration of the second- and third-excited stateson one hand, and the fourth- and fifth-excited states on the other hand. In this case, the strong degeneration caused bythe Rashba spin-orbit interaction induces a smaller heat current in the dips compared to the current dip at α = 6.0 meV nm.The results here are very interesting because the nonzero heat current at the resonant energy levels can only be obtained in systems with high density of states (or near degeneracy)offered by the ring structure, the small the Zeeman spin-splitting, or the Rashba spin-orbitinteraction.We also notice that in the all aforementioned cases the two-electron states are active in the transportin such away, that one fourth of the heat current is carried by the two-electron states.It should be mentioned that the mechanism of transferred heat current through the one- and two-electron states is different here. The heat current flows from the left lead to the ring system through the one-electron states, but the opposite mechanism happens in the case of the two-electron states, where the heat current is transferred from the ring to the left lead. Therefore, the contribution of the two electron-states reduces the “total” heat current in the system for the range of the chemical potential used here. Now, we consider the ring to be coupled to a cavity with x-polarized photon field,and initially no photon in the cavity. To see the influences of the photon field on the heat current, we tune the electron-photon coupling strength g_γand fix the Rashba coupling constant at α = 12.0 meV nm (degenerate states). Figure <ref>(a) displays the heat current as a function of thechemical potential for different values of the electron-photon coupling strength g_γ.It is clearly seen that in the presence of the photon field the heat current is almost unchangedaround μ = 1.112 meV (left green arrow) corresponding tothe degeneracy point of the second- and third-excited states at E_μ = 1.112 meV since they are off-resonancestateswith respect to the photon field. But, the heat current is suppressed at the degenerate energy of the fourth- and fifth-excited states E_μ = 1.473 meVdue to the activated photon replica states in the transport. The photon energy is ħω_γ = 0.55 meV which is approximatelyequal to the energy spacing between the ground state/first-excited state andthe fourth-/fifth-excited states, respectively. However the contribution of the photon replica states is weak here becausethe cavity contains no photon initially, but it influences the heat current in the system. In addition, the contribution of the one-electron (two-electron) states to the transportin the presence of the photon field is decreased (increased), respectively. As a result, the heat current is suppressed in the system. We tune the photon polarization to the y-direction and see the contribution of the two-electron statesto the transport is further enhanced. Thus, the heat current is again suppressed asis shown in fig04(b) (blue rectangles). The temperature dependence of the heat current for all three considered electron-photon coupling strengths is shown in fig05 when the Rashba coupling constant is α = 12.0 meV nm and μ = 1.112 meV (at thedegenerate energy level of the second- and the third-excited states). We fix the temperature of the right lead at T_ R = 0.01 K and tune the temperatureof the left lead. The heat current increases by enhancement of the temperature because the electronscarry more thermal energy. As is expected the heat current is not significantly changed by tuningthe strength of the electron-photon coupling because the energy states at E_μ = 1.112 meV are outof resonance with respect to the photon field as we mentioned above. §.§ Thermoelectric current A temperature gradient causes a current to flow along a quantum ring. The electrons movefrom the hot lead to the cold lead, but also from the cold to the hot one,depending on the position of the chemical potential relatively to the energy spectrum of the central system.Both electron and energy are transported in this case <cit.>. The movement of electrons under the temperature gradient induces the TEC.We show how the TEC defined in TEC is influenced by the Rashba spin-orbit interaction and the photon field. Figure <ref>(a) indicates the TEC versus the chemical potential of the leads forthree different values of the Rashba coupling constant: α = 0.0 (blue squares), α = 6.0 (red circles), and α = 12.0 meV nm (green diamonds). The TEC is essentially governed by the difference between the two Fermi functions of the external leads.The TEC is generated when the Fermi functions of the leads havethe same chemical potential but different width.One can explain the TEC of the system when α = 0.0 meV nm in the following way:The TEC becomes zero in two cases. First, when the two Fermi functions or the occupations are equal to 0.5 (half filling),and in the second one, both Fermi functions imply occupations of 0 or 1 (integer filling),as is shown in fig06(b) <cit.>. Consequently, the TEC is approximately zero at μ≈ 1.112 meV (blue squares) correspondingto half filling of the degenerate energy levels of the second- and the third-excited state <cit.>.The TEC is approaching to zero at μ≤ 1.0 and μ≥ 1.4 meV for the integer filling of0 and 1, respectively. The same mechanism applies to the ring system including the spin-orbit interactionwhen the Rashba coupling constant is α = 6.0 (red circles) and 12.0 meV nm (green diamonds).But in the presence of the Rashba spin-orbit interaction, the TEC and the half filling is slightly shiftedto the left side because the energy states are shifted down for the higher value of the Rashba coupling constant (see fig02(a)).We should mention that all the states contributing to the TEC are one-electron states, while theone- and two-electron states participated in the creation the heat current.The effects of the photon field on the TEC in the system should not be neglected.Figure <ref> displays the TEC versus the chemical potential of the leads. In fig07(a) the TEC is plotted for different values of the electron-photon coupling strength.Assuming the photon energy is ħω_γ = 0.55 meV, the Rashba coupling constant is fixed at α = 12 meV nm, and the photon field is polarized inthe x-direction. We can clearly see that the TEC is not efficiently influenced bytuning the electron-photon coupling strength because the cavity is initially empty of photons.The TEC is drastically changed by tuning the number of photon initially in the cavity as shown in Ref. <cit.>. In addition, the TEC is not significantly affected by the direction of the photon polarization in the cavityas is shown in fig07(b).Variation of the TEC and the occupation with the Rashba coupling constant α are shownin fig08 for the system without a photon field (w/o ph) (blue rectangles)and with the photon field (w ph) (red circles). The chemical potential is fixed at μ_L = μ_R = 1.112 meV corresponding to the degeneration point ofthe first- and the second-excited states at α = 12.0 meV nm (see fig02(a)).The TEC here depends on the same mechanism, whether the occupation is an integer or a half integer. Therefore, the TEC is zero at the half integer occupation around α = 12.0 meV nm, andthe TEC is approximately zero at the integer occupation around α≤ 5.0 andα⩾ 15.0 meV nmas is seen in fig08(a) and (b). Opposite to the heat current shown in fig05, the TEC is slightly enhancedby a stronger electron-photon coupling strength g_γ = 0.15 meVat higher temperature gradient (see fig09).It also indicates that the characteristics of the TEC are almost the same even if the temperaturegradient is increasedin the system up to 2.0 K. § CONCLUSIONSWe have investigated the thermal properties of a quantum ringcoupled to a photon field, and two electron reservoirs for sequential tunneling through the system.We focused on the quantum limit where the energy spacing between successive electroniclevels is larger than the thermal energy Δ E_μ > k_BΔ T. A generalmaster equation is used to study the time-evolution of electrons in the system.Although, one expects the heat current to be nearly zero at resonant energy levels withrespect to the leads <cit.>, our study shows that the heat current has nonzero valuesin the presence of the high density of states or near degeneracies caused by the ring structure and a tiny Zeeman spin splitting. A Rashba spin-orbit interaction can be used to fine tune the degeneracies and the thermal transport properties of the system. 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"authors": [
"Nzar Rauf Abdullah",
"Chi-Shung Tang",
"Andrei Manolescu",
"Vidar Gudmundsson"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726125814",
"title": "Spin-dependent heat and thermoelectric currents in a Rashba ring coupled to a photon cavity"
} |
University of the Witwatersrand, School of Physics,Private Bag 3, WITS-2050, Johannesburg, South Africa [email protected] work outlines the novel application of the empirical analysis of causation, presented by Kutach, to the study of information theory and its role in physics. The central thesis of this paper is that causation and information are identical functional tools for distinguishing controllable correlations, and that this leads to a consistent view, not only of information theory, but also of statistical physics and quantum information. This approach comes without the metaphysical baggage of declaring information a fundamental ingredient in physical reality and exorcises many of the otherwise puzzling problems that arise from this view-point, particularly obviating the problem of `excess baggage' in quantum mechanics.This solution is achieved via a separation between information carrying causal correlations of a single qubit and the bulk of its state space. § INTRODUCTION Fundamental or provisionally fundamental physical laws have no causal character. This curiosity, prominently remarked upon by Russell <cit.>,necessitates some elaboration: a notion of causation requires both that there is an asymmetry in determination(if A causes B then B cannot be said to cause A) and that causesare well-defined local factors <cit.>.Physical law cannot be said to be causal as it does not satisfy either requirement: initial/boundary conditions specified at a given time t canbe said to determine the infinite past of the system's state as much as they determine its infinite future. Moreover, causal sequences can be falsified without the falsification of any physical law <cit.>, thus divorcing any notion of causation from laws themselves. What we are safe in declaring is that, using just their mathematical structure, all ostensibly fundamental physical laws deal in correlations. That is, physical laws answer the question: `if we take some event C, what further events (or past events) E could we expect with what probabilities?'. The inclusion of probability here is for the sake of generality, we need not insist on whether or not probabilities could be universally applicable. The sequential associations of events or the evolution of the physical state of some system are then the correlations referred to. Despite this, notions of causal correlation are deeply useful from a practical standpoint, as their importance in the work of many scientific fields, as well as common sense, attests to. The primitive notion of causation is that two events are more explicitly linked than merely occurring often together. The `cause' event in fact directly induces the `effect' event (or at least makes it significantly more likely to occur than other outcomes), in a manner we can see isquite opposed to the ambiguity of fundamental physical laws, which uniformly and symmetrically relate a very large set of events indeed.In order to reconcile these differences we can turn to the empirical analysis theory of causality due toKutach <cit.>, and take causality to be how we, as scientists and humans, distinguish controllable correlations from those that we cannot control.Thus, `cause-effect' relations are excised from our metaphysics, but can still be described in terms of physical laws. This statement of causality is a highly simplified one and it is worth spending some time in explanation. However, for a truly thorough treatmentthe reader is invited to consult the aforementioned work of Kutach.This empirical analysis of causation is premised on two core principles: that there is a distinction between some fundamental reality and a derivative level, there is also a distinction between metaphysical and non-metaphysical aspects of causation. This results in a division of reality into three layers for the purpose of this analysis: the first being fundamental reality that is only concerned with the metaphysics of causation, the other two being derivative realities relating each to metaphysical and non-metaphysical aspects of causation. We will only be interested in those layers discussing the metaphysics of causation and will simply refer to them as fundamental and derivative as a result. Fundamental Reality (FR) is characterised by four simple points: 1. FR is how things really are 2. FR is the real basis for events in derivative reality 3. FR is as determinate as reality gets 4. FR is consistent.The reason these distinctions are useful is that the usual `cause-effect' relation between bits of reality, each with certain characteristics, will be relevant within the derivative levels of reality, it will not be part of the fundamental metaphysics, but is ultimately predicated on the laws of fundamental reality. It will be seen in the following that, `cause-effect' relations depend only on a part of the full fundamental situation, this being selected by means of derivative phenomena. Implying that `cause-effect' relations belong to no one layer of reality, so they are safely excised from our metaphysics, obviating many problems that arise when `cause-effect' relations are held to be metaphysical necessities. This is justified on the basis of the following consideration proposed in <cit.>: a ferromagnet consisting of spins is moved near a device that detects currents via the deflection of a needle. The `cause-effect' relation will consist of the motion of the magnet causing the deflection of the needle. However, the movement of the spin system is only one event in a very large set of fundamental events that result in the movement of the needle. We single out the movement event because we believe it more significant (in a counterfactual sense) to the resulting effect than the other related fundamental events. To illustrate the distinction of fundamental and derivative realities we can follow <cit.> in considering the framework of classical physics (as this was once thought fundamental). This consists of a limited set of fundamental objects needed to define the laws: particles, properties of mass and charge, and a spacetime equipped with a distance relation. These ingredients are a minimal set of all that is needed to explain fundamental classical reality. Other objects that exist or occur as a result of these fundamental existents are derivative as they are neither part of the framing of the fundamental laws nor assumed in them <cit.>. An example of this is velocity in classical physics, as this corresponds to no fundamental structure, needing a frame of reference (not part of the fundamental set of existents) to have any definite value. There are, of course, a wide variety of both one <cit.> and two <cit.> core concept analyses of causation where `cause-effect' is taken to be a metaphysical necessity. However, the difference here that results in causation being merely an empirically useful derivative phenomenon is that no one of the three layers of reality hosts the whole of a core concept of causation or entirely encapsulates a `cause-effect' relation. In this way, causation can be described in terms of `effective strategies', that is, a causal relation is one that some agent could in principle use to bring about a desired effect. Note that this does not tie causation to agency, as this is merely used to analyse what is permitted by physical law and no agent/agency is necessary to the existence of causal relations. The agency can be better understood as the freedom allowed by physical laws for any process to realise some outcome in principle.This approach to the study of causation will be formalised in this work, and the mathematical formalism will then be leveraged to address several key problems inthe overlap of physics and information theory. This will be done by demonstrating that the notion of information used in physics is identical to the definition of causation in terms of empirical analysis. Once this identification has been made it becomes possible to obviate the circularity in Shannon's original definition of distinct states <cit.> and information. This is because the notion of what states can carry information is now formulated in terms of available causal correlations, which depend only on the possible counterfactuals as well as the prescriptions of fundamental physical laws in a manner similar to that recommended in <cit.>. The identity between causation and information is then used to address the problem of `excess baggage' in quantum mechanics <cit.>, where states (if they are in some way ontic) must seemingly carry both finite and infinite amounts of classical information <cit.>, in contravention of the established finite limits of the Holevo theorem <cit.>. The correspondence of information and causation is used to tease out this problem and demonstrate that only the finite information carried by the quantum state actually corresponds to causal correlations of a single qubit, the infinite component is accommodated by an ensemble of qubits only. This is of particular importance to the discussion of the ontic status of the quantum wavefunction, as it would otherwise seem impossible for an ontic wavefunction, as discussed in <cit.>, to contain infinite information and still obey the Holevo theorem. Here the full utility of Kutach's account of causation is realised, in that this solution is only available when information can be regarded as an operational tool only and not being part of our metaphysics. As is shown here, this can be achieved through the identification of causation and information that is possible through Kutach's empirical account. This paper is structured as follows: In section <ref> the causal formalism is layed out and discussed. Section <ref> outlines the link between causation and information in physics, and section <ref> addresses the solution of excess baggage problem.§ CAUSATIONThe first ingredient in this analysis of causation is the distinction between fundamental and derivative reality.The latter of the two refers to the realm of experience similar to the `classical world' of common physical terminology. Whereas fundamental reality is the domain of the basic forms ofphysical law, quantum theory or any provisionally fundamental theory for instance. The distinction is simply that the laws of derivative reality must be said to be determined by those of the more fundamental underlying reality. This distinction informs the way we talk about events within these two realities, those that occur in derivative reality being the only cases where we must explain why causality might be useful or apparent. To this end we classify the events of derivative reality as course-grained,such an event E is a collection of fundamental events that might correspond to the derivative event, while a contextualised eventis course-grained with a reasonable probability distribution over its members. In Kutach's presentation, this distribution need not be empirical or rigorous in any way (in order to accommodate informal notions of causality),barring that it must satisfy the axioms of probability theory. In order to determine if two time-ordered events, A and B are causallycorrelated we will employ the following terminology.The protrast of the ordered pair (A,B) is a set of fundamental events in an event A that would correlate with B being said to occur, whereas thecontrast of (A,B) is a set of events in a reasonably chosen contrastingevent C that correlate with B occurring, equivalent to imagining the causal pair (C,B) instead. We can also, of course, consider causation in terms of a derivative event A and the probability it evolves into B under apparent laws of derivative reality.At this point it is worth remarking upon the fact that this formalism illustrates the very close linkage of causality and counterfactuals. We note that Lewis <cit.> championed analysis in terms of counterfactuals as giving a complete account of causality, but the literature is littered withdifficulties that this program encounters (as described in <cit.> for example). We will see, in the course of this paper, that the empirical account of causation employed here has far greater similarity to the approach advocated by Maudlin <cit.>. Whereby, physical law provides the connection between counterfactuals and causation, and is thus the vital ingredient needed to give a complete account. Thus, the notion of a `reasonable choice' of the contrasting event C is necessarily bound up in what counterfactuals would be allowed by physical law. Since this work is focussed upon causation within formal systems of physics and information, all counterfactual choices will not just be predicated on physical laws, but rather admissible counterfactualsare in fact given directly by physical laws of the system in question. In this sense, the framework for causation here is more strict than it's parent in Kutach's work. However, this is just as a result of a narrowing of focus for application to particular issues, rather than the general explication of the metaphysics of causation that is the object of the empirical framework itself. This will prove important as this restriction is vital to be able to define some notion of `counting' of causal correlations within a given physical system.In the context of an ordered pair of contextualised events C and E, the probability ofE given C is written as P(E|C). The nature of these probabilities will be taken as the proportion of fundamental events in C which will evolve into those within E, meaning there is a robust link between these probabilities and objective frequency measurements. We can naturally expect a fully general discussion to be couched in terms of probabilities as not all of the fundamental events in our course grained event C will necessarily correlate with members of the same event E. The use of probability is then motivated to account for the fact that an event C may result in several different possible consequent events and the frequency of differing outcomes will depend upon the strength of the underlying fundamental state correlations.Kutach then invokes the notion of promotion: the degree to which C_1 promotes E is given by the difference between the propensity with which events in C_1 and those in a contrasting event C_2 would evolve into those in E. This propensity can be determined either through fundamental or derivative laws (as these should agree on matters of empirical outcome promotion). We then propose to formalise the degree to which C_1 promotes E as the logarithmic difference (we note that <cit.> employs a linear difference)𝒞(C_1,E) = log(P(E|C_1)/P(E|C_2)) . This can be generalised to a larger set of events that might cause E, rather than simply the two in {C_1,C_2}.To do so consider a set { C_i|i ∈ [1,N]}, then 𝒞(C_n,E) = ∑_in^N w_ilog(P(E|C_n)/P(E|C_i)) ,where w_i is a weight, given by 1/N if we cannot define the probability P(C_i|ℒ_phys), where this is conditioned on the relevant set of physical laws ℒ_phys or if we consider a scenario such as a symbols transmitted freely on some communication system. The function 𝒞(E,C) is then proposed as a measure of the causal association of theevents C and E. Under this assumption, if 𝒞(C,E) is positive-definite we may conclude that it is reasonable to state that E is causally correlated with C. However, if 𝒞(C,E) ≤ 0 then we must conclude that on average other events that promote E equally or to a greater degree than C does, making causal claims about (C,E) weak. The relative magnitude of the 𝒞 function will also dictate the extent of the causal association between the two events. This causal association can be understood as follows: two events C and E are causally associated if there exists a physical scenario whereby the event C occurring would offer a preponderant probability (over and above most other strategies) of E being a consequence of C.There are two approaches to continuous families of events, the first is a simple generalisation of Eq. (<ref>),𝒞(E,C) = ∫ dclog(P(E|C)/P(E|c)) .However, this presents a difficulty in enumerating causal correlations. Therefore, the continuum should be reduced to a discrete case by identifying causal classes of events. In general an event A in a continuous family can be characterised bysome set of parameters η, it seems appropriate to determine whether A(η_1) and A(η_2) are contrasting events by the `outcome continuity' of η. Thus if one can continuously deform η_1 to obtain η_2, without causing a change in the most probable outcome of A(η), then the two events must be seen to belong to the same `contrast class', meaning they cannot be chosen as contrasting events because they are causally equivalent.For a larger set of outcomes a contrast class with constitute all events A(η) that preserve the same probability heirachy. Continuity is important because the value of η will serve to demonstrate control of a particular correlation, you can `tune' η to more strongly promote a given outcome while conducting whatis ostensibly the same experiment. We can appreciate the use of `outcome continuity' if we view a lack of this as signallingthat these events assign the greatest probability to differing causal histories, making η_1 and η_2 causally distinct. We note that η_1 and η_2 might anyway result in the same event history but this does not damage the use of `outcome continuity' as the argument is probabilistic in nature.In order to take the concept of the contrast class into account, any probability P(E|A), where A ≡ A(η), will be assumed to be averaged over the relevant continuous region of η unless otherwise stated.The reason for this is to make causal arguments robust and not simply dependent on the choice of parameters, which is of particular concern in the contrasting causes, as these might otherwise be chosen to minimise their association with a given outcome. This attempt to characterise contrasting events is aimed at allowing physical law and operational considerations to determine our contrasting event classes, in keeping with the important role of physical law in determining the counterfactuals necessary in causation. It is very clear that if we are to make an empirical analysis of causation that we should only admit causal counterfactualsthat would stand up to empirical inspection. In order to make full use of the function 𝒞 we can make use of a causal table. Illustrated below for a system with two events C_1 and C_2 with two possible outcomes E_1 and E_2. E_1 E_2 C_1 𝒞(E_1|C_1) 𝒞(E_2|C_1) C_2 𝒞(E_1|C_2) 𝒞(E_2|C_2) In this table we can scan down column i to pick out possible causes for event E_i. These can then be tested by scanning across the row of a favoured cause to see that it does not uniformally promote multiple outcomes.The method of enumerating causal correlations in a given system requires remarking upon. For a given correlation we might always pick the largest 𝒞 value and decide it is the only causal correlation. However, this cannot be correct, as a simple example can show. Consider a configuration of N molecules that results in a total energy E, which correlates with some additional observable values. In principle there are many such configurations, each of which has probability 1 of associating itself with measurements of the observables that correlate to E. Thus, among this set of correlations all have 𝒞 = 0 as we cannot prefer any of them. Moreover, we might consider all these states with energy E as one single causal class. It is evident that this difficulty arises due to the determinism of the problem, in that states have either P = 1 or P = 0. Additionally, one can appreciate that P = 0 states cannot be considered valid counterfactuals, so they cannot be included to make 𝒞 non-zero for those with P = 1. However, this difficulty can be resolved simply because of the determinism, all of the P = 1 correlations are causal, as though they have 𝒞 = 0, they are deterministic and there are no other valid contrasts to consider. Thus, the process of enumeration must be cautious, for a given effect we will take the causal classes with positive 𝒞 values to be the set of causal correlations that produce this effect, in the case that all have the same 𝒞, or there appear to be no causal correlations at all, we must carefully inspect the P values to confirm any conclusions.Thus, we aim to present the causality measure 𝒞as a formal and mathematical realisation of Kutach's promotion causality, allowing it to be used in more specialised physical discussion as well as assessments of the general use of causality. It is evident that Kutach's theory gives riseto the notion of causality as a functional tool used to make the distinction between correlations we can control and those which we cannot. Thus, we cansee that causality can be apparent within derivative reality without being a necessary ingredient of fundamental reality. What must also be clear is that our causal expectations will be recovered only if we select counterfactuals that are allowed by physical law and assign them physically sensible weightings,arbitrary fantasy counterfactuals could, for obvious reasons, easily undo our reasonable causal expectations (as argued by Maudlin <cit.>).§ INFORMATION To study the notion of information we will approach it from the perspective of communication and how this relates to the underlying physics.To do so we must define some terms, the first being a dictionary: this is a set of symbols which are assigned to some states within a physical system, each symbol having some meaning which we are free to choose when specifying the dictionary.For example we can encode a binary dictionary onto a current being measured in a wire, no current detected is assigned the symbol 0 while detection above a given threshold is assigned 1.If we then measure the current for a length of time we can translate this into a serious of 0'sand 1's which may be interpreted as awhich may contain information.By determining the number of distinguishable states available to our transmitting system we can calculate the Shannon entropy of a communication channel, in the binary case we have two available states and thus a message of N symbols on our channel has S ∝ Nlog(2) which provides an information measurewhen the results of measurements on our system are treated as values of a random variable. In this regard, the information ascribed to a message sent on some physical channelcan be considered as the apriori degree of unpredictability of the constituent symbols in the message. To discuss the relation of causality and information let us first enumerate the correlations in our binary system:we can have generation of current by some process at one end of the wire that correlates with detection of a similar amplitude current at the other end,or the case of no current being generated being correlated with below-threshold current detected at the other end.However, there are other correlations available to the binary system: these being the cases of `mis-correlation', where we detect a current withoutbeing correlated to generation or where we detect no current despite a current being generated. At least some of these `mis-correlations' would be physically justifiable, so we must ask under what conditions can we use the distinguishable states of a system to encode information.It is then clear that if we wish to use the distinguishable states of a system to transmit information then we must be able to reliably induce particular correlations in that system. Otherwise, the very notion of the `distinguishability' of these states is lost. In the binary system for instance, if we cannot reliably induce a currentthat is found upon subsequent measurement to be above the given threshold then we are in danger of scrambling any message on our channel, as our 1's might frequently appear as 0's (and thus no longer be clearly distinguished between).A particular physical system X can then be said to be capable of transmitting/containing information if we can reliably map some dictionary onto a subset of possible correlations within that system. We contend that this subset is composed of only the causal correlations of the system in question.This can be demonstrated using the terminology established in the previousdiscussion. Let A be the process we use to attempt to induce a particular physical state x in the system X and B be the realisation/measurementof that state. Then, if we have a case where 𝒞(A,B) ≤ 0, or |𝒞|≤ϵ, we must conclude that whether or not we canrealise our desired state x through the process A is highly mis-correlated and cannot be said to produce a distinguishable state.Therefore, if we attempt to transmit/store a sequence of symbols with X it will be akin to a stream of bits where 1 and 0 are frequently interchanged, garbling any message we might send and thus preventing us from transmitting any desired information. Clearly if all correlations available to the system have 𝒞 = 0 (and are confirmed non-causal) we must conclude that there can be no informationtransmitted/stored. Every added correlation class with 𝒞 significantly greater than zero must therefore expand the possible information content of messages realised within the system, as we can reliably expand our dictionary with each added causal correlation. We can therefore conclude that for a correlation to contain information it must necessarily be a causal correlation and conversely that any causal correlationmay store or transmit information. Thus we propose that the information capacity of a physical channel is framed in terms of the channel's causal correlations rather than distinguishable states. Importantly this definition of information clears up the problem of circularity in Shannon's original definition: Information can only be encoded in/transmitted via causal correlations of some physical system and these are defined by the measure 𝒞 and thus by the possible counterfactuals and probabilities derived from our theory of physical law. This means that, although causal correlations are those that can carry information, they are not defined as such and thus any circularity is obviated. Additionally, the definition of causal classes provides a natural way to obviate problems of distinguishability in continuous variable systems, by differentiating between them via their promotion of outcomes. Furthermore, we also find that the Shannon entropy can be determined through the counting of causal correlations available to the system. This is because this counting is degenerate with that of distinguishable physical states when these are members of derivative reality(this being the domain of `classical' information theory). However, the examples presented in the remainder of this work will demonstrate that the counting of causal correlations provides a far more robust andconsistent measure of information content. Generalising the counting of causal correlations, by analogy with Shannon theory <cit.>, the entropy becomesS = -∑_i=1^N p_i log(p_i) ,where the sum runs over the causal correlation classes corresponding to the contextualised pairs (A_i,B_i).The weight p_i of each correlation class is the probability of the correlation being realised.The preceding arguments suggest a striking agreement between the notions of derivative empirical causation and information. It follows that this equivalence implies that there is nothing `informational' in the lawsthat govern our fundamental reality, just as these laws are not causal. This must follow from the notion that causality is not so much a property of any fundamental reality as it is a tool for its analysis, so information is not a physical property of correlations as much as it is a flag of the controllability of said correlations. This stands at odds with a prevailing school of thought within the physical sciences which champions the `information is physical' view-point, notably articulated by Brillouin <cit.> and Deutsch <cit.> among others.In this paradigm information is a fundamental ingredient in laws of physics, and that information itself is an essentially physical quantity.For this reason we must supply an argument as to why information seems sufficiently significant in physics as to warrant such extraordinary metaphysical assertions, while being simply a tool of studying reality. It must be immediately apparent that this is answered by the entire premise of the presented model of causality, or simply put:information theory is so applicable in sciences because its very formulation guarantees it to be so.In fact the nature of information capacity as a demarcation between useful and non-useful correlations makes it impossible that it would not be applicableto the study of correlation and regularity that tends to compose the majority of sciences. We note that this does not justify the scientific status of the use of causality/information, but merely explains it. The justification of the use of causation in scientific endeavour will be examined in future work.In this particular work we apply this causal account of information to problems in quantum information theory. This means we must ask if this notion of information is adequate to the task. We can immediately see that this causal account of information, being structured on the division between fundamental and derivative reality,is immediately well suited to quantum mechanical problems which involve extracting information via measurements conducted in derivative (classical) reality from where it is transmitted/storedin states that are members of fundamental (quantum) reality.§ EXCESS BAGGAGE AND QUANTUM INFORMATION In quantum mechanics it has been demonstrated that a `qubit' system with two measurable states,referred to as `up' or |↑⟩ and `down' or |↓⟩,possesses a vast space of possible quantum states <cit.>. Commonly this is interpreted as meaning we should be able to encode a huge amount of retrievable/`classical' information in such a system <cit.>as this should scale with the size of the state space (according to the notion that information is a property of distinguishable states).This becomes remarkable when it is observed that we cannot retrieve any more than one bit from such a system, as argued by the Holevo theorem <cit.>. This is immediately problematic if the state of the system is viewed in a realist fashion as the apparently real state space does not correspond to real retrievable information. All the non-retrievable information is thus referred to as `excess baggage' by Hardy <cit.> and it must be explained why such a vast state space can offer up so little information. This problem can be viewed as follows: the states of a qubit can be expressed as the points on the surface of asphere of unit radius, known as the Bloch sphere. Thus, they are a function of two continuous parameters. This means that, for a given qubit, the probabilities of measuring `up' or `down' vary continuously depending on the basis we choose to measure the qubit in <cit.>. This suggests that for every possible basis the qubit represents a different statistical preparation of a classical bit (referred to as a `bit' hereafter). Therefore, a qubit can be represented by a continuous infinity of bits and must contain an arbitrarily large amount of information. Clearly then there is some disagreement between how much information a given qubit preparation actually contains, and it is necessary to provide an information measure that is consistent with both the Holevo bound and the argument outlined above. One resolution to this is to assert that the quantum state has no direct relation to any underlying ontology. However, in this work we will consider a solution independent of ontic or epistemic assertions. Despite this, a solutionto the problem of excess baggage is of prime importance to ontic formulations of quantum mechanics, where the quantum state ψ is a part of physical reality and not merely a summary of information available to some observer (as in an epistemic formulation). It may, of course, be possible to avoid the problem of excess baggageby suggesting that although a physical system may occupy a superposition state (in an ontic view of states) it does not immediately entail that the property values of the qubit are in reality superposed as well. As this would mean that we need not account all the possible property values of the qubit when we construct our information measure and the infinite component of the information thus vanishes. However, the solution we present here is available without need to further justify considerations such as the aforementioned. The value of our approach is then that it is more strongly general and made available via the empirical causal analysis of Kutach. Having established the problem we can mobilise the machinery developed earlier by realising that our coarse-grained events arethe preparation and subsequent measurement of the qubit. This means that we will look at what causal correlations or information transmission is possible with the set of events: preparation ofsome FR state with desired measurement probabilities and measuring one of two qubit values with some classical apparatus. Additionally, our dictionaries are being chosen as follows: bit value `1' maps to |↑⟩ and `0' maps to |↓⟩. This means that the dictionary is actually chosen when we choose basis, as this decides what we are measuring when we speak of |↑⟩ and |↓⟩. Thus, rather than being represented by a continuous infinity of bits, a qubit is in fact compatible with a continuous infinity of seemingly independent dictionaries. It must be remarked upon that the same case cannot be made for a bit, as even though we could describe a given bit in any basis we please, the commutativity and non-contextuality of it's algebraic description lead all bases to map unambiguously betweentheir particular |↑⟩ and |↓⟩ states. The qubit, however, exhibits both non-commutativity of it's observables and the measured values of “up" and “down" are strongly basis dependent (if we consider “up" and “down" as poles on a sphere, a choice of basis corresponds to a choice of the polar axis). The properties of the operator algebra thus lead to a lack of unique mappings between |↑⟩ states in different bases. In other words, a measurable |↑⟩_1 state in a given basis could map to a linear combination of the measurable |↑⟩_2 and |↓⟩_2 states in another basis. This means that measuring |↑⟩_2 cannot be unambiguously mapped to the result of a counterfactual measurement in the basis defined by |↑⟩_1 and |↓⟩_1. This would seem to suggest that we should be able torepresent the qubit as an infinite set of independent bits (one per basis). However, the lack of consistent counterfactual statements regarding the choice to measure in different bases will inevitably lead us to find that causal correlations in one basis will be incompatible with those in other bases and, by implication, the information content of a qubit is basis-dependent. Therefore, as a first step we can establish that the reason the qubit appears to be represented by infinite classical bits stems from the non-commutativity of its observable operators and the resulting basis-dependence of what we are measuring when we speak of ↑ and ↓ values.If we examine the causal correlations of a qubit in a given basis then we can see that the Holevo bounds emerges as follows:the causal correlations of a qubit in some basis depend on continuous parameters and can be divided into two equally weighted classes,`prepare mostly up and measure up' and `prepare mostly down and measure down', these are separated by a discontinuity, in the form of a class of random correlations that cannot belong to either of the aforementioned causal classes, as they do not differ in their promotion of contrasting outcomes. This means that the causal correlation space of a qubit in a given basismatches that of the statistical preparation of a bit. Therefore, since we must choose a basis to measure in, we will always find that the qubit can at most yield up one bit of information upon measurement. If we choose to measure in a different basiswe will find that the causal correlation spaces of different bases arenot necessarily compatible. For instance, if we define two bases β_1 and β_2 such that β_2 is rotated by an angle θ along one of the Bloch sphere directions. We then prepare a qubit so that we can transmit one bit via the causal correlations of β_1. The |↑⟩_1 and |↓⟩_1, which are causally associated with our preparation,each correspond to superpositions of |↑⟩_2 and |↓⟩_2.This means that we find that it becomes highly unreliable to retrieve the β_1 bit by measuring in β_2 but alsothat encoding a β_2 correlation with our β_1 preparation is just as unreliable (in the sense that we lose distinguishability of states). This is an important issue, the two bits in bases 1 and 2 are not truly independent, so the infinite set of dictionaries do not in fact encode an infinite set of bits.This can be fully illustrated by considering a state |ψ⟩_1 = |↓⟩_1. In basis 2 there is a probability ∝sin(θ)^2 of a measurement yielding |↑⟩_2. Thus, we can see that as we increase θ we are merely travelling through the causal class `prepare mostly down and measure down'. As we reach some θ^⋆ we transition into the random class and afterwards proceed into the `prepare mostly up and measure up' class. Thus it is clear that the bases 1 and 2 do not posses different causal correlations, they just represent a rotation of the causal class chosen in the preparation basis. Thus, regardless of what basis we choose, we never increase the number of causal classes available to the qubit, there are always just two.Effectively the basis-dependent behaviour of the qubit observables both seems to add the potential to set up simultaneous `multi-bit' causal correlations as well as providingthe linkage between bases that prevents any attempt to do so. This makes it clear that we cannotindependently encode multiple bits upon a single qubit,and that the basis-dependence of the qubit observables leads it to appear to be composed of infinite classical bits while still obeying the Holevo bound. It is worth noting that we could encode multiple independent bits in the superposition structure of a qubit state in some β_*, though doing so does nothing to change the number of causal correlations available to the qubit, as our argument above does not depend upon superposition details. However, this preparation does change the causal correlations available to an ensemble of such qubits. This is the case because we cannot extract superposition structure in a single measurement without a Holevo violation. In practice we must perform quantum state tomography and make many measurements to reconstruct the wavefunction. The amplitudes of the states in superposition can then be used to carry information in the same manner as a string of digits. However, these strings are not accessible without an ensemble of qubits to perform tomography upon. In this sense we would not truly increase the information content of a single qubit, as its causal space is unchanged, merely we have exploited the larger causal space of a qubit ensemble. Why is this the case if the information is in ψ which describes the qubit? It is because ψ describes the situation of a qubit with a given preparation in a given measurement process, i.e. it details the entire experimental arrangement and the extra information is being encoded in the statistical relationship between qubits in the ensemble. We note that there is a strong similarity between our solution here and that discussed by Timpson <cit.>. In that Timpson suggests the excess baggage arises from the difference in the amount of information needed to fully specify the state of a quantum system and the amount of information accessible via measurement (this distinction is argued not to arise in the classical case). It is clear that the solution presented here realises a very similar scenario but does so via the use of the empirical analysis of causation to justify the distinction and simultaneously why excess baggage does not arise in a classical context. An important aspect of the resolution of the excess baggage problem is that it is completely independent of ontological preference. Thus it obviates the difficulties experienced by ψ-ontology in this regard <cit.>. However, it is worth noting that it does not then favour any particular ontological/epistemological view-point in quantum mechanics. The empirical/operational nature of the approach makes it agnostic towards interpretation or metaphysics. This mode of explanation seems to break down when we include quantum entanglement, in which casewe can perform super-dense coding with shared entanglement <cit.> and can retrieve 2 N bits from N qubits. However, the shared entangled state has merely increased the number of exploitable correlationsand this scenario remains within the remit of the interpretation of information given here. In this scenario we have two sets of qubits, one held by the receiver and one by the transmitter. These two sets are entangled, with the sender and receiver both knowing the nature of the entanglement. The receiver decodes two bits when sent a single qubit because the entangled state basis has four states and four causal correlations. The need for shared entanglement means that we never expand the causal classes of a single qubit, as the qubit itself does not carry the information of the shared entanglement. This is the case because, without apriori knowledge of the entanglement, we cannot deduce it's presence/nature from a single set of transmitted qubits, we need to compare ensembles to identify the entanglement. In this regard it is similar to the superposition structure case above, the extra information is being carried in the correlations between different sets of qubits. But, in this case, we are also given knowledge of how our qubits correlate with the qubits that are transmitted to us. Thus, our causal state space is built up from two qubits and their correlations. The information content of a single qubit system is unchanged and the addition of extra causal correlations is in keeping with our causal explanation of information transmission. What becomes evident is that the causal correlation view of information applies to all retrievable information, as observable correlations all live within the realm of derivative reality, in the vocabulary of quantum mechanics they are `classical' objects. This illustrates that the term `quantum information' arises as a result of the asymmetry between the parameter space of quantum states and that of causal correlations associated with those states. It is evident that this view can still be reconciled with those expressed by Cerf and Adami <cit.>: that quantum correlations, being members of fundamental reality and possessing `quantum information'give rise to correlations with `classical information' within derivative reality, although it can only lead to their conclusion that quantum information gives rise to classical information in the sense that some quantum correlations can map directly onto causal relations. § FUNDING The author acknowledges support, through a post-doctoral grant, by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation and by the Square Kilometre Array (SKA). § ACKNOWLEDGEMENTS I would like to thank Jacques Naude (as well as other `Shannon Day' regulars), Justine Tarrant, Simon Beck, and Douglas Kutach for their input, discussion, and suggestions. unsrt | http://arxiv.org/abs/1707.08778v3 | {
"authors": [
"Geoff Beck"
],
"categories": [
"physics.hist-ph",
"quant-ph"
],
"primary_category": "physics.hist-ph",
"published": "20170727084348",
"title": "Causation, Information, and Physics"
} |
Wilczek Quantum Center, Zhejiang University of Technology, Hangzhou, Zhejiang, China University of Pittsburgh, Pittsburgh, PA, U.S.A.Department of Applied Physics, School of Science, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, China Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, ChinaUnconventional superconductivity or superfluidity are among the most exciting and fascinating quantum states in condensed matter physics. Usually these states are characterized by non-trivial spatial symmetry of the pairing order parameter, such as in ^3He and high-T_c cuprates. Besides spatial dependence the order parameter could have unconventional frequency dependence, which is also allowed by Fermi-Dirac statistics. For instance, odd-frequency pairing is an exciting paradigm when discussing exotic superfluidity or superconductivity and is yet to be realized in the experiments. In this paper we propose a symmetry-based method of controlling frequency dependence of the pairing order parameter via manipulating the inversion symmetry of the system. First, a toy model is introduced to illustrate that frequency dependence of the order parameter can be adjusted by controlling the inversion symmetry of the system. Second, taking advantage of the recent rapid developments of shaken optical lattices in ultracold gases, we propose a Bose-Fermi mixture to realize such frequency dependent superfluids. The key idea is introducing the frequency-dependent attraction between Fermions mediated by Bogoliubov phonons with asymmetric dispersion. Our proposal should pave an alternative way for exploring frequency-dependent superconductors or superfluids with cold atoms. 74.20.Rp, 67.85.PqEngineering frequency-dependent superfluidity in Bose-Fermi mixtures Bo Liu 31 July 2017 ====================================================================§ INTRODUCTION Symmetry plays an important role in physics of superconductors or superfluids and influences their properties in a profound way. For example, research on unconventional superconductors where the pairing order parameter has non-trivial spatial symmetry, such as spin-singlet d-wave in cuprates and spin-triplet p-wave in Sr_2RuO_4 has generated tremendous interest in strongly correlated electron systems. In addition to spatial dependence the order parameter may also depend on frequency. Indeed, in models describing the electron-electron attraction induced by interaction with lattice phonons (Frohlih Hamiltonian) the effective electron-electron attraction, and therefore the gap, are frequency-dependent <cit.>. The gap happens to be even in frequency but this does not have to be the case in general. In fact the idea of pairing order parameter with odd frequency dependence was proposed by V.L. Berezinskii in 1974 <cit.>. This Berezinskii conjecture has attracted considerable attention, in particular with respect to some exotic phenomena like the anomalous proximity effect in superconductor-ferromagnet junctions <cit.>.However the question of controlling the frequency dependence of the pairing order parameter remains unresolved. The idea of symmetry breaking may offer a clue: indeed, it has been skillfully used recently to obtain exotic superconducting and superfluid ordering in the ultracold atomic setting <cit.>. In this paper we report on a method of engineering frequency-dependent superconductors or superfluids based on manipulating the inversion symmetry. The main idea can be understood through the following symmetry argument. The superconducting order parameter is defined by a non-vanishing value of the anomalous correlator F(α,α')=⟨ψ(α)ψ(α')⟩ ,where α,α' stand for the internal degrees of freedom, such as coordinate, time, spin and band indices of the Fermi field ψ and ⟨ ...⟩ stands for the ground state or thermal state average. Let us assume that fermions pair up in a spin-singlet state (as is the case for the BCS model <cit.>). According to Fermi-Dirac statistics F(α,α') should be antisymmetric with respect to exchange of α and α'. Spin-singlet state is already antisymmetric therefore the remaining part of the anomalous correlator should satisfyF(q,ω_n)=F(-q,-ω_n),where q and ω_n are the momentum and Matsubara frequency. In order to control the frequency dependence of the pairing we suggest manipulating the inversion symmetry of the system. Indeed, if the system is inversion-symmetric thenF(q,ω_n)=F(-q,ω_n),which according to Eq. (<ref>) makes it also even in frequency, F(q,ω_n)=F(q,-ω_n).By introducing the inversion symmetry breaking the odd-frequency superconductivityF(q,ω_n)=-F(-q,ω_n), F(q,ω_n)=-F(q,-ω_n)may be feasible. In the following we will first use a toy model to show how this idea works. Then a cold atom based experiment is proposed. § TOY MODEL Let us first consider a 1D toy model of a Bose-Fermi mixture:Ĥ=Ĥ_F+Ĥ_B+Ĥ_BF.Here for simplicity we assume that Ĥ_F describes non-interacting spin-1/2 fermions,Ĥ_F=∑_k,σ(ϵ_F(k)-μ_F)f̂_k,σ^†f̂_k,σ,and Ĥ_B describes spinless bosons,Ĥ_B=∑_q(ϵ_B(q)-μ_B)b̂_q^†b̂_q.The key idea for controlling the frequency dependence of superconducting order parameter relies on introducing the inversion symmetry breaking via asymmetric energy dispersion of bosons, ϵ_B(-q)≠ϵ_B(q), that will be illustrated in detail below. Finally, Ĥ_BF describes the fermion-boson interaction,Ĥ_BF=U_BF/√(N)∑_qρ̂_q(b̂_-q+b̂_q^†)where ρ̂_q=N^-1∑_k,σf̂_k,σ^†f̂_q+k,σ stands for the Fourier transform of fermion density and N is the total number of lattice sites. In this toy model we can integrate out the bosons exactly using the path integral formalism, see Appendix A for details, and the resulting induced density-density type interaction between fermions is given byV_Ind(q,ω_n)=-U_BF^2/2ϵ_B(q)+ϵ_B(-q)-2μ_B/ω_n^2+(ϵ_B(q)-μ_B)(ϵ_B(-q)-μ_B)+iω_n(ϵ_B(q)-ϵ_B(-q)),where ω_n=(2n+1)π T are Matsubara frequencies and ħ=1, k_B=1 throughout the paper.To demonstrate how the inversion symmetry breaking can be used to control frequency dependence of the order parameter we assume the following dispersion for bosonsϵ_B(q)= c_R|q|, q≥0c_L|q|, q<0as a concrete example. We employ Eliashberg equations <cit.> to find the superconducting gap Δ(k,ω_n) as a function of momentum and frequency:Δ(k,ω_n)=-T/N∑_k',n'V_Ind(k-k',ω_n-ω_n')Δ(k',ω_n')/ω_n'^2Z^2(k',ω_n')+ξ^2(k',ω_n')+|Δ(k',ω_n')|^2, (1-Z(k,ω_n))iω_n=T/N∑_k',n'V_Ind(k-k',ω_n-ω_n')iω_n'Z(k',ω_n')/ω_n'^2Z^2(k',ω_n')+ξ^2(k',ω_n')+|Δ(k',ω_n')|^2, χ(k,ω_n)=T/N∑_k',n'V_Ind(k-k',ω_n-ω_n')ξ(k',ω_n')/ω_n'^2Z^2(k',ω_n')+ξ^2(k',ω_n')+|Δ(k',ω_n')|^2,where ξ(k,ω_n)=ϵ_F(k)+χ(k,ω_n), Z(k,ω_n) is the fermion mass renormalization and χ(k,ω_n) the renormalization of the chemical potential. To simplify the problem, here we consider 1D case and assume the fermions to have s-band-like dispersion, ϵ_F(k)=-t_Fcos(ka), k being lattice momentum and a the lattice constant. As shown in Fig. 1, when bosons have asymmetric dispersion (i.e. c_R/c_L≠1) and the inversion symmetry is broken, the odd-frequency component of the superconducting gap emerges. Furthermore, as shown on Fig. 2, the frequency-dependence of the gap can be controlled by manipulating the degree of asymmetry, i.e. the ratio c_R/c_L. It shows that the stronger the boson dispersion asymmetry the more favorable is the odd-frequency component of the superconducting gap. We note that the superconducting pairing considered here is in the spin-singlet channel, since the spin-singlet pairing always has lower free energy compared to the spin-triplet counterpart, as confirmed numerically.§ SUPERCONDUCTING ORDERING VIA PAIRING WITH SHAKEN BOSONS In the following we explore the possibility of realizing frequency-dependent superfluidity via a cold-atom based system. Let us consider a Bose-Fermi mixture in a 1D optical superlattice as shown in Fig. 3 and further consider the bosons to be in a shaken lattice. Lattice shaking will change the energy dispersion of bosons to a double-dip shape, see Appendix B. By further assuming that the bosons are in the plane-wave state (i.e. that condensation in one of the minima of the double-dip dispersion has occurred), the induced attraction between fermions can be obtained as follows. We start with the Hamiltonian which is a generalization of the Bose-Hubbard model where the momentum-space dispersion of free particles ϵ_B(k) has a double-dip (see details in Appendix B):Ĥ_B=∑_k(ϵ_B(k)-μ_B)b̂_k^†b̂_k+g/2N∑_k,k',qb̂_k+q^†b̂_k'-q^†b̂_k'b̂_k,k is lattice momentum in 1D and b̂_k^†, b̂_k-s are the bosonic creation/annihilation operators. The fermions are described byĤ_F=∑_k,σ(ϵ_F(k)-μ_F)f̂_k,σ^†f̂_k,σ,where possible existing repulsion between the fermions is treated at the mean-field level and is absorbed into the chemical potential. The interaction between bosons and fermions has the formĤ_BF=U_BF/N∑_qn̂_-qρ̂_qwhere ρ̂_q has the same definition as given below Eq. (<ref>) and, similarly, n̂_q is the bosonic density operator.We assume that bosons condense in the state with momentum k_0, corresponding to one of the minima of the double-dip dispersion ϵ_B(k). The effective attraction between fermions mediated by Bogoliubov phonons is derived with the help of the path integral formalism. Details are shown in Appendix C and here we only state the result. The induced density-density interaction between fermions is given by V_Ind(k,ω_n)=-U_BF^2n_0/2ϵ_B(k_0+k)+ϵ_B(k_0-k)-2ϵ_B(k_0)/(iω_n+ϵ_B(k_0+k)-ϵ_B(k_0)+gn_0)(-iω_n+ϵ_B(k_0-k)-ϵ_B(k_0)+gn_0)-g^2n_0^2,where n_0=N_0/N is the dimensionless “density” of the condensate (N_0 is the number of bosons in the condensate, N is the total number of lattice sites, as before). We now proceed to solve the Eliashberg equations for the superconducting order parameter and accompanying quantities but this time with the attraction given by Eq. (<ref>). Below we show sample order parameters calculated based on the inter-fermion attraction plotted on Fig. 4. As shown on Fig. 5 the frequency-dependent superfluid is obtained. We find that the real and imaginary parts of the order parameter have even- and odd-frequency dependence, respectively. This is in line with the structure of induced attraction Eq. (<ref>). Namely, V_Ind(k,ω_n) is even in frequency and V_Ind(k,ω_n) is odd, see Fig. 4. By tuning the shaking frequency it is possible to go from the case of symmetric (ϵ_B(k_0+k)=ϵ_B(k_0-k)) to asymmetric (ϵ_B(k_0+k)≠ϵ_B(k_0-k)) Bogoliubov dispersion, and hence to introduce an odd-frequency component in the superconducting order parameter. § CONCLUSIONS We have demonstrated the theoretical feasibility of fermions forming mixed-frequency superconducting order parameter by means of interacting with bosons with non-isotropic dispersion. To achieve such non-isotropy we considered bosons condensing in one of the minima of the double dip dispersion formed by (quasi) 1D shaking. We have shown that by tuning the aforementioned parameters it is possible for the superconducting order parameter to be 10% odd-frequency or greater. As a final note we want to mention that shaking the lattice is not the only possibility for creating non-isotropic and/or the double-dip dispersion, see for example recent work on spin-orbit coupled BEC <cit.>, and our scheme can be perfected and/or modified to achieve realization of a mixed-frequency superconductor in ultracold atomic settings.The authors would like to thank W. V. Liu and H. Xiong for discussions and hospitality and X. Lin and X. Yue for discussions. § INTEGRATING OUT THE TOY MODEL BOSONS Toy model Hamiltonian Eq. (<ref>) is quadratic in bosonic operators. This makes integrating out bosons a straightforward exercise in Gaussian integration. Using the path integral formalism we go from HamiltonianĤ=∑(ϵ_B(q)-μ_B)b̂_q^†b̂_q+U_BF/√(N)∑ρ̂_q(b̂_-q+b̂_q^†)to partition functionZ=∫ Db̅Db exp(-∑_0^βdτ b̅_q(τ)(∂_τ+ϵ_B(q)-μ_B)b_q(τ)-U_BF/√(N)∑_0^βdτ ρ_q(τ)(b_-q(τ)+b̅_q(τ)))where Db̅Db=∏ db̅_qdb_qup to an arbitrary normalization factor and β=1/T is the inverse temperature (throughout this paper k_B=1, ħ=1). Instead of operators, b̅_q(τ), b_q(τ)-s now stand for complex-valued bosonic fields and ρ_q(τ)-s are fields corresponding to Fourier components of fermion density. In Eq. (<ref>) ρ_q(τ)-s are treated as external fields. We will now integrate out the bosons and obtain the induced interaction between the fermions. To this end we employ the Matsubara frequency representation, b_q(τ)=√(T)∑_nb_q(ω_n)e^-iω_nτ, b̅_q(τ)=√(T)∑_nb̅_q(ω_n)e^iω_nτ and ρ_q(τ)=T∑_nρ_q(ω_n)e^-iω_nτ, where ω_n=2nπ T, n being integers. We re-express the imaginary time integrals in Eq. (<ref>) as sums over Matsubara frequencies ω_n,Z=∫ Db̅Db exp(-∑_q,nb̅_q(ω_n)(-iω_n+ϵ_B(q)-μ_B)b_q(ω_n)-U_BF√(T)/√(N)∑_q,nρ_q(ω_n)(b_-q(-ω_n)+b̅_q(ω_n))),then integrate out b̅_q(ω_n), b_q(ω_n) with the help of the following identity:∫ dz̅dz e^-z̅G^-1z+J̅z+z̅J=e^J̅GJ/ G^-1.This results in the following induced fermion-fermion interaction actionS_Ind=-U_BF^2T/2N∑_q,nρ_-q(-ω_n)ϵ_B(q)+ϵ_B(-q)-2μ_B/ω_n^2+(ϵ_B(q)-μ_B)(ϵ_B(-q)-μ_B)+iω_n(ϵ_B(q)-ϵ_B(-q))ρ_q(ω_n),from which V_Ind(k,ω_n) Eq. (<ref>) is obtained. The minus sign in front of the sum in Eq. (<ref>) ensures that induced interaction is attractive. § SHAKEN LATTICE DISPERSION Here we outline the physics of shaken lattices <cit.>. Consider two lowest bands (s- and p-) of a 1D optical lattice and their hybridization when the shaking frequency matches the band gap. The Hamiltonian isH(t)=k_x^2/2m+Vcos^2(k_0x+θ(t)/2),where θ(t)=fcos(Ω t) describes the periodic shaking of the lattice. Because of its special form the time-dependent Hamiltonian Eq. (<ref>) can be expanded using the properties of Bessel functions. Write the potential termVcos^2(k_0x+f/2cos(Ω t))=V/2(1+cos(2k_0x+fcos(Ω t)))ascos(2k_0x+fcos(Ω t))=1/2(e^i2k_0xe^ifcos(Ω t)+e^-i2k_0xe^-ifcos(Ω t)),then use the Jacobi-Anger expansione^izcos(ϕ)=∑_n=-∞^∞i^nJ_n(z)e^inϕ,where J_n(z) are the n-th order Bessel functions of the first kind. Applying the expansion and using the symmetry of Bessel functions,J_n(-z)=(-1)^nJ_n(z), e^ifcos(Ω t)=∑_n=-∞^∞i^nJ_n(f)e^inΩ t, e^-ifcos(Ω t)=∑_n=-∞^∞(-i)^nJ_n(f)e^inΩ tend up with the following expansion for the driving term:cos(2k_0x+fcos(Ω t))=J_0(f)cos(2k_0x)-2J_1(f)sin(2k_0x)sin(Ω t)+...where terms with frequencies 2Ω t and higher have been neglected. Therefore, up to a constant energy shift, the shaken Hamiltonian takes the formH(t)=k_x^2/2m+VJ_0(f)cos^2(k_0x)-VJ_1(f)sin(2k_0x)sin(Ω t)+...The above exercise in algebra allows us to connect physics of shaken lattice and band mixing. The s- and p- bands of the time-averaged HamiltonianH_av=1/T_0^TH(t)dt=k_x^2/2m+VJ_0(f)cos^2(k_0x)mix under the influence of periodic perturbation V(t)=-VJ_1(f)sin(2k_0x)sin(Ω t)+...when the shaking frequency matches the inter-band spacing, forming two Floquet bands. It was shown in <cit.> that for a certain parameter window of V, f, Ω one of the Floquet bands possess two degenerate minima at k_min,-k_min≠0,π (e.g. see Fig. 1b in <cit.>). To find Floquet bands of the shaken lattice we can follow <cit.> and project the periodic driving term Eq. (<ref>) on the lowest two bands (s- and p-) of the time-averaged Hamiltonian Eq. (<ref>) which results inV(t)=-VJ_1(f)([0 C(k,V,f); C^*(k,V,f)0 ])sin(Ω t)where C(k)=⟨Ψ_sk|sin(2k_0x)|Ψ_pk⟩ and the diagonal terms vanish due to symmetry considerations. |Ψ_sk⟩ and |Ψ_pk⟩ are the Bloch waves corresponding to the s- and p- bands of the cosine lattice and can be either computed numerically using the plane wave expansion or expressed exactly in terms of Mathieu functions <cit.>. When shaking frequency Ω is close to the value of the band gap (the resonance condition) it is justified to use the rotating wave approximation <cit.> which results in the following approximate Floquet Hamiltonian: H(k)=([ E_p(k)-Ω/2-iVJ_1(f)C(k)/2; iVJ_1(f)C^*(k)/2 E_s(k)+Ω/2 ]),where E_s(k)=⟨Ψ_sk|H_av|Ψ_sk⟩ and similarly for E_p(k). All the terms in Eq. (<ref>) can be obtained numerically exactly in terms of Mathieu functions and by diagonalizing Eq. (<ref>) the emergent shaken (Floquet) bands can be found. Including the inter-boson interactions will result in Bose condensation around either k_min or -k_min and the spectrum of excitations above that state (the Bogoliubov phonons) will end up not being inversion-symmetric <cit.>.§ INTEGRATING OUT THE SHAKEN BOSONS The bosonic part of the partition function takes the formZ= Db̅Db exp[-S_B-S_BF],whereS_B+S_BF=_0^βdτ ∑_kb̅_k(τ)(∂_τ+ϵ_B(k)-μ_B)b_k(τ)+g/2N_0^βdτ ∑_k,k',qb̅_k+qb̅_k'-qb_k'b_k+ +U_BF_0^βdτ ∑_ib̅_ib_iρ_iand Db̅Db=∏_kdb̅_kdb_k.In the above b̅_i, b_i are the Bose fields at lattice site i and b̅_k, b_k are their momentum-space counterparts. Assuming condensation in the state with momentum +k_0, b_k_0 and b̅_k_0 acquire non-zero mean-field values which can be accounted for by a shift of variables,b_i→√(N_0/N)e^ik_0r_i+b_i, b̅_i→√(N_0/N)e^-ik_0r_i+b̅_i,where r_i stands for i-th lattice site position. We also assume that the number of atoms above the condensate, N_B-N_0, is small, where N_0=b̅_k_0b_k_0=N_B-∑_k≠ k_0b̅_kb_k.Then substituting the mean-field value Eq. (<ref>) into the action Eq. (<ref>) and expanding in creation/annihilation operators up to the second order the following effective quadratic action is obtained:S_Bog=_0^βdτ ∑_kb̅_k(τ)(∂_τ+ϵ_B(k)-μ_B+2gn_0)b_k(τ) +gn_0/2_0^βdτ ∑_kb̅_k_0+k(τ)b̅_k_0-k(τ)+gn_0/2_0^βdτ ∑_kb_k_0+k(τ)b_k_0-k(τ)where in all the sums k runs over the first BZ momenta and in the second and third sums it is understood that if k_0+k becomes greater than π its value “wraps around” the 1BZ edge and becomes k_0+k-2π. n_0 is the condensate density, n_0=N_0/N, and can be determined from the equation of state, namely the equation that demands that the linear terms in the action vanish, gn_0=μ_B-ϵ_B(k_0).The final step is integrating out the bosons which is made easier by introducing the particle-hole notation:S_Bog=_0^βdτ 1/2∑_k([ b̅_k_0+k(τ)b_k_0-k(τ) ])([∂_τ+ϵ_B(k_0+k)-μ_B+2gn_0gn_0;gn_0 -∂_τ+ϵ_B(k_0-k)-μ_B+2gn_0 ])([b_k_0+k(τ); b̅_k_0-k(τ) ]),or, in momentum-frequency space,S_Bog=1/2∑_k,ω_n([ b̅_k_0+k(ω_n) b_k_0-k(-ω_n) ])G^-1(k,ω_n)([ b_k_0+k(ω_n); b̅_k_0-k(-ω_n) ])whereG^-1(k,ω_n)=([iω_n+ϵ_B(k_0+k)-μ_B+2gn_0 gn_0; gn_0 -iω_n+ϵ_B(k_0-k)-μ_B+2gn_0 ]).We then expand the boson-fermion interaction term Eq. (<ref>) to the first order in b̅_k, b_k:S_BF=U_BF_0^βdτ ∑_iρ_in_0+U_BF√(n_0)/√(N)_0^βdτ ∑_i,k≠ k_0ρ_i(b̅_ke^i(k_0-k)r_i+b_ke^-i(k_0-k)r_i).The first term renormalizes the fermion chemical potential, μ_F→μ_F+U_BFn_0,whereas the second-order terms in b̅_k, b_k (not shown in Eq. (<ref>)) will produce higher-order contributions to inter-fermion interactions (three-body interactions and higher). It is the first-order term in Eq. (<ref>) that produces the attractive density-density interaction between fermions. To see this make the change of variables in the sum, k_0-k→-k. Then the second term of Eq. (<ref>) can be represented as 1/2_0^βdτ ∑_kJ^†(k,τ)([b_k_0+k(τ); b̅_k_0-k(τ) ])+([ b̅_k_0+k(τ)b_k_0-k(τ) ])J(k,τ)withJ(k,τ)=([ ρ_k(τ); ρ_k(τ) ]), J^†(k,τ)=([ ρ_-k(τ) ρ_-k(τ) ]).Comparing with Eq. (<ref>) and employing the rules of Gaussian integration Eq. (<ref>) we obtain the induced fermion density-density interactionV_Ind(k,ω_n)=-U_BF^2n_0/2ϵ_B(k_0+k)+ϵ_B(k_0-k)-2ϵ_B(k_0)/(iω_n+ϵ_B(k_0+k)-ϵ_B(k_0)+gn_0)(-iω_n+ϵ_B(k_0-k)-ϵ_B(k_0)+gn_0)-g^2n_0^2 unsrtnat 99 MahanG. D. Mahan, Many-particle physics (Plenum Press, 2nd edition, 1990).BerezinskiiV. L. Berezinskii, JETP Lett. 20(9), 287 (1974).NagaosaY. Tanaka, M. Sato, and N. Nagaosa, J. Phys. Soc. Jpn. 81, 011013 (2012).IsaacssonGirvinA. Isaacsson and S. Girvin, Phys. Rev. A 72, 053604 (2005).LiuWuW. V. Liu and C. Wu. Phys. Rev. A 74, 013607 (2006).HemmerichSmithM. Olschlager, T. Kock, G. Wirth, A. Ewerbeck, C. Morais Smith and A. Hemmerich, New J. Phys. 15, 083041 (2013).BCSJ. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 1175 (1957).AlexandrovA. S. Alexandrov, Theory of Superconductivity. From Weak to Strong Coupling (IOP Publishing, 2003).OrbitalFeshbach1R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, Phys. Rev. Lett. 115, 135301 (2015).OrbitalFeshbachExp1G. Pagano, M. Mancini, G. Capellini, L. Livi, C. Sias, J. Catani, M. Inguscio, and L. Fallani. Phys. Rev. Lett. 115, 265301 (2015).OrbitaFeshbachExp2M. Hofer, L. Riegger, F. Scazza, C. Hofrichter, D. R. Fernandes, M. M. Parish, and J. Levinsen. Phys. Rev. Lett. 115, 265302 (2015).SpielmanReviewV. Galitski and I. B. Spielman. Nature 494, 49 (2013).ChinShakenLatticeC. V. Parker, L.-C. Ha, and C. Chin. Nat. Phys. 9, 769 (2013).NIST NIST Handbook of Mathematical Functions (NIST and CUP, 2010).ShirleyJ. H. Shirley, PhD thesis, California Institute of Technology, 1963.ChinMaxonRotonL.-C. Ha, L. W. Clark, C. V. Parker, B. M. Anderson, and C. Chin. Phys. Rev. Lett. 114, 055301 (2015). | http://arxiv.org/abs/1707.08331v2 | {
"authors": [
"Maksims Arzamasovs",
"Bo Liu"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170726091704",
"title": "Engineering Frequency-dependent Superfluidity in Bose-Fermi Mixtures"
} |
Bandit Convex Optimization forScalable and Dynamic IoT ManagementTianyi Chen and Georgios B. Giannakis Work in this paper was supported by NSF 1509040, 1508993, and 1711471.T. Chen and G. B. Giannakis are with the Department of Electrical and Computer Engineering and the Digital Technology Center, University of Minnesota, Minneapolis, MN 55455 USA. Emails: {chen3827, georgios}@umn.edu December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================= New technologies for recording the activity of large neural populations during complex behavior provide exciting opportunities for investigating the neural computations that underlie perception, cognition, and decision-making. Nonlinear state space models provide an interpretable signal processing framework by combining an intuitive dynamical system with a probabilistic observation model, which can provide insights into neural dynamics, neural computation, and development of neural prosthetics and treatment through feedback control. It brings the challenge of learning both latent neural state and the underlying dynamical system because neither is known for neural systems a priori. We developed a flexible online learning framework for latent nonlinear state dynamics and filtered latent states. Using the stochastic gradient variational Bayes approach, our method jointly optimizes the parameters of the nonlinear dynamical system, the observation model, and the black-box recognition model. Unlike previous approaches, our framework can incorporate non-trivial distributions of observation noise and has constant time and space complexity. These features make our approach amenable to real-time applications and the potential to automate analysis and experimental design in ways that testably track and modify behavior using stimuli designed to influence learning. § INTRODUCTION Discovering interpretable structure from a streaming high-dimensional time series has many applications in science and engineering. Since the invention of the celebrated Kalman filter, state space models have been successful in providing a succinct, hence more interpretable, description of the underlying dynamics that explains the observed time series as trajectories in a low-dimensional state space. Taking a step further, state space models equipped with nonlinear dynamics provide an opportunity to describe the latent “laws” of the system that is generating the seemingly entangled time series <cit.>. Specifically, we are concerned with the problem of identifying a continuous nonlinear dynamics in the state space (t) ∈ℝ^d that captures the spatiotemporal structure of a noisy observation (t): = F_θ((t), (t)) (state dynamics) (t) ∼ P((t) | G_θ((t), (t))) (observation model) where F and G are continuous functions that may depend on parameter θ, (t) is control input, and P denotes a probability distribution that captures the noise in the observation; e.g., Gaussian distribution for field potentials or Poisson distribution for spike counts.In practice, the continuous-time state dynamics is more conveniently formulated in discrete time as,_t+1= f_θ(_t, _t) + ϵ_t (discrete time state dynamics)where ϵ_t is intended to capture the unobserved (latent) perturbations of the state _t. Such (spatially) continuous state space models are natural in many applications where the changes are slow and the underlying system follows physical laws and constraints (e.g., object tracking), or where learning the laws are of great interest (e.g. in neuroscience and robotics) <cit.>. Specifically, in the context of neuroscience, the state vector _t represents the instantaneous state of the neural population, while f captures the time evolution of the population state. Further interpretation of f can provide understanding as to how neural computation is implemented <cit.>.If the nonlinear state space model is fully specified, Bayesian inference methods can be employed to estimate the current state <cit.>. Conventionally, the estimation of latent states using only the past observation is referred to as filtering – inference of the filtering distribution, p(_t |_≤ t). If both past and future observations are used, then the quantity of interest is usually the smoothing distribution, p(_≤ t|_≤ t)). We are also interested in predicting the distribution over future states, p(_t:t+s|_≤ t) and observations, p(_t+1:t+s|_≤ t) for s > 0. However, in many applications, the challenge is in learning the parameters θ of the state space model (a.k.a. the system identification problem). We aim to provide a method for simultaneously learning both the latent trajectory _t and the latent (nonlinear) dynamical and observational system θ, known as the joint estimation problem <cit.>.Expectation maximization (EM) based methods have been widely used in practice <cit.>, and more recently variational autoencoder methods <cit.> have been proposed, all of which are designed for offline analysis, and not appropriate for real-time applications. Recursive stochastic variational inference has been successful in streaming data assuming independent samples <cit.>, however, in the presence of temporal dependence, proposed variational algorithms (e.g. <cit.>) remain theoretical and lack testing.In this study, we are interested in real-time signal processing and state space control setting <cit.> where online algorithms are needed that can recursively solve the joint estimation problem on streaming observations. A popular solution to this problem exploits the fact that online state estimators for nonlinear state space models such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) can be used for nonlinear regression formulated as a state space model.By augmenting the state space with the parameters, one can build an online dual estimator using nonlinear Kalman filtering <cit.>. However, they involve coarse approximation of Bayesian filtering, involve many hyperparameters, do not take advantage of modern stochastic gradient optimization, and are not easily applicable to arbitrary observation likelihoods. There are also closely related online version of EM-type algorithms <cit.> that share similar concerns.In hopes of lifting these concerns, we derive an online black-box variational inference framework, referred to as variational joint filtering (VJF), applicable to a wide range of nonlinear state dynamics (dynamic models) and observation models, that is, the computational demand of the algorithm is constant per time step. Our approach aims at * Online adaptive learning: Our target application scenarios are streaming data. This allows the inference during an experiment or as part of a neural prosthetics. If the system changes, the inference will catch up with the altered system parameters. * Joint estimation: The proposed method is supposed to simultaneously learn the latent states p(_t |_≤ t), state dynamics f(_t, _t) and the observation model G(, ). No offline training is necessary to learn the system parameters. * Interpretability: Under the framework of state space modeling, rather than interpret the system via model parameters, we employ the language of dynamical systems and capture the characteristics of the system qualitatively via fixed point, limit cycle, strange attractor, bifurcation and so on which are key components of theories of neural dynamics and computation.We focus on low-dimensional latent dynamics that often underlie many neuroscientific experiments and allow for producing interpretable visualizations of complex collective network dynamics in this study.§ VARIATIONAL PRINCIPLE FOR ONLINE JOINT ESTIMATION The crux of recursive Bayesian filtering is updating the posterior over the latent state one step at a time: p( _t |_≤ t ) = p( _t |_t )_likelihood p( _t |_<t )_prior at time t /p(_t |_<t)_marginal likelihood where the input _t and parameters θ are omitted for brevity. Unfortunately, the exact calculations of Eq. (<ref>) are not tractable in general, especially for nonlinear dynamic models and/or non-conjugate distributions. We thus turn to approximate inference and develop a recursive variational Bayesian filter by deriving an evidence lower bound for the marginal likelihood as the objective function. Let q(_t) denote an arbitrary probability measure which will eventually approximate the filtering density p(_t |_≤ t). From Eq. (<ref>), we can rearrange the marginal log-likelihood aslog p(_t|_<t)= logp(_t|_t) p(_t|_<t)/p(_t|_≤ t)for any _t = _q(_t)[logp(_t|_t) p(_t|_<t)/p(_t|_≤ t)] the marginal is constant to q(_t) = _q(_t)[ logp(_t|_t) p(_t|_<t) q(_t)/p(_t|_≤ t) q(_t)] = _q(_t)[ log p(_t|_t) ]_reconstruction log-likelihood - (q(_t)p(_t|_<t)) + (q(_t)p(_t|_≤ t))_variational gap #1≥_q(_t)[ log p(_t|_t) ] + (q(_t))_entropy + _q(_t)[ log p(_t|_<t) ] = _q(_t)[ log p(_t|_t) ] + (q(_t)) + _q(_t)[ log_p( _t-1|_<t)[ p(_t |_t-1) ] ] = _q(_t)[ log p(_t|_t) ] + (q(_t)) + _q(_t)[ _p(_t-1 | _<t)[ log p(_t|_t-1) ]] + _q(_t)[ (p(_t-1|_<t)p(_t-1|_t,_<t)) ] _variational gap #2≥_q(_t)[ log p(_t|_t) ] + (q(_t)) + _q(_t)_ p( _t-1|_<t) [ log p(_t |_t-1) ]wheredenotes Shannon's entropy anddenotes the Kullback-Leibler (KL) divergence <cit.>. Maximizing this lower bound would result in a variational posterior q(_t) ≈ p(_t |_≤ t) w.r.t. q(_t). Naturally we plug in the previous step's solution to the next time step, obtaining a loss function suitable for recursive estimation:ℒ _q(_t)[ log p(_t|_t) ] + (q(_t)) + _q(_t)_ q( _t-1 ) [ log p(_t |_t-1) ] _dynamics log-likelihoodThis also results in consistent q(_t) for all time steps as they are in the same family of distribution.Meanwhile, as it is aimed to jointly estimate the observation model p(_t |_t) and state dynamics p(_t |_t-1), we achieve online inference by maximizing this objective ℒ w.r.t. their parameters (omitted for brevity) and the variational posterior distribution q(_t) simultaneously provided that q(_t-1) takes some parameterized form and has been estimated from the previous time step. Maximizing the objective ℒ is equivalent to minimizing the two variational gaps: (1) the variational filtering posterior has to be close to the true filtering posterior, and (2) the filtering posterior from the previous step needs to be close to p(_t-1|_t, _<t). Note that the second gap is invariant to q(_t) if p(_t-1|_t, _<t) = p(_t-1|_<t), that is, the one-step backward smoothing distribution is identical to the filtering distribution.On the flip side, intuitively, there are three components in ℒ that are jointly optimized: (1) reconstruction log-likelihood which is maximized if q(_t) concentrates around the maximum likelihood estimate given only _t, (2) the dynamics log-likelihood which is maximized if q(_t) concentrates at around the maximum of _q(_t-1)[ log p(_t|_t-1) ], and (3) the entropy that expands q(_t) and prevents it from collapsing to a point mass.In order for this recursive estimation to be real-time, we choose q(_t) to be a multivariate normal with diagonal covariance 𝒩(μ_t, _t) where μ_t is the mean vector and _t is the diagonal of the covariance matrix in this study. Moreover, to amortize the computational cost of optimization to obtain the best q(_t) on each time step, we employ the variational autoencoder architecture <cit.> to parameterize q(_t) with a recognition model. Intuitively, the recognition model embodies the optimization process of finding q(_t), that is, it performs an approximate Bayesian filtering computation (in constant time) of Eq. (<ref>) according to the objective function ℒ. We use a recursive recognition model that maps q(_t-1) and _t to q(_t). In particular, the recognition model takes a deterministic recursive form:[μ_t, _t] = h(_t, u_t-1, μ_t-1, _t-1)Specifically h takes a simple the form of the multi-layer perceptron (MLP) <cit.> in this study, and we refer to its parameters as the recognition model parameters. Note that the recursive architecture of the recognition model reflects the Markovian structure of the assumed dynamics (c.f., smoothing networks often use bidirectional recurrent neural network (RNN) <cit.> or graphical models <cit.>).The expectations appearing in the reconstruction log-likelihood and dynamics log-likelihood are not always tractable in general. For those intractable cases, one can use the reparameterization trick and stochastic variational Bayes <cit.>: rewriting the expectations over q as expectation over a standard normal random variable, i.e., μ_t + _t^1/2𝒩(0, 1), and using a single sample for each time step. Hence, in practice, we optimize the following objective function (the other variables and parameters are omitted for brevity),ℒ̂ = log p(_t|_t, θ) + _q(_t)log p(_t |_t-1, θ) + H(q(_t))where _t and _t-1 represent random samples from q(_t) and q(_t-1) respectively. Note that the remaining expectation over q(_t) has closed form solution under our Gaussian state noise, ϵ_t, assumption. Thus, our method can handle arbitrary observation and dynamic model unlike dual form nonlinear Kalman filtering methods that usually suffer difficulties in sampling, e.g. transforming Gaussian random numbers into point process observations.The schematics of the proposed inference algorithm is summarized by two passes in Figure <ref>. In the forward pass, the previous latent state generates the new state through the dynamic model, and the new state transforms into the observation through the observation model. In the backward pass, the recognition model recovers the current latent state from the observation, and the observation model, recognition model and dynamic model, are updated by backpropagation. Algorithm <ref> is an overview of the recursive estimation algorithm. Denoting the set of all parameters by Θ of the observation model, recognition model and dynamic models, the objective function in Eq. (<ref>) is differentiable w.r.t. Θ, and thus we employ empirical Bayes and optimize it through stochastic gradient ascent (using Adam <cit.>). We outline the algorithm for a single vector time series, but we can filter multiple time series with a common state space model simultaneously, in which case the gradients are averaged across the instantiations. Note that this algorithm has constant time and space complexity per time step.In practice, the measurements _t and input u_t are sampled at a regular interval. Algorithm <ref> is called after every such observation event, which will return the state estimate along with the parameters and the dynamical system. One can visualize these for real-time for monitoring, and/or have it streamed to another system for further automated processing (e.g. detect anomalies and raise an alarm or deliver feedback controls).§ APPLICATION TO LATENT NEURAL DYNAMICS Our primary applied aim is real-time neural interfaces where a population of neurons are recorded while a low-dimensional stimulation is delivered <cit.>. State-space modeling of such neural time series have been successful in describing population dynamics <cit.>. Moreover, models of neural computation are often described as dynamical systems <cit.>. For example, attractor dynamics where the convergence to one of the attractors represents the result of computation <cit.>. Here we propose a parameterization and tools for visualization of the model suitable for studying neural dynamics and building neural interfaces <cit.>. In this section, we provide methodological details for the results presented in the next section.§.§ Parameterization of the state space model Having in mind high-temporal resolution neural spike trains where each time bin has at most one action potential per channel, we describe the case for point process observation. However, note that our method generalizes to arbitrary observation likelihoods that are appropriate for other modalities, including calcium imaging or local field potentials. The observed point process time series _t is a stream of sparse binary vectors. All experiments of point process observation were binned finely so that the time bins contain one event each at most in this study.Our observation model, Eq. (<ref>), assumes that the observation vector _t is sampled from a probability distribution P determined by the latent state _t though a linear-nonlinear map possibly together with extra parameters at each time t,_t ∼ P(g(_t + ))where gℝ→ℝ is a point-wise map. We use the canonical link g(·) = exp(·) for Poisson likelihood and identity for Gaussian likelihood in this study. Note that this observation model is not identifiable since _t = () (^-1_t) whereis an arbitrary invertible matrix.We normalize the loading matrixin each iteration. It is straightforward to include more additive exogenous variables, history-filter for refractory period, coupling between processes, and stimulation artifacts <cit.>.For state dynamic model, we propose to use a specific additive parameterization withstate transition function and input interaction as a special case of Eq. (<ref>), _t+1= _t + f(_t) + _t _t + ϵ_t+1 f(_t) = ϕ(_t)_0, ϵ_t∼𝒩(0, σ^2 ) where ϕ(·) is a vector of r continuous basis functions, i.e. ϕ(·) = (ϕ_1(·), …, ϕ_r(·))^⊤,is the weight matrix of the radial basis functions, and _t is the interaction with the input _t. The interaction _t can be globally linear, parameterized as a matrix independent from _t, or locally linear, parameterized as a matrix-valued function of _t using also RBF networks. i.e. ((_t)) = _B ϕ(_t) where _B is the respective weight matrix. In this study, we used squared exponential radial basis functions <cit.>,ϕ_i() = exp(-γ_i ‖ - _i ‖_2^2)with centers _i and corresponding inverse squared kernel width γ_i. Though the dynamics can be modeled by other universal approximators such as percepton and RNN, we chose the radial basis function network for the reasons of non-wild extrapolation (zero velocity when the state is far away from data) and fast computation.The time complexity of our algorithm is 𝒪(mpr + n(m + p + q)) where n, m, p, q, r denote the dimensions of observation, latent space, input, the numbers of hidden units and radial basis functions for this specific parameterization. Practically to achieve realistic computation time for real-time applications in neuroscience, the number of radial basis functions and hidden units are constrained by the requirement. Note that the time complexity does not grow with time that enable efficient online inference.If we compare this to an efficient offline algorithm such as PLDS <cit.> run repeatedly for every new observation (“online mode”), its time complexity is 𝒪(t · (m^3 + mn)) per time step at time t which grows as time passes, making it impractical for real-time application.§.§ Phase portrait analysisPhase portrait displays key qualitative features of dynamics, and with a little bit of training, it provides a visual means to interpreting dynamical systems. The law that governs neural population dynamics captured in the inferred function f() directly represents the velocity field of an underlying smooth dynamics (<ref>) in the absence of input <cit.>. In the next section, we visualize the estimated dynamics as phase portrait which consists of the vector field, example trajectories, and estimated dynamical features (namely fixed points) <cit.>. We can numerically identify candidate fixed points ^∗ that satisfy f(^∗) ≈ 0. For the synthetic experiments, we performed an affine transformation to orient the phase portrait to match the canonical equations in the main text when the simulation is done through the proposed observation model if the observation model is unknown and estimated.§.§ PredictionFor state space models, we can predict both future latent trajectory and future observations. The s-step ahead prediction can be sampled from the predictive distributions: p(_t+1:t+s|_≤ t) =𝔼_q(_t) [p(_t+1:t+s|_t)] p(_t+1:t+s|_≤ t) =𝔼_p(_t+1:t+s|_≤ t) [p(_t+1:t+s|_t+1:t+s)] given estimated parameters by current time t without seeing the data _t+1:t+s during these steps. In the figures of experiments, we plot the mean of the predictive distribution as trajectories.§ EXPERIMENTS ON THEORETICAL MODELS OF NEURAL COMPUTATION We demonstrate our method on a range of nonlinear dynamical systems relevant to neuroscience. Many theoretical models have been proposed in neuroscience to represent different schemes of computation. For the purpose of interpretable visualization, we choose to simulate from two or three dimensional dynamical systems. We apply the proposed method to four such low-dimensional models: a ring attractor model as a model of internal head direction representation, an nonlinear oscillator as a model of rhythmic population-wide activity, a biophysically realistic cortical network model for a visual discrimination experiment and a chaotic attractor.In the synthetic experiments, we first simulated state trajectories by respective differential equations, and then generated either Gaussian or point process observations (to mimic spikes) via Eq. (<ref>) with corresponding distributions. The parametersandwere randomly drawn, and they were constrained to keep firing rate <60 Hz on average for realistic spiking behavior. All observations are spatially 200-dimensional unless otherwise mentioned. We refer to their conventional formulations under different coordinate systems, but our simulations and inferences are all done in Cartesian coordinates. Note that we focus on online learning in this study and always train our model with streaming data, even while comparing with offline methods.The approximate posterior distribution is defined recursively in Eq. (<ref>) as diagonal Gaussian with mean and variance determined by corresponding observation, input and previous step via a recurrent neural network. We used a one-hidden-layer MLP in this study. Typically the state noise variance σ^2 is unknown and has to be estimated from data. To be consistent with Eq. (<ref>), we set the starting value of σ^2 to be 1, and hence μ_0 = 0, _0 =. We initialize the loading matrixby factor analysis, and column-wisely normalize it by ℓ_2 norm every iteration to keep the system identifiable.§.§ Ring attractor Continuous attractors are often used as models for neural representation of continuous variables <cit.>. For example, a bump attractor network with ring topology is proposed as the dynamics underlying the persistently active set of neurons that are tuned for the angle of the animal's head direction <cit.>. Here we use the following 2-variable reduction of the ring attractor system: First, we study the following two-variable ring attractor system:τ_r ṙ= r_0 - rτ_φφ̇= Iwhere φ represents the direction driven by input I, and r is the radial component representing an internal circular variable, such as head direction. We simulated 100 trajectories (1000 steps) with step size Δ t = 0.1, r_0=1, τ_r = 1, τ_φ=1 with Gaussian state noise (std=0.005) added each step. Though the ring attractor is defined in polar coordinate system, we transformed it into Cartesian system for simulation and training. In simulation we used strong input (tangent drift) to keep the trajectories flowing around the ring clockwise or counter-clockwise. The point process observations were generated by passing the states through a linear-nonlinear map (Eq. (<ref>)) and sampling from a Poisson distribution. We streamed the observations into the proposed algorithm that consists of point process likelihood, dynamic model with 20 radial basis functions and locally linear input interaction in Eq. (<ref>) and a recognition MLP with 100 hidden units.Figure <ref>A illustrates one latent trajectory (black) and its variational posterior mean (blue). These two trajectories start at green circle and diamond respectively and end at the red markers. The inference starts near the center (origin) that is relatively far from the true location because the initial posterior mean is set at zero. The final states are very close which implies that the recognition model works well. Figure <ref>B shows the reconstructed velocity field by the model. We visualized the velocity as colored directional streamlines. We can see the velocity toward the ring attractor and the speed is smaller closer to the ring. The model also identifies a number of fixed points arranged around the ring attractor via numerical roots finding. Figure <ref>C shows the distribution of posterior means of all data points in the state space. We have more confidence of the inferred dynamical system in the denser area.Figure <ref>D shows the three components of Eq. (<ref>) and the objective lower bound clearly, demonstrating the convergence of the algorithm. We can see each component reaches a plateau within 400 sec. As the reconstruction and dynamics log-likelihoods increase, the recognition model and dynamical model are getting more accurate while the decreasing entropy indicates the increasing confidence (inverse posterior variance) on the inferred latent states. The average computation time of a joint estimation step is 1.1 ms (hardware specification: Intel Xeon E5-2680 2.50G Hz, 128GB RAM, no GPU). §.§ Nonlinear oscillator Dynamical systems have been a successful application in the biophysical models of single neuron in neuroscience. We used a relaxation oscillator, the FitzHugh-Nagumo (FHN) model <cit.>, which is a 2-dimensional reduction of the Hodgkin-Huxley model: We used awith the following nonlinear state dynamics:v̇= v(a - v)(v - 1) - w + I,ẇ= bv - cw,where v is the membrane potential, w is a recovery variable and I is the magnitude of stimulus current in modeling single neuron biophysics. This model was also used to model global brain state that fluctuates between two levels of excitability in anesthetized cortex <cit.>. We use the following parameter values a = -0.1, b = 0.01, c = 0.02 and I = 0.1 to simulate 100 trajectories of 1000 steps with step size 0.5 and Gaussian noise (std=0.002). At this regime, unlike the ring attractor, the spontaneous dynamics is a periodic oscillation, and the trajectory follows a limit cycle. The point process observations were also sampled via the observation model of the same parametric form as that of the ring attractor example. We used 20 radial basis functions for dynamic model and 100 hidden units for recognition model. While training the model, the input was clamped to zero, and expect the model to learn the spontaneous oscillator.We compare the state estimation with the standard particles filtering (PF) which are powerful online methods theoretically capable of producing arbitrarily accurate filtering distribution. We run two variants of the particle filter with different proposal distributions. One used diffusion as the proposal, i.e. _t = _t-1 + ϵ_t whereis the vector of state variables v and w, and the other, a.k.a. bootstrap particle filter <cit.>, used the true dynamics in Eq. (<ref>). We provided the true parameters for the observation model and noise term to PF which gives them an advantage. Both particle filters and VJF were run on 50 realizations of 5000-step long observation series. Figure <ref> shows the root mean squared deviations (RMSE) (mean and standard error over 50 realizations). It is expected that the bootstrap particle filter outperformed the diffusion particle filter since the former utilized the true dynamics. One can see the state estimation by VJF improved as learning carrying on and eventually outperformed both particle filters. Note that VJF had to learn the parameters of likelihood, dynamic model and recognition model during the run. We varied the number of RBFs (20 and 30) but the results are not substantially different.We also reconstructed the phase portrait (Fig. <ref>B) comparing to the truth (Fig. <ref>C). The two dashed lines are the theoretical nullclines of the true model on which the velocity of corresponding dimension is zero. The reconstructed field shows a low speed valley overlapping with the nullcline especially on the right half of the figure. At the intersection of the two nullclines there is an unstable fixed point. We can see the identified fixed point is close to the intersection. As most of the trajectories lie on the oscillation path (limit cycle) with merely few data points elsewhere, the inferred system shows the oscillation dynamics similar to the true system around the data region. The difference mostly happens in the region far from the trajectories because of the lack of data.We run a long-term prediction using VJF without seeing the future data _t+1:T during these steps (T=1000 steps = 1 sec) beginning at the final state of training data. We show the truth and prediction in figure <ref>D. The upper row is the true latent trajectory and corresponding observations. The lower row is the filtered trajectory and prediction by the proposed method. The light-colored parts are the 500 steps of inference before prediction and the solid-colored parts are 1000 steps truth and prediction. We also show the sample observations from the trained observation model during the prediction period.One of the popular latent process modeling tools for point process observation that can make prediction is the Poisson Linear Dynamical System (PLDS) <cit.> which assumes latent linear dynamics. We compared PLDS fit with EM on its long-term prediction on both the states and spike trains (Fig. <ref>). This demonstrates the nonlinear dynamical model outperforming the linear model even in the unfair online setting.To compare to the methods with nonlinear dynamical models, we also run latent factor analysis via dynamical systems (LFADS) <cit.> offline using the same data. LFADS implements its dynamical model with the gated recurrent unit (GRU) <cit.> that requires high dimensions. For this 2D system, we tried different GRU dimensionalities. We made minimal changes to its recommended setting including only the generator dimensionality, batch and no controller. The result shows that LFADS requires much higher dimension than the true system to capture the oscillation (Fig. S1). (The figure of its inferred trajectories is shown in the supplement.) We report the fitted log-likelihood per time bin -0.1274, -0.1272 and -0.1193 for 2D, 20D and 50D GRU respectively. In comparison, the log-likelihood of the proposed approach is -0.1142 with a 2D dynamical model (higher the better).§.§ Fixed point attractor for decision-making Perceptual decision-making paradigm is a well-established cognitive task where typically a low-dimensional decision variable needs to be integrated over time, and subjects are close to optimal in their performance. To understand how the brain implements such neural computation, many competing theories have been proposed <cit.>. We test our method on a simulated biophysically realistic cortical network model for a visual discrimination experiment <cit.>. In the model, there are two excitatory subpopulations that are wired with slow recurrent excitation and feedback inhibition to produce attractor dynamics with two stable fixed points (Fig. <ref>A). Each fixed point represents the final perceptual decision, and the network dynamics amplify the difference between conflicting inputs and eventually generates a binary choice.Note that, different from the former examples that use a linear-nonlinear map of latent states, the point process observations (spikes) of this experiment were directly sampled from the spiking neural network [The detail of the spiking neural network can be found in <cit.> and the code can be found at <https://github.com/xjwanglab/book/tree/master/wang2002>.](1 ms binwidth) that was governed by its own high-dimensional intrinsic dynamics. It is filling the gap between fully specified state space models and real neuron populations.We subsampled 480 selective neurons out of 1600 excitatory neurons from the simulation to be observed by our algorithm. The simulated data is organized into decision-making trials where each trial lasts for 2 sec and with different strength of visual evidence, controlled by “coherence”. Our method with 20 radial basis functions learned the dynamics from 140 training trials (20 per coherence level c, c=-1, -0.2, -0.1, 0, 0.1, 0.2, 1).Figure <ref>C shows the velocity field at zero coherence stimulus as colored streamlines. Note that our approach did not have prior knowledge of the network dynamics as the mean-field reduction <cit.> in Figure <ref>B. Although the absolute arrangement is dissimilar, the topology and relation of the five identified fixed points show correspondence with the mean-field reduction. The inference was completely data-driven (partial observation of spike trains) while the mean-field method required knowing the true dynamical model of the network and careful approximation by <cit.>. We showed that our method can provide a qualitatively similar result to the theoretical work which reduces the dimensionality and complexity of the original network.§.§ Chaotic dynamics Chaotic dynamics (or edge-of-chaos) have been postulated to support asynchronous states in the cortex, and neural computation over time by generating rich temporal patterns <cit.>. We consider the 3-dimensional standard Lorenz attractor as an example chaotic system to demonstrate the flexibility of our method. We simulated 216 latent trajectories from:ẋ = 10(y - x), ẏ = x(28 - z) - y, ż = x y - 8/3 z.The each coordinate of the initial states are on the uniform grid of 6 values in [-50, 50] inclusively, of which the combination results in 216 unique states. We discarded the first 500 transient steps of each trajectory and then use the following 1000 steps. We generated 200-dimensional Gaussian observations driven by the trajectories. Figure <ref>A shows estimated latent trajectory and the ground truth. One can see that the estimation lies in a similar manifold. In addition, we predicted 500 steps of future latent states without knowing the respective observations. Figure <ref>B shows 4 predicted trajectories starting from different initial states. One can see that the inferred system could generate qualitatively similar trajectory at most initial states but not perfectly for the true system is chaotic. §.§ Nonstationary systemAnother feature of our method is that its state dynamics estimate never stops. As a result, the algorithm is adaptive, and can potentially track slowly varying (nonstationary) latent dynamics. To test this feature, we compared a dual EKF and the proposed approach on nonstationary linear dynamical system. A spiral-in linear system was suddenly changed from clockwise to counter-clockwise at the 2000th step and the latent state was perturbed (Fig. <ref>). To adapt EKF, we used Gaussian observations that were generated through linear map from 2-dimensional state to 200-dimensional observation with additive noise (𝒩(0, 0.5)). To focus on the dynamics, we fixed all the parameters except the transition matrix for both methods, while our approach still has to learn the recognition model in addition. Figure <ref> shows that our approach achieved better online performance as dual EKF in this experiment. § REAL NEUROPHYSIOLOGICAL APPLICATION We applied the proposed method to a large scale recording to validate that it picks up meaningful dynamics. The dataset <cit.> consists of 148 simultaneously recorded single units from the primary cortex (V1) while directional drifting gratings were presented to an anesthetized monkey for around 1.3s per trial (Fig. <ref>A). We used the spike trains from 63 well-tuned units. The spike times were binned with a 1ms window (max 1 spike per bin). There is one continuous circular variables in the stimuli space: temporal phase of oscillation induced by the drifting gratings.A partial warm-up helps with the training. We chose a good initialization for the observation model, specifically the loading matrix and bias. There are 72 motion directions in total, each repeated 50 trials. We used the trials corresponding to 0 deg direction to initialize the observation model with dimensionality reduction methods such as variational latent Gaussian processes <cit.>, and then trained VJF with a 2D dynamic model fully online on the trials corresponding to 180 deg direction that it had not seen before. Since we do not have long enough continuously-recorded trials, we concatenated the trials (equivalent to 500s) as if they were continuously recorded to mimic an online setting. As expected, Figure <ref>B and C shows the inferred dynamical system is able to implement the oscillation. The two goodness of fit measures (log-likelihood and ELBO) in Figure <ref>D shows that our method benefits from but does not necessarily require such a warm-up. The model with warm-up initialization has better starting goodness of fit than the random initialized model but the random initialized model eventually achieved similar goodness of fit with adequate amount of data.§ DISCUSSION Neurotechnologies for recording the activity of large neural populations during meaningful behavior provide exciting opportunities for investigating the neural computations that underlie perception, cognition, and decision-making. However, the datasets provided by these technologies currently require sophisticated offline analyses that slow down the scientific cycle of experiment, data analysis, hypothesis generation, and further experiment. Moreover, in closed-loop neurophysiological setting, real-time adaptive algorithms are extremely valuable <cit.>.To fulfill this demand, we proposed an online algorithm for recursive variational Bayesian inference that simultaneously performs system identification and state filtering under the framework of state space modeling, in hope that it can greatly impact neuroscience research and biomedical engineering. There is no other method capable of all features, hence we compared several methods in different measures, often giving them the advantage. We showed that our proposed method consistently outperforms the state-of-the-art methods.Using the language of dynamical systems, we interpret the target system not via model parameters but via dynamical features: fixed points, limit cycles, strange attractors, bifurcations and so on. In our current approach, this interpretation heavily relies on visual inspection of the qualitative nonlinear dynamical system features. In contrast, most popular state space models assume linear dynamics <cit.> which is appropriate for smoothing latent states, but not expressive enough to recover the underlying vector field. Recently the Koopman theory that allows representation of general nonlinear dynamics as linear operators in infinite dimensional spaces <cit.> has gained renewed interest in modeling nonlinear dynamics. Although elegant in theory, in practice, however, the Koopman operators need to be truncated to a finite dimensional space with linear dynamics <cit.>. We note that the resulting linear models do not allow for topological features such as multiple isolated fixed points, nonlinear continuous attractors, stable limit cycles—features critical for non-trivial neural computation. Our algorithm is highly flexible and general—it allows a wide range of observation models (likelihoods) and dynamic models, is computationally tractable, and produces interpretable visualizations of complex collective network dynamics. Our key assumption is that the dynamics consists of a continuous and slow flow, which enable us to parameterize the velocity field directly. This assumption reduces the complexity of the nonlinear function approximation, thus it is easy to identify the fixed/slow points. Specifically we chose the radial basis function network to model the dynamics for our experiments, which regularizes and encourages the dynamics to occupy a finite phase volume around the origin.Our method has a number of hyperparameters. In the experiments, the differentiable hyperparameters were learnt via gradient descent while the selection of the other hyperparameters were made simple. In general, our method was robust; Perturbing the number of RBFs did not produce qualitatively different results (Fig. <ref>). <cit.> discussed growing radial basis function network adaptively which could be incorporated in our method to enable online tuning of the number of RBFs. The depth and width of neural networks were chosen empirically to improve the interpretability of resulting dynamical systems, but tuning did not result in large changes in the results. This work opens many avenues for future work. One direction is to apply this model to large-scale neural recording from a behaving animal. We hope that further development would enable on-the-fly analysis of high-dimensional neural spike train during electrophysiological experiments. Clinically, a nonlinear state space model provides a basis for nonlinear feedback control as a potential treatment for neurological diseases that arise from diseased dynamical states.hunsrt | http://arxiv.org/abs/1707.09049v5 | {
"authors": [
"Yuan Zhao",
"Il Memming Park"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170727211530",
"title": "Variational online learning of neural dynamics"
} |
apsrev#1⟨⟩#1√( #1) Tr^1Department of Physics, Universidad Simón Bolívar,Caracas 1080, Venezuela ^2 Department of Physics, Florida State University,Tallahassee, FL 32306-4350,USA We present a numerical and theoretical study that supports and explains recent experimentalresults on anomalous magnetization fluctuations of auniaxial ferromagnetic film in its low-temperature phase, which isforced by an oscillating field above the critical period of the associateddynamic phase transition (DPT)[P. Riego, P. Vavassori, A. Berger, Phys. Rev. Lett.118, 117202 (2017)].For this purpose, we perform kinetic Monte Carlosimulations of a two-dimensional Ising model with nearest-neighbor ferromagnetic interactionsin the presence of a sinusoidallyoscillating field, to which is added a constant bias field.We study a large range of system sizes and supercritical periods and analyze the data using a droplet-theoretical description of magnetization switching.We find that the period-averaged magnetization, which plays the role of the order parameterfor the DPT, presents largefluctuations that give rise to well-defined peaks in its scaled variance and itssusceptibility with respect to the bias field. The peaks are symmetric withrespect to zero bias and located at values of the bias field thatincrease toward the field amplitude as an inverse logarithm of thefield oscillation period.Our results indicate that this effect is independent of the system sizefor large systems, ruling out critical behavior associated with a phase transition.Rather, it is a stochastic-resonance phenomenon that has no counterpart in thecorresponding thermodynamic phase transition, providing a reminder that the equivalenceof the DPT to an equilibrium phase transition is limited to the critical region near thecritical period and zero bias. Fluctuations in a model ferromagnetic film driven by a slowly oscillatingfield with a constant bias Gloria M. Buendía^1 and Per Arne Rikvold^2 Received: date / Accepted: date ===================================================================================================== § INTRODUCTIONThe hysteretic response when a uniaxial spin system with long-range order( i.e., below its critical temperature) is subject to a symmetrically oscillating field of amplitude H_0 and period P,depends crucially on P. If P is muchlonger than the response time of the system (which depends on thetemperature and H_0), a symmetric hysteresis loop centered on zero results.If P is much shorter than the response time, asymmetric hysteresis loopscentered around the values of the system's static order parameter are observed.Numerical studies in the 1990's showed that the transition between thesetwo regimes is not smooth. Rather, there is a critical period P_c, where theperiod-averaged order parameter ⟨ Q ⟩(see formal definition in Sec. <ref>)vanishes in a singular fashion. This phenomenon was first observed byTomé and de Oliveira <cit.> in a kinetic mean-field study of an Ising model,followed by kinetic Monte Carlo (MC) simulations by Rao, Krishnamurthy, and Pandit <cit.>and Lo and Pelcovitz <cit.>. Early work in the field was reviewedby Chakrabarti and Acharyya in Ref. <cit.>. Kinetic MC combined with finite-size scaling analysis<cit.>, as well as furthermean-field studies of Ising and Ginzburg-Landau models<cit.>, confirmed not only that this isa true, dynamic phase transition (DPT), but also that it is in the sameuniversality class as the corresponding equilibrium Ising model. The DPT hasbeen confirmed experimentally in [Co/Pt]_3 magnetic multilayers<cit.> and uniaxial Co films <cit.>.With all the attention that has been given to the DPT and its universalityclass, one might lose sight of the fact that the equivalence between thecritical properties of the equilibrium Ising model and the DPT of the samemodel in an oscillating field does not necessarily amount to equivalence outside the critical region. A warning was provided very recently by Riego,Vavassori, and Berger <cit.>. These authors fabricated Co filmswith (1010) crystallographic surface structure with a single,in-plane magnetic easy axis, which they subjected to a sinusoidally oscillating,in-plane magnetic field plus a constant bias field H_b. Such a constant biasfield has previously been shown by MC simulations and finite-size scaling to be(at least a significant component of) the field conjugate to ⟨ Q ⟩ in thecritical region near P_c<cit.>, and this has later been confirmed formean-field models <cit.> and in experiments <cit.>.It therefore seemedsurprising that, in the experiments reported in Ref. <cit.>,both the fluctuations in the order parameter and its derivative with respect toH_b, for P ≫ P_c, behaved quite differently from the dependence of theequilibrium susceptibility on the applied static field at temperatures abovecritical. Instead of the wide, smooth, unimodal maximum of the supercritical equilibriumsusceptibility of the Ising model,two distinct peaks were observed at nonzero values of H_b, symmetricalabout zero<cit.>. In their article the authors also presented kinetic mean-field results that corroborate the presence ofthese peaks, which they dubbed “sidebands."The purpose of the present paper is to investigate the long-period parameterregime with kinetic MC simulations of a two-dimensional Ising model withnearest-neighbor ferromagnetic interactions.To match the experimental conditions of Ref. <cit.> as closely as possible, wechoose the oscillating field to have a sinusoidal waveform.We are not aware that systematic simulations in this regime have been performed previously.Our study reveals “sidebands" analogous to the experimental results.We thus conclusively confirm that the experimentally observed phenomenon is not caused byresidual magnetostatic long-range interactions.Using simulations for a range of field periods and system sizestogether with knowledge of the kinetics of magnetization switching by homogeneousnucleation and growth of antiphase droplets <cit.>,we demonstrate that the “sidebands"result from noncritical fluctuations during thehalf-cycles when the sign of the oscillating field is opposite to that of the bias field.This is essentially a stochastic resonance phenomenon <cit.>. The rest of this paper is organized as follows.In Sec. <ref> we describe the model and details of the simulation method, and we definethe appropriate observables to be measured.Our numerical results are presented in Sec. <ref>.In Sec. <ref>we present numerical observation of sidebands for a single, supercriticalvalue of the field period.In Sec. <ref> we present short time series ofthe system magnetization for several values of bias and period,which enable us to propose a simple approximation for ⟨ Q ⟩ in the limits ofweak bias and long period. In Sec. <ref> we present numerical results for⟨ Q ⟩ vs H_b for a wide range of supercritical periods, as wellas the sideband positions H_b^ peak as functions of period and system size.The latter are analyzed using results from the droplet theory of magnetization reversal.Our conclusions are given in Sec. <ref>. A short summary of pertinent results from the droplet theory of magnetization reversal is given in Appendix <ref>, and the case of extremely long periods is discussed in Appendix<ref>. A brief discussion of the mathematicallysimpler case of a square-wave oscillating field is presented in Appendix <ref>.§ MODEL AND MONTE CARLO SIMULATIONWe consider a kinetic S=1/2 Ising model with a time-dependent externalfield and ferromagnetic nearest-neighbor interactions.Its Hamiltonian is ℋ=-J∑_⟨ ij ⟩ s_is_j - [ H(t)+H_b] ∑_i s_i ,where J>0, s_i=±1, the first sum runs over all nearest-neighbor pairs,and the second one over all sites. H_b is a constant “bias field," and H(t)is a symmetrically oscillating external field of period P. Here we chooseH(t)=H_0cos( 2 π/P t ).The system is simulated on a square lattice of N=L × L sites withperiodic boundary conditions. We perform Glauber single-spin-flip dynamics in a heat bath at temperature T. A spin at a randomly chosen site i is allowed to flip froms_i to -s_i with probability W(s_i→ -s_i)=1/1+exp(βΔ E_i) ,where Δ E_i is the change in the system energy associatedwith flipping the spin i, and β=1/k_ BT where k_ B isBoltzmann's constant. The time unit is one MC step per site (MCSS),during which, on average, each site is visited once. Hereafter, H_0, H_b, and T areall given in units of the interaction constant J (i.e., J=k_ B=1),and P is given in units of MCSS. The Glauber dynamic can be derived asthe weak-coupling limit of the quantum-mechanical Hamiltonian of a collectionof quasi-free Fermi fields in thermal equilibrium with a heat bath<cit.>. However, the DPT with H_b=0 has been shown to be universalwith respect to dynamics that obey detailed balance in equilibrium, includingMetropolis <cit.> and “soft Glauber" <cit.>, as well as differentforms of H(t) including square-wave <cit.> and sawtooth<cit.>.We calculate the time dependent, normalized magnetization per site,m(t)=1/L^2∑_is_i(t),and by integrating it over each cycle of themagnetic field, we obtain the average magnetization during the kthcycle of the field,Q_k=1/P∫_(k-1)P^kP m(t)dt.The dynamic order parameter of the modelis the period-averaged magnetization, ⟨ Q ⟩,defined as the average of Q_k over many cycles. Its fluctuations are measured by the scaled variance,χ_L^Q=L^2(⟨ Q^2⟩ -⟨ Q ⟩^2),and its dependence on the bias field is measured by the susceptibility with respect to H_b,χ_L^b=d⟨ Q ⟩/dH_b . In order to take advantage of temperature and field dependent parameters measuredwith high precision in previous MC simulations <cit.>,our calculations are performed with H_0=0.3 at T=0.8T_c, whereT_c=2/ln(1+√(2))≈ 2.269 is the critical temperature of the standard,square-lattice Ising model in zero field. In the absence of a bias field, at thistemperature, and for sufficiently large L,switching between the equilibrium values of m, following field reversal from-H_0 to +H_0, occurs via anearly deterministic and L-independent multi-droplet mechanism <cit.>.In Ref. <cit.>, the characteristic switching timescale (the time from the fieldreversal until the system magnetization reaches zero) under Glauber dynamics with the sameparameters as we use here was measured byMC simulations as τ_0 ≈ 74.6.In the same work, the critical period in a sinusoidal field of amplitude H_0 with zero biaswas measured as P_c ≈ 258. The cycle-averaged magnetization⟨ Q ⟩ vanishes for P ≥ P_c and H_b=0.Near criticality, the constant bias field H_b is the field conjugate to ⟨ Q ⟩, andthe period P mimics the temperature in the equilibrium phase transition.Simulations were performed for periods between P = 258 and 28,000 andsystem sizes between L = 32 and 1024. Except for the smallest values of P, the measurements were obtained by averagingover 800 field cycles, after discarding 200 cycles. This means thatat least 800 × P MCSS were performed for each measurement. § NUMERICAL RESULTS AND ANALYSIS§.§ Observation of “sidebands"Results of simulations with P = 1000 ≈ 3.9P_cfor several values of L are displayed inFig. <ref>. “Sidebands" are observed, consistent with theexperiments reported in Ref. <cit.>. The dependence ofthe order parameter ⟨ Q ⟩ on the bias H_b is shown in Fig. <ref>(a).For weak H_b, ⟨ Q ⟩ increases almost linearly with H_b, but the slope of the curve increases considerably around |H_b| ≈ 0.09, followed by saturation of⟨ Q ⟩ for |H_b| ≳ 0.15.This behavior is reflected in the bimodal shape of the susceptibility χ_L^b,shown by the lower set of curvesin Fig. <ref>(b). Between the two peaks lies a flat-bottomed valley correspondingto the linear regime in part (a), and a rapid approach to zero for large |H_b| mirrors thesaturation of ⟨ Q ⟩ also seen in (a).The scaled variance χ_L^Q also displays peaks, whose positions coincide with those ofχ_L^b. However, the ratio χ_L^Q / χ_L^b for fixed Pdepends quite strongly on H_b withmaximum values near the peaks. This variable ratio precludes a straightforward interpretationin terms of an effective, nonequilibrium fluctuation-dissipation relation with P playing therole of “temperature." For these values of L and P, finite-size effects are seen to be negligible,ruling out critical behavior associated with a phase transition.The relationships between system size, field period, and finite-size effects will be discussedin further detail below.§.§ Magnetization time seriesTo gain a more detailed understanding of the relationships between bias, period, system size,and the order-parameter fluctuations, we present in Fig. <ref> short time series of the normalized magnetization, m(t).The total applied field, H(t) + H_b, is shown as an orange curve.In this figure we set H_b > 0, so that the up-spin phase is favored andthe down-spin phase is disfavored. Figure <ref>(a) shows data forP=1000 and H_b=+0.10, just on the strong-bias side of the fluctuationpeak for this period length.For the smaller system sizes (L=32 and 64), the switching from the favored(up-spin) tothe disfavored (down-spin) magnetization is stochastic and abrupt(mediated by a single or a few droplets of the down-spin phase <cit.>)and occurs only in narrow time windows near the negative extrema of the total appliedfield. For the larger systems, the switching becomes more deterministic and gradual(multi-droplet <cit.>).However, the growing down-spin phase does not have timeto completely fill the system before the fieldagain becomes positive. For the largest system studied, L=1024, the extremenegative magnetizations during a period are close to -0.2.Figure <ref>(b) shows data forP=1000 and H_b=+0.0915, at the maximum of the fluctuation peak.The switching behavior for L=32remains stochastic. However, the larger systems appear more deterministic, and their extremenegative magnetizations during a period are close to -0.4. Figure <ref>(c) shows data forP=1000 and H_b=+0.08, just on the weak-bias side of the fluctuation peak.The switching for L=32 remains stochastic. The larger systems behave more deterministically,and the extreme negative magnetizations during a period approach -0.8. These results illustrate how the switching behavior in the peak region crosses over froma stochastic single-droplet mechanism for small L to a nearly deterministic multidropletmechanism for larger L, in agreement with known results for field-driven magnetizationswitching by homogeneous nucleation and growth of droplets of the stable phase <cit.>. Figure <ref>(d) shows data forL=128 with a weak bias, H_b = +0.04, and two different period lengths, P=1000and 14,000. In both cases, the switching is nearly deterministic and complete, so that theperiod-averaged magnetization ⟨ Q ⟩ depends mostly on therelative amounts of time thesystem spends in the two phases. As P increases, the switching occurs earlier in the half-period. The differences between the single-droplet and multidroplet switchingmodes are further illustrated in Fig. <ref>. In Fig. <ref>(a), time series for m(t) over five cycles with P=1000at the corresponding peak position, H_b^ peak = +0.0915show data for L=32 and 1024. All the parameters are the same as inFig. <ref>(b), except the seed for the random number generator. When the total applied field, H(t) + H_b, is negative,the down-spin phase, which isdisfavored by the positive bias, is the equilibrium phase.Nucleation and growth of this phase may only occur during the time intervals ofnegative total applied field.Snapshots captured at m(t) = +0.1 during these growth periods,corresponding to a down-spin fraction of 0.45, are shown inFig. <ref>(b) for L=32 and in Fig. <ref>(c) forL=1024. For L=32 we see a single down-spin droplet which, as seen fromthe time series in Fig. <ref>(a), nucleated during thethird period shown, near the time when thefield had its largest negative value. It barely reached the capture thresholdof m=+0.1 before the field again became positive and caused it todecay. The stochastic natureof this single-droplet switching mode is alsoclearly reflected by the time series.During the five periods shown, the capture threshold was only reached twice.And only once, during the fifth period, do we see full saturation of thedown-spin phase before the field again becomes positive.For L=1024 the picture is quite different. In the snapshot we seea large number of growing clusters that have nucleated at different timesduring the negative-field time interval. Some of these have already coalescedby the time the snapshot was captured,while others are still growing independently. From the time series it is seen that this multi-dropletswitching mode leads to a nearly deterministic evolution of the totalmagnetization, withthe underlying stochasticity only evident in the slight variations of theminimum magnetization values from period to period. This switching process is well described by theKolmogorov-Johnson-Mehl-Avrami (KJMA)approximation <cit.>. Magnetization reversal from the favored to the disfavored direction is only possible whilethe total applied field,H(t) + H_b, has the opposite sign of the bias, H_b. This implies that-1 < H_b/H(t) ≤ 0. Switching from the favored phase to the disfavored one on average takeslonger time than switching in the opposite direction.Thus, the time the system can spend in the disfavored phaseduring each period must be less than or equal to the time that the field has thedisfavored direction, t_ Dmax= P/2[ 1 - 2/πsin^-1( |H_b|/H_0) ].In this limit of long period and weak bias, ⟨ Q ⟩ is simply determined bythe sign of H_b and the difference between the fractions of the period that the total fieldhas the same and the opposite sign as H_b, respectively. This yields⟨ Q ⟩≈2 m_0/πsin^-1( H_b/H_0),which is symmetric under simultaneous reversal of H_b and ⟨ Q ⟩. Here, m_0 is the magnitude of the magnetization in the favored phase.This approximation represents a lower bound on the magnitudes of ⟨ Q ⟩and χ^b<cit.>. The formeris included as a dashed curve in Fig. <ref>(a).However, the bounds depend on the waveform of the oscillating field, and as weshow in Appendix <ref>, they vanish in the case of a square-wave field. The corrections to this approximation are of O (t_ FD(H_b,H_0)/P ), wheret_ FD(H_b,H_0) is the average time it takes the magnetization to switch to thedisfavored direction,after the total applied field has changed sign. For |H_b| ≪ H_0, thecorrection vanishes as 1/P, as seen in Fig. <ref>(a).However, for larger |H_b|, t_ FD(H_b,H_0) ∼ P, andthe “correction" becomes the dominant part of ⟨ Q ⟩, determining the sidebandpeak positions, H_b^ peak.The details are discussed below in Sec. <ref>.§.§ Dependence on H_b, P, and L Results for L=128 and a range of periods between P_c = 258 and P = 28,000 are shown in Fig. <ref>. In the critical region, H_b is the field conjugate to⟨ Q ⟩<cit.>. At P = P_c, ⟨ Q ⟩ therefore vanishes in a singular fashionas H_b approaches zero. On the scale of Fig. <ref>(a),this singularity appears as a jump in ⟨ Q ⟩ atH_b = 0 for P=P_c, resulting in very narrow central peaksin both χ_L^b and χ_L^Q.We also found broad central peaks in both quantities for P = 400, which are due tofinite-size broadening of the critical region for this relatively modest system size.For clarity, these central peaks are not included in Fig. <ref>(b).Beyond P = 500, ⟨ Q ⟩ becomes linear for small H_b, with a slope thatapproaches that of the asymptotic approximation in Eq. (<ref>) as P increases.Simultaneously, the peaks in χ_L^b and χ_L^Q increase in height, and their positionsH_b^ peak move in the directions of ± H_0, as seen in Fig. <ref>(b).[For clarity, some of the values of P included inFig. <ref>(a) are excluded from Fig. <ref>(b).]The magnitudes of the peak positions, |H_b^ peak|, are plotted vs P for differentvalues of L in Fig. <ref>(a). We note two main features. First,|H_b^ peak| increases quite rapidly with P for relatively short periods, andmuch more slowly for longer periods.This behavior is consistent with the experimental data shown in Fig. 2 of Ref. <cit.>.Second, finite-size effects are essentiallynegligible for P ≲ 2000, as already shown inFig. <ref> for P=1000. For longer periods, |H_b^ peak|increases with L for smaller sizes, and then becomes size independent for larger L. In order to explain this behavior quantitatively, we first recall from the time seriesshown in Fig. <ref> that for bias near |H_b^ peak|, the time it takesm(t) to change significantly toward the disfavored sign is on the order of a finite fraction of P.For stronger bias, the total field driving the magnetization toward the disfavored sign is too weak and consequently the time required for switching is much longer than P,so that reliablemagnetization reversal does not occur. For weaker bias, the field in the disfavored direction is relatively strong, andcomplete and reliable magnetization reversal takes place on a timescale significantly shorter thanP.In other words, the peak positions correspond to bias values that producemagnetization reversal on a timescale of P. Equationsfor magnetization switching rates by the stochastic single-particle mechanism thatdominates for small systems[Eq. (<ref>)] and the nearly deterministic multidroplet mechanismthat dominates for large systems [Eq. (<ref>)]are found in Appendix <ref>. The nucleation rate for droplets of the disfavored phasevaries very strongly with the oscillating field, having appreciable valuesonly in a narrow window near the maximum field in the disfavored direction,|H| = H_0 - |H_b^ peak|.Using this value of |H| and ignoring less important prefactors, we canuse these equations to write the following requirement for |H_b^ peak|: L^-aexp( 1/bΞ_0/H_0 - |H_b^ peak| ) ∼ P,with a=2 and b=1 for single-droplet switching, and a=0 and b=3 formultidroplet switching.The meaning of the constant Ξ_0 ≈ 0.506 is explained in Appendix <ref>.In either case, this equation is equivalent to a statement that |H_b^ peak|should approach H_0 asymptotically as 1/ log P for long periods.(A caveat to this statement for the case of extremely long periods is discussed inAppendix <ref>.)Plotting 1/(H_0 - |H_b^ peak|)vs log P therefore should produce straight lines for large values of P.The ratio between the slopes of the lines representing multidroplet switching for large Land those representing single-droplet switching for small L should be 3/1.Such a plot is presented in Fig. <ref>(b). The slope ratio between the curvesrepresenting L=256 and L=32 in the long-P regime is approximately 2.867,consistent with the theoretical prediction. This conclusion is confirmed by the shorttime series of m(t) for P=20,000 for these two system sizes, shown inFig. <ref>. In the switching regions, the smaller system displays thestochastic, square wave form characteristic of single-droplet switching <cit.>,while the larger system shows the continuouswave form characteristic of multidroplet switching <cit.>. To further support our conclusions, we calculated the transition times and the order parameterin the multidroplet regime for the mathematically simpler case,in which the sinusoidally oscillating field hasbeen replaced by a square-wave field. The details of the calculations are given in Appendix<ref>. In Fig. <ref> we show that there is very good agreement between the theoreticallycalculated ⟨ Q ⟩ and the simulations, particularly when|H_b| ≲ |H_b ^ peak|. § SUMMARY AND CONCLUSION Riegoet al.<cit.> recently presented experimental data on Co films with asingle, in-plane magnetic easy axis, which were subjected to a slowly oscillating magneticfield with an added constant bias. In this paper we have presented kinetic MC simulations andtheoretical analysis of a two-dimensional Ising ferromagnet with onlynearest-neighbor interactions, designed to closely mimic the experimental setup.At zero bias, such systems exhibit a dynamic phase transition (DPT) at a critical period P_c,where the period-averaged magnetization ⟨ Q ⟩ vanishes in a singular fashion.It has previously been shown that the DPT belongs to the equilibrium Ising universality class,with P playing the role of temperature and the bias H_b being the field conjugate to⟨ Q ⟩.Following Riegoet al.<cit.>,we studied the dynamics of the system at values of P aboveP_c, and in agreement with the experiments we found that ⟨ Q ⟩ exhibits a strong bias dependence and fluctuation peaks at nonzero values ofH_b, symmetrically located around zero bias. Since the simulated system has only nearest-neighbor interactions, our results show that theexperimental results arenot due to any residual magnetostatic interactions.The simulational approach also enables studies of the effects of finite system size.We found that, at fixed P, finite-size effects saturate beyond a P-dependent size limit.Using the droplet theory of magnetization switching, we conclude that this saturation occursat the crossover between two different dynamic regimes. For small systems, the magnetizationswitching from the favored to the disfavored direction occurs by astochastic single-droplet mechanism. For large systems, the switching occurs by the size-independentand nearly deterministic KJMAmechanism, which involves a largenumber of simultaneously nucleating and growing droplets. We therefore conclude that this “sideband" phenomenon forsupercritical values of P isnot a critical phenomenon, but rather a stochastic-resonancephenomenon.We believe these insights will be important for the design and analysis of devices that involvemagnetization reversal by time-varying fields, such as memory elements, switches, and actuators.§ ACKNOWLEDGMENTSWe thank A. Berger for providing data from Ref. <cit.> before publication,and for useful comments on an earlier version of this paper.G.M.B. is grateful for the hospitality of the Physics Department at Florida State University,where her stay was supported in part by the American Physical Society InternationalResearch Travel Award Program (IRTAP). P.A.R. acknowledges partial support by U.S. National Science Foundation Grant No. DMR-1104829. § MECHANISMS OF MAGNETIZATION REVERSALWhen a d-dimensional Ising ferromagnet below its critical temperature is subjected tothe reversal of anapplied field of magnitude |H|, the homogeneous nucleation rate per unit system volumefor droplets of the new equilibrium phase is givenby <cit.> I(H) ≈ B(T) |H|^K exp[ - Ξ_0 (T)/|H|^d-1],where B(T) is a non-universal function of T. For d=2, K=3, and Ξ_0(0.8 T_c) ≈ 0.506 (which includes a factor of 1/T)<cit.>.The argument of the exponential function is the negative of thefree energy of a critical droplet of the equilibrium phase, divided by T.The inverse of L^d I(H) isthe average time between random nucleation events for a system of size L.Single-droplet reversal mechanism: Under conditions of small system and/or moderately weak field,the time it takes for the first nucleated droplet to grow to fill the systemis much shorter than the average nucleation time. As a result, the magnetization reversal iscompleted by this single, first droplet.Multidroplet reversal mechanism: Under conditions of large system and/or moderately strong field,the average time between nucleation events is less than the time it would take the firstnucleated droplet to grow to fill the system. Therefore, many droplets nucleate and growindependently in different parts of the system until they coalesce and collectively fill the system.The result is a gradual and nearly deterministic growth of the new phase through a multidropletprocess, well described by the KJMAapproximation <cit.>.The characteristic reversal time is independent of the system size and given by ⟨τ(H) ⟩∝[ v^d I(H) ] ^-1/(d+1) , where the propagation velocity ofthe droplet surface, v,is proportional to |H| in this parameter range<cit.> as expected from the Lifshitz-Allen-Cahn approximation<cit.>. § EXTREMELY LONG PERIODS If the radius of the critical droplet reaches a size of about L/2, it will not fit in the L × Lsystem, and a new regime, called thecoexistence regime, is entered <cit.>. In thisregime, the droplet is replaced by a slab of the equilibrium phase, and the nucleation time no longerdepends on |H|, but increases exponentially with L^d-1.The critical droplet radius in d dimensions is given by <cit.>,R_c ≈( (d-1) T Ξ_0/2 m_0 Ω_d)^1/d1/|H| ,where Ω_d is the volume of the critical droplet, divided by R_c^d.Numericalvalues for the constants with d=2 atT=0.8T_c ≈ 1.815 are found in Table I of Ref. <cit.>:Ξ_0 ≈ 0.506 and Ω_2 ≈ 3.152.(The factor T is included in the numerator to cancel the factor 1/T in Ξ_0.) Thus we have R_c ≈0.388/|H|≈L/2 .Replacing |H| by H_0 - |H_b| and setting L=32, we thus find1/(H_0 - |H_b|) ≈ 41.3. Finally, linearly extrapolating the large-P data for L=32 inFig. <ref>(b), we find that the single-droplet result from Eq. (<ref>)should remain valid for periods up to approximately 10^19 ± 2.[The uncertainty in the exponent is the result of assuming a 10% uncertainty in the estimate of1/(H_0 - |H_b|).] Beyond this limit, H_0 - |H_b| should remain independent of P, at a value of O(1/L).For larger L, the single-droplet result should be valid up to even longer periods.We do not expect that these extremely long periods should be of great experimental relevance formacroscopic systems. However, for nanoscopic systems the coexistence regime may beobservable with experimentally accessible periods. § SQUARE-WAVE OSCILLATING FIELDNow, instead of a sinusoidally oscillating field, consider a square-wave field, such thatH(t) = +H_0 during one half-period, and -H_0 during the other.Since the times that the total field is parallel and antiparallel to H_b now each equal P/2,the equivalent of the long-period, weak-bias approximation of Eq. (<ref>) becomes⟨ Q ⟩≈ 0. Therefore, the value of ⟨ Q ⟩ for finite P andweak H_b is determined by the difference between the average magnetization reversal timesfollowing a change of the total field from the favored to the disfavored direction, and theopposite. Since the total field now has its full favored or disfavored strength during the wholehalf-period, these average switching times will be shorter than the corresponding times in thesinusoidally oscillating field case. With a square-wave fieldof amplitude H_0 = 0.3 at 0.8 T_c under Glauber dynamics, the critical period hasbeen measured by MC simulations as P_c ≈ 137<cit.>.To calculate the transition times for a two-dimensional system in the multidroplet regime,we will again assume H_b ≥ 0 for concreteness. From Eqs. (<ref>) and (<ref>) with |H| = H_0 - H_b,we obtain the characteristic timescale fortransitions from the favored (parallel to the bias field) to the disfavoredmagnetization direction, after the total applied field has changed sign as t_ FD(H_b,H_0) = τ_0 ( 1/1 - H_b/H_0)^5/3exp(Ξ_0/3 H_0H_b/H_0/1 - H_b/H_0)≥τ_0,where τ_0 is the magnetization reversal time for H_b = 0. Analogously, the switching time from the disfavored to the favored magnetization direction is t_ DF(H_b,H_0) = τ_0 ( 1/1 + H_b/H_0)^5/3exp( - Ξ_0/3 H_0H_b/H_0/1 + H_b/H_0)≤τ_0.Both t_ FD and t_ DF reduce to τ_0 ≈ 74.6<cit.>for H_b = 0. 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1Benoziyo Center for Astrophysics, Weizmann Institute of Science, 76100 Rehovot, IsraelThe localization of the repeating fast radio burst (FRB), FRB 121102, suggests that it is associated with a persistent radio-luminous compact source in the FRB host galaxy. Using the FIRST radio catalog, I present a search forluminous persistent sources in nearby galaxies, with radio luminosities >10% of the FRB 121102 persistent source luminosity. The galaxy sample contains about 30% of the total galaxy g-band luminosity within <108 Mpc, in a footprint of 10,600 deg^2. After rejecting sources likely due to active galactic nuclei activity or background sources, I remain with 11 candidates that are presumably associated with galactic disks or star formation regions. At least some of these candidates are likely to be due to chance alignment. In addition, I find 85 sources within 1” of galactic nuclei. Assuming the radio persistent sources are not related to galactic nuclei and that they follow the galaxy g-band light, the 11 sources imply a 95% confidence upper limit on the space density of luminous persistent sources of 5×10^-5 Mpc^-3, and that at any given time only a small fraction of galaxies host a radio luminous persistent source (10^-3 L_*^-1). Assuming persistent sources life time of 100 yr, this implies a birth rate of 5×10^-7 yr^-1 Mpc^-3. Given the FRB volumetric rate, and assuming that all FRBs repeat and are associated with persistent radio sources, this sets a lower limit on the rate of FRB events per persistent source of 0.8 yr^-1. I argue that these 11 candidates are good targets for FRB searches and I estimate the FRB event rate from these candidates. § INTRODUCTIONFast radio bursts are dispersed millisecond-duration pulses observed at GHz frequencies (Lorimer et al. 2007; Thornton et al. 2013). Spitler et al. (2016) reported the discovery of a repeating FRB, and follow-up observations using the Very Large Array (VLA) were able to provide an arcsecond localization of the source of the bursts (Chatterjee et al. 2017). The localization is consistent with the position of a star formation region in a faint galaxy (r=25.1 mag) at a redshift of 0.19273 (Tendulkar et al. 2017; Bassa et al. 2017). Furthermore, Chatterjee et al. (2017) reported the existence of a persistent radio source within 1” of the FRB location, and Marcote et al. (2017) showed that the FRB and the persistent source are separated by less than 12 mas (40 pc projected). The radio source has a mean flux density of about 0.18 mJy, it is variable, presumably due to scintillations, and unresolved with an angular size smaller than about 1.7 mas.There are many suggestions for the FRB mechanism, including soft gamma-ray repeats (e.g., Popov & Postnov 2010; Kulkarni et al. 2014; Lyubarsky 2014; see however Tendulkar et al. 2016), Galactic stars (e.g., Maoz et al. 2015), pulsars (e.g., Lyutikov et al. 2016; Katz 2017a), magnetars (e.g., Metzger et al. 2017), pulsar wind nebulae (e.g., Dai, Wang & Yu 2017), active galactic nuclei (Katz 2017b), and more.Based on the FRB 121102 persistent source luminosity, spectrum, angular size, and the FRBs constant dispersion measure (DM) over a year time scale, Waxman (2017) inferred the properties of the emitting region. He concluded it is a mildly relativistic, 10^-5 M_⊙ shell which propagates into an ambient medium. The energy of this expanding shell is of the order of 10^49 erg, and its lifetime is of order 300 yr. Waxman (2017) also suggested an efficient mechanism that produces the FRBs within this persistent source via synchrotron maser. An intriguing property of the FRB 121102 persistent source is that it is radio bright. With an isotropic radiative energy of ν L_ν≈3×10^38 erg s^-1, at 1.4 GHz, it is over an order of magnitude brighter than the brightest known supernova (SN) remnants (e.g., Lonsdale et al. 2006; Parra et al. 2007; Chomiuk & Wilcots 2009). However, its luminosity is comparable with the brightest young (∼1 yr old) SNe (e.g., Weiler et al. 2002) Spitler et al. (2016) showed that the FRBs' arrival times are highly non Poissonic (see also e.g., Wang & Yu 2017). This fact makes it difficult to estimate the actual mean rate of the FRB 121102 events.Here, I report on a search for persistent radio sources in galaxies in the nearby Universe (z<0.025). I find 11 candidates, with luminosity greater than 10% of the FRB121102 persistent source.Regardless of the nature of these luminous radio sources, and their relation to FRBs, by estimating the completeness of the survey I was able to place an upper limit on the number density of such bright compact persistent sources in the local Universe. Furthermore, assuming that all the FRBs repeat and associated with persistent radio sources, I set a lower limit on the rate of FRB events per persistent source. I also discuss the volumetric rate of FRBs and the implications for FRB searches.The structure of this paper is as follows: In <ref> I describe the search for persistent sources in the nearby Universe, while in <ref> I analyze the results. The nature of the persistent radio sources is discussed in <ref>, and in <ref> I summarize the results and discuss the implications for FRB searches. § SEARCH FOR FRB PERSISTENT SOURCESHere I present a search for persistent radio-luminous compact sources in the nearby Universe. In <ref> I present the nearby galaxy sample, and in <ref> I describe the cross matching of the galaxy list with the FIRST radio catalog. In <ref> I estimate chance coincidence probabilities, while in <ref> I discuss the radio sources' variability. All the steps and analysis were performed using tools and catalogue available in the MATLAB Astronomy & Astrophysics Toolbox[https://webhome.weizmann.ac.il/home/eofek/matlab/] (Ofek 2014). §.§ The nearby galaxy sample and its completenessI compiled a catalog of nearby galaxies within 108 Mpc. The catalog is based on combining the HyperLEDA galaxies[http://leda.univ-lyon1.fr/] (Paturel et al. 2003; Makarov et al. 2014) with the NASA Extragalactic Database (NED[https://ned.ipac.caltech.edu/]) redshifts, and the Sloan Digital Sky Survey (SDSS; York et al. 2000) galaxies with known redshifts. Both catalogs are restricted to the FIRST[FIRST catalog version 2017 Dec 14.] radio survey footprint (Becker, White, & Helfand 1995). This catalog is far from being completeand I estimate its completeness below. I note that the total area of the FIRST footprint is about 10,600 deg^2.The HyperLEDA catalog lists galaxies brighter than 18 mag, without redshifts. I compiled a catalog of all redshifts available in NED[This catalog, with 3.27×10^6 redshifts, is available as part of the MATALB Astronomy & Astrophysics Toolbox (Ofek 2014), under cats.galaxies.NED_z.]. I cross-matched galaxies in the HyperLEDA catalog with the redshift catalog. The association radius was set to 15”. Next, I selected only galaxies with redshifts in the range 0 to 0.025 that are found in the FIRST footprints. Specifically, I demand that the FIRST coverage maps at the galaxy position have rms below 0.26 mJy and above zero. The resulting catalog has 16,152 entries.I supplemented this catalog with SDSS galaxies in the FIRST footprint[The SDSS catalog contains many HyperLEDA galaxies.], with redshifts above 0 and below 0.025 that are not listed in the HyperLEDA catalog (5” association). This catalog contains an additional 12,663 galaxies. All the galaxies' magnitudes were corrected for Galactic extinction (Cardelli et al. 1989; Schlegel et al. 1998). The luminosity functions of the two galaxy catalogue are shown in Figure <ref>. The luminosity functionshows a steep drop for galaxies fainter than about 0.1 L_*. I note that the FRB 121102 host galaxy luminosity is near the peak of the luminosity function in Figure <ref>.Next, I estimate the completeness of the combined galaxy catalog in terms of total g-band luminosity in the nearby Universe. Figure <ref> shows the galaxy sample g-band luminosity per unit volume as a function of distance.I also integrated the total B-band luminosity in the Karachentsev et al. (2004) catalog of galaxies within 10 Mpc. In order to correct for the galaxy zone of avoidance, I restricted the search again to the FIRST footprints. Since the Karachentsev et al. catalog lists the B-band magnitude, while our catalog uses the SDSS g-band, I use a B(Vega)-g(AB)=0.18 mag correction[Assuming a black-body spectrum with an effective temperature of 6000 K.].I find that the g-band luminosity density in the Karachentsev et al. (2004) catalog is about 0.055 L_* Mpc^-3, where here L_* is the luminosity corresponding to g-band absolute magnitude of -19. This is close to the measured L_* at z=0.1 (≈-18.8; e.g., Montero-Dorta & Prada 2009). Assuming that the Karachentsev et al. (2004) catalog is nearly complete in terms of the total luminosity and star formation rate[If this assumption was wildly incorrect, then we would expect that blind SN surveys will find many more nearby SN associated with faint dwarf galaxies.], and that the galaxy luminosity density within 10 Mpc is representative of the galaxy luminosity density at larger distances, I calculate how much luminosity is missing from of our HyperLEDA+SDSS catalog per unit volume as a function of distance, and integrate. I find that the total luminosity in our catalog has a completeness of 30%. I note that this factor is uncertain due to the fact that the galaxy distribution is correlated. Given the galaxy correlation function (Tucker et al. 1997; Zehavi et al. 2002) I estimate the completeness to have about 10% uncertainty.§.§ Search for radio sources associated with nearby galaxiesI cross-matched our galaxy catalogs with FIRST point sources. I defined point sources to have a major axis smaller than 3 times the uncertainty[The size uncertainty is calculated using the formula in: http://sundog.stsci.edu/first/catalogs/readme.html] in the radio size measurement.The HyperLEDA-catalog search radius, for each galaxy, was set to the galaxy's 25 mag arcsec^-2 surface brightness semi-major axis listed in the catalog, while for the SDSS sample I used the SDSS Petrosian radius[http://skyserver.sdss.org/dr7/en/help/docs/algorithm.asp?key=mag_petro].In the cross-matching step, I selected only sources with intrinsic radio luminosity (assuming at the galaxy redshift) of >10%of the luminosity of the persistent source associated with FRB 121102. For this, I used the FRB 121102 persistent source mean radio flux density of 0.18 mJy and luminosity distance of 938 Mpc (i.e., ν L_ν≈3×10^38 erg s^-1).The cross-matching yielded 122 possible matches, of which 91 are from the HyperLEDA catalog and 31 are from the SDSS catalog. The candidates are listed in Table <ref>, and in <ref> I discuss the chance coincidence probability for these sources.llllllllllll 12 0ptLuminous persistent radio source candidatesR.A.Dec.f_ pΔf_ p L/L_ persz θf_ NVSSΔf_ NVSSχ_ NVSS-FIRSTX_ c Comment mJy mJy arcsecmJy mJy ct/ks 09:27:58.282 -02:25:58.95 2.1 0.14 0.14 0.02314.32 Spiral arm + IR source10:47:26.693 +06:02:47.72 2.9 0.14 0.13 0.019 5.872.40.4-1.1 Off galaxy center; passive galaxy23:53:51.412 +07:58:35.91 4.2 0.13 0.16 0.01842.68 Near spiral arm; near red+IR source14:10:43.667 +08:59:29.96 3.2 0.15 0.21 0.02317.664.00.4 1.8 Edge of spiral disk; red faint source?10:25:26.189 +17:15:47.97 2.8 0.13 0.11 0.018 7.08 Spiral arm10:58:23.641 +24:13:55.32 2.3 0.15 0.12 0.02129.79 Spiral arm13:14:41.932 +29:59:59.19 2.2 0.14 0.14 0.02320.584.00.5 3.4 Edge of spiral galaxy; IR source16:22:44.571 +32:12:59.28 2.0 0.15 0.11 0.022 0.892.70.4 1.7 Small blue galaxy; near center14:00:38.929 -02:51:22.79 1.5 0.15 0.11 0.02526.4118.8 Elliptical galaxy halo; no vis/IR source11:45:29.346 +19:23:27.46 3.5 0.20 0.26 0.02533.352.40.4-2.3 Edge of galaxy; No optical or IR source14:19:18.855 +39:40:36.0321.1 0.15 0.95 0.020 0.50 18.51.0-2.2 Compact blue star forming galaxy00:17:27.498 -09:34:26.56 2.5 0.15 0.15 0.023 0.51 Center of galaxy00:17:59.547 -09:16:00.89 1.6 0.15 0.10 0.02319.062.90.5 2.5 Halo of galaxy, likely IR source02:52:42.189 -08:48:15.76 3.1 0.15 0.12 0.018 0.184.40.5 2.4 Center of galaxy11:24:03.341 -07:47:01.13 2.1 0.14 0.16 0.025 0.503.70.6 2.5 Center of galaxy01:44:43.099 -04:07:46.25 5.2 0.13 0.21 0.01834.81 10.90.6 9.0 Halo of galaxy, blue source + IR sourceA list of radio sources that spatially coincide with nearby galaxies. Sources above the horizontal-line separator are the 11 candidates. The full table is available electronically, and here I present only the first several entries. R.A. and Dec. are the J2000.0 right ascension and declination, respectively, of the radio source, f_ p is its peak radio flux, L/L_ pers is its luminosity in units of the mean luminosity of the FRB 121102 persistence source, z is the spatially coincident galaxy redshift, and θ is the angular separation between the galaxy and radio source. f_ NVSS and Δf_ NVSS are the NVSS flux density and its uncertainty, respectively, χ_ NVSS-FIRST is the FIRST to NVSS variability in units of the 1-σ uncertainty (Equation <ref>), and X_ c is the ROSAT counts per kilo-second, from the ROSAT bright and faint source catalogs (Voges et al. 1999; 2000). For the ROSAT catalogue, I used a 45” search radius. Entries with no data indicate non detection. I inspected the SDSS and WISE (Wright et al. 2010) 4.6 micron-band images of all candidates. The most common candidates are associated with galactic nuclei to within 1 arcsec (85 sources). In these cases I assumed that the radio source is due to active galactic nuclei (AGN) activity in the galaxy center. As most radio sources above a flux density of a few mJy are AGNs, this assumption is probably reasonable. I further note that at a redshift of 0.025 one arcsecond corresponds to about 0.5 kpc, which is an order of magnitude smaller than the typical size of galaxies. This gives some confidence that even if we remove from our sample persistent sources which are spatially coincidence with their galaxy's center, the number of non-AGN sources missed is small.Many other sources seem to be projected on a galaxy, but outside any star formation region or light associated with the galaxy – these sources are sometimes associated with an unresolved source (likely a background quasar). Finally, 11 sources were found to coincide with galactic disks light, or compact star forming galaxies (similar to FRB 121102; Bassa et al. 2017). I regard these 11 sources as the persistent-source candidates, and they are listed at the top part of Table <ref>. However, if FRBs are related to AGN activity (e.g., Katz 2017; Vieyro et al. 2017), then the number of candidates changes to 85.Figure <ref> shows the SDSS images of the 11 candidate galaxies, with markers showing the radio source position and galaxy center. §.§ Chance coincidenceGiven the total sky area of galaxies in our catalogs (≅5.3 deg^2), it is likely that at least some of the candidates are due to chance coincidence of background high-redshift sources with nearby galaxies. Here, I estimate the probability that the sources I find are background sources, unrelated to the spatially associated low-redshift galaxy.For each galaxy in the catalogs (HyperLEDA and SDSS) I calculate its area on the celestial sphere (given its radius). Furthermore, given the galaxy distance, I calculate the number density of sources on the sky that are compact radio sources with luminosity >10% of the FRB 121102 persistent radio-source luminosity. By multiplying the area of each galaxy by the corresponding surface density of radio sources and summing, I find that the expectancy value for the number of chance-coincidence background sources is 37.7.Excluding the 85 sources found within 1” from galactic nuclei, there are 37 sources spatially associated with nearby galaxies (i.e., =122-85). The 1” exclusion is small compared with the typical galaxy dimension and has a negligible effect on the expectancy value for chance coincidence.Given that the expectancy is 37.7 and that the observed number is 37, it is likely that at least some of the 11 candidates are due to chance coincidence.Assuming that the selection process (e.g., association with galaxy light in images) selects the best candidates, thanthe probability that all the 11 candidates are associated with their galaxy (rather than background sources) is 2.9%. The probability that at most 5, 2, 1, 0 of the 11 candidates are associated with their galaxy is 20%, 40%, 43%, and 50%, respectively.This analysis suggests that some, or even all, of the 11 candidates I found are background sources unrelated to the spatially associated galaxy. If follow-up observations will indicate that all 11 candidates are unrelated to FRBs this will improve the upper limit on the luminous radio source space density (see <ref>) by a factor of about 7.I note that a reasonable follow-up prioritization of the 11 candidates in Table <ref> is by the inverse galaxy size and association with star-forming regions. One reason is that smaller galaxies have lower probability for chance coincidence with background objects. In this respect, the most interesting candidate in the list is J141918+394036. This source has the highest luminosity of all candidates and it is associated with a small-area blue galaxy.§.§ Source variabilityI cross-matched the sources in Table <ref> with the NVSS catalog (Condon et al. 1998) with a 15” match radius. The NVSS flux and error are listed in Table <ref>. I further calculated, and list in Table <ref>,χ_ NVSS - FIRST = f_ NVSS - f_ FIRST/√(Δf_ NVSS^2 + Δf_ FIRST^2 + (ϵ_ cosf_ FIRST)^2 ).Here f_ NVSS is the NVSS[The NVSS and FIRST measurements each give a weighted mean flux over several epochs.] peak-flux density, f_ FIRST is the FIRST peak-flux density, Δf_ NVSS, and Δf_ FIRST are the NVSS and FIRST flux errors, respectively, and ϵ_ cos is a calibration error assumed to be 0.03 (Condon et al. 1998). I note that Ofek et al. (2011) measurements of a few calibration sources suggest that theVLA calibration error may be smaller. Furthermore, Ofek & Frail (2011) found that any systematic offset between the FIRST and NVSS fluxes is small compared with the typical flux errors. The FIRST-NVSS variability search indicates that these sources are roughly constant. However, I cannot rule out small amplitude variability due to scintillation or some long term decrease or increase in flux.§ ANALYSIS §.§ Persistent source volumetric densityI found 11 persistent source candidates. This sets a 95% (1-sided) upper limit of 18.2 sources in the searched volume (Gehrels 1986). Assuming the persistent source density is proportional to the g-band luminosity, andgiven the sky area and completeness of the catalog (<ref>), this gives 95% confidence-level upper limit on the number density of luminous persistent sources of:ρ_pers5×10^-5Mpc^-3. Using the total galaxy luminosity per unit volume I found in <ref>,this is equivalent to a number of persistent sources per L_* galaxy of 10^-3 L_*^-1.However, if FRBs are related to galactic nuclei, then I find 85 candidates and the limit in Equation <ref> will change to 3×10^-4 Mpc^-3 (5×10^-3 L_*^-1).I note that this estimate ignores any possible evolution of persistent sources with redshift. However, the expected increase in the star formation rate between z=0 to z=0.025 is only 8% (e.g., Wyder et al. 2005; Schiminovich et al. 2005).Given the density in Equation <ref>, Figure <ref> presents the upper limit on the sky surface density of radio luminous persistent sources, as a function of redshift.It is not yet clear whether the assumption that FRBs follow the galaxy light is correct. For example, based on the low luminosity of the FRB 121102 host (extinction corrected r-band abs. mag. -16.8), Nicholl et al. (2017) claimed that FRBs prefer dwarf galaxies.Using the Karachentsev et al. (2004) catalog, I calculate the fraction of B-band luminosity in galaxies fainter than the host galaxy of FRB 121102 (i.e., g-band abs. mag. -16.6; assuming g-r≈0.2 mag). I find that ≈ 3% of the luminosity is in galaxies fainter than abs. mag. -16.6. Since it is expected that there is some incompleteness even in the Karachentsev et al. catalog, the actual fraction can be a little bit higher. Therefore, I cannot rule out the possibility that the FRB 121102 host galaxy luminosity follows the general galaxy luminosity function, at more than 97% confidence. I note that the luminosity of the FRB 121102 host galaxy is near the peak of the luminosity function of galaxies in my catalog. Therefore, even if the assumption that persistent radio sources follow the g-band light is incorrect, then my upper limit on the space density of radio persistent sources is likely still valid to an order of magnitude.The upper limit on the FRB 121102 persistent source age estimated by Waxman (2017; see also Nicholl et al. 2017) is 300 yr, while the lower limit on its age is 5 yr (Metzger et al. 2017; Waxman 2017). Adopting the persistent source age of τ_ pers≈100 yr, our number density implies a birth rate of ρ̇_pers,birthrate 5×10^-7(t_ age/100yr)^-1yr^-1Mpc^-3.This suggests that the origin of FRB persistent sources is some sort of rare phenomenon (e.g., explosions). It is tempting to relate this birth rate to that of known events (e.g., Super Luminous Supernovae [SLSN] or Gamma-Ray Bursts [GRB]; e.g., Nicholl et al. 2017). Howevre, SLSN and GRB can be seen to large distances, and it is very likely that other fainter, yet unknown, transient classes exist. In this context I note that Waxman (2017) found that the total energy of the nebula associated with the FRB 121102 persistent source is of the order of 10^49 erg. This is low relative to the energetics of known rare events. §.§ The rate of FRBsIn order to constrain the rate of FRB events per persistent source, we need an estimate of the FRBs rate. There are many FRB rate estimates (e.g., Deneva et al. 2009; Burke-Spolaor et al. 2014; Champion et al. 2016; Lawrence et al. 2016; Vander Wiel et al. 2016). They are typically reported for different parameters (e.g., fluence limit and FRB duration), and therefore a comparison between these rates requires caution.Furthermore, I note that FRB searches may be slightly biased by several reasons. For example: (1) Usually FRB searches are done up to some limiting dispersion measure (DM). The DM threshold may evolve with time (e.g., whenever a new record in DM is found, the DM threshold is updated). This may result in an incompleteness for bright/far FRB events. (2) FRB searches are performed using non-coherent de-dispersion. This may bias against FRBs with durations shorter than a fraction of a millisecond. I note that in Zackay & Ofek (2016) we suggested a computationally efficient method for calculating the coherent de-dispersion.In order to avoid some of these complications, I prefer an estimate based on a single instrument.Therefore, for the rate estimate I adopt the Cahmpion et al. (2016) analysis of ten FRBs detected by the Parkes telescope. Cahmpion et al. (2016) report an FRB all-sky rate of R(F>0.13mJy ms)=7000_-3000^+5000 day^-1 (95% confidence errors) above fluence of 0.13 Jy ms for a minimal 0.128 ms pulse duration.I note that since the radio telescope beam is not uniform and the observed cumulative flux density function of sources (so called log(N)–log(S)) is some power-law (-3/2 for Euclidean Universe; see however Vedantham et al. 2016), this may affect the effective beam size of the survey (used in the rate calculation). Assuming a cumulative flux density power-law of -3/2 and assuming a Gaussian beam with a size equal to the beam full-width half maximum, the correction factor is 1 (see appendix C in Ofek et al. 2011). Therefore, here I adopt the Cahmpion et al. (2016) rate. §.§ The FRB volumetric rateAssuming that all FRBs repeat, that they are associated with radio persistent sources, their emission is isotropic, and using the FRB rate (<ref>), I use the upper limit on the persistent source number density (Eq. <ref>) to put a lower limit on the rate of FRB events per persistent source.The first step is to estimate the volumetric rate of FRBs. To estimate the effective volume of the Parkes search, I use all the 16 FRBs found by the Parkes radio telescope[Adopted from the FRB catalog: http://www.astronomy.swin.edu.au/pulsar/frbcat/]. For each FRB DM I removed the estimated Milky Way DM, and I attributed a fraction[Attributing a fraction of the DM to intergalactic dispersion is an approximation – in practice the host galaxy DM is coming from some unknown probability distribution of host-galaxy DM.] f_ DM,cosmo of the remaining DM to intergalactic dispersion. The analysis is performed for f_ DM,cosmo in the range of 0.1 to 1. I converted the remaining intergalactic DM to redshift via the formulae in Zheng et al. (2014) assuming Planckcosmological parameters (Ade et al. 2015). Each redshift was transformed to a cosmological volume. Given a flux limited survey, Euclidian universe, and no cosmological evolution, the expectation value of the survey volume is two times the average of the volumes enclosed by the sources (e.g., similar to the argument of the V/V_max test; e.g., Schmidt 1968). Given the Universe geometry and cosmological evolution in the star formation rate, the ratio between the effective survey volume and the mean volume (corresponding to the redshift) of sources is somewhat smaller than 2 (i.e., star formation increases with redshift). In the calculation I assume the FRB rate follows the star formation rate and take into account the star-formation evolutionfrom the compilation based on[https://ned.ipac.caltech.edu/level5/March14/Madau/Madau5.html] Wyder et al. (2005), Schiminovich et al. (2005), Robotham & Driver (2011), and Cucciati et al. (2012). In any case, the effect of evolution is considerably smaller than the uncertainty in the rate estimate. This allows us to calculate theFRB volumetric rate as a function of f_ DM,cosmo. Figure <ref> shows the FRB rate as a function of f_ DM,cosmo.Regardless of f_ DM,cosmo, and assuming cosmological evolution in the FRB rate that follows the star formation rate, I find an FRB volumetric rate ofρ̇_FRB (3.7±2.4) ×10^-5Mpc^-3yr^-1.This estimate takes into account the effect of cosmological time dilation by multiplying the observed rate by (1+<z>), where <z> is the volume-weighted mean redshift of the FRBs. I note that this estimate depends on the unknown luminosity function of FRBs and should be regarded as an order of magnitude estimate. Furthermore, if the FRB emission is beamed then the rate at Equation <ref> is, again, only a lower limit.§.§ FRB rate per persistent sourceAssuming there is a steady state of FRB persistent source densitythat follows the star foramtion rate, and assuming that all FRBs are associated with persistent sources, I divide the lower limit on the FRB rate per unit volume (Eq. <ref>) by the upper limit on the persistent sources space density to derive a lower limit on the rate of FRBs per persistent source.Figure <ref> presents the 95% confidence lower limit on rate of FRB events per persistent source as a function of f_ DM,cosmo.For f_ DM,cosmo=1I find an FRB rate per persistent source of R_FRB,pers0.8 yr^-1. This lower limit on the rate corresponds to FRB events for which the intrinsic luminosity is as high as the FRBs seen by Parkes. §.§ Predictions and implications for observing strategyAn interesting implication is that if all FRBs repeat and are associated with luminous radio sources, then searches for FRBs in nearby galaxies (e.g., M31) using small radio dishes are likely to fail as on average only one in 1000 galaxies hosts a luminous radio source. An observing strategy that is favored by my findings is to monitor for FRBs among the 11 candidates listed in Table <ref>. Figure <ref> shows the predicted mean number of FRB events per persistent source per day that may be detected using a Parkes-like telescope if directed to an FRB-emitter source (i.e., presumably a persistent radio source) at a distance of 108 Mpc. This plot is shown for an FRB cumulative luminosity function ∝ L^-2/3, L^-1, and L^-5/3. The FRB rate per such source is estimated by R_ Parkes,108 Mpc R_ FRB, pers(<d_ Parkes>/108 Mpc)^-2γ. Here R_ FRB, pers is the lower limit on the FRB rate per persistent source (<ref>; Fig. <ref>), <d_ Parkes> is the mean distance of the Parkes detected FRBs (<ref>; i.e., z≈0.7 for f_ DM, cosmo=1), and γ is the assumed power-law index of the FRB cumulative luminosity function.I note that if FRBs emission is beamed in a constant direction then Figure <ref> is correct on average for a population. However, in this case, some persistent radio sources will show no FRB emission.Furthermore, it is important to note that since the FRB 121102 events are not generated by a Poisson process, large deviations from the average expected rate are possible.§ THE NATURE OF THE LUMINOUS RADIO SOURCES IN NEARBY GALAXIESAn intriguing question is what is the origin of the luminous radio sources reported here? It is likely that at least some of of these sources are background objects unrelated to a nearby galaxy. Better estimate of the fraction of background objects requires follow-up observations. However, assuming that at least some of these sources are related to their spatially coincident galaxy, there are still several physical explanations.First, it is possible that these sources are similar in nature to the radio persistent source associated with FRB 121102. This possibility can be further tested using follow-up radio spectral, temporal and interferometric observations.Second, I would like to consider the possibility that some of these sources are just the bright end of the supernova (SN) remnants (SNR) luminosity function, or young SN with considerable circumstellar material that convert their kinetic energy to radiation on short time scales (∼1 yr). This hypothesis can be tested by long-term monitoring of the radio sources, and a search for flux variability. I note that the brightest known radio SNe (e.g., Weiler et al. 2002) are as bright as the FRB 121102 persistent radio source. However, these SNe are variable on time scales of about one year, which is not consistent with the non-detection of variability between the NVSS (when available) and FIRST epochs which are typically several years apart.Another possibility is gamma ray bursts (GRBs). In GRBs the ejecta velocity is much higher and therefore can produce luminous events (e.g., Levinson et al. 2002). In fact, in the past, the non-detection of transient radio sources was used to set an upper limit on the GRB space density, which can be translated to a lower limit on the GRBs beaming factor (Levinson et al. 2002; Gal-Yam et al. 2006).Levinson et al. (2002) estimated the number of GRB radio afterglows in a flux-limited survey. However, our survey is both volume limited and luminosity limited. Therefore an upper limit on the number of expected GRB afterglows in our survey is given by their rate in a volume limited surveyR_GRB 0.2 (d/108Mpc)^3(f_b/1/75)^-1ρ̇_ GRB/0.5Gpc^-3yr^-1yr^-1.Here f_b is the GRB beaming factor (e.g., Gueta, Piran & Waxman 2005), and ρ̇_ GRB is their rate per unit volume.Assuming GRB afterglows can be observed for 100 yr (afterwards their luminosity is equivalent to that of SN remnants), and given the survey completeness, I conclude that there are 1.5GRB afterglows in my sample.In any case, there are two differences between GRB afterglows and the FRB persistent sources: The first is that while the FRB persistent sources are presumably expanding only mildly relativistically (Waxman 2017), GRB afterglows are expected to expand relativistically. This may result in some differences in the angular size of FRB persistent sources vs. GRB radio afterglows and the scintillation induced variability (e.g., different frequency-dependent variability). Furthermore, there may be some differences in the radio spectrum. I suggest that radio follow-up observations are required in order to reveal the nature of the luminous radio sources reported in this paper. § SUMMARYTo summarize, I present a survey aimed at searching for luminous compact radio sources in galaxies with z<0.025.* I find 11 sources with radio luminosity, at 1.4 GHz, ofν L_ν>3×10^37 erg s^-1 (i.e., >10% of the FRB 121102 persistent source luminosity) which are spatially associated with disks or star-forming regions of galaxies. Here I exclude sources that are spatially coincidence with galactic centers. * Given the completeness estimate for the galaxy catalog, and the FIRST survey area, I place an upper limit on the density of luminous persistent sources in the nearby Universe (5×10^-5 Mpc^-3). This upper limit assumes that luminous persistent sources follow the g-band luminosity of galaxies. If FRBs are related to galactic nuclei this limit will be changed to 3×10^-4 Mpc^-3. * Such luminous radio sources are rare – about 10^-3 per L_* galaxy. * Assuming a persistent source life time of t_ age=100 yr, their birth rate is 5×10^-7(t_ age/100 yr)^-1 yr^-1 Mpc^-3. * Assuming all FRBs repeat and are associated with persistent radio sources, I set a lower limit on the FRB rate per persistent source of 0.8 yr^-1. * About 3% of the galaxy-population integrated luminosity is in galaxies fainter than the absolute mag. of the FRB 121102 host (g≈-16.6). This suggests that it is too early to conclude that FRBs prefer dwarf galaxies. * If some of the candidates in Table <ref> are associated with FRBs then a few-days observation with sensitive (i.e., Parkes-like) radio telescopes may reveal FRB events from these sources. 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"authors": [
"Eran O. Ofek"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170727205805",
"title": "A search for FRB 121102-like persistent radio-luminous sources -- Candidates and implications for the FRB rate and searches"
} |
Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, JapanCondensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, JapanSpin nematic phase is a quantum magnetic phase characterized by a quadrupolar order parameter.Since the quadrupole operators are directly coupled to neither the magnetic field nor the neutron,currently, it is an important issue to develop a method for detecting the long-range spin nematic order.In this paper we propose that electron spin resonance (ESR) measurements enable us to detect the long-range spin nematic order.We show that the frequency of the paramagnetic resonance peak in the ESR spectrumis shifted by the ferroquadrupolar order parameter together with other quantities.The ferroquadrupolar order parameter is extractable from the angular dependence of the frequency shift. In contrast, the antiferroquadrupolar order parameter is usually invisible in the frequency shift.Instead, the long-range antiferroquadrupolar order yields a characteristic resonance peak in the ESR spectrum,which we call a magnon-pair resonance peak. This resonance corresponds to the excitation of the bound magnon pair at the wave vector k= 0.Reflecting the condensation of bound magnon pairs, the field dependence of the magnon-pair resonance frequencyshows a singular upturn at the saturation field.Moreover, the intensity of the magnon-pair resonance peak shows a characteristic angular dependenceand it vanishes when the magnetic field is parallel to one of the axes that diagonalize the weak anisotropic interactions.We confirm these general properties of the magnon-pair resonance peak in the spin nematic phaseby studying an S=1 bilinear-biquadratic model on thesquare lattice in the linear flavor-wave approximation.In addition, we argue applications to the S=1/2 frustrated ferromagnets andalso the S=1/2 orthogonal dimer spin system SrCu_2(BO_3)_2,both of which are candidate materials of spin nematics.Our theory for the antiferroquadrupolar ordered phaseis consistent with many features of the magnon-pair resonance peak experimentally observed in thelow-magnetization regime of SrCu_2(BO_3)_2.Electron spin resonance for the detection of long-range spin nematic order Tsutomu Momoi December 30, 2023 ==========================================================================§ INTRODUCTION Spin nematic phase is a hidden ordered phase of quantum magnets where the spin rotation symmetry is spontaneously broken but, different from the ferromagnetic and the antiferromagnetic phases, the time reversal symmetry is kept intact. The spin nematic phase is characterized by a quadrupolar order parameter made of a symmetric pair of electron spins <cit.>. The emergence of the spin nematic phase requires the absence of the spontaneous dipolar orders, implying interplay and frustration of spin-spin interactions behind the order. Until today, much effort has been made to explore the spin nematic phase in S=1/2 frustrated ferromagnets both theoretically <cit.> and experimentally <cit.>, in the S=1/2 orthogonal dimer spin system <cit.>, and also in S≥ 1 spin systems with biquadratic interactions <cit.>. Recently, the research field of the spin nematic order expands to the field of an iron pnictide superconductor, FeSe <cit.>. In the current situation surrounding researches of the spin nematic phase, one of the most important problems is to develop a method for detecting the spin nematic order in an experimentally feasible way. The difficulty in the problem is that the quadrupole operators are directly coupled to neither the magnetic field nor the neutron. Several theoretical proposals were recently made by studying theories of the nuclear magnetic resonance (NMR) <cit.>, inelastic neutron scattering <cit.>, inelastic light scattering <cit.>, resonant inelastic x-ray scattering <cit.>, and electron spin resonance (ESR) <cit.>. As pointed out in Ref. <cit.>, various quantities related to ESR are naturally coupled to quadrupole operators and thus ESR is a promising way for detecting the hidden spin nematic order. In ESR experiments, bound triplon-pair modes were observed in the spin gapped phase of the orthogonal dimer spin system SrCu_2(BO_3)_2 <cit.> and a bound magnon-pair mode was also in the fully polarized phase of Sr_2CoGe_2O_7 <cit.>. Bound magnon pairs can give the instability to the spin nematic ordering when they close the energy gap <cit.>. The behavior of these bound pair modes in ESR spectrum is however not yet understood inside the spin nematic phase from both the theoretical and experimental sides. In view of the current situations, an ESR theory for identification of the quadrupolar order in the spin nematic phase is called for.In this paper, we theoretically study ESR in the spin nematic phase to elucidate how to identify the quadrupolar order from the ESR spectrum. We propose two methods of detecting spin nematic orders. In the first part of the paper, we demonstrate for quite a generic model that ESR measurements enable us to extract the ferroquadrupolar (FQ) order parameter (i.e. the spin nematic order parameter developed at the wave vector k= 0) from the frequency shift of the electron paramagnetic resonance (EPR) peak in the ESR spectrum, as one of the authors briefly mentioned in Ref. <cit.>. Incidentally the antiferroquadrupolar (AFQ) order parameter is not detectable in this method. In the latter part of the paper, we propose a complementary ESR measurement suitable for identification of the AFQ order, taking an example of an S=1 spin model. In this method, we focus on resonance of bound magnon-pair excitations, which characteristically appear in quadrupolar order phases signaling the condensation of bound magnon pairs. The corresponding resonance peak, which we call the magnon-pair resonance peak, appears at a finite frequency in the ESR spectrum in the AFQ phase. This resonance peak can be clearly distinguished from others since the peak intensity has a characteristic angular dependence and the peak frequency also has a distinctive field dependence. A typical field dependence of the magnon-pair resonance frequency is shown as the solid curve in Fig. <ref> for the AFQ phase in an S=1 bilinear-biquadratic model. Having the two methods of ESR for the FQ and the AFQ orders, we can characterize the FQ and the AFQ phases clearly. We can apply our theory to S=1/2 candidate materials of spin nematics, such as S=1/2 frustrated ferromagnets and S=1/2 orthogonal dimer spin compound SrCu_2(BO_3)_2. Our theory for the AFQ ordered phase is consistent with many features of the experimentally observed magnon-pair resonance peak <cit.> in SrCu_2(BO_3)_2, which suggests the existence of a spin nematic order.This paper is organized as follows. In Sec. <ref>, we review briefly an important identity [Eq. (<ref>)] that we rely on in this paper. Using the identity, we show in Sec. <ref> that the EPR frequency is shifted by the FQ order parameter depending on the direction of the sample. We use a little trick in order to extract the FQ order parameter from the frequency shift. Since the EPR frequency is usually insensitive to the AFQ order parameter, in Sec. <ref>, we focus on a low-energy bosonic excitation that qualifies as evidence of the presence of the long-range AFQ order. In Sec. <ref>, we briefly explain the concept of how to detect the boson in ESR experiments in general. To flesh out the general discussion, we take an example of an S=1 bilinear-biquadratic model on the square lattice (Secs. <ref> and <ref>) and study its ESR with the aid of the linear flavor-wave theory. The linear flavor-wave theory in the fully polarized phase (Sec. <ref>) and in the AFQ phase (Sec. <ref>) shows that the boson corresponding to the bound magnon-pair excitation yields a resonance peak in the ESR spectrum in addition to the EPR one. The magnon-pair resonance is closely investigated in the fully polarized phase in Sec. <ref>. Section <ref> is devoted to the investigation of additional resonances in the AFQ phase, where we find the magnon-pair resonance and another unpaired magnon resonance. Some related discussions are addressed in Sec. <ref>. Section <ref> contains applications to the S=1/2 frustrated ferromagnets and the S=1/2 orthogonal dimer spin system. Finally, we summarize the paper in Sec. <ref>.§ FRAMEWORK Here we describe a generic model that we deal with in the paper and an important identity that we rely on throughout our discussions. We consider quantum spin systems described by the following Hamiltonian:ℋ = ℋ_ SU(2) -H S^z + ℋ',where ℋ_ SU(2) denotes interactions that respect the SU(2) rotation symmetry of the spin, the second term is the Zeeman energy, S^z=∑_i S_i^z is the z component of the total spin S = ∑_iS_i, and ℋ' is an anisotropic interaction that breaks weakly the SU(2) symmetry of ℋ_ SU(2). For simplicity, we take ħ=k_B=a_0=1 hereafter, where a_0 is the lattice constant. We regard ℋ' as a perturbation to the Hamiltonianℋ_0 = ℋ_ SU(2) - HS^z.We emphasize that, in this paper, we consider the cases where the spin nematic phase emerges in the unperturbed system with the Hamiltonian (<ref>) and the perturbation ℋ' affects neither the existence of the quadrupolar order nor the magnitude of the quadrupolar order parameter. On the other hand, when we consider ESR of quantum spin systems, however small ℋ' may be, we must take the anisotropic interaction ℋ' into account.The ESR absorption spectrum is given by the imaginary part of the retarded Green's function of the total spin. The ESR spectrum I(ω) in the Faraday configuration is related to 𝒢^R_S^+S^-(ω), where S^± = S^x ± i S^y are the ladder operators of the total spin, 𝒢^R_O_1O_2(ω) is the retarded Green's function 𝒢^R_O_1O_2(ω)=-i∫_0^∞ dt e^iω t⟨[O_1(t), O_2(0)]|$⟩.In fact, when the applied electromagnetic wave is circularly polarized, the ESR spectrumI(ω)is given by <cit.> I(ω) =ω H_R^2/8[-𝒢^R_S^+S^-(ω)]within the linear response theory, whereH_Ris the amplitude of the external oscillating magnetic field. On the other hand, when the applied electromagnetic wave is completely unpolarized, the ESR spectrum is given by (appendix <ref>)I(ω) = ω H_R^2/8[ -𝒢^R_S^+S^-(ω) - 𝒢^R_S^-S^+(ω) ].The result (<ref>) is invariant under any rotation around thezaxis, the direction along which the electromagnetic wave propagates. Since𝒢^R_S^-S^+(ω) = -𝒢^R_S^+S^-(-ω), the ESR spectrum (<ref>) is fully determined from the retarded Green's function,𝒢^R_S^+S^-(ω). Here, we note that the frequencyωis positive since it represents the energy of the photon. In the remainder of the paper, we deal with the ESR spectrum of Eq. (<ref>) because inclusion of𝒢^R_S^-S^+(ω)has no impact on the conclusions of this paper. The Green's function𝒢^R_S^+S^-(ω)satisfies the identity <cit.> 𝒢^R_S^+S^-(ω) =2⟨S^z|⟩/ω - H - ⟨[𝒜, S^-]|⟩/(ω-H)^2 + 1/(ω-H)^2𝒢^R_𝒜𝒜^†(ω),whereω-His shorthand forω-H + i0and𝒜is an operator defined by𝒜 = [ℋ', S^+].Equation (<ref>) is an exact relation that simply results from the equation of motion ofS^±<cit.>. Now it is clear that if there was no anisotropy, i.e.ℋ'=0, the ESR spectrumI(ω)would contain only the single EPR peak,I(ω)=I_ EPR(ω) = π H_R^2H/4⟨S^z|δ⟩(ω-H).The anisotropic interactionℋ', however small it may be, is always present in the materials, giving rise to the finite linewidth to the EPR peak (<ref>). It also yields other resonance peaks. The𝒜operator determines qualitative and quantitative properties of the ESR spectrum. In particular, Eq. (<ref>) shows that if there is an additional resonance peak well isolated from the EPR one, it must come from the retarded Green's function𝒢^R_𝒜𝒜^†(ω). In fact, many interesting resonance peaks are found in ESR experiments and explained on the basis of the identity (<ref>) <cit.>.Let us split the ESR spectrum (<ref>) into two parts: the EPR peakI_EPR(ω)and the other peaksI'(ω)well isolated from the former, if they exist,I(ω) = I_ EPR(ω) + I'(ω).In the leading order of the perturbation, the additional peakI'(ω)is given byI'(ω)= ω H_R^2/8(ω-H)^2[ -G^R_𝒜𝒜^†(ω)],where the full retarded Green's function𝒢^R_𝒜𝒜^†(ω)in Eq. (<ref>)is replaced tothe unperturbed retarded Green's functionG^R_𝒜𝒜^†(ω)of the unperturbed system (<ref>). As far asI'(ω)is concerned, measuring ESR is equivalent to measuring the resonance of the complex operator𝒜given in Eq. (<ref>). We emphasize that the resonance peak position ofI'(ω)is determined fully in the unperturbed system. While the intensity ofI'(ω)is proportional to the square of the small coupling constant ofℋ', the frequency dependence is fully determined from the unperturbed Green's function. § FREQUENCY SHIFT BY THE FQ ORDER In this section, we discuss effects of the FQ order on the EPR absorption peak, showing that the FQ order parameter can be extracted experimentally from the EPR frequency shift. §.§ Generic caseWe start with a general discussion. Let us consider a perturbative expansion of the identity (<ref>) up to the first order of the perturbation,𝒢^R_S^+S^-(ω) ≃2⟨S^z|⟩/ω- H( 1- ⟨[𝒜, S^-]|_⟩0/2⟨S^z|_⟩01/ω-H),where⟨O|_⟩0denotes the average taken in the unperturbed system (<ref>)[We note that, when a certain symmetry is broken spontaneously in the unperturbed system and it is also broken by the anisotropy in the full Hamiltonian ℋ, one must choose a proper direction of the infinitesimal auxiliary symmetry breaking field in the unperturbed system so that the ground state is smoothly deformed by imposing weak perturbation.]. The approximation (<ref>) implies that the resonance frequency of the EPR peak is shifted fromω_r=Htoω_r ≃ H - ⟨[𝒜, S^-]|_⟩0/2⟨S^z|_⟩0.The formula (<ref>) is valid as long as the anisotropic interaction is perturbative, the paramagnetic resonance peak is not split into plural peaks by the perturbationℋ'<cit.>, and⟨[𝒜, S^-]|_⟩0is real. An example of the splitting is found in one-dimensional quantum antiferromagnets with a uniform Dzyaloshinskii-Moriya (DM) interaction <cit.>. While it is nontrivial whether⟨[𝒜, S^-]|_⟩0is real in the presence of the FQ order, we show in Sec. <ref> that it is indeed real on the basis of a specific model. In this paper we denote the frequency shift byδω_r=ω_r-H. The formula (<ref>) is rephrased asδω_r = - ⟨[𝒜, S^-]|_⟩0/2⟨S^z|_⟩0. A key observation is to notice a fact that a quadratic interactionℋ'usually makes the commutator[𝒜, S^-]quadratic in the formula (<ref>). To see it, we take a generic example,ℋ' = ∑_p=x,y,z∑_⟨i,j|_⟩nδ^(n)_p S_i^p S_j^p,whereS_j^p(p=x,y,z) denotes thep-component of spinSoperators onjth site,⟨i,j|_⟩nrepresents a pair of thenth neighbor sitesiandj. In particular,⟨i,j|_⟩0meansi=j. The interaction (<ref>) covers quite a wide range of anisotropic spin-spin interactions found in quantum magnets. In the following discussions, we choose the most dominant anisotropic interaction. For quantum spin systems withS=1/2, the most likely anisotropic interaction is the anisotropic exchange interaction on the nearest-neighbor bond (n=1). For spin systems withS≥1, the most likely one is the single-ion anisotropy (n=0). In principle, the anisotropic exchange interaction on the nearest-neighbor bond can be the dominant one even forS≥1. Because these two types of anisotropy give rise to resemblant frequency shifts, it is difficult to judge whether the dominant anisotropic interaction lives in single-spin sites or in bonds connecting two spin sites. The resemblance was demonstrated in theS=1Heisenberg antiferromagnetic chain <cit.>. We assume the following inequalities|δ^(n)_z|>|δ^(n)_x|≥ |δ^(n)_y|=0,without loss of generality, for the most dominant anisotropy. From Eq. (<ref>), the anisotropic interaction (<ref>) leads to the frequency shiftδω_r =- 2δ^(n)_z - δ^(n)_x/2⟨S^z|_⟩0∑_⟨i,j|_⟩n⟨(3S_i^zS_j^z- S_i · S_j)|_⟩0 -δ^(n)_x/2⟨S^z|_⟩0∑_⟨i,j|_⟩n⟨S_i^- S_j^-|_⟩0.This equation contains the operatorS_i^-S_j^-, which creates or annihilates a pair of magnons. Since bound magnon pairs condense in the spin nematic phase,∑_⟨i,j|_⟩n⟨S_i^-S_j^-|_⟩0is proportional to the condensed amount of bound magnon pairs at the wave vectork=0, that is, the FQ order parameter.The average⟨(3S_i^zS_j^z - S_i ·S_j)|_⟩0on the first line of Eq. (<ref>) is also nonzero simply because of the magnetic field along thezdirection. In fact, the frequency shift in several one-dimensional quantum magnets, where the long-range spin nematic order is absent, was explained on the basis of Eq. (<ref>) without the second line <cit.>.Let us introduce a little trick in order to get rid of the contribution unrelated to the spin nematic order. Here, we rotate the material around theyaxis by an angleθ. The rotation only affects the form of the anisotropic interaction (<ref>) asℋ' = ∑_⟨i,j|_⟩n[ (δ^(n)_z cos^2θ + δ^(n)_xsin^2θ) S_i^zS_j^z+ (δ^(n)_z sin^2θ + δ^(n)_x cos^2θ) S_i^xS_j^x - (δ^(n)_z-δ^(n)_x) sinθcosθ (S_i^zS_j^x + S_i^xS_j^z)].The interactions in the unperturbed systemℋ_0is invariant under the rotation. The rotated anisotropic interaction leads to the frequency shift,δω_r (θ) = (δ^(n)_z-δ^(n)_x)(3cos^2θ -1) +δ^(n)_x/2⟨S^z|_⟩0∑_⟨i,j|_⟩n⟨(3S_i^zS_j^z- S_i · S_j)|_⟩0 + (δ^(n)_z-δ^(n)_x)sin^2θ +δ^(n)_x/2⟨S^z|_⟩0∑_⟨i,j|_⟩n⟨S_i^-S_j^-|_⟩0-(δ^(n)_z-δ^(n)_x)sinθcosθ/2⟨S^z|_⟩0∑_⟨i,j|_⟩n⟨{2(S_i^zS_j^++S_i^+S_j^z) + 3(S_i^zS_j^-+S_i^-S_j^z)}|_⟩0,where we added the argument toδω_ron the left hand side to clarify that the frequency shift is a function ofθ. If the rotation does not affect the direction where the FQ order grows, the angular dependence comes only out of the coefficients of those averages. The validity of the assumption is to be confirmed in the next subsection for a specific example.The EPR frequency shift (<ref>) consists of three parts: (i) the uniaxial part coupled to the average⟨(3S_i^zS_j^z-S_i ·S_j)|_⟩0, (ii) the FQ order part coupled to the FQ order parameter⟨S_i^-S_j^-|_⟩0, and (iii) the off-diagonal part coupled to⟨{2(S_i^zS_j^++S_i^+S_j^z) + 3(S_i^zS_j^-+S_i^-S_j^z)}|_⟩0. Note again that the operator3S_i^zS_j^z-S_i ·S_jis not an FQ order parameter although it is a quadrupole operator. That operator has a nonzero expectation value when thezaxis is inequivalent to thexyplane by applying the magnetic field along thezaxis. Looking at the angular dependence of the frequency shift (<ref>), we can separate the FQ order parameter from the other terms as follows. Let us focus onδω_r(0)andδω_r(π/2)because the second line of Eq. (<ref>) vanishes atθ=0 π/2. Those frequency shifts are rewritten asδω_r(0) + δω_r(π/2)= δ^(n)_z+δ^(n)_x/2⟨S^z|_⟩0(∑_⟨i,j|_⟩n⟨(3S_i^zS_j^z -S_i · S_j)|_⟩0 + ∑_⟨i,j|_⟩n⟨S_i^-S_j^-|_⟩0),δω_r(0) - δω_r(π/2)= δ^(n)_z-δ^(n)_x/2⟨S^z|_⟩0(3∑_⟨i,j|_⟩n⟨(3S_i^zS_j^z -S_i · S_j)|_⟩0 - ∑_⟨i,j|_⟩n⟨S_i^-S_j^-|_⟩0).Sinceδ^(n)_p(p=x,z) are known parameters from ESR measurements at high enough temperatures out of the FQ phase and the magnetization⟨S^z|_⟩0is also known independently of the ESR experiments. Thus, combiningδω_r(0)andδω_r(π/2), we can obtain the FQ order parameter∑_⟨i,j|_⟩n⟨S_i^-S_j^-|_⟩0. In the uniaxially anisotropic case ofδ^(n)_z≠0andδ^(n)_x=δ^(n)_y=0, the procedure is simplified thanks to the following simple relation,1/2δω_r(0) + δω_r(π/2) = δ^(n)_z/2⟨S^z|_⟩0∑_j ⟨S_i^-S_j^-|_⟩0.§.§ Specific case The important remaining tasks in this section are to confirm that the frequency shift (<ref>) is real and that the averages in Eq. (<ref>) is invariant under the rotation. To do so, we take a specific example of anS=1bilinear-biquadratic model with the single-ion anisotropy,ℋ = ∑_n=1,2∑_⟨i,j|_⟩n[ J_nS_i · S_j + K_n ( S_i · S_j)^2] - H(S^z cosθ + S^x sinθ) + D ∑_j (S_j^z)^2 + E∑_j {(S_j^x)^2-(S_j^y)^2},on the square lattice. Rotating the system abouty-axis by angleθ, we can redefine the Hamiltonian asℋ =∑_n=1,2∑_⟨i,j|_⟩n[ J_nS_i · S_j + K_n ( S_i · S_j)^2]- HS^z+ ∑_j [(Dcos^2θ + Esin^2θ) (S_j^z)^2+ (Dsin^2θ + E cos^2θ) (S_j^x)^2 - E (S_j^y)^2 - (D-E)sinθcosθ (S_j^zS_j^x+S_j^xS_j^z)].The latter representation is easier to handle. Here we assume, without loss of generality, that bothDandEhave the same sign. Note that the form of the anisotropic interaction of Eq. (<ref>) is a special case of Eq. (<ref>). In the language ofδ^(n)_pin Eq. (<ref>), the parametersDandEcorresponds to them asδ^(0)_z = D+E, δ^(0)_x = 2E.Note that we also imposed the condition (<ref>). We assume thatJ_n>0andK_n<0forn=1,2and that the single-ion anisotropy can be seen as a perturbation.It was shown at the mean-field level that the ground state of the unperturbed model is in the FQ phase whenboth|K_1|/J_1and|K_2|/J_2are large enough <cit.>. The mean-field FQ ground state|ψ_0⟩is represented as a product state,|ψ_0⟩ = ∏_j |ϕ_0(θ_H, φ)⟩_j,of the local state|ϕ_0(θ_H, φ)⟩_j,|ϕ_0(θ_H, φ)⟩_j = i (e^iφcosθ_H |1⟩_j - e^-iφsinθ_H |-1⟩_j),where|m⟩_jis the eigenstate ofS_j^zwith the eigenvaluem. The anglesθ_Handφ, which are real, are determined so that the ground-state energy is minimized. The ground state has the FQ order∑_j⟨ψ_0 | (S_j^-)^2 |ψ_0|=⟩ -N e^i2φsin 2θ_H,which is in general complex. Here,Nis the number of spins. In the following, we show thate^i2φis real in the presence of the single-ion anisotropy as long as it is perturbative. To discuss the ground state energy of the mean-field FQ state (<ref>), we introduce the quadrupole operatorsQ_j =[Q_j^x^2-y^2; Q_j^3z^2-r^2; Q_j^xy; Q_j^yz; Q_j^zx ]=[ (S_j^x)^2 - (S_j^y)^2; [2(S_j^z)^2 -(S_j^x)^2 - (S_j^y)^2 ]/√(3); S_j^x S_j^y + S_j^y S_j^x; S_j^y S_j^z + S_j^z S_j^y; S_j^z S_j^x + S_j^x S_j^z ]and rewrite the Hamiltonian (<ref>) asℋ = ∑_⟨i,j|_⟩n[ J_n/2( S_i · S_j +Q_i · Q_j) + J_n-K_n/2 ( S_i · S_j -Q_i · Q_j)]-HS^z + 2ND/3 + D(3cos^2θ-1)+Esin^2θ/2√(3) Q^3z^2-r^2 + Dsin^2θ + E(cos^2θ+1)/2 Q^x^2-y^2 - (D-E)sinθcosθ Q^zxwithQ ≡∑_j Q_j. Writing the local state as|ϕ(θ_H, φ)⟩_j=e_1|1⟩_j + e_0 |0⟩_j + e_-1|-1⟩_j, we can represent the ground-state energyE_FQ = ⟨ψ_0|ℋ|ψ_0|$⟩ as <cit.> E_ FQ/N = 2(J_1+J_2)| e ·e̅|^2- 2(J_1+J_2-K_1-K_2) |2e_1e̅_-1 - (e_0)^2|^2- H (|e_1|^2-|e_-1|^2) + 2D/3+ D(3cos^2θ-1)+Esin^2θ/6(|e_1|^2+|e_-1|^2-2|e_0|^2)+ Dsin^2θ + E(cos^2θ + 1)/2√(2)(e_-1e̅_1+ e_1 e̅_-1)-(D-E)sinθcosθ(e_1e̅_0 + e_0 e̅_1 - e_0 e̅_-1 - e_-1e̅_0), where e̅_a is the complex conjugate of e_a (a=1,0,-1). Plugging e_1=ie^iφcosθ_H, e_0 = 0, and e_-1=-ie^-iφsinθ_H into Eq. (<ref>), we obtainE_ FQ/N = 2(J_1+J_2) -2(J_1+J_2-K_1-K_2)sin^2 2θ_H -H cos 2θ_H + 3D(cos^2θ + 1)+Esin^2θ/6 +Dsin^2θ + E(cos^2θ+1)/√(2)cos2φsin 2θ_H.Let us determine θ_H and φ in the spirit of the perturbation theory. First, when D=E=0, the ground-state energy becomesE_ FQ^0/N = 2(K_1+K_2)- H^2/2H_ sat + 2(J_1+J_2-K_1-K_2)(cos 2θ_H - H/H_ sat)^2with the saturation field H_ sat = 4(J_1+J_2-K_1-K_2). Since K_n<0<J_n for all n=1,2, the angle θ_H is determined to beθ_H^0 = 1/2cos^-1(H/H_ sat)in 0≤θ_H^0≤π/2 when 0≤ H<H_ sat and θ_H^0=0 when H_ sat<H. We restrict ourselves to the former case where the system is in the FQ phase. The ground state |ψ_0⟩ has the FQ order,∑_j ⟨ψ_0 |(S_j^-)^2|ψ_0|=⟩ - Ne^i2φ√(1-(H/H_ sat)^2),where the angle φ determines the direction of quadrupolar directors. Next, we treat the single-ion anisotropy perturbatively. Weak anisotropies D and E do not modify the solution θ_H = θ_H^0 at the lowest order of the perturbation. The anisotropy, however small it may be, serve as a symmetry breaking field and determines φ. From the energy in Eq. (<ref>) the angle φ is chosen, for arbitrary angle θ, as cos 2φ = 1 when both D and E are negative, and cos 2φ = -1 when both are positive. In any case, since sin 2φ=0, we thus find e^i2φ = cos 2φ∈ℝ and that the FQ order parameter ∑_j ⟨(S_j^-)^2|$⟩ is real, satisfying the relation∑_j ⟨(S_j^-)^2|_⟩0=∑_j ⟨Q^x^2-y^2_j|_⟩0,for the model (<ref>) at low enough temperatures. The FQ order parameter is thus given by⟨Q^x^2-y^2|_⟩0≡∑_j ⟨Q^x^2-y^2_j|_⟩0 = - N cos 2φ√(1-(H/H_ sat)^2). At the same time, we can also conclude that the obtained solutions ofφandθ_Hare independent of the angleθof the rotation. That is, the FQ order parameter is invariant under the rotation around theyaxis. Also, the averages∑_j ⟨(3(S_j^z)^2-2)|_⟩0=Nand∑_j ⟨2(S_j^zS_j^++S_j^+S_j^z)+3(S_j^zS_j^-+S_j^-S_j^z)|_⟩0=0turn out to be independent ofθ. In particular, the latter average vanishes because it involves the creation and the annihilation of the gapped unpaired magnon. The assumption made in deriving Eq. (<ref>) in the previous subsection is thus justified.Finally, the frequency shift (<ref>) of the model (<ref>) at zero temperature becomesδω_r ≃(D-E)(3cos^2θ-1)+2E/2⟨S^z|_⟩0/N - (D-E)sin^2θ + 2E/2⟨S^z|_⟩0⟨Q^x^2-y^2|_⟩0.We emphasize that the value of the FQ order parameter is determined independent of perturbative anisotropiesDandE. Those anisotropies merely fix the angleφ, which was spontaneously determined in the unperturbed system.Although we do not show that the frequency shift is real for generic cases, it is reasonable to expect that the result holds true generally as long as the mean field approximation is applicable and the perturbative anisotropic interaction is given by either the single-ion anisotropy or the anisotropic exchange interaction (<ref>). §.§ Experimental applications For the analysis of experimental data, we comment on the comparison between the frequency shift and the linewidth of the EPR peak. Should the linewidth be larger than the magnitude of the EPR frequency shift, the frequency shift would be undetectable. However, this is not the case as long as the anisotropic interaction is small enough compared to the isotropic ones. According to the generic perturbation theory (<ref>), the EPR frequency shift is of the first order of the perturbative anisotropic interaction, whereas the linewidth is of the second order because it is proportional toImG^R_𝒜𝒜^†(ω=H)<cit.>. Therefore, given the long-range FQ order is well developed, the linewidth is small enough not to mask the frequency shift (<ref>).We conclude the section, referring to chromium spinel oxides. Chromium spinelsACr_2O_4(A=Zn, Cd, Hg) are considered as anS=3/2pyrochlore Heisenberg antiferromagnet with biquadratic interactions <cit.>. Recently it was pointed out that those chromium spinels can have the FQ phase next to the fully polarized phase <cit.>. Indeed, high-field measurements discovered the presence of a classically unexpected phase just below the fully polarized phase <cit.>. We emphasize that the result (<ref>) is applicable to those interesting compounds. In fact, in deriving Eq. (<ref>), we only specified the anisotropic interactionℋ'and specified neither the spin quantum number nor the form of the SU(2)-symmetric interaction in the Hamiltonian (<ref>). § EMERGENCE OF ANOTHER PEAK BY THE AFQ ORDER: MAGNON-PAIR RESONANCE The frequency shift (<ref>) is insensitive to the AFQ order since the AFQ order parameter has alternating sign depending on the position. In this section, we develop an alternative way for detecting the AFQ order, studying the additional absorptionI'(ω)in ESR spectrum (<ref>). Here, the point is that the𝒜operator creates a magnon pair excitation. §.§ Concept Let us explain the concept of our method, taking the following example of the anisotropic interaction:ℋ' =δ' ∑_⟨i,j|_⟩1 (S_i^x S_j^z + S_i^z S_j^x).Here, we do not specify the precise form ofℋ_SU(2). We just assume that the unperturbed system (<ref>) has the AFQ ground state. The𝒜operator for Eq. (<ref>) is given by𝒜 = δ' ∑_⟨i,j|_⟩1 (S_i^+S_j^+ -3S_i^z S_j^z +S_i · S_j).The first term of Eq. (<ref>) creates or annihilates the magnon pair formed on the nearest-neighbor bond. This shows that dynamics of the bound magnon pair is directly observable in ESR experiments throungh the relation (<ref>). In the spin nematic phase with the AFQ order, the bound magnon pair can acquire an excitation gap, sayΔ, atk=0, though it becomes gapless at the wave vector of the AFQ ordered ground state. If so, the ESR spectrum in the spin nematic ordered phase contains a sharp resonance peak whose frequency corresponds to the excitation gap of the bound magnon pair atk=0. As shown in Eq. (<ref>), the ESR spectrum (<ref>) contains a contribution of the𝒜operator through the imaginary part of the retarded Green's function of𝒜. Since𝒜involves the creation and annihilation operators of the bound magnon pair, the ESR spectrum will have a resonance peak atω=Δ, i.e.I'(ω)∝𝒜_MPRδ(ω- Δ). We call this peak the magnon-pair resonance peak. In Secs. <ref> and <ref>, we confirm that the ESR spectrum indeed yields the sharp magnon-pair resonance by taking an example of anS=1spin model on the square lattice. The interaction (<ref>) results from the rotation of a spin anisotropy, e.g.δ^(z)∑_⟨i,j ⟩_1S_i^z S_j^z,around theyaxis as we did in Eq. (<ref>). For a general rotation angleθ, the resultant magnon pairing operator of the rotated interaction (<ref>) shows the angular dependence ofsinθcosθ. This gives a characteristic angle dependence in the intensity𝒜_MPRof the magnon-pair resonance peak. [See Eq. (<ref>).] This angular dependence was a key to characterize qualitatively the quadrupolar liquid state by using ESR <cit.>. §.§ Model To flesh out the general discussion described in Sec. <ref>, we take an example of anS=1spin model on the square lattice given by the following unperturbed Hamiltonianℋ_0= ∑_⟨i,j|_⟩1 J_11[ S_i · S_j + ( S_i · S_j)^2] - ∑_⟨i,j|_⟩2{ J_12[ S_i · S_j + ( S_i · S_j)^2]+J_22( S_i · S_j)^2} -HS^zand the perturbative single-ion anisotropyℋ'= ∑_i[(Dcos^2θ + E sin^2 θ) (S_i^z)^2 + (Dsin^2 θ + E cos^2 θ) (S_i^x)^2 - E(S_i^y)^2- (D-E)sinθcosθ (S_i^z S_i^x + S_i^x S_i^z)].This form of the anisotropy is the same as in Eq. (<ref>), which is obtained by rotating the common formD(S_i^z)^2 + E {(S_i^x)^2 - (S_i^y)^2}abouty-axis by angleθ. We assume that the couplingsJ_11,J_12, andJ_22are all positive. The bilinear-biquadratic interaction in Eq. (<ref>) has the SU(3) symmetry whenJ_22=0and its low-energy behavior was closely investigated in Ref. <cit.>. It is easy to confirm that the unperturbed Hamiltonian (<ref>) is written asℋ_0 = J_11/2∑_⟨i,j|_⟩1( Q_i · Q_j +S_i · S_j) - ∑_⟨i,j|_⟩2[ J_12/2 ( Q_i · Q_j +S_i · S_j) +J_22/2 ( Q_i · Q_j -S_i · S_j)] - HS^z,by using the quadrupole operatorsQ_jof Eq. (<ref>).We employ the model (<ref>) for the following reasons. First, the model (<ref>) does not suffer from a known technical problem of the linear flavor-wave theory, that is, violation <cit.> of the frequency sum rule <cit.> of the dynamical structure factor ∫_0^∞ dωωS^αα(k=0,ω)=0, which holds true for the unperturbed Hamiltonian (<ref>)at zero magnetic field H=0. Here,S^αα(k=0,ω)=-Im G^R_S^αS^α(ω)/N. The linear flavor-wave theory does not always satisfy the exact sum rule but can be recovered by including three- and four-particle interactions <cit.>. It is the great advantage of the model (<ref>) that we can omit such a complicated procedure. This sum rule at zero magnetic field is related to the exact result of the EPR frequencyω=Hin the ESR spectrum (<ref>). We will briefly comment about an example of the violation of the exact result (<ref>) in the linear flavor-wave theory in the last part of Sec. <ref>. Second, the model (<ref>) exhibits the AFQ phase in quite a wide field range up to the saturation field [H_c^AFQin Eq. (<ref>)]. We will come back to those points in Secs. <ref> and <ref>. Last but not least, theS=1model is closely related to anS=1/2frustrated ferromagnetic model on a square lattice <cit.>. We emphasize that the results about the AFQ phase obtained in the present paper also hold true for theS=1/2model. Here, the single-ion anisotropy (<ref>) in theS=1model is translated into anisotropic exchange interactions in theS=1/2model.See Sec. <ref> for further discussions. Following the general discussion in Sec. <ref>, we study the additional peakI'(ω)given by Eq. (<ref>). Here we only need to derive the unperturbed retarded Green's functionG^R_𝒜𝒜^†(ω)of the operator𝒜=[ℋ',S^+]. For that purpose, we use the linear flavor-wave theory <cit.> for the unperturbed system (<ref>) at zero temperature.§.§ Mean-field ground state As well as the spin-wave theory, the flavor-wave theory is developed by taking into account quantum fluctuations around the ordered state. We need to start with deriving the AFQ ground state of the bilinear-biquadratic model (<ref>) in a site-decoupled semi-classical approximation. Let us denote its mean-field ground state by|ψ_0⟩. Under a strong enough magnetic field,|ψ_0⟩is in the fully polarized phase and exactly given by|ψ_0⟩ = ∏_j i |1⟩_j.As the magnetic field is decreased, the ground state of the model (<ref>) enters into a partially polarized phase. The mean-field ground state of the partially polarized phase has AFQ order, which is given in the form|ψ_0⟩= ∏_j |ϕ_0(e^i k_M · r_jθ_H, φ)⟩_jwith|ϕ_0(θ_H, φ)⟩_j = i (e^iφcosθ_H |1⟩_j - e^-iφsinθ_H |-1⟩_j)for0 ≤θ_H ≤π/2and0 ≤φ< π. Herek_M = (π, π)is the wave vector where the AFQ order grows,r_jspecifies the location of the spinS_j, ande^ik_M ·r_j=±1gives the staggered sign depending on the sublattice. The factorsinθ_Hrepresents the fraction of the condensed bound magnon pair. The state|ϕ_0(θ_H,φ)⟩_jis also given by an SU(3) rotation of the polarized state,|ϕ_0(θ_H, φ)⟩_j=i exp(i φ S_j^z)exp(i θ_H Q_j^xy)|1⟩_j.As we did in Sec. <ref>, we determineθ_Hfor the unperturbed system and then studyφdependence perturbatively. For the unperturbed Hamiltonianℋ_0, the mean-field ground-state energy of the AFQ state is given by <cit.> E^0_ AFQ/N =-2(J_12+J_22)+ 2(J_11+J_22) cos^2 2θ_H- H cos 2θ_H,whereNis the number of sites. This energy is minimized atθ_H=0forH≥H_c^AFQ, whereH_c^AFQdenotes the saturation fieldH_c^ AFQ = 4(J_11+J_22),and atθ_H =1/2cos^-1(H/H_c^ AFQ)forH<H_c^AFQ. In the AFQ phase, the saturation fieldH_c^AFQis the critical field where bound magnon pairs start condensing when the field is decreased. Another mean-field solution is an antiferromagnetically ordered state given by a staggered SU(2) rotation of the polarized state|ϕ_0(e^i k_M · r_jθ_H,φ)⟩_j=i exp(i φ S_j^z) exp( i e^i k_M · r_jθ_H S_j^y)|1⟩_j.In this solution, the saturation field isH_c^ AFM = 2(6J_11+J_22),where the single magnon closes the energy gap. WhenHis decreased, ifH_c^AFQ> H_c^AFM, bound magnon pairs condense below the saturation field before the unpaired magnon does. The comparison betweenH_c^AFQandH_c^AFMshows that the AFQ phase is realized for4J_11 < J_22at least near the saturation. In Sec. <ref>, we see that inclusion of the quantum fluctuation relaxes the condition (<ref>) toJ_11 < J_22.The anisotropic perturbationℋ^'[Eq. (<ref>)] can make the ground-state energy depend onφ. In the mean-field calculation, the first-order perturbation changesE^0_AFQ/NtoE_ AFQ/N = -2(J_12+J_22) +2(J_11+J_22)cos^2 2θ_H-H cos 2θ_H + 3D(cos^2θ + 1)+Esin^2θ/6.In contrast to the case of the FQ state (<ref>), the mean-field energy (<ref>) of the AFQ statedoes not depend on the angleφdespite the equivalent form of the anisotropic interaction in these two systems. Thisφindependence shows that the angleφis undetermined in the AFQ state at the mean-field level. To determineφ, we need to go beyond the mean-field level. We leave it undetermined sinceφis insignificant for our calculations below. §.§ Linear flavor-wave theory in the fully polarized phase In this subsection we discuss the flavor-wave theory <cit.> in the fully polarized phase above the saturation fieldH_c^AFQof the AFQ phase. Later in Sec. <ref> we discuss the flavor-wave theory in the AFQ phase belowH_c^AFQ.The flavor-wave theory is formulated in terms of Schwinger bosons. Let us denote the creation and annihilation operators of the Schwinger bosons at thejth site byb_j,m^†andb_j,m, respectively. The flavor indexm=1,0,-1corresponds to the eigenvalue ofS_j^z. Using these bosons, an operatorO_jat thejth site is written asO_j = ∑_m, m'=1,0,-1 b_j,m^†Õ_j^mm' b_j,m'whereÕ_jdenotes a3 ×3matrix whose element is given byÕ_j^mm'=_j⟨m|O_j|m' ⟩_j. The Schwinger bosons are subject to the constraint∑_m=1,0,-1 b_j,m^†b_j,m = 1for every sitej. The3×3matrixÕ_jis easily found by writing|1⟩=(10 0)^T,|0⟩=(010)^T, and|-1⟩=(001)^T. For example, the matrices for the spin operatorsS_j^α(α=x,y,z)are given byS̃_j^x= 1/√(2)[ 0 1 0; 1 0 1; 0 1 0 ],S̃_j^y = i/√(2)[0 -10;10 -1;010 ],S̃_j^z= [100;000;00 -1 ].The matrix representation ofQ_iis easily obtained by combining the matrices ofS_i(<ref>). The fully polarized phase can be seen as a condensation phase of theb_j,1bosons. We can replace the operatorsb_j,1andb_j,1^†by ac-number,the fraction of the condensed boson,b_j,1=b_j,1^† = √(1-a_j^† a_j - b_j^† b_j).Here,we rewrote boson operators asb_j,0 = b_j,b_j,-1 = a_j,to simplify the notation. We thus end up with the Schwinger boson representation of the spin operator,S_j^x = 1/√(2)(√(1-a_j^† a_j - b_j^† b_j)b_j + b_j^†√(1-a_j^† a_j - b_j^† b_j)+ a_j^† b_j + b_j^† a_j), S_j^y = -i/√(2)(√(1 - a_j^† a_j - b_j^† b_j)b_j - b_j^†√(1 - a_j^† a_j - b_j^† b_j)+ b_j^† a_j - a_j^† b_j), S_j^z = 1 - 2a_j^† a_j - b_j^† b_j.As it is expected, the ladder operator S_j^- = S_j^x - i S_j^y involves the creation operator b_j^†. Likewise, (S_j^-)^2 = Q_j^x^2-y^2 - i Q_j^xy involves the creation operator a_j^†. In fact, the Schwinger boson representation of Q_j is Q_j^x^2-y^2 = √(1-a_j^† a_j^† -b_j^† b_j)a_j + a_j^†√(1-a_j^† a_j - b_j^† b_j), Q_j^3z^2-r^2 = 1/√(3) (1-3b_j^† b_j), Q_j^xy = i ( a_j^†√(1-a_j^† a_j - b_j^† b_j) - √(1-a_j^† a_j - b_j^† b_j)a_j), Q_j^yz = - i/√(2)( √(1-a_j^† a_j - b_j^† b_j)b_j - b_j^†√(1-a_j^† a_j - b_j^† b_j) + a_j^† b_j - b_j^† a_j), Q_j^zx = - 1/√(2)[ - (√(1-a_j^† a_j - b_j^† b_j)b_j + b_j^†√(1-a_j^† a_j - b_j^† b_j)) +a_j^† b_j + b_j^† a_j ].Up to the linear order of creation and annihilation operators, onlyQ_j^x^2-y^2≃a_j^†+ a_jandQ_j^xy≃i(a_j^†- a_j)can create the “a” boson that corresponds to the bound magnon pair. Up to the quadratic order of the creation and annihilation operators, the unperturbed Hamiltonian (<ref>) turns effectively intoℋ_0 = ∑_ k[ ω_a( k) a_ k^† a_ k + ω_b( k)b_ k^† b_ k],wherea_kandb_kare the Fourier transforms ofa_jandb_j, respectively, andω_a,b(k)are given byω_a( k) = -4 [J_11(1-) - (J_12+J_22)(1-)] - 8J_22 + 2H, ω_b( k) = -4J_11 (1-) + 4J_12 (1-) + H,with= 1/2(cos k_x + cos k_y),= cos k_x cos k_y.In the parameter rangeJ_22 > J_11,the singlea-boson state atk=k_Mhas the lowest eigenenergy when the magnetic field is close to the saturation field, while theaboson has the larger excitation energy than thebboson at extremely strong fields.Two kinds of bosons have the following excitation gaps:ω_a( k_M) = 2(H-H_c^ AFQ), ω_b( k_M) =4(J_22-J_11) + H-H_c^ AFQ.Thus, the bound magnon pair (aboson) condenses atH=H_c^AFQwhile the unpaired magnon (bboson) remains gapped when the condition (<ref>) is satisfied. In contrast, they both have excitation gaps atk=0, which we denote byΔ_aandΔ_b,Δ_a≡ω_a( 0)= 8J_11 + 2(H-H_c^ AFQ), Δ_b≡ω_b( 0)= H.Within the framework of the linear flavor-wave theory, the EPR (<ref>) of the unperturbed system (<ref>) at temperaturesT≪His understood as the excitation of thebboson from the ground state. §.§ Magnon-pair resonance in the fully polarized phase Here we study the additional ESR spectrumI'(ω), given by Eq. (<ref>), in the fully polarized phase. Using the linear flavor-wave theory (<ref>), weevaluate the retarded Green's functionG^R_𝒜𝒜^†(ω). The𝒜operator determined from the rotated single-ion anisotropy (<ref>) is𝒜 =∑_j [ (Dcos^2θ + E sin^2 θ) (Q_j^zx+iQ_j^yz)- (Dsin^2θ + Ecos^2θ) Q_j^zx+i E Q_j^yz - (D-E) sinθcosθ (Q_j^x^2-y^2+iQ_j^xy -√(3)Q_j^3z^2-r^2)].Up to the linear order of the creation and the annihilation operators, it is approximated as𝒜/√(N) ≃√(2)(Dcos^2θ + Esin^2θ) b_ k= 0^†- 1/√(2)(Dsin^2θ + Ecos^2θ) (b_ k= 0+b_ k= 0^†)+E/√(2) (b_ k= 0 - b_ k= 0^†) - 2(D-E)sinθcosθa_ k= 0.All the terms in the first line of Eq. (<ref>) yield the EPR peak. The term in the second line, containing theaboson operator, yields the delta-function magnon-pair resonance peakδ(ω-Δ_a)atω=Δ_a. According to Eq. (<ref>), the slope of the resonance frequencyω=Δ_aas a function ofHis double of that of the EPR one (<ref>) because the “a” boson creates the magnon pair and the “b” boson creates the single unpaired magnon.Equation (<ref>) also indicates that the intensity of the magnon-pair resonance peak shows the angular dependence ofsin^2θcos^2θ:I'(ω) ≃N(D-E)^2H_R^2/2Δ_a/(Δ_a-H)^2sin^2θcos^2θδ(ω-Δ_a).§.§ Linear flavor-wave theory in the AFQ phase We move on to the discussion of the linear flavor-wave theory in the AFQ phase <cit.>. Since the angleφof the AFQ directors is not pinned by the anisotropy in the mean-filed approximation, we consider the AFQ state for the generalφ. We note that this degeneracy is not lifted even by the first order perturbation of the anisotropy in the linear flavor-wave approximation as shown in Appendix <ref>. We leave it as an open question to determineφbecause the determination ofφhas little impact on our conclusions in this paper, as shown in Sec <ref>. In the fully polarized phase, we took into account low-energy excitations from the fully polarized state by replacing a local base|1⟩with either|0⟩or|-1⟩. In the AFQ phase, the mean-field ground state [Eq. (<ref>)] is obtained from the fully polarized state by performing an alternate SU(3) rotation (<ref>),|ψ_0⟩ = ∏_j iR̃(e^i k_M · r_jθ_H, φ) |1⟩_j,where the matrixR̃(e^ik_M ·r_jθ_H, φ)is expressedasR̃(e^i k_M · r_jθ_H, φ) = exp(iφS̃_j^z)exp(ie^i k_M · r_jθ_H Q̃_j^xy)withexp(iφS̃_j^z) =[e^iφ 0 0; 0 1 0; 0 0 e^-iφ ],exp(ie^i k_M · r_jθ_H Q̃_j^xy)= [ cosθ_H0e^i k_M · r_jsinθ_H;010; -e^i k_M · r_jsinθ_H0 cosθ_H ].In this representation, excitations above the AFQ state are formally described by local replacements of|1⟩to either|0⟩or|-1⟩, similar to the case of the fully polarized phase. As well as the ground state (<ref>), the Schwinger boson representation of an operatorO_jis given by the SU(3) rotation of Eq. (<ref>),O_j = ∑_m,m' b_i,m^† [R̃^†(e^i k_M · r_jθ_H) Õ_j R̃(e^i k_M · r_jθ_H)]^mm' b_i,m'.Forφ=0, the spin operatorS_jand the quadrupole operatorQ_jin the AFQ phase are related to theSchwinger boson representation in the FP phase, given in Eqs. (<ref>)–(<ref>), as follows:([S_j^x; Q_j^zx;])= 1/√(2)([ cosθ_H -e^i k_M · r_jsinθ_H;e^i k_M · r_jsinθ_H cosθ_H;]) ([ b_j+b_j^† + b_j^† a_j+ a_j^† b_j; b_j+b_j^† - b_j^† a_j- a_j^† b_j;]),([S_j^y; Q_j^yz;])= 1/√(2)i([ cosθ_He^i k_M · r_jsinθ_H; -e^i k_M · r_jsinθ_H cosθ_H;]) ([ b_j - b_j^† + b_j^† a_j - a_j^† b_j; b_j - b_j^† - b_j^† a_j + a_j^† b_j; ]),([ S_j^z; Q_j^x^2-y^2; ])=([cos 2θ_H e^i k_M · r_jsin 2θ_H; - e^i k_M · r_jsin 2θ_Hcos 2θ_H; ]) ([ 1-2a_j^† a_j - b_j^† b_j;a_j + a_j^†;]),([ Q_j^xy; Q_j^3z^2-r^2;])= ([ -ia_j + i a_j^†; 1/√(3) (1-3b_j^† b_j); ]).Forφ≠0, they are rotated as( [ S_j^x (φ); S_j^y (φ); ]) =( [cosφsinφ; -sinφcosφ; ]) ( [ S_j^x (0); S_j^y (0); ]),( [S_j^z (φ); Q_j^3z^2-r^2 (φ);]) =( [S_j^z (0); Q_j^3z^2-r^2 (0);]),( [ Q_j^x^2-y^2 (φ);Q_j^xy (φ); ]) =( [cos 2φsin 2φ; -sin 2φcos 2φ; ]) ( [ Q_j^x^2-y^2 (0);Q_j^xy (0); ]),( [ Q_j^zx (φ); Q_j^yz (φ);]) =( [cosφsinφ; -sinφcosφ; ]) ( [ Q_j^zx (0); Q_j^yz (0);]).Up to the quadratic terms, the unperturbed Hamiltonian (<ref>) is split into two parts,ℋ_0≃ℋ_0^a + ℋ_0^b,whereℋ_0^a = ∑_ k[ A_ k a_ k^† a_ k + B_ k/2 (a_ k^† a_- k^† + a_ ka_- k)], ℋ_0^b = ∑_ k[ C_ k b_ k^† b_ k+ D_ k/4 (b_ k+ k_M^† b_- k^†+b_ k+ k_Mb_- k)].The parametersA_k,B_k,C_k, andD_kare given byA_ k = 4J_11sin^2 2θ_H - 4J_11 (1-) cos^2 2θ_H+ 4J_12(1-) + 4J_22sin^2 2θ_H- 4J_22(1+) cos^2 2θ_H + 2H cos 2θ_H, B_ k = -4J_11sin^2 2θ_H + 4J_22sin^2 2θ_H, C_ k = - 4J_11cos^2 2θ_H + 4J_11cos 2θ_H + 4J_12 (1-) + 4J_22sin^2 2θ_H +Hcos 2θ_H, D_ k = -4J_22sin 2θ_H. To diagonalize the Hamiltonians (<ref>) and (<ref>), we perform the following Bogoliubov transformations,[a_ k; a_- k^† ] =[coshΘ_ k^a -sinhΘ_ k^a; -sinhΘ_ k^acoshΘ_ k^a ][α_ k; α_- k^† ], [ b_ k+ k_M; b_- k^† ] =[ coshΘ_ k^b - sinhΘ_ k^b;-sinhΘ_ k^b coshΘ_ k^b ][ β_ k+ k_M; β_- k^† ].The parametersΘ_k^aandΘ_k^bare determined in order to eliminate the off-diagonal terms:Θ_ k^a= 1/2tanh^-1( B_ k/A_ k), Θ_ k^b= 1/2tanh^-1( 2D_ k/C_ k+C_ k+ k_M).The Bogoliubov transformations diagonalize the Hamiltonian (<ref>) toℋ_0 =∑_ k[ ω_a( k)α_ k^†α_ k +ω_b( k) β_ k^†β_ k]with the following dispersion relations,ω_a( k) = √(A_ k^2 - B_ k^2), ω_b( k) = C_ k- C_ k+ k_M/2 + √((C_ k+C_ k+ k_M/2)^2 - D_ k^2).Inheriting the terminology in the fully polarized phase, we call the bosons created byα_k^†andβ_k^†as the “a” boson and the “b” boson, respectively, also in the AFQ phase. Atk=k_M, the “a” boson corresponding to the bound magnon pair is gapless, whereas the “b” boson corresponding to the unpaired magnon is gapped:ω_a( k_M)= 0,ω_b( k_M)= J_22-J_11/J_22+J_11H.The “a” boson is the characteristic Nambu-Goldstone boson that accompanies the AFQ ordered ground state. Atk=0, both excitations are gapped:Δ_a= ω_a( 0) = 8J_11[ 1+J_22-J_11/J_11{1-(H/H_c^ AFQ)^2}]^1/2, Δ_b= ω_b( 0) = H. The linear flavor-wave theory reproduces the exact EPR frequencyω=H[Eq. (<ref>)] of the unperturbed system (<ref>) both in the fully polarized phase and in the AFQ phase. The reproduction of the exact result is an important criterion of appropriateness of the low-energy effective theory. The criterion is akin to the sum rule mentioned in Ref. <cit.>. If we include an SU(2)-symmetric but SU(3)-asymmetric interaction,ℋ_ a = -J_21/2∑_⟨i,j|_⟩1(Q_i · Q_j -S_i · S_j )to the unperturbed Hamiltonian (<ref>), the linear flavor-wave theory fails to reproduce the exact EPR frequencyω=HbecauseΔ_bin the AFQ phase is modified toΔ_b = 4J_11cos 2θ_H + 4J_22√(1-(J_22-J_21/J_22)^2 sin^2 2θ_H).This technical problem is an artifact of the linear flavor-wave theory and not essential to our purpose of demonstrating the general properties of the magnon-pair resonance in the AFQ phase. Thus, we put aside this probably complicated discussion, restricting ourselves to the model withJ_21=0. §.§ Magnon-pair resonance in the AFQ phase Here we study the additional ESR spectrumI'(ω), given by Eq. (<ref>), in the AFQ phase. We approximate the operator𝒜up to the quadratic order of the spin operators, that is, the quadratic order of theβoperators and the linear order of theαoperators. The𝒜operator (<ref>) is represented as𝒜/√(N)≃ 3(D-E)cos2θ + D+3E/2√(2) e^-iφ{(cosθ_H coshΘ_ 0^b -sinθ_H sinhΘ_ 0^b)β_ 0 + (-cosθ_HsinhΘ_ 0^b + sinθ_H coshΘ_ 0^b)β_ k_M^†} -(D-E)cos2θ - (D+3E)/2√(2)e^iφ{ (cosθ_H coshΘ_ 0^b -sinθ_H sinhΘ_ 0^b)β_ 0^† + (-cosθ_HsinhΘ_ 0^b + sinθ_H coshΘ_ 0^b)β_ k_M} - 2(D-E)sinθcosθ e^-i2φ{cos^2θ_H (α_ 0coshΘ_ 0^a-α_ 0^†sinhΘ_ 0^a)-sin^2θ_H(α_ 0^†coshΘ_ 0^a - α_ 0sinhΘ_ 0^a)} -(D-E)sinθcosθ∑_ k{e^-i2φsin2θ_H (β_ k+ k_M^†coshΘ_ k^b - β_- ksinhΘ_ k^b)(β_ kcoshΘ_ k^b - β_- k+ k_M^†sinhΘ_ k^b) + 3(β_ k^†coshΘ_ k^b - β_- k+ k_MsinhΘ_ k^b)(β_ kcoshΘ_ k^b - β_- k+ k_M^†sinhΘ_ k^b)}.The term containing theαoperators creates theaboson atk=0, whereas the linear terms of theβoperators create thebboson at eitherk=0ork=k_M. The quadratic term of theβoperators contributes to the two-magnon continuum made of two scatteringbbosons. While thebboson excitation atk=0is involved in the EPR peakI_EPR(ω)with the resonance frequencyω= H, theaboson excitation atk=0gives rise to the magnon-pair resonance peakI_MPR(ω). Thebboson excitation atk= k_Mand two-unpaired-magnon excitation result in unpaired magnon resonance peakI_k_M(ω)and broad two-magnon continuumI_2-mag(ω), respectively. In total, the ESR spectrumI(ω)=I_EPR(ω)+I'(ω)contains three sharp peaks and a broad continuum:I(ω) = I_ EPR(ω) + I_ MPR(ω) + I_ k_M(ω) + I_2-mag(ω),which are given at the leading order of the perturbation byI_ EPR(ω) ≃π H/4⟨S^z|_⟩0δ(ω-H), I_ MPR(ω) ≃𝒜_ MPRδ(ω - Δ_a), I_ k_M(ω) ≃𝒜_ k_Mδ(ω-ω_b( k_M)), I_2-mag(ω) ≃∑_ kF( k) [sin^22θ_H {δ(ω-ω_b(- k)-ω_b( k)) +δ(ω-ω_b(- k)+ω_b(- k+ k_M))} + 18δ(ω-ω_b(- k+ k_M)-ω_b( k))].The intensities𝒜_MPRand𝒜_k_Mare given by𝒜_ MPR/N =π/8(D-E)^2sin^2θcos^2θΔ_a/(Δ_a-H)^2 (cos^2θ_H coshΘ_ 0^a +sin^2θ_H sinhΘ_ 0^a)^2, 𝒜_ k_M/N = π/16ω_b( k_M)/2(ω_b( k_M)-H)^2((D-E)cos2θ-D-3E/2)^2 (sinθ_HcoshΘ_ 0^b-cosθ_HsinhΘ_ 0^b)^2 ,and the factor F( k) isF( k)/N = 1/8(D-E)^2sin^2θcos^2θ(sinh2Θ_ k^b/2)^2within the lowest-order perturbation theory. Note that these are independent of the angle φ of the quadrupolar order. This φ independence comes as a consequence of the linear flavor-wave approximation and the first-order perturbation. Higher-order processes can induce φ dependent corrections to the above results. The intensities 𝒜_ MPRand 𝒜_ k_M can be rephrased as𝒜_ MPR/N = π/8(D-E)^2sin^2 θcos^2 θΔ_a/(Δ_a-H)^2{2H/H_c^ AFQ + [(1+(H/H_c^ AFQ)^2)(J_22+J_11/2J_11 - J_22-J_11/2J_11(H/H_c^ AFQ)^2)+4(J_22-J_11)/Δ_a(1-(H/H_c^ AFQ)^2)^2 ]}, 𝒜_ k_M/N = π/16((D-E)cos2θ - D-3E/2)^2J_22^2-J_11^2/2J_11^2H(H_c^ AFQ/H - H/H_c^ AFQ).The presence of the magnon-pair resonance peakI_MPR(ω)is a direct consequence of the quadrupolar order in the ground state. We found that the magnon-pair resonance peak appears at the finite frequencyω=Δ_ain the AFQ phase, which is continuously connected to the magnon-pair resonance peak found in the fully polarized phase. Reflecting the condensation of bound magnon pairs, the field dependence of the resonance frequencyω=Δ_ashows a singular upturn at the saturation fieldH=H_c^AFQ(Fig. <ref>). In addition, there is another peakI_k_M(ω)[Eq. (<ref>)] which is absent in the fully polarized phase. This peakI_k_M(ω)corresponds to creation of the single “b” boson atk=k_M. Although ESR usually involves excitations atk=0only as Eq. (<ref>) shows, the AFQ order with the wave vectork_Mmakes the resonance atk=k_Mpossible.In general, the magnon-pair resonance peakI_MPR(ω)could be masked by the broad two-magnon continuum. However there is a better chance to observe this resonance peak near the saturation field. The lowest energy of the continuum takes the highest value2ω_b (k_M)=2(J_22-J_11)H_c^AFQ/(J_22+J_11)at the saturation field. For the parameter rangeJ_22 > 1.702 J_11, this lower edge of the continuum is well above the magnon-pair resonance frequencyΔ_a=8J_11at the saturation. It is also worth mentioning the field dependence of the intensities. The intensity of magnon-pair resonance peak inI_MPR(ω)remains finite the near the saturation field. In contrast, the intensities of the unpaired magnon peak inI_k_M(ω)and the two-magnon continuumI_2-mag(ω), both of which originate from the unpaired magnon excitations, vanish near saturation as𝒜_ k_M ∝H_c^ AFQ -H F(k)∝H_c^ AFQ -HforH<H_c^AFQ. Hence the continuumI_2-mag(ω)disappears around the saturation fieldH_c^AFQ. Therefore, our method is more effective under the high magnetic field. The peak intensity𝒜_MPRof the magnon-pair resonance has a strong field dependence, showing a divergence at a certain fieldH_∗belowH_c^AFQ[see Fig. <ref>(a)]. This divergence occurs when the magnon-pair resonance peak merges into the EPR one atH≃H_∗. The divergence comes from the factorΔ_a/(Δ_a-H)^2in Eq. (<ref>); as Eq. (<ref>) and Fig. <ref> show, the excitation gapΔ_aof the bound magnon pair equals toHatH_∗= J_22/[J_22-J_11+J_11(H_c/8J_11)^2]. The intensity𝒜_k_M[Eq. (<ref>)] of the unpaired magnon resonance also shows the divergence atH=0(Fig. <ref>), because of merging of the peak into the EPR one atH≃0. We note that the EPR and the MPR peaks can be mixed under certain interactions. The EPR peak and the MPR peak are generated by application ofβ_k=0^†andα_k=0^†to a given eigenstate, respectively. To mix those resonances, an anisotropic interaction including a term such asβ_kα_k'^†orβ_k^†α_k'is necessary. For example, a uniform DM interactionwithDvector parallel to thexaxis can generate effectively such an interaction,∑_⟨i,j|_⟩1 D ·S_i ×S_j=∑_⟨i,j|_⟩1 |D| [(b_i-b_i^†)(a_j+a_j^†) -(a_i+a_i^†) (b_j-b_j^†)]. If an anisotropic interaction allows the mixing, it will be difficut to distinguish the MPR peak from the EPR one when their resonance frequencies are close, because the EPR peak has a finite linewidth in the presence of anisotropic interactions. When they are apart, the mixing is not important as long as the anisotropic interactins are perturbative. Although the intensities suffer from the insignificant divergences, the resonance frequencies are free from any singular behavior in the linear flavor-wave approximation except for the singular bent point at the saturation fieldH_c^AFQdue to physically reasonable characteristic upturn forH>H_c^AFQ(Fig. <ref>). Using the magnon-pair resonance frequency and the unpaired-magnon resonance frequency atk_M, we can identify the AFQ order phase in quite a wide field range.Angular dependence of the intensity of the magnon-pair resonance (<ref>) enables another method of identification free from the technical problems. The magnon-pair resonance peak shows a characteristic angular dependence𝒜_ MPR∝sin^2 θcos^2θ,which makes the magnon-pair resonance peak vanish when the magnetic field is parallel to thex,y, orzaxes [as shown in Fig. <ref>(b)]. The angular dependence (<ref>) holds true independent of the model and the theoretical technique, as we pointed out in Sec. <ref>. This angular dependence reflects the fact that the operatorsQ_j^yzandQ_j^zxneither create nor annihilate the bound magnon pair, different fromQ_j^x^2-y^2andQ_j^xy. Thus, the characteristic angular dependence of𝒜_MPR(<ref>) qualifies as an evidence of the formation of the bound magnon pair in the system (<ref>) with the single-ion anisotropy. We note that the linewidthof the EPR peak ofS=1/2frustrated ferromagnetic chain compounds is also expected to show the angular dependence ofsin^2θcos^2θ<cit.>. That angular dependence of the linewidth comes from the same root as the intensity of the magnon-pair resonance peak. § DISCUSSIONS Here, we discuss some issues related to the magnon-pair resonance peak in the ESR spectrum shown in Sec. <ref>. We also discuss applications of our theory toS=1/2spin systems. §.§ Effects of other anisotropic interactions In Sec. <ref>, we assumed the single-ion anisotropy (<ref>) as an example of the perturbative anisotropic interaction. Since ESR depends crucially on the form of the perturbative anisotropic interactionℋ', it is necessary to confirm that our results obtained in Sec. <ref> is robust against inclusion of other kinds of anisotropic interactions.The anisotropic exchange interaction (<ref>) on thenth neighbor bond leads to the same result because the bound magnon pair is not localized at a single site but spread around bonds <cit.>. If the𝒜operator (<ref>) contains some of operators that have the same symmetry as the wavefunction of two-magnon bound state, it generates the magnon-pair resonance to the ESR spectrum through the formula (<ref>). In contrast, the DM interactionℋ'_DM=∑_⟨i,j|_⟩nD_ij ·S_i ×S_jis irrelevant to the magnon-pair resonance (<ref>) and to the unpaired magnon resonance (<ref>) because of the symmetry; the DM interaction neither create nor annihilate the bound magnon pair on the bond because it is antisymmetric with respect to the bond-centered inversion, whereas the wavefunction of the bound magnon pair on the bond is symmetric.§.§ Applications to the spin nematic order in S=1/2 spin systemsIn Sec. <ref>, we studied theS=1spin model to demonstrate the magnon-pair resonance. We can apply this result to the spin nematic order inS=1/2spin systems performing an appropriate mapping of low-energy degrees of freedoms.§.§.§ S=1/2 frustrated ferromagnetsIn the case of spin-1/2 frustrated ferromagnets, the spin nematic order parameter, defined on the nearest neighbor bonds, hask=0wave vector, in which the sign of it alternates inside the unit cell of the crystal structure <cit.>. For example, on the square lattice, the two quadrupolar directors on different bonds along two unit vectorse_1ande_2are orthogonal to each other <cit.>. Because of this sign change, theS=1/2spin nematic order parameter is not captured into the frequency shift discussed in Sec. <ref>, even though it hask=0wave vector.Low-energy degrees of freedom in theS=1/2spin nematic systems are given byS=1spin degrees of freedom formed on the nearest neighbor bonds <cit.>. These excitation modes are effectively related to the excitations of theS=1AFQ state through a mapping between bond degrees of freedom inS=1/2spin systems and on-site spin degrees of freedom inS=1spins <cit.>, where theS=1spins are assigned on the center points of the nearest-neighbor bonds ofS=1/2spins. On the square lattice, the gapless excitations withk=k_Min theS=1AFQ state correspond to the excitations withk=0wave vector andB_1irreducible representationof the space groupC_4vin theS=1/2spin nematic states. Above the saturationH>H_c^AFQ, this mode is the lowest excitation which closes the gap at the saturation field. However this mode is inaccessible in ESR measurements, since ESR is directly accessible only to thek=0wave vector modes with theA_1(trivial) irreducible representation. Only the gapped excitation modes withk=0can be observed among the bound magnon pair excitations as same as in theS=1AFQ state discussed in Sec. <ref>.§.§.§ S=1/2 orthogonal dimer spin system The spin nematic phase can also appear in spin-gapped systems when bound magon (triplon) pairs close the energy gap in an applied field <cit.>. In theS=1/2Heisenberg model on the Shastry-Sutherland lattice <cit.>, which is also called an orthogonal dimer spin model, it was theoretically demonstrated thatthe ground state is an exact dimer state with a finite energy gap <cit.> and bound two-triplon excited states <cit.> are stabilized at zero field by the correlated hopping process <cit.>. Theoretical calculations <cit.> pointed out that a two-triplon bound state withS^z=2can have a lower energy than two triplon continuum above the gapped ground state and that the energy-gap closing in an applied magnetic field leads to the condensation of bound triplon pairs. Since the lowest energy state of the bound pair in theS^z=2sector has the wave vectork=k_M, the field-induced condensed phase becomes an AFQ phase [The field-induced antiferroquadrupolar (AFQ) phase in the spin-gapped systems is also described as a low-density condensate of bound magnon pairs, as same as the AFQ phase near the saturation field. The vacuum is a singlet ground state and the bosonic particles are bound triplon pairs in the former case, whereas the vacuum is the fully polarized state and the bosons are bound magnon pairs in the latter. ].In this system, anisotropic interactions between two orthogonal dimers cause the operatorS_i^+ S_j^+on the inter-dimer bonds in the𝒜operator. This operator creates a triplon pair on a nearest-neighbor pair of dimers, which gives rise to a triplon-pair resonance peak in the ESR spectrum. In the spin gap phase, the resonance frequency behaves asω=Δ_a+2(H_c-H),whereH_cdenotes the onset-field of the magnetization process andΔ_athe energy gap of the bound triplon pair atk=0at the critical fieldH=H_c. We note that the bound pair closes the gapatk=(π,π)and this excitation is well dispersive, i.e.Δ_a>0<cit.>. In the magnetic phase aboveH_c, we expect that this peak continuously connects to the triplon pair resonance peak in the AFQ phase showing a singularity in the field dependence of the frequency at the critical field.In an ESR study <cit.>, bound triplon-pair resonance peaks were indeed observed in theS=1/2orthogonal dimer spin compound SrCu_2(BO_3)_2. The lowest-energy resonance peak of the bound triplon pairs shows the field dependence (<ref>), having a strong intensity aroundH=H_c. Even after the peak frequency changes the slope as a function of a field aroundH=H_c, the resonance peak remains with strong intensity inside the magnetic phase between the spin gapped and the 1/8-plateau phases when the magnetic field is parallel toaaxis. The implication of this resonance peak inside the magnetic phase has not been properly understood until now. Our research elucidates that this ESR result has already suggested the appearance of a spin nematic order in the ground state of the field-induced magnetic phase below the 1/8-plateau. This system deserves further investigations.To compare the field dependence of the resonance peak with observed results in real compounds, we need to include mixing between the ground state and excited states. For example, DM interaction induces a mixing between the singlet ground state and triplon excited states <cit.>. In the case of the bound two-triplon excited state, anisotropic interactions can induce mixing with the singlet ground state. This can be easily seen by considering an anisotropy on the inter-dimer bonds2δ (S_i^x S_j^x - S_i^y S_j^y) = δ (S_i^+ S_j^+ + S_i^- S_j^-),which mixes the bound two-triplon state with the singlet dimer state. This effect might smear the singularity in the field dependence of the resonance frequency atH=H_c. §.§ Magnon-pair resonance in the case of ferroquadrupolar orderLastly we comment on the additional peaks in the ESR spectrum in the FQ phase.In the FQ phase, only the EPR peak will be found in the ESR spectrum. The bound magnon-pair excitation in the FQ phase is gapless atk=0, i.e.Δ_a=0, but the resonance atω=0is invisible in the ESR spectrum for the factorωin Eq. (<ref>).Note that the unpaired magnon resonance peak atk=k_Min the AFQ phase corresponds to the EPR peak in the FQ phase since the FQ order grows atk = 0.If the magnetic field is above the saturation fieldH_c^FQ, the bound magnon pair excitation opens a gap, showing a characteristic field dependenceΔ_a=2(H-H_c^FQ). This peak is observable in the ESR measurements as it comes from thek=0modes. This gives a clear difference from the case of the AFQ order; the lowest excitation which closes the gap as2(H-H_c^AFQ)above the AFQ phase has thek=k_Mwave vector [Eq. (<ref>)] and it cannot be observed in ESR measurements. Thus, the appearance of this peak in the ESR spectrum above the saturation fieldH_c^FQand the disappearance belowH_c^FQsignal the emergence of the FQ phase belowH_c^FQ. § SUMMARY In this paper we showed that the FQ and the AFQ orders are distinguishable in ESR experiments. We studied both the frequency shift of EPR resonance and the frequencies of the additional resonance peaks in the ESR spectrum.For the generic spin model (<ref>), the FQ order parameter turned out to shift the resonance frequency of the EPR peak in the ESR spectrum. The EPR frequency shift shows a characteristic angular dependence [as shown in Eq. (<ref>)] on a rotation of the material around theyaxis keeping the magnetic field parallel to thezaxis. Here we determined theyandzaxes so that the anisotropic spin interactions in these spin components are, respectively, weakest and strongest. Thus the FQ order parameter can be extracted from the frequency shift. For example, as Eqs. (<ref>) and (<ref>) show, the frequency shifts atθ=0andπ/2enable us to determine the FQ order parameter⟨S_i^-S_j^-|_⟩0experimentally because only two quantities⟨3S_i^zS_j^z-S_i ·S_j|_⟩0and⟨S_i^-S_j^-|_⟩0are the undetermined variables in these equations. In particular, when the perturbative anisotropic interaction is uniaxial, the FQ order parameter is simply derived from the single equation (<ref>). The unexplained high field phase of chromium spinels is an interesting target to which this method is applicable.In the case of the AFQ order, though the EPR frequency shift is usually insensitive to the order parameter, fingerprints of the AFQ order appear in the additional resonance peaks other than the EPR peak in the ESR spectrum. As far as the resonance peaks well isolated from the EPR one are concerned, the ESR spectrum, as shown in Eq. (<ref>), is derived from the spectrum of the retarded Green function of the operator𝒜=[ℋ',S^+]given by the small anisotropic interactionℋ'. The operator𝒜is usually quadratic containing the magnon pair creation operator [see Eqs. (<ref>) and (<ref>)]. This is one of the most interesting properties of the ESR spectrum. Except for the vicinity of the EPR peak atω=H, measuring the ESR spectrum is effectively equivalent to measuring the spectrum of theoperator𝒜. We note that, in our pertubative analysis, the anisotropic interaction plays no role of yielding the spin nematic phase and of making the magnitude of the quadrupolar order parameter grow. Those are fully determined in the unperturbed system. The long-range AFQ order yields two additional sharp resonance peaks in the ESR spectrum. One is attributed to the resonance of the bound magnon-pair excitation, which we called the magnon-pair resonance. The magnon-pair resonance is also found in the fully polarized phase adjacent to the AFQ phase, where the resonance frequency shows the linear field dependence whose slope is double of that of the EPR frequency, as was experimentally observed in Ref. <cit.>. (A similar triplon-pair resonance peak was also experimentally observed in Ref. <cit.>.) With decreasing the magnetic field, the system enters into the AFQ phase, where the magnon-pair resonance frequency shows the singular upturn as a function of the magnetic field, reflecting the condensation of bound magnon pairs (Fig. <ref>). The other resonance peak is attributed to the excitation of the unpaired magnon at the wave vectork_M=(π,π). Usually, ESR detects excitations at the wave vectork=0. In the AFQ phase, since the ground state structure has the wave vectork_M, the excitation gap of the magnon atk_Mbecomes visible in the ESR spectrum as an independent resonance peak. Our results on the FQ and the AFQ orders are valid as long as (1) the anisotropic interaction is small enough to be seen as a perturbation to the system and (2) the anisotropic interaction is governed mainly by the single-ion anisotropy or the anisotropic exchange interaction. The weak DM interaction has no impact on the result obtained in this paper because the DM interaction is antisymmetric with respect to the bond-centered inversion unlike the spin nematic order parameter. Though we restricted ourselves to the cases of weak anisotropic interactions in this paper, it is also interesting to investigate cases governed by a large anisotropic interaction such as the case of Ref. <cit.>. While the formula of the frequency shift [Eq. (<ref>)] is invalid in such cases, the discussion of the sharp magnon-pair resonance peak isolated from the EPR one will be qualitatively valid even in the case of large anisotropic interactions though we need to derive the full Green's function𝒢^R_𝒜𝒜^†(ω)instead.§ ACKNOWLEDGMENTS We thank Akira Furusaki, Masayuki Hagiwara, Hiroyuki Nojiri, Nic Shannon, and Shintaro Takayoshi for helpful discussions. The present work is supported by JSPS KAKENHI Grant Nos. 16J04731 and 16K05425.§ POLARIZATION INDEPENDENCE OF THE ESR SPECTRUMIn this Appendix, we derive the ESR spectrum of the unpolarized microwave. To discuss it, we review the linear response theory of the ESR absorption spectrumI(ω)in a generic spin system described byℋ̃(t) = ℋ - ∫_-π^π dθ∫_-π^π dϕA(θ) X(t,θ, ϕ),whereℋis the Hamiltonian of the spin system of our interest,A(θ)is a total spin operatorA(θ) = S^xcosθ + S^y sinθ,andX(t,θ, ϕ)is the oscillating magnetic field,X(t,θ, ϕ) = h_R(θ, ϕ) cos(ω t + ϕ).The amplitudeh_R(θ, ϕ)follows a distribution which can be random or not. For example, the Hamiltonian under a circularly polarized magnetic fieldℋ̃(t) = ℋ - H_R/2 (S^+e^iω t + S^- e^-iω t)is a special case of Eq. (<ref>) withh_R(θ, ϕ) = H_R/2[ δ(θ)δ(ϕ) + δ(θ - π2)δ(ϕ - π2)]. The ESR absorption spectrumI(ω)is give by the energy absorption rate per a period of the oscillating field,I(ω) = ω/2π∫_0^2π/ω dtd/dt[ρ(t)ℋ̃(t)],whereρ(t)is the density matrix in the Heisenberg picture,ρ(t) =U(t)exp(-ℋ̃(t)/ T)/[exp(-ℋ̃(t)/T)] U^†(t)withU(t) = exp(i∫_0^t dt' ℋ̃(t')).Ifh_R(θ, ϕ)follows a random distribution, we replace Eq. (<ref>) withI(ω) = ω/2π∫_0^2π/ω dtd/dt[ρ(t)ℋ̃(t)],whereOdenotes the average of the quantityOwith respect to the random distribution. Let us first consider the case thath_R(θ, ϕ)is uniquely determined such as Eq. (<ref>). The energy absorption rate (<ref>) is written asI(ω)=- ω/2π∫_0^2π/ω dt∫_-π^π dθ dϕ [ρ(t) A(θ)] ∂ X(t,θ, ϕ)/∂ t.Within the linear response, the trace [ρ(t)A(θ)] is approximated as[ρ(t)A(θ)] ≃⟨A(θ)|+⟩i∫_0^∞ dt' ∫_-π^π dθ' dϕ'X(t-t',θ', ϕ')⟨[A(t',θ), A(0,θ')]|.⟩Here, the average ⟨·|$⟩ is taken with respect to the HamiltonianℋandA(t, θ)is defined asA(t', θ) = e^it'ℋ A(θ) e^-it'ℋ.Taking these relations into account, we find that the ESR energy absorption rate (<ref>) is given byI(ω) ≃ -iω/2π∫_0^2π/ω dt∫_0^∞ dt'∫_-π^π dθ dϕ dθ' dϕ'∂ X(t,θ, ϕ)/∂ tX(t-t',θ',ϕ') ⟨[A(t',θ), A(0,θ')]|⟩=iω/8∫_0^∞ dt' ∫_-π^π dθ dϕ dθ' dϕ' h_R(θ, ϕ)h_R(θ', ϕ')sin(ω t' + ϕ - ϕ'){⟨[S^+(t'), S^-(0)]|e⟩^-i(θ - θ') + ⟨[S^-(t'), S^+(0)]|e⟩^i(θ-θ')+ ⟨[S^+(t'), S^+(0)]|e⟩^-i(θ + θ') + ⟨[S^-(t'), S^-(0)]|e⟩^i(θ + θ')}.Whenh_R(θ, ϕ)is given by Eq. (<ref>), the ESR absorption rate (<ref>) becomesI(ω) = ω H_R^2/8[ -𝒢^R_S^+S^-(ω)],which reproduces Eq. (<ref>).We next consider the case thath_R(θ, ϕ)follows a random distribution. Applying the linear response theory (<ref>) to the random averaged energy absorption rate (<ref>), we find I(ω)=- ω/2π∫_0^2π/ω dt∫_-π^π dθ dϕ [ρ(t) A(θ)] ∂ X(t,θ, ϕ)/∂ t≃iω/8∫_0^∞ dt' ∫_-π^π dθ dϕ dθ' dϕ'h_R(θ, ϕ)h_R(θ', ϕ')sin(ω t' + ϕ - ϕ'){⟨[S^+(t'), S^-(0)]|e⟩^-i(θ - θ') + ⟨[S^-(t'), S^+(0)]|e⟩^i(θ-θ')+ ⟨[S^+(t'), S^+(0)]|e⟩^-i(θ + θ') + ⟨[S^-(t'), S^-(0)]|e⟩^i(θ + θ')}.If the applied electromagnetic wave is “white”, that is, if the random averageh_R(θ, ϕ)h_R(θ', ϕ')satisfiesh_R(θ, ϕ)h_R(θ', ϕ') = H_R^2/(2π)^2δ(θ-θ')δ(ϕ-ϕ'),the ESR energy absorption rate (<ref>) is simplified asI(ω) = ω H_R^2/8[ -𝒢^R_S^+S^-(ω) - 𝒢^R_S^-S^+(ω)]. § PERTURBATIONS OF THE GROUND STATE ENERGY IN THE AFQ PHASE This appendix is devoted to estimation of the ground state energy in the AFQ phase of theS=1model on the square lattice described by the Hamiltonianℋ = ℋ_0 + ℋ',whereℋ_0is the unperturbed Hamiltonian (<ref>). We take the single-ion anisotropy (<ref>) as the perturbationℋ'.As we discussed in Sec. <ref>,the angleφthat specifies the direction of the AFQ order growing is not determined at the mean-field level. Here, we estimate theφdependence of the ground-state energy using the linear flavor-wave theory explained in Sec. <ref>. The perturbative single-ion anisotropy shifts the ground state energy from its unperturbed value by an amountδ E_0 = ⟨ GS|ℋ'| GS|,⟩up to the first order of the perturbation. Here,| GS⟩is the ground state in the AFQ phase of the unperturbed Hamiltonian, or the vacuum annihilated byα_ kandβ_ kfor all k:α_ k| GS⟩ = β_ k| GS⟩ = 0. The energy shift (<ref>) in the Schwinger boson language is given byδ E_0 = const. - ( D - 3(D-E)/2sin^2θ) ∑_ k⟨ GS|b_ k^† b_ k| GS|⟩ +1/2{ Dsin^2θ + E(1+cos^2θ)}cos 2φsin 2θ_H ∑_ k⟨ GS|(2a_ k^† a_ k+ k_M + b_ k^† b_ k+k_M)| GS|.⟩Using the Bogoliubov transformations (<ref>) and (<ref>) and the property (<ref>) in the ground state, we can further reduceδ E_0toδ E_0 = const. - ( D - 3(D-E)/2sin^2θ) ∑_ ksinh^2Θ_ k^b.The shift (<ref>) is independent of the angleφand hence leavesφundetermined. 73 fxundefined [1]ifx#1 fnum [1] #1firstoftwosecondoftwo fx [1] #1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0] ` 12 `$12 `&12 `#12 `1̂2 `_12 `%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty [Blume and Hsieh(1969)]blume_hsieh authorauthorM. Blume and authorY. Y. Hsieh,http://dx.doi.org/10.1063/1.1657616journaljournalJ. Appl. Phys. volume40, pages1249 (year1969)NoStop[Chen and Levy(1971)]chen_leby authorauthorH. H. Chen and authorP. M. 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Totsuka, 10.1103/PhysRevB.76.214428journaljournalPhys. Rev. B volume76, pages214428 (year2007)NoStop[Shindou and Momoi(2009)]shindou2009 authorauthorR. Shindou and authorT. Momoi,10.1103/PhysRevB.80.064410journaljournalPhys. Rev. B volume80,pages064410 (year2009)NoStop[Shindou et al.(2011)Shindou, Yunoki, and Momoi]shindou2011 authorauthorR. Shindou, authorS. Yunoki, and authorT. Momoi,10.1103/PhysRevB.84.134414journaljournalPhys. Rev. B volume84, pages134414 (year2011)NoStop[Heidrich-Meisner et al.(2006)Heidrich-Meisner, Honecker, and Vekua]heidrichmeisner2006 authorauthorF. Heidrich-Meisner, authorA. Honecker,and authorT. Vekua,10.1103/PhysRevB.74.020403journaljournalPhys. Rev. B volume74,pages020403 (year2006)NoStop[Vekua et al.(2007)Vekua, Honecker, Mikeska, and Heidrich-Meisner]vekua_1dnematic authorauthorT. Vekua, authorA. Honecker, authorH.-J. Mikeska,andauthorF. Heidrich-Meisner, 10.1103/PhysRevB.76.174420journaljournalPhys. Rev. B volume76, pages174420 (year2007)NoStop[Kecke et al.(2007)Kecke, Momoi, and Furusaki]kecke_1dnematic authorauthorL. Kecke, authorT. Momoi,andauthorA. Furusaki,10.1103/PhysRevB.76.060407journaljournalPhys. Rev. B volume76, pages060407 (year2007)NoStop[Hikihara et al.(2008)Hikihara, Kecke, Momoi, and Furusaki]hikihara_1dnematic authorauthorT. Hikihara, authorL. Kecke, authorT. Momoi,and authorA. Furusaki,10.1103/PhysRevB.78.144404journaljournalPhys. Rev. B volume78, pages144404 (year2008)NoStop[Sudan et al.(2009)Sudan, Lüscher, and Läuchli]sudan_1dnematic authorauthorJ. Sudan, authorA. Lüscher, and authorA. M. Läuchli, 10.1103/PhysRevB.80.140402journaljournalPhys. Rev. B volume80, pages140402 (year2009)NoStop[Ueda and Totsuka(2009)]ueda2009 authorauthorH. T. Ueda and authorK. Totsuka, 10.1103/PhysRevB.80.014417journaljournalPhys. Rev. B volume80, pages014417 (year2009)NoStop[Zhitomirsky and Tsunetsugu(2010)]zhitomirsky_j1j2 authorauthorM. E. Zhitomirsky and authorH. Tsunetsugu, http://stacks.iop.org/0295-5075/92/i=3/a=37001journaljournalEuro. Phys. Lett. volume92, pages37001 (year2010)NoStop[Sato et al.(2013)Sato, Hikihara, and Momoi]sato_q1dnematic authorauthorM. Sato, authorT. Hikihara, and authorT. Momoi,10.1103/PhysRevLett.110.077206journaljournalPhys. Rev. Lett. volume110, pages077206 (year2013)NoStop[Starykh and Balents(2014)]starykh_q1dnematic authorauthorO. A. Starykh and authorL. Balents,10.1103/PhysRevB.89.104407journaljournalPhys. Rev. B volume89, pages104407 (year2014)NoStop[Momoi et al.(2012)Momoi, Sindzingre, and Kubo]momoi2012 authorauthorT. Momoi, authorP. Sindzingre, and authorK. Kubo,10.1103/PhysRevLett.108.057206journaljournalPhys. Rev. Lett. volume108, pages057206 (year2012)NoStop[Ueda and Momoi(2013)]UedaMomoi2013 authorauthorH. T. Ueda and authorT. Momoi, 10.1103/PhysRevB.87.144417journaljournalPhys. Rev. B volume87, pages144417 (year2013)NoStop[Janson et al.(2016)Janson, Furukawa, Momoi, Sindzingre, Richter, and Held]janson2016 authorauthorO. Janson, authorS. Furukawa, authorT. Momoi, authorP. Sindzingre, authorJ. Richter,and authorK. Held,10.1103/PhysRevLett.117.037206journaljournalPhys. Rev. Lett. volume117, pages037206 (year2016)NoStop[Nawa et al.(2014)Nawa, Okamoto, Matsuo, Kindo, Kitahara, Yoshida, Ikeda, Hara, Sakurai, Okubo, Ohta, and Hiroi]nawa_nmr_exp authorauthorK. Nawa, authorY. Okamoto, authorA. Matsuo, authorK. Kindo, authorY. Kitahara, authorS. Yoshida, authorS. Ikeda, authorS. Hara, authorT. Sakurai, authorS. Okubo, authorH. Ohta,and authorZ. Hiroi,10.7566/JPSJ.83.103702journaljournalJ. Phys. Soc. Jpn. volume83, pages103702 (year2014)NoStop[Büttgen et al.(2014)Büttgen, Nawa, Fujita, Hagiwara, Kuhns, Prokofiev, Reyes, Svistov, Yoshimura, andTakigawa]buttgen_nmr_exp authorauthorN. Büttgen, authorK. Nawa, authorT. Fujita, authorM. Hagiwara, authorP. Kuhns, authorA. Prokofiev, authorA. P.Reyes, authorL. E. Svistov, authorK. Yoshimura,and authorM. Takigawa, 10.1103/PhysRevB.90.134401journaljournalPhys. Rev. B volume90, pages134401 (year2014)NoStop[Nawa et al.(2017)Nawa, Takigawa, Krämer, Horvati ćć, Berthier, Yoshida, and Yoshimura]nawa2017 authorauthorK. Nawa, authorM. Takigawa, authorS. Krämer, authorM. Horvati ćć, authorC. Berthier, authorM. Yoshida,andauthorK. Yoshimura,10.1103/PhysRevB.96.134423journaljournalPhys. Rev. B volume96, pages134423 (year2017)NoStop[Grafe et al.(2017)Grafe, Nishimoto, Iakovleva, Vavilova, Alfonsov, Sturza, Wurmehl, Nojiri, Rosner, Richteret al.]grafe_nmr_exp authorauthorH.-J. Grafe, authorS. Nishimoto, authorM. Iakovleva, authorE. Vavilova, authorA. Alfonsov, authorM.-I. Sturza, authorS. Wurmehl, authorH. Nojiri, authorH. Rosner, authorJ. Richter,et al.,10.1038/s41598-017-06525-0journaljournalSci. Rep. volume7, pages6720 (year2017)NoStop[Orlova et al.(2017)Orlova, Green, Law, Gorbunov, Chanda, Krämer, Horvati ćć, Kremer, Wosnitza, and Rikken]orlova authorauthorA. Orlova, authorE. L. Green, authorJ. M. Law, authorD. I. Gorbunov, authorG. Chanda, authorS. Krämer, authorM. Horvati ćć, authorR. K. Kremer, authorJ. Wosnitza,and authorG. L. J. A. Rikken,10.1103/PhysRevLett.118.247201journaljournalPhys. Rev. Lett. volume118, pages247201 (year2017)NoStop[Yoshida et al.(2017)Yoshida, Nawa, Ishikawa, Takigawa, Jeong, Krämer, Horvati ćć, Berthier, Matsui, Goto, Kimura, Sasaki, Yamaura, Yoshida, Okamoto, and Hiroi]yoshida2016 authorauthorM. Yoshida, authorK. Nawa, authorH. Ishikawa, authorM. Takigawa, authorM. Jeong, authorS. Krämer, authorM. Horvati ćć, authorC. Berthier, authorK. Matsui, authorT. Goto, authorS. Kimura, authorT. Sasaki, authorJ. Yamaura, authorH. Yoshida, authorY. Okamoto,and authorZ. Hiroi,10.1103/PhysRevB.96.180413journaljournalPhys. Rev. B volume96, pages180413 (year2017)NoStop[Momoi and Totsuka(2000a)]momoi2000b authorauthorT. Momoi and authorK. Totsuka,10.1103/PhysRevB.62.15067journaljournalPhys. Rev. B volume62, pages15067 (year2000a)NoStop[Nojiri et al.(2003)Nojiri, Kageyama, Ueda, and Motokawa]nojiri2003 authorauthorH. Nojiri, authorH. Kageyama, authorY. Ueda,and authorM. Motokawa,10.1143/JPSJ.72.3243journaljournalJ. Phys.Soc. Jpn. volume72, pages3243 (year2003)NoStop[Papanicolaou(1988)]papanicolaou2 authorauthorN. Papanicolaou,http://dx.doi.org/10.1016/0550-3213(88)90073-9journaljournalNucl. Phys. B volume305,pages367(year1988)NoStop[Tanaka et al.(2001)Tanaka, Tanaka, and Idogaki]tanaka2001 authorauthorK. Tanaka, authorA. Tanaka, and authorT. Idogaki, http://stacks.iop.org/0305-4470/34/i=42/a=304journaljournalJ. Phys. A: Math. Gen. volume34, pages8767 (year2001)NoStop[Harada and Kawashima(2002)]harada authorauthorK. Harada and authorN. Kawashima,10.1103/PhysRevB.65.052403journaljournalPhys. Rev. B volume65, pages052403 (year2002)NoStop[Tsunetsugu and Arikawa(2006)]tsunetsugu_nematic_tri authorauthorH. Tsunetsugu and authorM. Arikawa,10.1143/JPSJ.75.083701journaljournalJ. Phys. Soc. Jpn. volume75, pages083701 (year2006)NoStop[Läuchli et al.(2006)Läuchli, Mila, and Penc]lauchli_blbq_tri authorauthorA. Läuchli, authorF. Mila, and authorK. Penc,10.1103/PhysRevLett.97.087205journaljournalPhys. Rev. Lett. volume97, pages087205 (year2006)NoStop[Takata et al.()Takata, Momoi, and Oshikawa]takata_pyrochlore authorauthorE. Takata, authorT. Momoi, and authorM. Oshikawa,@noophttp://arxiv.org/abs/arXiv:1510.02373arXiv:1510.02373NoStop[Wang et al.(2015)Wang, Kivelson, and Lee]wang_fese_nat authorauthorF. Wang, authorS. A. Kivelson,and authorD.-H. Lee,10.1038/nphys3456journaljournalNat. Phys. volume11, pages959 (year2015)NoStop[Yu and Si(2015)]yu_fese_afq authorauthorR. Yu and authorQ. Si,10.1103/PhysRevLett.115.116401journaljournalPhys. Rev. 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Momoi,10.1103/PhysRevB.87.054429journaljournalPhys. Rev. B volume87, pages054429 (year2013)NoStop[Smerald et al.(2015)Smerald, Ueda, and Shannon]smerald_nematic authorauthorA. Smerald, authorH. T. Ueda, and authorN. Shannon,10.1103/PhysRevB.91.174402journaljournalPhys. Rev. B volume91, pages174402 (year2015)NoStop[Onishi(2015)]onishi_1dnematic authorauthorH. Onishi,10.7566/JPSJ.84.083702journaljournalJ. Phys. Soc. Jpn. volume84, pages083702 (year2015)NoStop[Michaud et al.(2011)Michaud, Vernay, and Mila]mila2011 authorauthorF. Michaud, authorF. Vernay, and authorF. Mila,10.1103/PhysRevB.84.184424journaljournalPhys. Rev. B volume84, pages184424 (year2011)NoStop[Savary and Senthil()]savary_rixs authorauthorL. Savary and authorT. Senthil, @noophttp://arxiv.org/abs/arXiv:1506.04752arXiv:1506.04752NoStop[Furuya(2017)]furuya_esr_1dnematic authorauthorS. C. Furuya,10.1103/PhysRevB.95.014416journaljournalPhys. Rev. B volume95, pages014416 (year2017)NoStop[Akaki et al.(2017)Akaki, Yoshizawa, Okutani, Kida, Romhányi, Penc, and Hagiwara]akaki_esr_quad authorauthorM. Akaki, authorD. Yoshizawa, authorA. Okutani, authorT. Kida, authorJ. Romhányi, authorK. Penc,and authorM. Hagiwara,10.1103/PhysRevB.96.214406journaljournalPhys. Rev. B volume96, pages214406 (year2017)NoStop[Kubo and Tomita(1954)]kubo_tomita authorauthorR. Kubo and authorK. Tomita, 10.1143/JPSJ.9.888journaljournalJ. Phys. Soc. Jpn. volume9, pages888 (year1954)NoStop[Oshikawa and Affleck(2002)]oshikawa_esr authorauthorM. Oshikawa and authorI. Affleck,10.1103/PhysRevB.65.134410journaljournalPhys. Rev. B volume65, pages134410 (year2002)NoStop[Ozerov et al.(2015)Ozerov, Maksymenko, Wosnitza, Honecker, Landee, Turnbull, Furuya, Giamarchi, and Zvyagin]ozerov_esr_dimpy authorauthorM. Ozerov, authorM. Maksymenko, authorJ. Wosnitza, authorA. Honecker, authorC. P. Landee, authorM. M. Turnbull, authorS. C. Furuya, authorT. Giamarchi,and authorS. A. Zvyagin,10.1103/PhysRevB.92.241113journaljournalPhys. Rev. B volume92, pages241113 (year2015)NoStop[Furuya and Oshikawa(2012)]furuya_esr_boundary authorauthorS. C. Furuya and authorM. Oshikawa,10.1103/PhysRevLett.109.247603journaljournalPhys. Rev. Lett. volume109, pages247603 (year2012)NoStop[Note1()]Note1 noteWe note that, when a certain symmetry is broken spontaneously in the unperturbed system and it is also broken by the anisotropy in the full Hamiltonian ℋ, one must choose a proper direction of the infinitesimal auxiliary symmetry breaking field in the unperturbed system so that the ground state is smoothly deformed by imposing weak perturbation.Stop[Maeda and Oshikawa(2005)]maeda_shift_jpsj authorauthorY. Maeda and authorM. Oshikawa,10.1143/JPSJ.74.283journaljournalJ. Phys. Soc. Jpn. volume74, pages283 (year2005)NoStop[Povarov et al.(2011)Povarov, Smirnov, Starykh, Petrov, and Shapiro]povarov_1d_dm authorauthorK. Y. Povarov, authorA. I. Smirnov, authorO. A. Starykh, authorS. V. Petrov,and authorA. Y. Shapiro, 10.1103/PhysRevLett.107.037204journaljournalPhys. Rev. Lett. volume107,pages037204 (year2011)NoStop[Furuya et al.(2011)Furuya, Suzuki, Takayoshi, Maeda,and Oshikawa]furuya_esr_haldane authorauthorS. C. Furuya, authorT. Suzuki, authorS. Takayoshi, authorY. Maeda,and authorM. Oshikawa,10.1103/PhysRevB.84.180410journaljournalPhys. Rev. B volume84, pages180410 (year2011)NoStop[Maeda et al.(2005)Maeda, Sakai, and Oshikawa]maeda_esr_shift authorauthorY. Maeda, authorK. Sakai,andauthorM. Oshikawa,10.1103/PhysRevLett.95.037602journaljournalPhys. Rev. Lett. volume95, pages037602 (year2005)NoStop[Furuya et al.(2012)Furuya, Bouillot, Kollath, Oshikawa,and Giamarchi]furuya_esr_bpcb authorauthorS. C. Furuya, authorP. Bouillot, authorC. Kollath, authorM. Oshikawa,and authorT. Giamarchi,10.1103/PhysRevLett.108.037204journaljournalPhys. Rev. Lett. volume108, pages037204 (year2012)NoStop[Mori and Kawasaki(1962)]mk authorauthorH. Mori and authorK. Kawasaki,10.1143/PTP.27.529journaljournalProg. Theor. Phys. volume27, pages529 (year1962)NoStop[Penc et al.(2004)Penc, Shannon, and Shiba]penc_pyrochlore authorauthorK. Penc, authorN. Shannon, and authorH. Shiba,10.1103/PhysRevLett.93.197203journaljournalPhys. Rev. Lett. volume93, pages197203 (year2004)NoStop[Kimura et al.(2011)Kimura, Hagiwara, Takeuchi, Yamaguchi, Ueda, Ueda, and Kindo]Hg authorauthorS. Kimura, authorM. Hagiwara, authorT. Takeuchi, authorH. Yamaguchi, authorH. Ueda, authorY. Ueda,and authorK. Kindo,10.1103/PhysRevB.83.214401journaljournalPhys. Rev. B volume83, pages214401 (year2011)NoStop[Miyata et al.()Miyata, Ueda, and Takeyama]Cd authorauthorA. Miyata, authorH. Ueda,andauthorS. Takeyama, @noop http://arxiv.org/abs/arXiv:1302.3664arXiv:1302.3664NoStop[Miyata et al.(2012)Miyata, Ueda, Ueda, Motome, Shannon, Penc, and Takeyama]Zn authorauthorA. Miyata, authorH. Ueda, authorY. Ueda, authorY. Motome, authorN. Shannon, authorK. Penc,and authorS. 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Lett. volume82,pages3701 (year1999)NoStop[Knetter et al.(2000)Knetter, Bühler, Müller-Hartmann, andUhrig]knetter2000 authorauthorC. Knetter, authorA. Bühler, authorE. Müller-Hartmann, and authorG. S. Uhrig,10.1103/PhysRevLett.85.3958journaljournalPhys. Rev. Lett. volume85, pages3958 (year2000)NoStop[Totsuka et al.(2001)Totsuka, Miyahara, and Ueda]totsuka2001 authorauthorK. Totsuka, authorS. Miyahara, and authorK. Ueda,10.1103/PhysRevLett.86.520journaljournalPhys. Rev. Lett. volume86, pages520 (year2001)NoStop[Momoi and Totsuka(2000b)]momoi2000a authorauthorT. Momoi and authorK. Totsuka,10.1103/PhysRevB.61.3231journaljournalPhys. Rev. B volume61,pages3231 (year2000b)NoStop[Note2()]Note2 noteThe field-induced antiferroquadrupolar (AFQ) phase in the spin-gapped systems is also described as a low-density condensate of bound magnon pairs, as same as the AFQ phase near the saturation field. The vacuum is a singlet ground state and the bosonic particles are bound triplon pairs in the former case, whereas the vacuum is the fully polarized state and the bosons are bound magnon pairs in the latter.Stop[Romhányi et al.(2011)Romhányi, Totsuka, and Penc]romhanyi2011 authorauthorJ. Romhányi, authorK. Totsuka,and authorK. Penc,10.1103/PhysRevB.83.024413journaljournalPhys. Rev. B volume83, pages024413 (year2011)NoStop | http://arxiv.org/abs/1707.08784v2 | {
"authors": [
"Shunsuke C. Furuya",
"Tsutomu Momoi"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727085754",
"title": "Electron spin resonance for the detection of long-range spin nematic order"
} |
http://arxiv.org/abs/1707.08579v2 | {
"authors": [
"C. Repellin",
"A. M. Cook",
"T. Neupert",
"N. Regnault"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726180005",
"title": "Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures"
} |
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Polarization Transfer Observables in Elastic Electron-Proton Scattering at 𝐐^2= 2.5, 5.2, 6.8 and 8.5 GeV^2 L. Zhu December 30, 2023 ===========================================================================================================In a black hole, hair and quantum information retrieval are interrelated phenomena. The existence of any new form of hair necessarily implies the existence of features in thequantum-mechanically evaporatedradiation. Therefore, classical supertranslation hair can be only distinguished from global diffeomorphisms if we have access to the interior of the black hole. Indirect information on the interior can only be obtained from the features of the quantum evaporation. We demonstrate that supertranslations (T^-,T^+) ∈ BMS_-⊗ BMS_+ can be used as bookkeepers of the probability distributions of the emitted quantawhere the first element describes the classical injection of energy and the second one is associated to quantum-mechanical emission. However,the connection between T^- and T^+ is determined by the interior quantum dynamics of the black hole.We argue that restricting to the diagonal subgroup is only possible for decoupled modes, which do not bring any non-trivial information about the black hole interior and therefore do not constitute physical hair.It is shown that this is also true for gravitational systems without horizon, for which both injection and emission can be described classically. Moreover, we discuss and clarify the role of infrared physics in purification. § INTRODUCTION AND SUMMARY §.§.§ The Puzzle of Black HolesA black hole is an extraordinary physical system. While in a classical theory, it is extremely simple for an outside observer, as a consequence of the no-hair-theorem (see <cit.>), its internal quantum complexity measured by the Bekenstein-Hawking-entropy <cit.> N=M^2/M_p^2 is enormous. Both properties are obviously interrelated. The black hole entropy appears because many different matter configurations can collapse into the same black hole geometry. The no-hair-theorem prevents an outside observer from resolving these differences which remain hidden behind the horizon. Quantum-mechanically, the black hole evaporates <cit.> and unitarity requires that along the evaporation process the black hole should deliver back the information which was classically hidden in its interior <cit.>. This means that although the classical metric has no hair, the evaporation products should have features which compensate for this lack of information. In other words, the quantum radiation emitted during the black hole evaporation should carry the same information which the classical no-hair-theorem prevents us to extract from the geometry. In the last twenty years it has become popular to use the AdS/CFT correspondence as strong indication of the unitarity of black hole evaporation. However, this hope will not be fulfilled until counting with the CFT dual of a small evaporating black hole has been achieved. More generally, we shall argue in this note that the solution to the evaporation problem requires to have a microscopic model of the black hole as a quantum system –whether obtained from AdS/CFT or differently. In <cit.>, we have developed such a model that, among other things, indicates the existence of forms of quantum hair effects of order 1/N. Moreover, and in a model independent way, it is easy to see that taking into account the change of the black hole mass due to Hawking evaporation leads to deviations from featureless emission on precisely this order of magnitude <cit.>.§.§.§ Classical BMS-HairRecently, a new way to attack the problem has been suggested <cit.> based on asymptotic BMS-symmetries <cit.>. This approach has received widespread attention (see <cit.>). In particular, a potential new form of classical hair for a black hole has been proposed <cit.>. The idea is simply the following. One starts with a black hole of mass M and injects an energy μ in the form of incoming radiation with some angular features.[All quantities will be properly defined at the beginning of section <ref>.]This incoming radiation can be associated with a supertranslation in BMS_- which we denote by T^-. Classically, the resulting system is a black hole with total mass M+μ but supertranslated by T^-. One can do the same construction with identical μ but with different angular features, different supertranslations T_i^-, to obtain a family of different metrics all of them with the same ADM-charges. Thus, it seems that one can indeed define classical hair if all these metrics sharing the same ADM-charges are physically inequivalent. At the classical level, this means that those metrics are not just the same metric written in different coordinate systems, that they are not related by a globally defined diffeomorphism. As we shall elucidate, the problem with this form of classical hair is that for an observer outside, there is no way to decide if all these metrics are different or simply the same metric in different coordinates. In order to decide that, the observer needs to have information about the interior of the black hole. In summary, defining hair by means of the classical gravitational memory associated to some incoming radiation is only operative if somehow we can have extra information about the memory effects in the interior of the black hole which is, in a different guise, the essence of the no-hair-theorem. Fortunately, there is an indirect way to decide from the outside whether two black hole metrics defined by injecting the same amount of energy but with different angular features are physically different or not. We can just wait until the black holes emit some radiation and compare the radiation produced by the two black holes. For simplicity, we restrict ourselves in our discussion to a pure theory of gravity in which only gravitational radiation can be emitted. The corresponding process is depicted in figure <ref>, where we distinguish the classical, semi-classical and purely quantum contributions. The first thing to be noticed is that this test is purely quantum in the sense that only quantum-mechanically, the black hole can emit radiation. The second thing is that the information we can get on the emitted quantum radiation by actual measurements is necessarily encoded in the form of probability distributions. Thus, if those black holes defined by different T^i_- are indeed different, we should expect that the corresponding quantum probability distributions are also different. §.§.§ Insufficiency of Calculation in Classical Background MetricIt is natural to expect that this difference has a non-trivial projection on deviations from isotropy, that the emitted quanta carry angular features. Then we obtain quantum probability distributions P_i(θ,ϕ) from the measurement of the radiated quanta. We can use those to define a classical supertranslation T_i^+. By that we simply mean a classical supertranslation with the flow of emitted radiation determined by the measured quantum probability functions P_i(θ,ϕ). In this sense, the former experiment produces a set of couples (T_i^-, T_i^+) where the first supertranslation in BMS_- is classical and the second one in BMS_+ is determined by the quantum probability distribution. From this point of view, if the classical T_i^- really implants hair, then the quantum T_i^+ should be non-trivial, contain spherical harmonics with l≥ 2.The crucial point is that this behavior cannot be achieved by the standard Hawking computation performed in a supertranslated Schwarzschild metric, as pair creation in the background vacua defined by the near horizon geometry. The reason is that the supertranslation acts as a diffeomorphism near the horizon and does not change the local geometry. Therefore, it does not suffice if the P_i only depend on the injected radiation and the geometry of the black hole. Instead, they must also depend on its internal dynamics. We can make the argument a bit more quantitative and assume that from the whole energy μ injected a fraction μ̃ is associated to angular features. This means that the part of the incoming classical flow ℱ_in with angular labels l≥ 2 contributes to ∫[2]Ωℱ_in with a value equal to μ̃. Clearly, μ̃=0 would correspond to the injection of featureless radiation.[In this case, the associated supertranslation will not support angular features and will only project on the l=0,1 spherical harmonics.] In order to parametrize how the P_i depend on μ̃ and the internal structure of the black hole, we shall use the typical number of quantum constituents of the black hole. In this sense, we expect P_i(θ,ϕ; μ̃, N) where the label i refers to the dependence on the incoming T_i^- and where we identify the number of quantum constituents of the black hole with the entropy N. Then the natural dimensionless parameter measuring the dependence of P_i on the internal structure is μ̃/√(N) where √(N) is the black hole mass in Planck units. In this setup, angular features in the evaporation, finite N effects in P_i(θ,ϕ; μ̃, N), depend on the black hole microscopic model. Those will define a couple (T_i^-, T_i^+)generically not in the diagonal subgroup BMS_0 of BMS_- ⊗ BMS_+.Thus, the first important message of our note will be that the information about T_i^- cannot determine the quantum probability distribution T_i^+, we cannot predict the quantum probability distribution solely from the incoming radiation implanting the hair. §.§.§ Subleading Soft Modes We can investigate how this situation changes in the semi-classical limit M→∞, N→∞, in which the Hawking computation becomes exact. In this case, the energy associated to features becomes zero so that angular features can only be encoded in zero-energy modes. The effective decoupling of these modes will lead to a P_i identical to the incoming radiation. This produces couples (T_i^-, T_i^+) in the diagonal subgroup BMS_0. In more concrete terms, the lim_N→∞ P_i(θ,ϕ; μ̃, N) will only capture local horizon physics or zero-energy modes.[It is important to stress that the pseudo Goldstone-Bogoliubov modes identified in <cit.> are not equivalent to near horizon diffeomorphisms and consequently are good microscopic candidates to describe the low energy effective changes of the microstate of the black hole during the process of absorption and evaporation.]This brings us to our second point, namely how the actual features of the quantum probability distribution P_i depend on infrared physics.[See <cit.> for a recent suggestion for purification by infrared modes.]We know that in gapless theories such as gravity, evaporation interpreted as a S-matrix process has a zero probability amplitude without any accompanying soft gravitons. In order to obtain a finite answer, one has to include the emission of a certain class of soft radiation, namely IR-modes. However, this fact by no means implies that this companion radiation should carry the angular features that we need to purify the evaporation.On the contrary, we know from infrared physics that IR-radiation is only sensitive to the initial and final scattering states. It is independent of the details of the process or in our case of the microscopic details of the black hole, cannot resolve the microstate. Independently of the question to what extent IR-radiation and hard quanta are correlated, we can quantitatively estimate the amount of information we could lose when we integrate over unresolved IR-modes. From well-known results of infrared physics it follows that their number only grows logarithmically with the resolution scale ϵ, n_soft∼ -lnϵ. However, what we have discussed implies that the natural resolution scale of features should be ϵ∼ 1/N. Thus, the second important message of our note is that unresolved IR-modes cannot account for thebulk of information in black hole evaporation, but could only contribute as a subleading logarithmic correction. The part which carries features is the part of the radiation that can be resolved and that depends not on the infrared divergences but on the inner structure of the black hole, or in scattering language, on the details of the scattering process. A possible candidate is soft non-IR radiation, which is independent of infrared divergences. As it should be, non-IR radiation depends on the details of the scattering process so that it cannot be predicted without a microscopic theory of the black hole. §.§.§ Summary and OutlineIn summary, non-trivial hair can be only defined by couples (T_i^-, T_i^+) ∈ BMS_-⊗ BMS_+ where the element in BMS_- is classicaland carries some finite energy and the element in BMS_+ is defined as a bookkeeper of the quantum probability distribution of the radiated quanta. What concrete element T_i^+ is associated with a given T_i^- cannot be derived solely from the classical geometry, but depends on the internal quantum structure of the black hole.This non-trivial mapping is precisely what makes the quantum hair informative. For a system with non-trivial dynamics, it is therefore impossible to restrict to a subgroup of BMS_- ⊗ BMS_+. Predictivity on this quantum output can be only achieved in the zero-energy (or equivalently N=∞ limit) where we only get elements in the diagonal subgroup BMS_0.[In <cit.> and <cit.> it is suggested to constraint the potential values of T_i^+ using an infinite set of conserved charges. Imposing these conservation laws makes the corresponding S-matrix completely insensitive to the internal structure of the black hole and consequently, in the language we are using here, can only capture unobservable zero-modes.]But since the soft modes are decoupled once the infrared divergences of the theory are properly taken into account <cit.>,they cannot lead to observable features.[This decoupling of soft modes is a quantum effect that should not be confused with the existence, for instance in asymptotically Minkowski space time, of a non-trivial family of asymptotically flat connections defining a representation of the BMS-group (see <cit.> and references therein). This multiplicity of classical inequivalent vacua is quantum-mechanically reabsorbed in the cancellation of infrared divergences.] The outline of the paper is as follows. In section <ref>, we first recap some properties of BMS-gauge. In particular, we show how angular features of radiation define a supertranslation, which can be measured as a memory effect. Moreover, we discuss the role of soft modes. Then we use a combination of injected and emitted radiation of the same total energy to define Goldstone supertranslations as element (T^-, T^+) ∈ BMS_-⊗ BMS_+.In section <ref>, we first concentrate on a gravitational system without horizon, which we shall call planet for concreteness, and show how we can use Goldstone supertranslations to change its angular distribution of mass. In doing so, the key point is that it is impossible to infer T^+ from T^- unless one knows the internal dynamics of the planet. Moreover, we highlight the importance of angular features by showing that it is impossible to determine the angular mass distribution of the planet without access to its interior. Subsequently, we apply Goldstone supertranslations to a black hole. We demonstrate how supertranslations can be used as bookkeeping tool for the emitted quanta. However, without knowledge of the microscopic dynamics of the black hole, they have no predictive power.We also point out how we can use Page's time to estimate the magnitude of deviations from featureless evaporation. After concluding in section <ref>, we provide a more detailed discussion of IR-physics in appendix <ref>. In appendix <ref>, we discuss the matching of the supertranslation field in advanced and retarded coordinates and finally we explicitly calculate a Goldstone supertranslation of a planet in appendix <ref>.§ QUANTUM HAIR§.§ Recap of BMS-Gauge and Memory Effect §.§.§ Retarded CoordinatesWe first recap some properties of BMS-gauge, which is defined by the four gauge conditions <cit.>g_11 = g_1A = 0, g_AB = r^2 sin^2θ , where A,B, … = 2,3. Typically, BMS-gauge is used to study a spacetime asymptotically, for r→∞, but it is possible to extend the metric to the bulk by imposing the conditions (<ref>) to all orders in 1/r. In a typical situation, however, a metric in BMS-gauge does not cover the whole spacetime. A metric in BMS-gauge exists both in retarded time u, which is suited to describe outgoing radiation, and in advanced time v, which is suited to describe incoming radiation. The matching between these two metrics will be crucial for our treatment. Explicitly, an asymptotically flat metric in retarded time takes the form <cit.>:s^2= (-1 + m^+_B/r + O(r^-2)) u^2 - (2+O(r^-2))ur + r^2 (γ_AB + C_AB^+ r^-1 + O(r^-2))x^Ax^B + O(r^-2)x^Au ,where the metric on the sphere has to fulfill the requirement g_AB = r^2sin^2 θ. Here m_B^+ is the Bondi mass, γ_AB the standard metric on the sphere and C^+_AB = (2 D_A D_B - γ_ABD^2 )C^+is determined by the supertranslation field C^+, where D_A is the covariant derivative on the sphere. It is helpful to expand the supertranslation field in spherical harmonics. Then the mode l=0 represents a time shift and the mode l=1 corresponds to spatial translations. Therefore, all modes with l≥ 2 define proper supertranslations.Metrics with different values of C^+ are connected via asymptotic diffeomorphisms, the choice of the supertranslation field constitutes a residual gauge freedom of BMS-gauge. These diffeomorphisms are the famous supertranslations. Therefore, we can define a supertranslation T^+ by the change it induces in the supertranslation field:T^+ := Δ C^+. In order to analyze the effect of supertranslations, we will need the constraint equation G_00=8π G T_00, whose leading order reads in BMS-gauge:∂_u m^+_B =1/4GD^2(D^2+2) ∂_u C^+ -ℱ_out ,where ℱ_out = 1/8(∂_u C^+_AB)(∂_u C^+AB) + 4πlim_r→∞ (r^2T_uu)is the total incoming null energy, composed of gravitational waves (first summand) and other forms of gravitating energy (second summand). §.§.§ Advanced CoordinatesThe situation in advanced coordinates is very similar. The metric takes the forms^2= (-1 + m^-_B/r + O(r^-2)) v^2 + (2+O(r^-2))vr + r^2 (γ_AB + C_AB^- r^-1 + O(r^-2))x^Ax^B + O(r^-2)x^Av ,where the supertranslation field and the supertranslations in advanced coordinates are defined as in (<ref>) and (<ref>). The constraint equation becomes∂_u m^-_B =1/4GD^2(D^2+2) ∂_u C^- +ℱ_in ,where ℱ_in is the incoming energy, in analogy to (<ref>).§.§.§ Measurement of the Supertranslation Field: Memory EffectAs already discussed, one can change the value of the supertranslation field by a diffeomorphism. Therefore, it follows by general covariance that the value of the supertranslation field cannot have in general any experimental implication. However,since it corresponds to physical outgoing or ingoing radiation, the difference of the supertranslation field at different times does have experimental implications: It describes the memory effect caused by the radiation, a permanent displacement of test masses after the radiation has passed <cit.>. We will restrict ourselves to a simple situation in which we start with some stationary metric g^1_μν and we finish in a different stationary metric g^2_μν. In between, there is a radiation epoch, ℱ_in/out only has support during this time span. Asymptotically on 𝒥^±, the process defines a non-stationary metric interpolating between g^1_μν and g^2_μν which should be a solution to the Einstein equations. Since Birkhoff's theorem implies that we can set ∂_A m_B^±= 0 in a stationary metric, we can single out the zero-mode from (<ref>) by integrating over the sphere:μ^+ = -∫u∫[2]Ωℱ_out/4π ,where we first consider retarded time and μ^+ = m^+_B, 2- m^+_B, 1 is the total change of Bondi mass due to the radiation epoch. This formula shows explicitly that the Bondi mass m^+_B is monotonically decreasing, it measures the energy which has not yet left the bulk.Defining the emitted energy with non-trivial angular distribution as Δℱ̃_out := ∫uℱ_out - μ^+, the constraint (<ref>) becomes0 =1/4GD^2(D^2+2) T^+ - Δℱ̃_out . Thus, angular features in the outgoing radiation induces a supertranslation T^+=Δ C^+. Note that it is independent of the total emitted energy μ^+. In advanced coordinates, we get from the constraint (<ref>):μ^- = ∫v∫[2]Ωℱ_in/4π . The advanced Bondi mass m^-_B is monotonically increasing, it measures the energy which has already entered the bulk. Defining Δℱ̃_in := ∫vℱ_in - μ^-, the constraint (<ref>) becomes0 =1/4GD^2(D^2+2) T^- + Δℱ̃_in . This formula implies that an advanced supertranslation T^- tracks angular features in the incoming radiation.§.§ Goldstone SupertranslationsAs already pointed out, we shall define hair on the basis of scattering processes where some injected gravitational energy is radiated back by the system. The hair will be encoded in the angular features of the injected radiation and the outgoing radiation. In this sense, we define hair as a typical response function. Through these formal scattering processes we define a map relating gravitational systems, black holes or planets, in different states sharing the same values for all the ADM-conserved quantities. We denote this induced map a Goldstone supertranslation since it relates states which are degenerate in energy. Note that this scattering definition of hair is tied to the mechanism of radiation whatever it could be. §.§.§ Relationship to Antipodal Matching As a first step, it is important to discuss whether there are general constraints on this scattering process. Namely, it has been suggested in <cit.> that any gravitational S-matrix in an asymptotically flat spacetime must satisfy the following relation for an arbitrary initial quantum state |α⟩:S T^- |α⟩ = P(T^+) S |α⟩ , where (T^-, T^+) ∈ BMS_- ⊗ BMS_+ and P is the antipodal map on the sphere:P(T^+)(θ, φ) = T^+(P(θ, φ)). Imposing this invariance implies that if a matrix element ⟨β| S |α⟩ is non-vanishing, then⟨α| T^- |α⟩ = ⟨β|P(T^+) |β⟩ .This means that the memory effect of the outgoing wave, parameterized by T^+, must match the memory effect of the incoming wave, parameterized by T^-, antipodally at each angle. Because of the constraints (<ref>) and (<ref>), this is equivalent to the statement that the outgoing energy Δℱ̃_out matches the ingoing energy Δℱ̃_in antipodally at each angle, in particular that Δℱ̃_out is fully determined in terms of Δℱ̃_in.This criterion has a very interesting connection to IR-physics. As discussed in appendix <ref>, we know that in a gapless theory such as gravity most process in which no soft modes are emitted have zero probability <cit.>. In order to obtain a finite answer, one has to include the emission of a certain class of soft radiation, namely IR-modes. The sole exception are processes for which the kinematical factor B_α,β defined in <cit.> is zero. The crucial point is that this happens if and only if the ingoing energy matches the outgoing energy antipodally at each angle, as discussed in detail in <cit.>. Thus we conclude that[We will elaborate on this point in <cit.>.]⟨α| T^- |α⟩ = ⟨β|P(T^+) |β⟩⇔B_α, β=0.This means that restricting to processes which fulfill the condition (<ref>) is equivalent to only considering processes that are IR-finite even without including IR-emission.A priori, there is nothing wrong with solely considering such processes. However, they only form a set of measure zero of those processes that occur in reality. Namely any realistic scattering is accompanied by the emission of soft IR-modes. Once we include soft IR-emission, we know that all processes – with an arbitrary non-zero value of B_α,β – are IR-finite. Thus in reality, any process can occur, also ones that do not fulfill the antipodal matching condition (<ref>). For this reason, we will not restrict ourselves to processes that obey (<ref>). §.§.§ Role of Soft IR-Gravitons Since we consider processes that include the emission of soft IR-modes, it is natural to ask if those modes could carry information about the black hole state and if they could even suffice to purify black hole evaporation. This is only possible if two conditions are fulfilled. First, IR-modes would have to be sensitive to the microstate of the black hole. We expect this not to be the case since they only depend on the initial and final scattering state, but not on the details of the process. While we leave the above question for future work, we now focus on the second condition, namely that the number of resolvable IR-modes would have to be big enough to be able to carry the whole black hole entropy.In contrast to the proposal made in <cit.>, we argue that generic properties of IR-physics imply that this is not the case. As follows from equation (<ref>),the number of unresolved soft modes scales logarithmically with the IR-resolution scale. Thus, when we lower the energy scale of resolution from ϵ_1 to ϵ_2, the number of additional IR-modes that we can resolve is:n^res_soft∼ B_α,βlnϵ_1/ϵ_2 ,where B_α,β∼ G s is determined by the energy scale s of the process.We apply this formula to the single emission of a Hawking quantum of energy r_g^-1. It will be crucial in this argument that Hawking radiation gets softer for bigger black holes. The worst resolution scale compatible with observing this process is ϵ_1=r_g^-1. The key point is that the resolution scale in this process cannot be arbitrarily good. Namely, it is set by the time-scale of the process, ϵ_2∼ t_b-h^-1. Since the life-time of a black hole scales as t_b-h∼ N r_g, we get n^res_soft≲1/Nln N.Thus, after the black hole has evaporated by emitting N Hawking quanta, the maximal entropy contained in the soft IR-modes isS_soft≲ln N.Independently of the question whether IR-modes are strongly correlated with the Hawking quanta, this shows that they cannot account for the whole entropy of the black hole, but could only give a logarithmic correction. Of course, this leaves open the possibility that non-IR soft modes could account for the bulk of black hole information. However, since they are independent of IR-divergences and accompanying dressing tools, the results of infrared physics do not constrain them. §.§.§ Role of Zero-Energy GravitonsFinally, we briefly discuss the role of zero-energy gravitons. To this end, we consider the process of a Goldstone supertranslation in the limit of zero energy injected and zero energy radiated. This is equivalent to the scattering with a graviton of zero energy.Since those carry no energy, they cannot emit IR-modes and therefore obey the antipodal matching condition (<ref>). This fact simply reflects the well-known decoupling of soft modes <cit.>. The physical interpretation of this phenomenon is that any bulk configuration is transparent for decoupled soft modes so that the energy profile of the outgoing wave is antipodally related to that of incoming energy. But when the emitted/injected radiation does not carry energy, μ^±=0, then the constraint (<ref>) (or respectively (<ref>)) implies that ∫[2]Ωℱ_in/out =0. Since ℱ_in/out represents real gravitational radiation, it follows by the requirement of positive energy that ℱ_in/out=0. Thus, only supertranslations with D^2(D^2+2) T^± = 0 can occur in such a zero-energy process. This means that only the angular modes l=0,1 are left, time- and space-translation. Hence zero-energy radiation cannot lead to a physical memory effect that is observable in finite time. In other words, it is impossible to measure a zero-energy graviton in finite time. The upshot is that predicting T^+ from the knowledge of T^- is only possible for zero-energy modes. Those are, however, unphysical since they cannot be measured in finite time. So we will only consider processes of non-zero energy in our paper. As explained, it is not possible for them to constrain or even predict T^+ from T^- without detailed knowledge of the dynamics in the bulk. The response function, which determines T^+ in terms of T^-, is trivial only for modes of zero energy.§.§.§ Physical Hair With Non-Zero EnergySo from here on, we consider the case where after we inject radiation ℱ_in of non-zero total energy μ, the system radiates back the same total amount of energy, but with a possibly different distribution ℱ_out.[We recall that we restrict ourselves for now to a pure gravitational radiation, which propagates along null geodesics. Therefore, all emitted energy is bound to reach future null infinity 𝒥^+.]While such systems are of course special, we will see that black holes can be one of them. This is a zero-energy process in the sense that the total energy of the system does not change. Thus, this process, which is depicted in figure <ref>, constitutes a transformation between degenerate systems and therefore defines hair. As far as we reduce ourselves to gravitational radiation, we can generically describe this process in terms of two supertranslations: At 𝒥^-, T^- is determined by the angular distribution Δℱ̃_in of incoming energy according to the constraint (<ref>) and at 𝒥^+, T^+ follows from the angular distribution Δℱ̃_out of outgoing energy via the constraint (<ref>).Thus, the whole process is associated to an element (T^-,T^+) ∈ BMS_-⊗ BMS_+. It describes a zero-energy transition which interpolates between two spacetimes of the same total energy, but contrary to the case of a zero-energy mode, this transformation is non-trivial and it is not decoupled.It is crucial to note that for an asymptotic observer, T^- and T^+ are independent. Whereas one is free to choose T^- by preparing an appropriate incoming radiation, T^+ is sensitive to the properties of the system in the bulk. In other words, T^+ is a response of the system which does not only depend on the ingoing radiation, parameterized by T^-, but also on the state of the system and its particular dynamics, which are not entirely visible asymptotically. In particular, there is no reason why (T^-,T^+) should be in any subgroup of BMS_- ⊗ BMS_+. §.§.§ Coordinate Matching In order to compare ingoing and outgoing radiation, T^- and T^+,we need to relate the supertranslation field C^- in advanced coordinates to the supertranslation field C^+ in retarded coordinates. Namely, we assume that we are given a classical spacetime whose asymptotic behavior is fully known to us. Then it is possible to describe this spacetime both in advanced and retarded BMS-gauge. Given an advanced coordinate system g_μν^v, we want to know if there is a unique retarded coordinate system g_μν^u we can associate to it. If we have such a mapping, it determines the relation of the advanced supertranslation field C^-, defined as the r^1 part of g^v_AB, and the retarded supertranslation field C^+, defined as the r^1 part of g^u_AB.Given g_μν^v, we therefore have to find a diffeomorphism 𝒟 such that g_μν^u:=𝒟(g_μν^v) is in retarded BMS-gauge. Then we can read off from g_μν^u the C^+ associated to C^-. However, we could have instead considered the diffeomorphism 𝒟'= T^+∘𝒟, where T^+ is a supertranslation diffeomorphism in retarded coordinates. Also 𝒟' transforms the metric in advanced BMS-gauge to a metric in retarded BMS-coordinates. Clearly, if T^+ is a nontrivial supertranslation, the supertranslation field in the resulting metric differs from the one in g^u. From this consideration it is obvious that the matching between the advanced and the retarded supertranslation field is in general not unique. The only hope we could have is that there is a natural way to identify C^- and C^+.In a static situation, a natural prescription is to require that the spatial part of the two metrics matches,g^u_AB = g^v_AB .As is shown explicitly in appendix <ref> for the example of the Schwarzschild metric, we can achieve this by identifying C^+(θ, φ) = - C^-(θ, φ), as also proposed in <cit.>. Up to a sign, we match the supertranslation field angle-wise. Consequently, the same matching holds for the supertranslations:T^+(θ, φ) = - T^-(θ, φ). There are several reason why the coordinate matching (<ref>) is natural. First of all, the prescription (<ref>) comes from a simple intuition. For an observer in a static spacetime who lives on a sphere of fixed radius, the description of the sphere should be the same independently of the choice of time coordinate. More generically, it is possible to require that the action of advanced and retarded supertranslations is the same in the bulk. This was done in <cit.> for the cases of Schwarzschild and Minkowski. Moreover, we can consider a detector at big radius which is sensitive to gravitational memory. Then we investigate a process of back scattering, in which the angular distributions of incoming and outgoing energy are identical at each angle. This corresponds to a wall in the bulk which reflects the wave without further modifying it. In this case, the memory effect the ingoing wave causes, parameterized by T^-, is exactly canceled by the memory effect of the outgoing wave, parameterized by T^+, so that there is no overall memory effect after the process. In that case, if we match T^- and T^+ at each angle as in (<ref>), it is possible to simply describe the overall memory effect as T^-+T^+.However, it is crucial to stress that the coordinate matching (<ref>) does not have any constraining power on the physical process. It does not predict outgoing from ingoing radiation, but only shows how one and the same setup can be described in different coordinates. This is also evident from figure <ref>. The matching condition at i^0 only relates the absolute values of the supertranslation fields. In contrast, processes of non-zero energy solely determine a change of the supertranslation field, as is clear from equations (<ref>) and (<ref>). Thus, radiation of non-zero energy is independent of the coordinate matching. § APPLICATION OF HAIR§.§ Planetary Hair In order to make the ideas presented above concrete, we discuss an explicit example, namely theapplication of a Goldstone supertranslation to a certain class of planets. We start from a spherically symmetric nongravitational source T_μν, which sources a spherically symmetric spacetime g_μν with ADM-mass M. In such a spacetime, we want to realize a Goldstone supertranslation, we send in a wave with total energy μ and angular distribution Δℱ̃_in in such a way that after some time, the planet emits a wave of the same energy μ but with a possibly different angular distribution Δℱ̃_out. Of course, only a special class of planets behaves in that way.We explicitly construct such spacetimes in appendix <ref>, to which we refer the reader for details of the calculation. First, we consider the incoming wave. As discussed, the angular distribution Δℱ̃_in of injected energy determines an advanced supertranslation T^-. As derived in equation (<ref>), we can use it to describe the change of the metric due to the injected radiation:δ g^v_μν = τ_v_0, v_1(v) s^-(r) (ℒ_ξ_v(T^-) g^v_μν + 2 μ G/rδ_μ^0 δ_ν^0 ),where ℒ_ξ_v(T^-) g^v_μν is an infinitesimal supertranslation which changes the supertranslation field by a small amount T^-. Whereas the asymptotic supertranslation T^- only depends on the leading part of the incoming energy, it is crucial to note that the transformation (<ref>) also depends on a careful choice of the subleading components of the incoming wave.[Subleading terms are the 1/r^3-term in T_00 and the whole T_0A in (<ref>). If one does not insist that the wave acts as a supertranslation also in the bulk, one is free to choose the coefficient of one of the two terms. The other one is determined by energy conservation: T_μν^μν;μ=0.]Only with a particular choice, the wave acts as a diffeomorphism not only asymptotically but also in the bulk outside the planet.We observe that the effect of the wave is twofold. First, it adds the total mass μ to the planet and secondly, it supertranslates the metric by T^-. However, these effects are localized both in space and time. The function τ_v_0, v_1(v) describes the smooth interpolation between g^v_μν and g^v_μν + δ g^v_μν, we have τ_v_0, v_1(v<v_0)=0 and τ_v_0, v_1(v>v_1)=1. The function s^-(r) describes the absorption of the wave, namely absorption takes place whenever s^- '(r)<0. There is no absorption outside the planet, s^-(r>R)=1, where R is the radius of the planet, and the wave is fully absorbed before it reaches the center, s^-(r=0)=0. It will be crucial to note that the transformation s^-(r)ℒ_ξ_v(T^-) g^v_μν only acts as a diffeomorphism when s^- '(r)=0. Moreover, the transformation (<ref>) shows that we focus on planets which have a second very special property aside from the fact that they emit as much energy as they receive: Namely there is no transport of energy between different angles. This means that the mass of the planet does not redistribute after absorption (the same will be true after emission).The fact that this assumption is unnatural and not true for generic systems will contribute to our conclusions.As a second step, we consider the emission of a wave by the planet.Of course, the properties of the emitted wave depend on the internal dynamics of the source T_μν. It is crucial to note we cannot resolve them in our purely gravitational treatment, we cannot predict what wave will be emitted. From the point of view of gravity, any emission process is possible as long as it respects energy-momentum-conservation. However, we can study the effect of a given emitted wave. As derived in equation (<ref>), it can be described in terms of the supertranslation T^+ induced by the angular distribution Δℱ̃_out of outgoing energy:δ g^u_μν = τ_u_0, u_1(u)s^+(r) (ℒ_ξ_u(T^+) g^u_μν - 2μ G/rδ_μ^0 δ_ν^0 ).As for the case of absorption, the emission has two effects: It decreases the total mass by μ and it supertranslates the metric by T^+. Moreover, it is localized in space and time in an analogous manner. We want to compare the planet before and after the Goldstone supertranslation, we are interested in the combined effect of the transformations (<ref>) and (<ref>). To this end, we have to specify a mapping between the advanced and retarded supertranslations. As explained in section <ref>, we employ the angle-wise matching (<ref>). Thus, we obtain the static final state of the planet:δ g^tot_μν=θ(r-R) ℒ_ξ_u(T^+-T^-) g_μν + θ(R-r)(s^+(r)ℒ_ξ_u(T^+)g_μν - s^-(r) ℒ_ξ_u(T^-) g_μν) . We get a planet which has the same ADM-mass but a different angular distribution of mass. This is clear from the fact that the transformation (<ref>) acts as a diffeomorphism only outside the planet. Since we used in our computation a planet with the special property that its angular distribution of energy is frozen, we can read off the distribution from difference of energy distributions of the injected and emitted wave. In this case, T^- - T^+ encodes all information about the angular energy distribution of the planet in the bulk.[For the planet with frozen energy distribution, there is also a very literal way in which one can interpret the quantity T^- - T^+: One can imagine a gedankenexperiment where a source of light is located in the interior of the planet after the Goldstone supertranslation and we collect the light rays on the sky. The light sent from this common center point determines in this way a section at infinity described by the supertranslation field T^- - T^+.Thus, the different redshift effects due to the inhomogeneities of the planet matter distribution define a supertranslated section in the sky as the one for which light rays originate from a common spacetime point. This is reminiscent of Penrose's concept of "good sections" <cit.>.]However, this is no longer true for generic systems which exhibit non-trivial dynamics after absorption and emission. In that case, T^- and T^+ merely encode the initial state. Only with full knowledge of the theory which governs the internal dynamics of the planet, we can infer the state of the planet at a later time from the asymptotic data T^- and T^+. §.§.§ The Role of Supertranslations In summary, we obtain the following key properties of a Goldstone supertranslation in the case of a planet: Outside the planet, it acts as a diffeomorphism. In particular, it does not change its ADM-mass. In contrast, it does not act as a diffeomorphism inside the planet where absorption takes place. Therefore, it is not a trivial global diffeomorphism but changes the spacetime physically. Thus, the Goldstone supertranslation encodes differences in the angular distribution among matter configurations degenerate with respect to the ADM-conserved quantities. It is crucial to discuss the role of supertranslations in this process: * For an asymptotic observer, (T^-, T^+) can be used as label for the angular features of ingoing and outgoing radiation. * An asymptotic observer, however, cannot infer T^+ from T^-. This is only possible with knowledge of internal dynamics of the planet. * Thus, (T^-, T^+) is a bookkeeping tool but without detailed information about the interior, it does not have predictive power.As we shall discuss in a moment, the same conclusions hold in the black hole case. The only difference is that the internal dynamics leading to emission are fully quantum mechanical for a black hole. This will mean that in any classical description, supertranslation cannot constrain or even predict black hole evaporation. Using the example of the planet, it is easy to convince ourselves that antipodal matching cannot play a role in processes of non-zero energy. Namely if it did, this would mean that the only planets which could exist would have the extremely special property that they emit all energy they receive from one side exactly on the other side. §.§.§ Hidden Angular FeaturesFinally, we discuss the transformation (<ref>) when we do not have access to (T^-, T^+), when we do not record ingoing and outgoing radiation but only compare the initial and final state of the planet. In that case, the planet possesses an interesting property, namely a special kind of no-hair-theorem. Concretely, we take the perspective of an observer who has no access to the interior of the planet and discuss the difference between two planets which have the same mass but a different angular mass distribution. As we have observed, the transformation (<ref>) acts as a diffeomorphism outside the planet. Therefore, an outside observer cannot distinguish the two following cases when he is given a supertranslated outside metric. First, it could be the result of the transformation (<ref>), where the planet was physically changed due to a Goldstone supertranslation. Secondly, however, one can also obtain the supertranslated metric by acting on the initial planet with a global diffeomorphism. In this case, clearly, the planet does not change. Thus, also for a planet, an outside observer is not able to resolve angular features. In order to decide whether two asymptotic metrics differing by a supertranslation describe two different distributions of matter or the same distribution of matter in different coordinates, one needs access to the whole spacetime, the interior of the source.We conclude that generic gravitational systems posses physical angular features which are inaccessible for an outside observer. This strengthens our believe that the microstates of a black hole have a non-trivial projection on angular features. The only difference is that while the restriction to outside measurements was artificial in the case of the planet, an outside observer has in principle no access to the interior of a black hole. As we will discuss in the next section, he can therefore never decide whether a supertranslated metric corresponds to a physical change of the matter inside the black hole or to a global and therefore meaningless diffeomorphism. This is the reason for the classical no-hair theorem of a black hole and why we assign an entropy to the black hole and not to the planet. §.§ Black Hole Quantum Hair §.§.§ Supertranslations as Bookkeeping DeviceNow we are ready to discuss the system of our interest, namely black holes. Since absorption and emission are of different nature in that case, we will discuss them separately. For absorption, we can proceed in full analogy to the planet and inject a wave with total energy μ and arbitrary angular distribution Δℱ̃_in. By Birkhoff's theorem, the spacetime outside the black hole is the same as for the planet so that the wave behaves identically. As in the case of the planet, the wave cannot be absorbed outside the horizonand acts as a diffeomorphism everywhere outside the black hole and also on the horizon. For the planet, we observed that the knowledge of injected energy alone does not suffice to predict what radiation the planet emits. Instead, this can only be done with knowledge of the interior dynamics of the planet. Those, however, can be described classically in the case of the planet. For the black hole, the situation is even worse. Not only do we not have access to any interior dynamics, but these dynamics are also fully quantum. It is impossible to describe them even with full classical knowledge of the interior of the black hole. Before we elaborate on this point, we first show how it is possible to use supertranslations as bookkeeping device for black hole evaporation. Unlike for the case of the planet, this is a non-trivial question since the evaporation products are generic quantum states.In order to define an associated supertranslation, we shall proceed as follows. We consider an ensemble of quantum-mechanically identical black holes of mass M.[Experimentally, we can realize this by preparing identical quantum states in such a way that they collapse and form black holes.] For each black hole, we wait until it has emitted exactly one Hawking quantum.We only record their angular features, the deviation from an isotropic emission. This means that we assume that the microstates of the black hole have a non-trivial projection on angular features of the evaporation products. As explained in the introduction and illustrated for the case of the planet, we believe this assumption to be natural. Thus, we record the Hawking quanta using a filter for angular features, where we use one for each spherical mode (l,m). This defines a probability distribution for the angular features of the ensemble: P(l,m). Obviously, the probability distribution (<ref>) only contains a part of the quantum-mechanically available information. However, we will only focus on it since it can be described in terms of classical supertranslations. At this point, it is crucial to point out that the probability distribution (<ref>) does not originate from a mixed state but as a result of an ordinary quantum measurement. Thus, unlike in a description in terms of a mixed state, it is not associated to any fundamental loss of information.Since we need to recover a featureless emission in the semi-classical limit, it follows thatP(0,0)=1-ϵ ,where ϵ→ 0 in the semi-classical limit. This means that only a fraction ϵ of the emitted quanta carries features. For l≥ 2, we consequently get P(l,m) = ϵ A_l,m , where ∑_l=2^∞∑_m=-l^m=l A_l,m =1. The information contained in the P(l,m) is purely quantum mechanical. At the semi-classical level, we have that P(l,m)=δ_l0 and in the classical limit, we have no emission at all. Using the quantum probability distribution (<ref>), we can associate to every Hawking quantum an average energy flux: ℱ_out = ħ r_g^-1∑_l=0^∞∑_m=-l^m=lP(l,m) Y_l,m , where Y_l,m are the standard spherical harmonics. Just like for the case of the planet, where we considered a classical process of emission, we can use the flux (<ref>) to define a classical supertranslation T^+. Of course, this is only possible as long as ħ≠ 0 since the energy flux is zero otherwise.When we record the quantum-mechanically emitted energy ℱ_out, we can proceed in analogy to the planet and use the supertranslation fields T^- and T^+ to track the evolution of the black hole. Concretely, in order to perform a Goldstone supertranslation, we first inject an energy μ and then we wait until n_H=μ/(ħ r_g^-1) quanta have evaporated, as is depicted in figure <ref>. Then we end up with a black hole of the same mass as before the process. Of course, the sensitivity of the final state on the initial state is suppressed by μ/M but unitarity dictates that the dependence is never trivial. §.§.§ Insufficiency of Supertranslation Hair However, it is impossible to predict T^+ solely from the knowledge of T^-. The reason is that the wave that we inject acts as a diffeomorphism outside the horizon and also on the horizon. Therefore, the geometry outside the black hole is unchanged after the wave has passed. Since the semi-classical Hawking calculation is only sensitive to the geometry on the horizon and outside the black hole, its result cannot change as a result of a supertranslation diffeomorphism. Therefore, additional knowledge about the interior is required to predict T^+.We can make this argument more concrete by taking the perspective of an observer who lives in a Schwarzschild metric supertranslated by T^-. The observer has no record of how the black hole was formed and is only allowed to make experiment outside the horizon. Her goal is to determine the microstate of the black hole. More specifically, she wants to know if the black hole is in the bald microstate, whose evaporation products are featureless and in particular perfectly isotropic, or in a non-trivial microstate, whose evaporation products carry some angular features. By our definition of microstate, one way to do so is to wait till the black hole has evaporated and to determine the properties of the evaporation products. The question we are asking is if there is another way to determine the microstate of a black hole. The answer is negative, for the following reason: When an outside observer finds herself in a black hole metric with supertranslation field T^-, this can happen because of two very distinct reason. Firstly, it could be the result of injecting a wave with a non-trivial angular distribution of energy into a black hole. In that case, the black hole is in a non-trivial microstate and T^- indeed characterizes the microstate. However, there is a second way in which we can obtain a supertranslated Schwarzschild metric. Namely, we can consider a featureless microstate, whose evaporation products are isotropic, and apply a supertranslation diffeomorphism to this setup. In this way, we do not change the physical state of the black hole but only describe it in a different metric. Thus, T^- can also correspond to a featureless microstate described in different coordinates.Without access to the evaporation products, the only way to distinguish those two cases – injection of wave with angular features versus global diffeomorphism – is to enter the black hole. There, the wave acts non-trivially, not as a diffeomorphism, whereas the global diffeomorphism still does.Since the same exterior metric can correspond to both a trivial and a non-trivial microstate, the metric alone cannot suffice to predict the evaporation products.From the outside, it is therefore impossible to distinguish classical supertranslation hair and global diffeomorphisms. In summary, as in the case of a planet, we can use (T^-, T^+) as a natural bookkeeping device for the black hole to track the angular features of ingoing and outgoing radiation. However, knowing T^- does not suffice to predict T^+, an observer outside the black hole cannot infer T^+ from T^-. This is only possible with a microscopic model of the interior dynamics of the black hole.§.§.§ Generalization to EvaporationHaving discussed how we can implant hair on a black hole with a Goldstone supertranslation, it is trivially to consider the case of pure evaporation. We obtain it if we just leave out the first part of the Goldstone supertranslation, namely the injection of a wave. Therefore, it suffices to consider 𝒥^+ as screen, where the constraint (<ref>) determines the retarded supertranslation field T^+ in terms of the angular distribution Δℱ̃_out. In that case, the metric outside the black hole changes according to (<ref>):δ g^u_μν = τ_u_0, u_1(u) (ℒ_ξ_u(T^+) g^u_μν - 2μ G/rδ_μ^0 δ_ν^0 ).This equation shows that the back reaction splits in two parts. First, energy conservation dictates that the mass of the black hole is reduced by the total emitted energy μ = ∫u∫[2]Ωℱ_out. This part of the back reaction is undebatable but does not suffice to ensure unitarity of the process. Fortunately, ℱ_out contains more information than just the emitted energy, namely the supertranslation T^+. Consequently, we obtain the back reacted black hole not only by reducing its mass, but by supertranslating it by T^+.This approach is only valid if the supertranslation acts non-trivially in the interior of the black hole, because it is induced by a physical wave. But in that case, the ability to associate hair to a black hole is equivalent to the ability to purify its evaporation.§.§ A Comment on Page's Time So far, we have not specified the magnitude of deviations from a thermal evaporation. We can estimate them by requiring that we reproduce Page's time in our approach. In its most basic formulation, Page's time is a direct consequence of describing the black hole evaporation in a Hilbert space of fixed dimension. In brief, if we keep the dimension of the full Hilbert space, which describes at any time both the black hole and the emitted radiation, fixed and equal to 2^N, then at t=t_P, which corresponds to the half life-time, the evaporation of ∼ N/2 quanta, there is no place to continue increasing the entanglement between the radiation and the black hole internal degrees of freedom. At this time, entanglement starts to decrease and information starts to be delivered. This makes clear why purification of black hole evaporation relies on fixing N and keeping it finite. Page's time can be defined as the time-scale for the emission of the order of N quanta. Therefore, we first consider an ensemble of N identical quantum mechanical black holes and for each of them, we record the first emitted quantum. For a measurement on a single black hole, the standard deviation isσ_1 ∼ O(1)since the quanta are distributed isotropically to leading order. However, when we average over N measurements, the standard deviation decreases asσ_N∼1/√(N) .Features become visible as soon as their strength becomes bigger or equal than the uncertainty of the measurement. After Page's time we can therefore resolve features with the relative amplitude ϵ∼1/√(N) . In the formulation of the probability distribution (<ref>), this means that after O(N) measurement, those features becomes visible which are only carried by a fraction 1/√(N) of the quanta.So far, we have only considered one emission for N identical black holes. If we consider instead O(N) emissions of a single black hole, the difference is that the probability distribution for each emission step is generically different. This is true because of the back reaction of the previously emitted quanta. However, the argument in terms of the resolution stays the same, after Page's time, we can still resolve those features which are only carried by a fraction ϵ∼ 1/√(N) of quanta. This argument provides evidence for the black hole N-portrait <cit.> where features are 1/N-effects with a resolution scale O(1/√(N)).[An interesting question that we shall not discuss in this note but that can be worth to mention is the possibility that a quantum computer designed using a Grover like algorithm <cit.> can reduce t_P from O(N) to O(√(N)).] In particular, in an S-matrix analysis along the lines of <cit.>, where black hole formation was studied as 2→ N-scattering process, angular features should appear as 1/N-correction to the leading amplitude.§ CONCLUSION The main message of this note is easy to summarize. Any form of black hole hair should imply the existence of features in the black hole evaporation products, in the emitted radiation. This obvious requirement immediately entails, given the intrinsically quantum nature of black hole radiation, that black hole hair should be defined quantum-mechanically and that such a definition is inseparable from the mechanism through which the black hole delivers, in the radiation, information about its internal structure.In this note, we have suggested to define hair on the basis of elementary processes of classical absorption followed by quantum emission. Moreover, we specialized to angular features in the radiation. This simplification has been done in order to use the asymptotic symmetry group and the corresponding supertranslations to parametrize both the incoming and the emitted radiation. Since we have complete control over the angular features of the injected radiation, we can define hair on the basis of the angular features of the quantum-mechanically emitted radiation. These features encode information about the internal structure of the black hole which can be measured by an external observer. In this sense, they provide an operative and intrinsically quantum definition of hair. In principle, we can imagine two different sources of those features of the emitted radiation. The first one is a classical modification of the near horizon geometry that will modify the corresponding semi-classical Bogoliubov transformations. The second one is a real quantum interaction of the injected radiation with the quantum constituents of the black hole. The first possibility requires to define local changes of the horizon geometry that preserve all the ADM-charges. Thus, locally, they can be always tuned to be equivalent to a diffeomorphism. Therefore, theycannot have observable consequences, classical supertranslations do not suffice to define observable black hole hair. So the only real possibility of quantum emitted radiation with features is having a non-trivial scattering between the injected radiation and the microscopic constituents of the black hole. This means that the features that define hair in the way we are suggesting depend on the microscopic quantum structure, which we can parametrize as a dependence on the black hole entropy N. Thus, the hair that we are defining vanishes in the limit N=∞. As it is clear from the discussion, this way of addressing the definition of hair is what we can call an S-matrix approach, where by that S-matrix we simply mean the dynamics involved in the complex process of actual absorption and quantum emission. If we focus on angular features, we can encode the properties of the hair in terms of the commutators, as operators, of this S-matrix and the generators of the asymptotic symmetry group. Associating with the injected energy a supertranslation T^- in BMS_-, a way to approach the existence of hair is by considering the commutator [S, T^-]. Generically, the non-trivial hair will be associated with the symmetry generators that are broken since those are the ones that will create net differences between the angular features of the injected and emitted radiation. Although the infrared dynamics of gravity selects the zero-energy modes as natural symmetries of S, they are not able to tell us anything about the internal structure of the black hole since they are decoupled. Zero-modes are unable to encode observable features.What we have presented in this note is just the general framework to address the problem of quantum hair. In order to go further, it is necessary to use a concrete model of the black hole interior. The model in <cit.> provides, in principle, the tools toaddress this questions in a quantitative way, something to which we hope to come back in the future.§ RECAP OF IR-EFFECTSIn this appendix, we shall collect some well-known facts about infrared physics which could be useful to clarify some controversial aspects on the meaning of soft modes. Some of these issues have been revisited recently in a series of papers <cit.>. * In QED, asymptotic physical states associated with freely moving charged particles should be dressed in order to satisfy the Gauss law constraints. This dressing simply adds to the freely moving charge its companion electrostatic field, the non radiative part of the retarded Lienard-Wiechert-field behaving at large distances as 1/r^2. In quantum field theory, this dressing can be defined using a coherent state of off-shell photons <cit.> with dispersion relation ω(k) = k⃗v⃗ for v⃗ the velocity of the charged particle. This coherent state dressing contains an infinite number of k=0 modes and it is identical to the dressing operator defined in <cit.>. In scattering theory, one can define physical asymptotic states and an IR-safe S matrix using this dressing operator. * Alternatively, one can use no dressing. Then, in perturbative QED as well as in perturbative gravity, we find IR-divergences due to virtual photon/graviton loops. These, after a careful analysis of overlapping divergences, can be resummed and exponentiated <cit.>.When we consider the transition from an initial state |α⟩ to a final state |β⟩, we obtainS_α, β^loop = ^B_α, βlnλ/Λ/2 S_α, β^0,where S_α, β^0 is the amplitude without taking into account soft loops whereas S_α, β^loop contains them. Here Λ is a UV-cutoff that defines what is soft, λ is a IR-cutoff and B_α, β is a non-negative number, which only depends on the initial state |α⟩ and the final state |β⟩. It is zero if and only if the ingoing current in |α⟩ matches the outgoing current in |β⟩ antipodally at each angle. In the case of gravity, it scales as B_α, β∼ G s, where s is the energy of the process.[That this scaling also holds for graviton scattering at an ultra-Planckian center of mass energy was shown in <cit.>.]For B_α,β≠ 0, soft loops clearly lead to a vanishing amplitude in the limit λ→ 0.* In order to cancel the IR-divergences due to virtual photons/gravitons, the Bloch-Nordsieck-recipe <cit.> requires to add a certain class of soft emission processes. Again the effects of emitting theses soft IR-modes of energies below ϵ can be resummed and exponentiated, yielding the rate <cit.> |S_α, β^full|^2 := ∑_γ|S_α, βγ^soft|^2 =^B_α, βlnϵ/λ f(B_α, β)|S_α, β^soft|^2, where f(B_α, β) is due to energy conservation and f(B_α, β)≈ 1 for small B_α, β. Combing the contribution from (<ref>) and (<ref>), one obtains a rate which is independent of the IR-cutoff λ and in particular finite for λ→ 0.This cancellation leads to the connection, highlighted in <cit.>, between the soft photon theorem and the electrostatic coherent state dressing. In QED, we do not have new symmetries besides the decoupling of zero-energy photons. The same is true in perturbative gravity. * In the correction factor ^B_α, βlnϵ/λ in (<ref>), the n^th summand of the exponential series comes from the emission of n IR-modes. Therefore, we can estimate the number of soft modes from the term which gives the biggest contribution in the series. This givesn_soft^unres∼ B_α, βlnϵ/λ .We conclude that the number of unresolved soft modes only scales logarithmically with the infrared resolution scale ϵ.§ MATCHING IN SCHWARZSCHILD COORDINATES In this section, we demonstrate explicitly how we can transform a Schwarzschild metric with non-trivial supertranslation field from advanced to retarded coordinates. In this way, we show how we can naturally identify the advanced supertranslation field C^- with the retarded one C^+. We start from the Schwarzschild metric g_μν^v ,0 in advanced coordinates without supertranslation field:d s^2 = -(1-2GM/r) v^2 + 2v r +r^2 Ω^2. The corresponding generators of supertranslations are ξ_v^v =f^-, ξ_v^r =-1/2 D^2 f^-,ξ_v^A = f^-, A/r , which are characterized by an arbitrary function f^- on the sphere. Thus, the supertranslated metric isg_μν^v(f^-) = g_μν^v ,0 + ℒ_ξ_v(f^-) g_μν^v ,0 . In retarded coordinates, the Schwarzschild metric g_μν^u ,0 without supertranslation field is:d s^2 = -(1-2GM/r) u^2 - 2u r +r^2 Ω^2. The corresponding generators of supertranslations are ξ_u^v =f^+, ξ_u^r = 1/2 D^2 f^+,ξ_u^A =-f^+, A/r , where it is important to note that the signs of ξ_u^r and ξ_u^A have changed with respect to (<ref>). The supertranslated metric is:g_μν^u(f^+) = g_μν^u ,0 + ℒ_ξ_u(f^+) g_μν^u ,0 .The task now is to transform g_μν^v to retarded coordinates. As explained in section <ref>, there can in general not be a unique way to match the advanced and retarded supertranslation fields. However, a natural choice in a static metric is to require that the spherical metrics match: g_AB^v(f^-) = g_AB^u(f^+). Therefore, we use the diffeomorphism 𝒟_m defined byv = u + 2 _r_0^r 1/1-2GM/rr' - D^2 f^-/1-2GM/r - 2 f^-.Then it turns out that𝒟_m(g_μν^v(f^-)) = g_μν^u ,0 - ℒ_ξ_u(f^-) g_μν^u ,0 = g_μν^u(-f^-) .Thus, we identifyf^+=-f^-.Up to a sign, the supertranslation field in advanced coordinates matches the retarded one angle-wise. With this choice, not only the spherical metrics match, but also the g_00-components, the Newtonian potentials.§ EXPLICIT SOLUTION FOR GOLDSTONE SUPERTRANSLATION OF A PLANET§.§.§ Step 1: AbsorptionThe Goldstone supertranslation consists of two steps: First, an initially spherically symmetric planet absorbs as wave. As is well-known (see (9.3) in <cit.>), the metric of a static spherically symmetric spacetime can be cast in the general formd s^2 = - A(r) t^2 + B(r)r^2 +r^2 Ω^2, where all physical information is contained in the tt- and rr-components. Since we want to describe a planet, there should neither be a surface of infinite redshift, A(r) > 0∀ r, nor an event horizon, B(r) < ∞ ∀ r. Furthermore, asymptotic flatness implies that A(r) r→∞⟶ 1 and B(r) r→∞⟶ 1 sufficiently fast. Using the transformation v = t + _r_0^rr' √(B/A) ,we obtain the metric g_μν^v in advanced BMS-gauge:s^2 = - Av^2 + 2 √(AB) vr + r^2 Ω^2,which is suited to describe incoming radiation. Note that this metric describes the whole spacetime and not only its asymptotic region, r→∞.We will restrict ourselves to infinitesimal supertranslations. In advanced time, these are generated by ξ_v^v =f^-, ξ_v^r =-1/2 r D_B ξ_v^B,ξ_v^A =f^-, A_r^∞ r' (√(AB) r'^-2), where an arbitrary function f^- on the sphere determines the change of the supertranslation field.We denote it by f^- instead of T^- in this appendix to avoid confusion with the energy-momentum-tensor of the wave. The minus-superscript indicates that we deal with a supertranslation in advanced coordinates. Our goal is to realize the infinitesimal diffeomorphism defined by (<ref>) in a physical process, outside the planet, we want to have the stationary metric g^v_μν before some time v_0 and after some point of time v_1, we want to end up in the stationary metric g^v_μν + ℒ_ξ_v(f^-) g_μν^v. For v_0<v<v_1, physical radiation interpolates between the two metrics. Inside the planet, the wave should be absorbed so that the transformation fades out and the metric around the origin remains unchanged. Adding as final ingredient a change of the Bondi mass μ, which is necessary to ensure the positive energy condition, we obtainδ g^v_μν = τ_v_0, v_1(v) s^-(r) (ℒ_ξ_v(f^-) g^v_μν + 2μ G/rδ_μ^0 δ_ν^0 ),where 0≤τ_v_0, v_1(v) ≤ 1 parameterizes the interpolation, τ_v_0, v_1(v<v_0)=0 and τ_v_0, v_1(v>v_1)=1. The function s^-(r) describes the absorption of the wave by the planet. It has the property that it is monotonically increasing with s^-(0)=0 and s^-(∞)=1, where s^-(0)=0 ensures that the wave is fully absorbed before the origin and no black hole forms. Moreover, s^- '(r)≠ 0 is only permissible whenever the local energy density of the planet is non-zero. The magnitude of s^- '(r) determines how much absorption happens at r. It is crucial to note that the transformation s^-(r) ℒ_ξ_v(f^-) g_μν is a diffeomorphism only where s^-(r) is constant. Thus, the transformation (<ref>) acts as a diffeomorphism only outside the planet, but not inside. This reflects the fact that we want to obtain a physically different planet. A transformation which acts as a diffeomorphism everywhere could not achieve this.Since we work with infinitesimal supertranslations, it is important that we stay within the regime of validity of this first-order approximation, that terms linear in f^- dominate. As it will turn out in the calculation, this is the case if max_(θ, φ) |f^-| ≪ v_1-v_0. This means that the time-shift induced by the supertranslation must be much smaller than the time-scale of the process, the supertranslation must be performed slowly. We will choose v_1-v_0 such that this is the case and so that we can neglect all higher orders in f^- when we calculate the Einstein equations.We have to show that the transformation (<ref>) leads to a valid solution of the Einstein equations. Thus, if we calculate the Einstein G_μν and consequently the new energy-momentum-tensor T_μν, we have to demonstrate that this is a valid source. To this end, we have to perform two checks. First of all, it must be conserved, T_μν^μν;μ = 0. This is trivially true in our construction because of the Bianchi identity, G_μν^μν;μ = 0. Secondly, we have to show that T_μν fulfills an appropriate energy condition. For that purpose, we first note that this perturbation only depends on the local geometry, except for ξ^μ_v and s^-(r), which also depend on spacetime points at bigger radii. Thus, outside the planet, we have the same solution as in <cit.>, except for the fact the we perform our supertranslation slowly: T_00 = 1/4π r^2[μ - 1/4 D^2(D^2 + 2) f^- + 3M/2r D^2 f^-] τ'_v_0, v_1(v), T_0A =3M/8π r^2 D_A f^- τ'_v_0, v_1(v), where we used that there is no absorption outside the planet: s^-(r>R)=1. Obviously, the energy condition is fulfilled. At this point, we remark that leaving out all subleading parts, which are proportional to M, would also lead to a valid wave in the metric (<ref>), i.e., T_μν^μν;μ would also be true to all orders if one only considered the leading order of (<ref>). This means that we add the subleading parts to (<ref>) not because of energy conservation but since we want to realize the transition (<ref>) not only to leading order in 1/r, but to all orders.[Of course, energy conservation relates the two subleading parts of T_00 and T_0A. When we choose one, it determines the other.]Fortunately, we do not either have to worry about the energy condition inside the planet. For a small enough perturbation, this is true since the energy condition inside a planet is not only marginally fulfilled. This means that s^-'(r) can be non-zero inside the planet: This corresponds to absorption of the wave by the planet.Lastly, we have to show that the wave is still a valid solution after it has been partly absorbed. For the purpose of illustration, we model the planet as a sequence of massive shell with vacuum in between, T_μν=0. In that case, the only non-trivial question is whether (<ref>) fulfills the energy condition after it passed some or all of the shells. Therefore, we calculate the energy-momentum-tensor in this region. By Birkhoff's theorem, the local geometry corresponds to a Schwarzschild solution with diminished mass M̃ (where M̃ can be zero). It only depends on the matter which it has passed via ξ_v^μ. We parameterize the difference of ξ_v^μ and the vector field one would get in a pure Schwarzschild geometry of mass M̃ by σ := _R_min^R_max r' ((√(AB)-1) r'^-2) ,wherewe have no matter for r>R_max and between r and R_min.[We use that AB=1 in a Schwarzschild geometry of arbitrary mass.]Explicitly, this means that we can writeξ_v^A =f^-, A(1/r + σ),where it is important that σ does not depend on r in our region of interest. Of course, σ=0 corresponds to the case when there is no matter outside.With the help of Mathematica <cit.>, we compute: T_00 = 1/4π r^2[μ̃ - (1+σ r)(1/4 D^2(D^2 + 2) f̃^- - 3M̃/2r D^2 f̃^-)] τ'_v_0, v_1(v), T_0A =[3M̃/8π r^2 D_A f̃^- +σ/16 π D_A (D^2 + 2)f̃^-] τ'_v_0, v_1(v), T_AB =-σ r/8 π[(2D_A D_B - γ_ABD^2 )f̃^- ]τ'_v_0, v_1(v), where f̃^-=s^-(r)f^- is the supertranslation which is attenuated because of absorption in the outer shells. It is crucial to note that s^- '(r)=0 in this calculation since we are not inside one of the shells of the planet and likewise μ̃ =s^-(r)μ. As we can estimate σ very crudely as σ < 1/R, we see that for sufficiently large μ, the energy condition is fulfilled. With a more accurate estimate, we expect that the freedom of choosing μ is not restricted when the wave passes a massive shell. In summary, we have shown that the metric (<ref>), which describes the dynamical transition from a spherically symmetric planet to a counterpart with nontrivial angular distribution of mass, is a valid solution.§.§.§ Step 2: EmissionThe second step is to describe the emission of the wave by the planet. Thus, our initial metric is the one after absorption, as determined by equation (<ref>):δ g^v_μν = s^-(r) (ℒ_ξ_v(f^-) g^v_μν + 2μ G/rδ_μ^0 δ_ν^0 ).As we want to consider emission, our first step is to transform it to retarded coordinates. Intuitively, it is clear that it should be possible to describe a slightly asymmetrical planet also in retarded coordinates. While it is generically hard to write down the corresponding diffeomorphism which connects the two metrics, we can use that the metric of a planet does not differ from Schwarzschild in the exterior region. Therefore, we can use the diffeomorphism (<ref>) to obtaing_μν^u = g_μν^u, 0 + s^-(r)(ℒ_ξ_u(-f^-) g_μν^u, 0 + θ(R-r)dev),where g_μν^u, 0 is the metric of the initial, spherically symmetric planet in retarded coordinates. This means that we apply a supertranslation in retarded coordinates which is defined by the function f^- used to defined the advanced supertranslation. The function dev accounts for the fact that we do not know the continuation of the matching diffeomorphism (<ref>) to the interior of the planet. Therefore, g_μν^u might deviate slightly from BMS-gauge but only in the interior. We expect, however, that the matching diffeomorphism can be continued such that dev=0. Finally, we want to point out that g_μν^u(r=0) = g_μν^u, 0(r=0) since s^-(r=0)=0, the wave does not reach the center and the mass distribution of the planet is still spherically symmetric around r=0.The case of the planet provides us with another justification why the matching (<ref>) is natural. With this identification, both the metric (<ref>) in advanced coordinates and the metric (<ref>) in retarded coordinates cover the whole manifold. Extrapolating the results of <cit.>, where finite supertranslations of Schwarzschild and Minkowski are discussed, we expect that for any other matching, for any other value of the supertranslation field, this is no longer the case. If this is true, the requirement that the BMS-coordinate system covers the whole manifold singles out a unique value of the advanced supertranslation field as well as a unique value of the retarded supertranslation field, and therefore a coordinate matching.Next, we want to describe how the metric (<ref>) emits a wave. This wave should realize a supertranslation described by f^+, which is generically different from f^-:δ g^u_μν = τ_u_0, u_1(u)s^+(r) (ℒ_ξ_u(f^+) g^u, 0_μν - 2μ G/rδ_μ^0 δ_ν^0 ),where we used that ℒ_ξ_u(f^+) g^u_μν = ℒ_ξ_u(f^+) g^u, 0_μν to first order in f^+ and f^-. Thus, working only to first order simplifies our calculations significantly since we can simply use the calculations for the absorption. The wave (<ref>) becomes: T_00 = 1/4π r^2[μ̃ - (1+σ r)(1/4 D^2(D^2 + 2) f̃^+ - 3M̃/2r D^2 f̃^+)] τ'_u_0, u_1(u), T_0A = -[3M̃/8π r^2 D_A f̃^+ +σ/16 π D_A (D^2 + 2)f̃^+] τ'_u_0, u_1(u), T_AB =-σ r/8 π[(2D_A D_B - γ_ABD^2 )f̃^+ ]τ'_u_0, u_1(u). As for the absorption, we have shown that we can realize the transformation (<ref>) with a physical wave.Finally, we analyze the joint effect of absorption and emission. Combining the transformations (<ref>) and (<ref>), we get total total change of the metric:δ g_μν^tot= θ(r-R) ℒ_ξ_u(f^+-f^-) g_μν^u, 0 + θ(R-r)(s^+(r)ℒ_ξ_u(f^+) g_μν^u, 0 - s^-(r) ℒ_ξ_u(f^-) g_μν^u, 0 + dev), where we used retarded coordinates. As desired, the mass of the planet stays invariant. Moreover, δ g_μν^tot acts as a diffeomorphism outside the planet, namely it is the difference of the advanced supertranslation, described by f^-, and the retarded supertranslation, described by f^+. If we furthermore assume that the term dev, which reflects our incomplete knowledge of the matching between advanced and retarded coordinates in the planet, is zero, we see that the metric does not change for f^-=f^+. We obtain a trivial transformation if the angular energy distribution of ingoing and outgoing radiation is angle-wise the same. § ACKNOWLEDGEMENTS We are happy to thank Gia Dvali for many stimulating discussions and comments. We also thank Artem Averin, Henk Bart and Raoul Letschka for discussions. The work of C.G. was supported in part by Humboldt Foundation and by Grants: FPA 2009-07908 and ERC Advanced Grant 339169 "Selfcompletion".tocsectionReferences 1 noHair P. T. Chrusciel, J. Lopes Costa, and M. 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Veneziano, Glimpses of black hole formation/evaporation in highly inelastic, ultra-planckian string collisions,http://dx.doi.org/10.1007/JHEP02(2017)111J. High Energy Phys. 2017 no. 2, 111, http://arxiv.org/abs/1611.03643arXiv:1611.03643 [hep-th].BN F. Bloch and A. Nordsieck, Note on the Radiation Field of the electron, http://dx.doi.org/10.1103/PhysRev.52.54Phys. Rev. 52 (1937) 54–59.hobson M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby, General Relativity: An Introduction for Physicists. Cambridge University Press, 2006. mathematica Wolfram Research, Inc., Mathematica 11.0,2016. | http://arxiv.org/abs/1707.08580v2 | {
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"published": "20170726180005",
"title": "Black Hole Evaporation, Quantum Hair and Supertranslations"
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[email protected] Federal University, Mira str. 19, 620002, Ekaterinburg, Russia2M. N. Mikheev Institute of Metal Physics UrB RAS, Kovalevskaya str., 18, 620990, Ekaterinburg, Russia A. A.Stepanenko1 D. O.Volkova1 P. A.Igoshev21 A. A.Katanin21 We study a question of presence of Kohn points, yielding at low temperatures non-analytic momentum dependence of magnetic susceptibility near its maximum, in electronic spectum of some three-dimensional systems. In particular, we consider one-band model on face centered cubic lattice with hopping between nearest and next-nearest neighbors, which models some aspects of the dispersion of ZrZn_2, and the two-band model on body centered cubic lattice, modeling the dispersion of chromium. For the former model it is shown that Kohn points yielding maxima of susceptibility exist in a certain (sufficiently wide) region of electronic concentrations; the dependence of the wave vectors, corresponding to the maxima, on the chemical potential is investigated. For the two-band model we show existence of the lines of Kohn points, yielding maximum of the susceptibility, which position agrees with the results of band structure calculations and experimental data on the wave vector of antiferromagnetism of chromium.Kohn anomalies in momentum dependence of magnetic susceptibility of some three-dimensional systems Rui Carlos Andrade MartinsDepartamento de Física, Universidade de Aveiro July 14, 2017 ==================================================================================================§ INTRODUCTION Pecularities of an electronic spectrum have strong impact on physical properties of strongly-correlated systems. In particular, peaks of the density of states (occuring, e.g., due to van Hove singularities) lead to ferromagnetism,while nesting of the Fermi surface results in a tendency to spin density wave formation. Although an “ideal” nesting is rather rare (especially in real magnetic substances), spin density waves can be caused by “local” nesting, which is present near certain points of Fermi surface, connected by a spin density wave vector Q and having opposite Fermi velocities. Study of the effect of this “local” nesting originates from W. Kohn's pioner paper <cit.>, which has shown that the susceptibility of systems with spherical Fermi surface of radius k_F depends non-analytically on momentum close to wave vectors of the length Q=2k_F, connecting points on the opposite sides of the Fermi surface, which have also opposite velocities (so-called Kohn points). Generally this non-analytical peculiarity is rather weak, leading, however, to a number of important phenomena, such as anomalies of phonon frequencies, Kohn-Luttinger mechanism of superconductivity<cit.>, etc. The strongest effect of Kohn anomalies of susceptibility on magnetic properties can be expected in the case when the wave vector 𝐐, connecting Kohn points, corresponds to a global (non-analytical) maximum of the susceptibility, such that 𝐐 coincides with a wave vector of spin- or charge density wave.While the property of a maximum of susceptibility being global depends on the whole electronic structure, and can be established, as a rule, only numerically or experimentally, the presence of a local maximum of susceptibility follows from local geometry of Fermi surface near Kohn points. In particular, L. M. Roth with co-authors <cit.>, and also T. M. Rice <cit.> have established that in one-band models the susceptibility has a local maximum at the wave vector 𝐐 if the Fermi surface near each of the Kohn points, separated by the vector Q, has opposite curvature in two perpendicular directions, or curvature in one of the directionsis equal to zero.The latter condition, in particular, is fulfilled for two-dimensional Fermi surfaces; the case of circular Fermi surface was explicitly studied by Stern <cit.>, later T. Holder and W. Metzner <cit.> have considered the effect of Kohn points for arbitrary two-dimensionalFermi surfaces, in particular they have investigated peculiarities of spin and charge susceptibilities, as well as self-energy of electrons.Although at finite temperatures T non-analytical behaviour of susceptibility, caused by Kohn points, takes place only at momenta |q-Q | ≳ T/v_F (v_F is the Fermi velocity at Kohn points), it becomes important in the vicinity of quantum phase transitions at T→ 0. Recently, T. Schäfer, et al. <cit.> investigated effect of Kohn points on quantum critical behaviour and have shown that corresponding critical exponents can strongly deviate from predictions of Hertz-Moriya-Millistheory because of the presence of Kohn anomalies. In that study, however, only simple cubic lattice with nearest neighbour hopping, for which there are lines of Kohn points, has been investigated.Depending on momentum dependence of susceptibility, two different cases, in which one can expect an essential effect of Kohn points near quantum phase transitions, can be distinguished. In the first casethe vector of a spin density wave is small, and the competition of ferromagnetism and spin densitywave takes place. This situation is possibly realised near quantum phase transition in ZrZn_ 2 <cit.>, where sharp change of a critical exponent of electrical resistivity under pressure is observed <cit.>.In the second case the wave vector of spin density is not small (as it happens, e.g., in chromium <cit.>), and gradual suppression of Neel temperature of an incommensurate order by external factors, e.g. pressure <cit.> or doping <cit.> takes place. The aim of the present paper is to consider electronic systems on various three-dimensional lattices, which lead to Kohn anomalies, and model the two above described types of momentum dependence of susceptibility. In particular, consideration of face-centered cubic (fcc) lattice with hopping between nearest- and next-nearest neighbours (Section 2 of the paper) allows to describe the situation with a small wave vector (caused by contribution of Kohn points), as well as its gradual evolution to a spin density wave vector, essentially different from zero. Although evolution of a wave vector of an incommensurate order with changing concentration of electrons on fcc lattice was investigated earlier <cit.>, the relation of the obtained wave vectors to the geometry of a Fermi surface, and also the effect of Kohn points on magnetic susceptibility was not considered. To model the Fermi-surfaces of chromium, the two-band dispersion on body-centered cubic (bcc) lattice is also considered (see Section 3 of the paper). For each of the these cases Kohn points are determined and their contribution to a static magnetic susceptibility is calculated.§ THE ONE-BAND MODEL§.§ Formulation of the model, corresponding Fermi surfaces and susceptibility We consider the Hamiltonian of the Hubbard modelĤ=-∑_i,j,σt_i,jĉ_iσ^†ĉ_jσ+U∑_in̂_i↑n̂_i↓,where ĉ_iσ^†(ĉ_iσ) are the Fermi-operators of creation (annihilation) of an electron at site i with spin projection σ=↑,↓, n̂_iσ=ĉ_iσ^†ĉ_iσ is the operator of number of electrons at site i, t_i,j is the hopping parameter, U characterizes the on-site repulsion of electrons and determines the strength of electron correlations. To calculate magnetic susceptibility, we use generalized random phase approximation (RPA), see e.g., Ref. <cit.>,χ_𝐪=χ_𝐪^0/1-U_ effχ_𝐪^0,whereχ_𝐪^0=-∑_𝐤f_𝐤-f_𝐤+𝐪/E_𝐤-E_𝐤+𝐪is the susceptibility of the system of non-interacting electrons,E_𝐤=∑_δ t_i,i+δexp(i 𝐤𝐑_δ) is the dispersion, f_𝐤≡ f(E_𝐤) is the Fermi function,q is the wave vector,k is the electronic quasimomentum, U_ eff is the value of an effective inter-electron interaction in the particle-hole spin channel. This approximation (with some parameter U_ eff<U) is qualitatively applicable for the considered three-dimensional systems due to sufficiently weak momentum dependence of the self-energy and vertices of electron-electron interaction, irreducible in the particle-hole channel <cit.>. The value U_ eff, which can be determined, e.g., from the sum rules,does not play an important role for further consideration; it is assumed only, that it is sufficient to form (weakly) magnetically ordered state, i.e. U_ effχ^0_ Q≈ 1. The applicability of qualitative results of RPA for studying Kohn anomalies in strongly correlated three-dimensional systems was shown, e.g., in Ref. <cit.>.Below we consider fcc lattice with hoppings between nearest neighbors t and next nearest neighbors -t', corresponding to the dispersionE_𝐤 = -4t(cosk_x/2cosk_y/2+cosk_x/2cosk_z/2+cosk_y/2cosk_z/2)+2t'(cosk_x+cosk_y+cosk_z)-μ,where μ is the chemical potential, the lattice constant is set equal to unity. Depending on the values of t'/t and μ/t, the Fermi surface can be either connected (for μ not too close to the upper edge of the band), or consist of several disconnected parts. We consider the former case, as the most physically interesting, examples of the Fermi surfaces for some values t'/t, μ/t are shown in Fig. <ref>. The momentum dependences of bare susceptibilities χ_ q^0, determined numerically from the equation (<ref>) for some values of the parameters t'/t and μ at the temperature T=0 are shown in Figs. <ref>,<ref>. It can be seen that these susceptibilities have nonanalytic dependences near maxima and inflection points, which connection with the Kohn points is discussed in the following subsections.§.§ The contribution of Kohn points to the non-uniform susceptibility In general case Kohn points 𝐊 and 𝐊+𝐐, leading to a nonanalytic momentum dependence of the susceptibility correspond to the points of the Fermi surface (E_ K=E_ K+Q=0), having opposite Fermi velocities. As L. M. Roth with coauthors <cit.> and T. M. Rice <cit.>have shown, under certain conditions on the curvature of the Fermi surface, the magnetic susceptibility can have a nonanalytic maximum on the wave vector, connecting two Kohn points. In order to formulate these conditions, we consider the expansion of the dispersion near the corresponding points by representing 𝐤=𝐊+𝐤_1. Introducing a local coordinate system in momentum space 𝐤_1=(k'_x,k'_y,k'_z), which axis k'_z is aligned along the Fermi velocity 𝐯_ K=(∇ E_𝐤)_𝐤=𝐊, and the axes k'_x,y are rotated around the new axis k'_z such that ∂^2E_𝐤/(∂ k'_x∂ k'_y)=0, we can express the dispersion in the form<cit.>E_𝐊+𝐤_1≃ vk'_z+(k'_x)^2/2m_x+(k'_y)^2/2m_y,E_𝐊+𝐐+𝐤_1≃-vk'_z+(k'_x)^2/2m_x+(k'_y)^2/2m_y,where v=| v_ K| and we assume that the Kohn points are connected by certain symmetry operations of the crystal lattice, so that the masses m_x,y are the same in the first and second lines of Eq. (<ref>). According to the results of Refs. <cit.>, the magnetic susceptibility will have a (local) maximum at the wave vector 𝐐, if one of the following conditions is fulfilled: m^-1_x m^-1_y<0 (so-called hyperbolic Kohn points) or m_x^-1 m_y^-1=0 (cylindrical Kohn points). Physically, the most interesting case corresponds to Kohn points, spaced by a wave vector along one of the symmetric directions (since only these directions can provide a global maximum of susceptibility).In the case of hyperbolic Kohn points, the momentum dependence of magnetic susceptibility at T=0 near the maximum has the form <cit.>χ_𝐪+𝐐^0=χ_𝐐^0-√(|m_x m_y|)/16π|q_z|,where q_z is the projection of the vector q on the axis k'_z.Let us consider, for example, the expansion of the dispersion on fcc lattice near Kohn points, which are located symmetrically with respect to the point X=(0,0,2π) and characterized by different directions of the spin-density wave vector 𝐐. In this case the Fermi surface can either be “closed” around the point Γ=0, see Fig. <ref>a(the point X is located outside the Fermi surface and E_X>0),or have a “window” near the point X (see Fig. <ref>b, c, E_X<0). In both cases we assume that the points {𝐊,𝐊+𝐐}=X±𝐐/2 belong to the Fermi surface. For the sake of simplicity we will assume that |t'|<t/2. We consider below different symmetric vectors Q that connect the Kohn points near the point X.1) For the direction 𝐐=(Q_1X,Q_1X,Q_1X), the condition E_ K=E_ K+Q=0 fixes the value Q_1X=2arccos[(-2t + μ)/(2 (t + 3 t'))]. The expansion of thedispersion near Kohn points yieldsv = √((6t'+μ)(4t+6t'-μ)(t^2+2tt'+3t'^2))/(t+3t'),m_x^-1 = -4t^3+t^2(10t'-μ)+4t'^2μ +t t'(10t'+μ)/4(t^2+2t t'+3t'^2),m_y^-1 = -(t+4t')μ -2t(2t+7t')/4(t+3t').From the condition of the existence of Kohn points and presence of the maximum of the susceptibility (the signs of the inverse masses must be different), we obtain the following inequality for the values of μ and t': μ<min[t(4t^2+10t t'+10t'^2)/t^2-t t'-4t'^2,t(4t+14t')/t+4t'].The result (<ref>), together with the restriction of the chemical potential within the band μ>min(E_X,E_L), where E_X=4t+6t', E_L=-6t' are the values of electronic dispersion (for μ=0) at the points X and L, respectively, is shown graphically in Fig. <ref>a. In the considered range t' (except t'≈-t/3) there is sufficiently wide range of chemical potentials μ where the Kohn points give the dominant contribution to the momentum dependence of the susceptibility. 2) For the direction 𝐐=(0,Q_2X,Q_2X) we have Q_2X=2arccos[(-2 t - 2 t'+ μ)/(2 (t + 2 t'))],v = √((-4t-6t'+μ)(4t^2 + tμ+2(t+μ)t'+2t'^2)/2t+4t'),m_x^-1 = -2t',m_y^-1 = 4t^3+6 t^2 t'+4t'^3-t^2μ-μ^2t' /4t^2+t(2t'+μ)+2t'(2t'+μ).From the condition for the existence of Kohn points and the presence of the anomaly (the signs of the inverse masses must be different), we obtain the following inequalities for the values of μ and t' for 0< t'/t <1/2: -2(2t^2+t t'+2t'^2)/t+2t'<μ<-t^2+√(t^4+16t^3 t'+24t^2 t'^2+16t'^4)/2t'.However, Kohn anomalies can appear also in case of the same sign of the inverse masses, provided by |m_x|≫ |m_y| or |m_x|≪ |m_y|. The ranges of t' and μ that fulfill these conditions are shown in Fig. <ref>b.3) For the direction (0,0,Q_3X) we findQ_3X=2arccos[(-2t ±√(4t^2 + (4t + μ) t' - 2 t'^2))/(2 t')].We have m_x=m_y by symmetry,and therefore in general case this direction does not lead to Kohn points that provide maximum of the susceptibility. However, the corresponding mass m_x^-1 = 2(t^2+t t' -2t'^2) ∓ t√(4t^2+t'(4t-2t'+μ))/2t'can be equal to zero for certain values of μ, t', providing a maximum of susceptibility.4) For the direction (Q_4X,0,0) we have Q_4X=2arccos[(-4t-4 t'+μ)/(2 t')],v = √((4t+6t'-μ)(-4t-2t'+μ)),m_x,y^-1 = -t-2t'∓ t√(-4t-2t'+μ/4t').The corresponding regions of existence of Kohns anomalies in the susceptibility are shown in Fig. <ref>c.Similarly we can consider Kohn points located symmetrically with respect to the point L: for the wave vectors(Q_1L,Q_1L,Q_1L), (Q_2L,Q_2L,0) and (Q_3L,0,0) we find Q_1L = 2arccos[(6t+μ)/(6t-6t')], Q_2L = 2arccos[(2t+2t'+μ)/(2t-4t')], Q_3L = 2arccos[(-4t'-μ)/(2t')],respectively; the masses m_x,y are always of the same sign;for the wave vector (Q_4L,-Q_4L,0) we have Q_4L=2arccos[(2t-2t'-μ)/(2t+4t')],the corresponding conditions that the masses are opposite are shown graphically in Fig. <ref>d.For Kohn points, which are symmetric with respect to the point W and connected by the wave vector (Q_1W,0,0), we find Q_1W=2arccos[(4t+4 t'-μ)/(2 t')]. In this case, the masses for the point (π-Q_1W/2,0,2π) will be the same as in the equation (<ref>) above for points, symmetric with respect to the point X, characterized by the vector (Q_4X,0,0),and they interchange their values at the point (π+Q_1W/2,0,2π). From these results it follows that for t'>0 Kohn points give a nonanalytic contribution to the susceptibilityin a wide range of -2t≲μ≲ 4t, and for t'<0 in the range t≲μ≲ 3t. Specific ranges of chemical potentials and concentrations at t '= 0.3t and t' = -0.45t are presented in Table. 1. §.§ The relation between the Kohn points and the momentum dependences of the susceptibility and effect of Kohn points on quantum phase transitions The analysis of the previous subsection allows us to identify peaks of susceptibility, found numerically in Sec. 2.1, with the contributions of various Kohn points, establishing the correspondence of these contributions with different types of magnetic order.For t'>0 the evolution of maxima of the irreducible susceptibility in Fig. <ref> with decrease of the chemical potential corresponds to a change of the type of magnetic order FM →(Q_W,Q_W,Q_W) →(Q_1X,Q_1X,Q_1X) →(0,0,Q_3X), with a possible narrow region of the phase (0,Q_2X,Q_2X)between phases (Q_1X,Q_1X,Q_1X) and (0,0,Q_3X). The phase (Q_W,Q_W,Q_W), which was not considered in Section <ref> because of very specific (disconnected) form of the Fermi surface, is present at large chemical potential near the upper edge of the band. Further types of order (Q_1X,Q_1X,Q_1X) and (0,0,Q_3X), which occur with decrease of the chemical potential are determined by the vicinity of the point Xand characterized by the wave vectors Q_1X and Q_3X, coinciding with those found in Section <ref>. Although the sequence of the phases, determined in this paper, coincides with previously obtained <cit.>, there are some differences due to more accurate (analytical) determination of wave vectors in Section <ref>. For example, the range of existence of the phase (π,π,π), which is the limiting case of the (Q_1X,Q_1X,Q_1X) phase for Q_1X→π, shrinks to the pointμ=2t.On the other hand, at t'<0 the decrease of the chemical potential leads to the following sequence of dominating phases: FM→(0,0,Q_W)→(Q_1X,Q_1X,Q_1X), the phase (Q_1X,Q_1X,Q_1X) is present at not too large |t'/t|.Although opening of Fermi surface “window” in the vicinity of point L at μ=μ_L=-6t' does not change thesymmetry of the wave vector, for which the global susceptibility maximum is achieved (by virtue of dominating contributions of the points X,W), the momentum dependence of χ_𝐪^0 in the vicinity of 𝐪 = 0 changes significantly (see Fig. <ref>b,c).In particular, for μ>μ_L the susceptibility rather weakly depends on 𝐪at small q (Fig. <ref>c), analogously to the case of small t' two-dimensionalt-t' Hubbard model slightly above van Hove filling <cit.>, so that in this case the competition of ferromagnetic and incommensurate correlations occurs.Dominating type of magnetic ordering in this situation can be easily changed by correlations and/or peculiarities of the Fermi surface away from the L point. At the same time, at μ<μ_L the susceptibility has the pronounced minimum at 𝐪=0 (Fig. <ref>b) indicating the absence of ferromagnetic correlations, as well as dominating incommensurate correlations.The change of susceptibility momentum dependence when the Fermi surface crosses L point is largely due to occurence of Kohn points connected by wave vectors (Q_1L,Q_1L,Q_1L) and (Q_2L,Q_2L,0)in the vicinity of the L pointat μ<μ_L.In all considered cases the susceptibility maximum is caused by hyperbolic Kohn points considered in Section 2.1.At finite temperatures the theory predictsat quantum phase transition point critical exponents of the susceptibility χ_ Q∝ T^-γ and the correlation length ξ∝ T^-ν equal to γ=ν=1, see Refs. <cit.>.§ THE ANTIFERROMAGNETISM OF CHROMIUM AND THE TWO-BAND MODELTo study the nature of antiferromagnetism of chromium we consider first the results of ab initio calculations within the local density approximation <cit.> using tight-binding linear muffin-tin orbital atomic spheresframework (LDA TB-LMTO-ASA) <cit.>. The von Barth-Hedin local exchange-correlation potential has been used <cit.>;we choose the lattice parameter a=2.8845 Å and the mesh 40×40×40in the reciprocal space.The chromium band structure and momentum dependence of the susceptibility, together with partial contributions of differtent d orbitals, are presented in Fig. <ref>.The susceptibility has a maximum at the incommensurate wave vector (0,0,Q) in Γ-H direction with Q close to 2π. To explain the obtained momentum dependence of the susceptibility and establish the connection with Kohn anomalies, we consider a simple two-band model, qualitatively describing the chromium Fermi surfaces, Ĥ=∑_ k,σE_m( k)ĉ_ k m σ^+ĉ_ k m σ, m=1,2 being the band index. The corresponding dispersion isE_𝐤^(m)= - 8t_1mcosk_x/2cosk_y/2cosk_z/2- 2t_2m(cosk_x+cosk_y+cosk_z)- 4t_3m(cosk_xcosk_y+cosk_xcosk_z+cosk_ycosk_z)+E_0m,where t_1m, t_2m, t_3m are nearest-, next-nearest and next-next-nearest neighbour hopping integrals of body centered cubic lattice for the first and second bands respectively, E_0m are energy levels of the bands. Hopping integrals and energy levels are determined by the coincidence of Fermi surface points along symmetric directions, obtained within the first-principle calculations and the model (<ref>). The resulting parameter values are presented in Table <ref>,the corresponding Fermi surfaces are shown in Fig. <ref>. It is worthwhile to note that the considered model does not assume approximate nesting between different sheets of Fermi surface and in this respect it is more realistic than models of antiferromagnetism of chromium considered earlier, see, e.g., Refs. <cit.>.At the same time, the absence of nesting in our model even improves qualitative applicabilityof random phase approximation, in comparison tothe case of perfect nesting, discussed previously in Refs. <cit.>.The dominant contribution to the susceptibility in the present model is expected from Kohn points on different sheets of the Fermi surface: electron-like sheet closed around the point Γ and hole-like sheet closed around the point H=(0,0,2π).Theexpansion of the spectrum in the vicinity of Kohn points on different sheets yields different magnitudes of Fermi velocity and inverse masses.To find the Kohn points we assume that the vector Q=(0,0,Q_z) is parallel to one of coordinate axis (which corresponds to the results of ab initio calculations and experimentally observed vector Q) and enforce the conditions that (a) the points K and K+Q belong to Fermi surfaces of first and second band respectively, (b) the corresponding Fermi velocities v_1( K) and v_2( K+Q) are antiparallel.Due to the symmetry it is sufficient to consider only the part K_z<0 of the Fermi surface of the first sheet, which can be parametrized by coordinates K_x,K_y of point K.The quantity Q_z(K_x,K_y), determined by the condition (a), is therefore also a single-valued function of these two coordinates of Kohn point on the first sheet, the result of calculation of this function is plotted in Fig. <ref>a. To verify the condition (b), we show the values of-cos(𝐯_1( K), 𝐯_2( K+Q))=-𝐯_1( K)·𝐯_2( K+Q)/(v_1( K) v_2( K+Q)) in Fig. <ref>b.One can see that Kohn points form two lines in the reciprocal space, characterized by valuesQ_1z≈ 1.83π and Q_2z≈ 1.90π.Moreover, the inner line of Kohn points with Q_z=Q_1z is a boundary of a region, in which Fermi velocities on different sheets are almost antiparallel.This region is, however, characterized by substantial dependence of Q_z on K_x,y and, therefore, does notsubstantially contribute to the momentum dependence of the susceptibility, which is confirmed by a numerical calculation of the susceptibility, see below.The positions of Kohn point lines on the Fermi surfacesare shown in Fig. <ref>. To obtain the momentum dependence of the susceptibility, we parametrize the position of Kohn points K_i(φ) on each line (enumerated by i=1,2) by an angle φ and introduce local rotated coordinate system in reciprocal space such that k'_zi axisis directed along the difference of Fermi velocities 𝐯_1( K_i(φ))-𝐯_2( K_i(φ)+ Q_i) at the corresponding Kohn points, while k'_xi axis is directed perpendicular to k'_zi and tangentially to i-th line ofKohn points on the first sheet.Moreover, on Kohn point lines, where Fermi velocities of the two sheets are almost opposite (see Fig. <ref>b), k'_xi axis lies in a tangential plane to the Fermi surface. The corresponding expansion of the spectrum in the vicinity of Kohn points has a formE_𝐊_i(φ)+𝐤^(1) ≃v_1i(φ)k'_zi+(k'_xi)^2/2m_1i(φ),E_𝐊_i(φ)+𝐐_i+𝐤^(2) ≃-v_2i(φ)k'_zi+(k'_xi)^2/2m_2i(φ),where v_1i(φ)=|𝐯_1( K_i(φ))|, v_2i(φ)=| 𝐯_2( K_i(φ)+ Q_i)|, and we neglect weak dependence 𝐐_i(φ), i. e. we assume that lines of Kohn points are approximately parallel. Due to the symmetry of the Fermi surfaces one can restrict oneself by angles φ∈[0,π/4].To reduce the dispersion to standard form (<ref>) we perform change of variablesk'_zi →k'_zi-(k'_xi)^2/2(v_1i(φ)+v_2i(φ))[1/m_1i(φ)-1/m_2i(φ)].Written in new variables, the spectrum has the formE_ K_i(φ)+𝐤^(1) ≃v_1i(φ)k'_zi+(k'_xi) ^2/2m_si(φ),E_ K_i(φ)+ Q_i+ k^(2) ≃-v_2i(φ)k'_zi+(k'_xi)^2/2m_si(φ),where1/m_si(φ) = 1/v_1i(φ)+v_2i(φ)[v_1i(φ)/m_2i(φ)+v_2i(φ)/m_1i(φ)]. The dependences of inverse massesm_ji(φ), m_si(φ) (i,j=1,2) are shown in Fig. <ref>.The signs of resulting masses m_si(φ) do not change with φ but opposite for the two Kohn point lines.The contribution of these points to interband part of the susceptibility at T=0 can be calculated analogously to Refs. <cit.>,χ_𝐪+𝐐^ ib, 0 ≡-∑_𝐤f(E_𝐤^(1))-f(E_𝐤+𝐪+𝐐^(2))/E_𝐤^(1)-E_𝐤+𝐪+𝐐^(2)≃ χ_𝐐^ ib, 0-4/π^2∑_i=1,2∫_0^π/4dφ/D_i(φ)1/v_1i(φ)+v_2i(φ){[ √(|V_si(φ)q'_zi(φ))|, m_si(φ)q'_zi(φ) <0,; V_si(φ)q'_zi(φ)/(πΛ), m_si(φ)q'_zi(φ) >0, ].whereV_si(φ) = 2m_si(φ)v_1i(φ)v_2i(φ)/v_1i(φ)+v_2i(φ),D_i(φ) = |∂(k_x,k_y,k_z)/∂(φ,k'_xi,k'_zi)|_k'_xi=k'_zi=0,q'_zi(φ) being the projection of vector q onto the axis k'_zi, Λ∼ 1 is the cutoff parameter in the reciprocal space.Therefore, in this case the square-root non-analytic momentum dependence of the susceptibility is expected at wave vectors close to 𝐐_1,2 and connecting Kohn point lines on different sheets.This conclusion is confirmed by the calculation of momentum dependence of interband contribution to the magnetic susceptibility within the two-band model: the result of numerical calculation of the integral in the first line of Eq. (<ref>) is presented in Fig. <ref>, showing that the local non-analytic maxima of magnetic susceptibility are exactly positioned at the wave vectors 𝐐_1,2.Moreover, the contribution of outer Kohn point line with wave vector Q_2, which is close to experimentally determined wave vector of spin density wave in chromium <cit.>, appears to be dominating.The analysis of the susceptibility at finite temperatures in the considered case of cylindrical Kohn points (under the assumption that the band structure is not substantially changed with pressure or doping) predicts the critical exponents γ=ν=1 of susceptibility and correlation length at the quantum phase transition pointwith weak logarithmic corrections, see Refs. <cit.>.§ CONCLUSION In this paper we have investigated the effect of Kohn points in electronic spectrumof three-dimensional systems on magnetic properties within a one-band model with hopping between the nearest- and next-nearest neighbors on fcc lattice and two-band model, containing hopping within three coordination spheres on bcc lattice and modeling some features of the electronic dispersion of chromium.For fcc lattice, we have investigated the effect of Kohn hyperbolic points on the magnetic susceptibility in a wide range of chemical potentials (concentrations) and determined the ranges of chemical potentials for symmetric directions, in which the effect of these points is expected. The obtained results are confirmed by the numerical analysis of the susceptibility of non-interacting electrons on fcc lattice. Near quantum phase transitions the studied Kohn points lead to a temperature dependence of the susceptibility and the correlation length χ_ Q∝ξ∝ 1/T. We have additionally investigated the effect of the opening of the window of Fermi surface near the point L, which may be important for explaining the properties of ZrZn_2. It is shown that when the point L of the Brillouin zone belongs to the filled states (E_L<0, the Fermi surface window is open near L), there is a competition of ferromagnetic and incommensurable correlations. Closing this window (E_L>0) leads to a drastic change of the momentum dependence of the magnetic susceptibility and substantial increase of the contribution of incommensurate correlations.On the basis of the analysis of the two-band model ofchromium, it is shown that two lines of Kohn points may be present in this substance, corresponding to close values of antiferromagnetic wave vector (0,0,Q_z) with Q_1z≈ 1.83π and Q_2z≈ 1.90π in units of the inverse lattice constant. These lines of Kohn points lead to a one-sided nonanalytic (square root) momentum dependence of the susceptibility at T=0, the contribution of the outer line of Kohn points with Q_z=Q_2z, which is close to the experimentally measured spin-density wave vector in chromium, dominates. At finite temperatures, temperature dependences of the susceptibility and the correlation length χ_ Q∝ξ∝ 1/T with weak logarithmic corrections are expected near quantum phase transition.The described approaches to study the contribution of Kohn points can further use realistic band stucture, obtained within ab initio investigations in the framework of the density functional method and the dynamic mean-field theory, which will allows one to investigate the effect of Kohn points in real substances. The application of the dynamic mean-field theory <cit.> and dynamical vertex approximation <cit.> will also allow one to investigate the effect of electronic interaction (including the non-local interactions) beyond random phase approximation. Although previous study of one-band model <cit.> showed that the electron-electron interaction does not change qualitatively the effect of Kohnanomalies near quantum phase transitions, this problem requires more detailed study in future.A detailed analysis of the available experimental data on magnetic properties of weak ferro- and antiferromagnets near quantum phase transitions will allow to analyze possible deviations from the predictions of the Hertz-Moriya-Millis theory caused by the effect of Kohn anomalies. Acknowledgments A. A. Katanin thanks T. Schäfer, A. Toschi, K. Held, and W. Metzner for valuable discussions of the effectof Kohn anomalies in strongly correlated systems. The work was carried out within the state assignment of FASO of Russia (theme “Electron” 01201463326), and supported in part by grant from the Russian Foundation for Basic Research 17-02-00942a and project of the Ural Branch of the Russian Academy of Sciences 15-8-2-9.Kohn W. Kohn, Phys. Rev. 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Takashima, M. Nohara, and H. Takagi, Nature 455, 1220 (2008). Cr E. Fawcett, Rev. Mod. Phys. 60, 209 (1988). Cr_QPT R. Jaramillo, Yejun Feng, J. Wangc, and T. F. Rosenbaum, PNAS 107, 13631 (2010). Cr_dop1 A. Yeh, Yeong-Ah Soh, J. Brooke, G. Aeppli, T. F. Rosenbaum, and S. M. Hayden, Nature 419, 459 (2002). Cr_dop2 M. Lee, A. Husmann, T. F. Rosenbaum, and G. Aeppli, Phys. Rev. Lett. 92, 187201 (2004). Cr_dop3 D. A. Sokolov, M. C. Aronson, L. Wu, Y. Zhu, C. Nelson, J. F. Mansfield, K. Sun, R. Erwin, J. W. Lynn, M. Lumsden, and S. E. Nagler, Phys. Rev. B 90, 035139 (2014). Igoshev P. A. Igoshev, M. A. Timirgazin, V. F. Gilmutdinov, A. K. Arzhnikov, V. Yu. Irkhin,J. Phys.: Condens. Matter, 27, 446002 (2015). Igoshev1M. A. Timirgazin, P. A. Igoshev, A. K. Arzhnikov, V. Yu. Irkhin, J. Phys.: Condens. Matter 28, 505601 (2016). Tremblay Y.M. Vilk, A.-M.S. Tremblay, J. Phys. I (France) 7, 1309 (1997). Onufr F. Onufrieva, P. Pfeuty, and M. Kiselev, Phys. Rev. Lett. 82, 2370 (1999); arXiv:cond-mat/9804191; J. Phys. Chem. Solids 59, 1853 (1999). OurBook A. A. Katanin, V. Yu. Irkhin, P. A. Igoshev Model approaches to magnetism of two-dimensional itinerant systems [in Russian], Fizmatlit, Moscow (2012).Jones_Gunnarsson_RMP_1989 R. O. Jones and O. Gunnarsson, Rev.Mod. Phys. 61, 689 (1989).Andersen_Jepsen_PRL_1984 O. K. Andersen and O. Jepsen, Phys.Rev. Lett. 53, 2571 (1984).Barth_Hedin_1972 U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).DMFT A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996); G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006).Lomer W. M. Lomer, Proc. Phys. Soc. 80, 489 (1962)Shibatani A. Shibatani, K. Motizuki, and T. Nagamiya, Phys. Rev. 177, 984 (1969)Dzyaloshinskii I. E. Dzyaloshinskii and E. I. Katz, Sov. Phys. JETP 35, 584 (1972).Machida K. Machida and M. Fujita, Phys. Rev. B 30, 5284 (1984)Dzyaloshinskii1 A. T. Zheleznyak, V. M. Yakovenko, I. E. Dzyaloshinskii, Phys. Rev. B 55, 3200 (1997) DGA1 A. Toschi, A. A. Katanin, K. Held, Phys. Rev. B 75, 045118 (2007).DGA2 G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, ArXiv: 1705.00024. | http://arxiv.org/abs/1707.08924v3 | {
"authors": [
"A. A. Stepanenko",
"D. O. Volkova",
"P. A. Igoshev",
"A. A. Katanin"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727161244",
"title": "Kohn anomalies in momentum dependence of magnetic susceptibility of some three-dimensional systems"
} |
An Improved ε-constrained Method in MOEA/D for CMOPs with Large Infeasible Regions[1]Department of Electronic Engineering, Shantou University, Guangdong, 515063, China[2]College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Jiangsu, 210016, China[3]School of Software Engineering, South China University of Technology, Guangdong, 515063, China[4]Department of Mathematics, Shantou University, Guangdong, 515063, China[5]BEACON Center for the Study of Evolution in Action, Michigan State University. East Lansing, Michigan, USA. An Improved Epsilon Constraint-handling Method in MOEA/D for CMOPs with Large Infeasible Regions Zhun Fan[1] Wenji Li[1] Xinye Cai [2] Han Huang [3] Yi Fang [1] Yugen You [1] Jiajie Mo [1] Caimin Wei [4]Erik Goodman[5]Received: date / Accepted: date ====================================================================================================================================================================================================== This paper proposes an improved epsilon constraint-handling mechanism, and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions (RFS) in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then the fourteen benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem.§ INTRODUCTION Real-world optimization problems usually involve the simultaneous optimization of multiple conflicting objectives with a number of constraints. Without loss of generality, a CMOP considered in this paper is defined as follows (<cit.>):𝐅(𝐱) = (f_1(𝐱),…,f_m(𝐱)) ^ T g_i(𝐱) ≥ 0, i = 1,…,qh_j(𝐱) = 0, j= 1,…,p 𝐱∈ℝ^nwhere F(𝐱) = (f_1(𝐱),f_2(𝐱), … ,f_m(𝐱)) ^ T ∈ℝ ^m is an m-dimensional objective vector, g_i(𝐱) ≥ 0 is an inequality constraint, and h_j(𝐱)=0 is an equality constraint. 𝐱∈ℝ^n is an n-dimensional decision vector. The feasible region S is defined as the set {𝐱 | g_i(𝐱) ≥ 0, i = 1,…,qand h_j(𝐱) = 0, j= 1,…,p}.In CMOPs, there are usually more than one constraint. The overall constraint violation is a widely used approach to deal with constraint violations, as it summarizes them into a single scalar, as follows: ϕ(𝐱) = ∑_i=1^q |min(g_i(𝐱),0)| + ∑_j = 1^p |h_j(𝐱)|If ϕ(𝐱) = 0, 𝐱 is feasible; otherwise, it is infeasible. Any solution in set S is feasible, and for any two solutions 𝐱^1 ∈ S and 𝐱^2 ∈ S, 𝐱^1 is said to dominate 𝐱^2 if f_i(𝐱^1) ≤ f_i(𝐱^2) for each i ∈{1,...,m} and f_j(𝐱^1) < f_j(𝐱^2) for at least one j ∈{1,...,m}, denoted as 𝐱^1 ≼𝐱^2. For a solution 𝐱^* ∈ S, if there is no other solution in S dominating 𝐱^*, then 𝐱^* is called a Pareto optimal solution. A set including all of the Pareto optimal solutions is called a Pareto optimal set (PS). Mapping the PS into the objective space obtains a set of objective vectors, which is called a Pareto optimal front (PF), and PF = {F(𝐱)| 𝐱∈ PS}. CMOEAs aim to find a representative set of Pareto optimal solutions. They have to tackle the multiple conflicting objectives with a number of constraints simultaneously, and to maintain a good balance between convergence and diversity of the achieved solutions. In CMOEAs, there are two basic components: one is the constraint-handling mechanism, and the other is the multi-objective evolutionary algorithm (MOEA).In terms of constraint-handling, many methods have been proposed in evolutionary optimization (<cit.>). They can be roughly divided into penalty function methods, special representations and operators, repair methods, separation of objectives and constraints and hybrid methods (<cit.>). The penalty function method is widely used due to its simplicity in the constraint handling (<cit.>). However, the ideal penalty factors cannot be known in advance for an arbitrary CMOP, and tuning the penalty factors can be a very tedious task. In recent years, a number of other constraint-handling techniques have had a relatively high impact in evolutionary optimization, including feasibility rules, stochastic ranking, ε-constrained method, novel penalty functions, novel special operators, multi-objective concepts and ensemble of constraint-handling techniques (<cit.>). However, most of them aim to solve constrained scalar optimization problems when they are first proposed.MOEAs can be classified into three different types according to their selection approaches. The first type is non-dominated-based methods, and representative examples include NSGA-II (<cit.>), PAES-II (<cit.>), SPEA-II (<cit.>), NSGA-III (<cit.>) and so on. The second type is decomposition-based approaches, and typicalexamples include MOEA/D (<cit.>), MOEA/D-DE (<cit.>), EAG-MOEA/D (<cit.>), MOEA/D-M2M (<cit.>), MOEA/D-SAS (<cit.>) and so on. Currently, MOEA/D is a popular algorithm to solve unconstrained multi-objective optimization problems (MOPs). MOEA/D decomposes a MOP into many scalar optimization subproblems, and optimizes them simultaneously in a collaborative way. The last type is indicator-based methods. This type of MOEAs selects solutions based on the improvement of a performance metric. Representative methods include IBEA (<cit.>), SMS-EMOA (<cit.>), HypE (<cit.>), FV-MOEA (<cit.>) and so on. There are two commonly used test suites of CMOPs, including CTP (<cit.>) and CF test instances (<cit.>). For CTP1-CTP5 and CF1-CF10, the feasible regions are relatively large, and a CMOEA can approximate their PFs without encountering any infeasible obstacles during the entire evolutionary process. Thus, CTP1-5 and CF1-10 are not good test problems to evaluate the performance of constraint-handling mechanisms. For the remaining test problems CTP6-8, the feasible regions are relatively large, and the population of a CMOEA can reach these regions with high probability. Thus, CTP and CF test suites can not effectively measure the performance of constraint-handling techniques. When solving CTP (<cit.>) and CF (<cit.>) test instances, the constraint dominance principle (CDP) (<cit.>) is good enough to handle the constraints.To overcome the shortcomings of the CTP and CF test suites discussed above, we propose a set of new CMOPs (named LIR-CMOP1-14). Each of them has a number of large infeasible regions, and the feasible regions are relatively small. The population of a CMOEA cannot easily discover these small feasible regions, which brings new challenges to the existing CMOEAs. In fact, many real-world optimization problems also have this characteristic. For example, the robot gripper optimization problem considered in this paper has large infeasible regions as illustrated in Section <ref>. Thus, it has important significance in practice to design specific mechanisms for solving CMOPs with large infeasible regions.In this paper, we propose an improved ε-constrained version of MOEA/D to deal with CMOPs. Compared with the original ε-constrained method (<cit.>), the proposed method can keep a good balance in the search between the feasible and infeasible regions. It uses the information of the feasible ratio of the population to dynamically balance the exploration between the feasible regions and infeasible regions.The remainder of the paper is organized as follows. Section <ref> introduces related work on MOEA/D and the existing CMOEAs based on MOEA/D. Section <ref> illustrates the improved epsilon constraint-handling method as here embedded in MOEA/D. Section <ref> designs a set of new CMOPs (LIR-CMOPs) with large infeasible regions. Section <ref> describes a comprehensive set of experiments to compare the proposed CMOEA (MOEA/D-IEpsilon) with four other CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D. In Section <ref>, a robot gripper optimization problem is used to test MOEA/D-IEpsilon and the other four CMOEAs. Finally, Section <ref> presents the conclusions.§ RELATED WORK§.§ MOEA/DMOEA/D (<cit.>) decomposes a MOP into a number of scalar optimization subproblems and optimizes them simultaneously in a collaborative way. Each subproblem is defined by a decomposition function with a weight vector λ^i. In MOEA/D, a set of N uniformly spread weight vectors λ^1,…,λ^N are adopted to formulate N subproblems. The weight vectors λ^i satisfy ∑_k = 1 ^ mλ_k^i = 1 and λ_k^i ≥ 0 for each k ∈{1,…,m}. In terms of decomposition methods, there are three commonly used approaches, including weighted sum (<cit.>), Tchebycheff (<cit.>) and boundary intersection approaches (<cit.>).In the weighted sum approach, each subproblem is defined by summing each objective weighted by a different weight. The j-th subproblem with the weighted sum decomposition method is defined as follows:g^te(𝐱|λ) = ∑_i = 1 ^mλ_i^jf_i(𝐱) 𝐱∈SFor a minimizing MOP, in the case of a convex PF, the weighted sum approach can work well. However, if the PF is non-convex, only a part of PF can be found by this approach.In the Tchebycheff decomposition method, the j-th subproblem is defined as follows: g^te(𝐱|λ,z^*) = max_1 ≤ i ≤ m{λ_i^j |f_i(𝐱) - z_i^*| } 𝐱∈Swhere z^* = (z_1^*,…,z_m^*) is the ideal point, and z_i^* = min{f_i(𝐱 | 𝐱∈ S}. The Tchebycheff method is a widely used decomposition approach. It can approximate both concave and convex parts of PFs.In the boundary intersection approach, two distances d_1 and d_2 are defined to evaluate the convergence and diversity respectively. The j-th subproblem is defined as follows: g^pbi(𝐱|λ^j,z^*) = d_1 + θ d_2 𝐱∈Sd_1 = ‖ (F(𝐱) - z^*)^Tλ^j ‖/‖λ^j ‖d_2 = ‖ (F(𝐱) - z^* ) - d_1 λ^j/‖λ^j ‖) ‖The boundary intersection method is able to solve MOPs with any shape of PFs. However, the penalty factor θ must be set in advance. §.§ Decomposition-based CMOEAsIn decomposition-based CMOEAs, a CMOP is decomposed into a set of constrained scalar optimization subproblems, and these subproblems are solved in a collaborative way simultaneously. Representative methods include C-MOEA/D (<cit.>), MOEA/D-Epsilon (<cit.>), MOEA/D-CDP (<cit.>) and MOEA/D-SR (<cit.>).C-MOEA/D (<cit.>) embeds an epsilon constraint-handling approach into MOEA/D, and the epsilon value is set adaptively. To be more specific, the epsilon level is set to CV_mean * FR. CV_mean denotes the mean value of the overall constraint violation in the current population, and FR (Number of feasible solutions/Population size) denotes the feasible ratio of solutions in the current population. For two solutions, if their overall constraint violations are both less than CV_mean * FR or their overall constraint violations are equal, the one with the better aggregation value is selected. Otherwise, the one with the smaller overall constraint violation is selected.MOEA/D-Epsilon (<cit.>) also adopts the epsilon method to handle constraints. Unlike C-MOEA/D, the epsilon value in MOEA/D-Epsilon is set dynamically with the increase of generation counter K. The detailed setting of the epsilon value can be found in (<cit.>).MOEA/D-CDP (<cit.>) adopts CDP (<cit.>) to deal with constraints in the framework of MOEA/D. There are three basic rules to select solutions. For two feasible solutions, the one with the better aggregation value is selected. For two infeasible solutions, the one with the smaller overall constraint violation is selected. For a feasible and an infeasible solution, the feasible one is selected.MOEA/D-SR (<cit.>) embeds the stochastic ranking method (SR) (<cit.>) in MOEA/D to deal with constraints. A parameter p_f ∈ [0,1] is set to balance the selection between the objectives and the constraints in MOEA/D-SR. For two solutions, if a random number is less than p_f, the one with the better aggregation value is selected into the next generation. If the random number is greater than p_f, the solutions selection is similar to that of MOEA/D-CDP. In the case of p_f = 0, MOEA/D-SR is equivalent to MOEA/D-CDP. In summary, C-MOEA/D and MOEA/D-Epsilon both adopt the epsilon constraint-handling approach to solve CMOPs. To get across large infeasible regions, ε should be increased at sometimes, and be greater than the maximum overall constraint violation in the current population. However, in C-MOEA/D, ε is always less or equal than CV_mean, and in MOEA/D-Epsilon, ε is always decreasing during the evolutionary process. In MOEA/D-CDP, feasible solutions are always better than infeasible solutions. Thus, the infeasible solutions which can help to get across large infeasible regions are difficult to survive. MOEA/D-SR applies a parameter p_f to balance the searching between the feasible and infeasible regions. In order to get across large infeasible regions, p_f should be set dynamically. However, p_f is a static parameter in MOEA/D-SR. To overcome the shortcomings of the four decomposition-based CMOEAs discussed above, an improved epsilon constraint-handling method embedded in MOEA/D is proposed.§ THE PROPOSED METHODIn this section, the concept of epsilon level comparison, the original epsilon level setting method and the improved epsilon level setting approach are described. §.§ Epsilon Level ComparisonIn the epsilon constraint handling approach (<cit.>), the relaxation of constraints is controlled by the epsilon level ε. For two solutions 𝐱^1 and 𝐱^2, their overall constraint violations are ϕ^1 and ϕ^2. Then, for any ε satisfying ε≥ 0, the epsilon level comparison ≼_ε is defined as follows: (𝐱^1,ϕ^1) ≼_ε(𝐱^2,ϕ^2)⇔ 𝐱^1 ≼𝐱^2, if ϕ^1, ϕ^2 ≤ε 𝐱^1 ≼𝐱^2, if ϕ^1 = ϕ^2ϕ^1 < ϕ^2, otherwiseIn Eq. (<ref>), the epsilon comparison approach is equivalent to CDP (<cit.>) when ε = 0. In the case of ε = ∞, it does not consider any constraints. In other words, the comparison between any two solutions is based on their non-dominated ranks on objectives when ε = ∞.§.§ Epsilon Level SettingIn the epsilon constraint-handling method, the setting of ε is quite critical. In (<cit.>), an epsilon level setting method is suggested as follows: ε(k) = ε(0)(1 - k/T_c)^cp, 0 < k < T_c, ε(0) = ϕ(𝐱^θ)0, k ≥ T_cwhere 𝐱^θ is the top θ-th individual of the initial population sorted by overall constraint violations in a descending order. cp is to control the speed of reducing relaxation of constraints. ε(k) is updated until the generation counter k reaches the control generation T_c. When k ≥ T_c, ε(k) = 0. The recommended parameter ranges in (<cit.>) are listed as follows: θ = (0.05 * N), cp ∈ [2,10] and T_c ∈ [0.1T_max, 0.8T_max]. N denotes the population size, and T_max represents the maximum evolutionary generation. §.§ Improved Epsilon Level SettingThe setting of ε(k) in Eq.(<ref>) is always decreasing during the evolutionary process, which may not be suitable to solve CMOPs with large infeasible regions. To overcome this problem, an improved epsilon setting approach is suggested as follows:ε(k) =rule 1:ϕ(𝐱^θ), ifk = 0rule 2:(1 - τ)ε(k-1), ifr_k < α andk < T_crule 3:(1 + τ)ϕ_max, ifr_k ≥α andk < T_c rule 4:0, ifk ≥ T_cwhere ϕ_k(𝐱^θ) is the overall constraint violation of the top θ-th individual in the initial population, r_k is the ratio of feasible solutions in the k-th generation. τ ranges between 0 and 1, and has two functions. One is to control the speed of reducing the relaxation of constraints, and the other is to control the scale factor multiplied by the maximum overall constraint violation. α is to control the searching preference between the feasible and infeasible regions, and α∈[0,1]. ϕ_max is the maximum overall constraint violation found so far.The ε(0) setting method in Eq. <ref> is sometimes the same as that in Eq. <ref>. If ε(0) = 0, ε(k) in Eq. <ref> is identically equal to zero, which tends to hinder a CMOEA's exploration of the infeasible regions. However, ε(k) in Eq. <ref> is not identically equal to zero when ε(0) = 0 according to the third rule of the proposed epsilon setting approach.In the case k > 0, three rules are adopted to control the value of ε in Eq. <ref>. Rule 2 is adopted to strengthen the searching in the feasible regions. Rule 3 is used to strengthen the exploration in the infeasible regions. The last rule 4 is same as in the CDP (<cit.>) constraint-handling method.Two parameters k and r_k are applied to choose the right control rule for ε(k). If k < T_c and r_k < α, rule 2 for setting ε(k) is adopted. In this circumstance, ε(k) is set to (1 - τ)ε(k-1), which has an exponential decreasing rate. It has a faster descent rate than the epsilon setting in Eq. (<ref>), which can help to enhance the searching in the feasible regions more effectively. If k < T_c and r_k ≥α, rule 3 for setting ε(k) is applied. In this situation, most solutions are feasible. Thus, strengthening the exploration in the infeasible regions may help a CMOEA to get across a number of large infeasible regions. In rule 3, ε(k) = (1 + τ)ϕ_max, which strengthens the exploration in the infeasible regions. Thus, the improved epsilon method has the balanced ability to explore the feasible and infeasible regions simultaneously. α is a critical parameter to balance the searching between the feasible and infeasible regions. If the RFS r_k is less than α, rule 2 is adopted to enhance the exploration in the feasible regions. Otherwise, rule 3 is applied to enhance the exploration in the infeasible regions. Thus, the proposed epsilon constraint method can keep a good balance of exploration between the feasible and infeasible regions. It utilizes the RFS to dynamically balance the exploration between the feasible regions and infeasible regions. Compared with the ε setting in Eq. (<ref>), the proposed method in Eq. (<ref>) has the ability to increase ε(k) during the evolutionary process, which can help to solve CMOPs with large infeasible regions. In the case of k ≥ T_c, rule 4 is applied. In this situation, ε(k) = 0, and the epsilon constraint-handling method exerts the highest selection pressure toward the feasible regions.§.§ Embedding the improved epsilon method in MOEA/D The proposed MOEA/D-IEpsilon integrates the improved epsilon constraint-handling method in Eq. <ref> into the framework of MOEA/D. In MOEA/D-IEpsilon, a CMOP is decomposed into a number of constrained scalar subproblems, and these subproblems are optimized simultaneously in a collaborative way. In our experimental studies, the Tchebycheff approach is adopted, and its detailed definition is listed in Eq. (<ref>).For a given weight vector λ, there exists an optimal solution of Eq. (<ref>), and this optimal solution is also a Pareto optimal solution of Eq. (<ref>). Therefore, we can achieve different Pareto optimal solutions of Eq. (<ref>) by setting different weight vectors. The psuecode of MOEA/D-IEpsilon is listed in Algorithm 1. It is almost the same as that of MOEA/D, except for the method of subproblem updating. Lines 1-6 initialize a number of parameters in MOEA/D-IEpsilon. First, a CMOP is decomposed into N subproblems which are associated with λ^1,…, λ^N. Then the population P, the initial epsilon value ε(0), the ideal point z^* and the neighbor indexes B(i) are initialized.Lines 11-22 generate a set of new solutions and update the ideal point z^*. To be more specific, a set of solutions which may be updated by a newly generated solution 𝐲^j is selected (lines 11-17). In line 18, the differential evolution (DE) crossover is adopted to generate a new solution 𝐲^j. The polynomial mutation operator is executed to mutate 𝐲^j in line 19. The ideal point z^* is updated (lines 20-22).Lines 23-30 implement the updating process of subproblems. In line 26, the subproblems are updated based on the improved epsilon constraint-handling approach, and the detailed procedures are listed in Algorithm 2. Finally, a set of non-dominated solutions (NS) is selected based on the non-dominated sort in line 33. In Algorithm 2, there are three basic rules to update a subproblem. For two solutions 𝐱^j and 𝐲^j, if their overall constraint violations are less than or equal to ε(k), and 𝐲^j has a smaller aggregation value (the value of the decomposition function) than that of 𝐱^j, then 𝐱^j is replaced by 𝐲^j (lines 3-7). If 𝐱^j and 𝐲^j have the same overall constraint violation, and 𝐲^j has a smaller aggregation value than that of 𝐱^j, then 𝐱^j is replaced by 𝐲^j (lines 8-12). Otherwise, if 𝐲^j has a smaller overall constraint violation than that of 𝐱^j, then 𝐱^j is replaced by 𝐲^j (lines 13-14). When the subproblem is updated, the function UpdateSubproblems(𝐱^j,𝐲^j,ε(k)) returns true, otherwise, it returns false. § TEST INSTANCESTo evaluate the performance of the proposed MOEA/D-IEpsilon, a set of new CMOPs with large infeasible regions (named LIR-CMOPs) is designed according to our previous work (<cit.>). In terms of constraint functions, all of them have large infeasible regions. In term of objective functions, there are two components:shape functions and distance functions (<cit.>).The shape functions are applied to set the shape of the PFs. In the LIR-CMOP test suite, two types of shape functions, including both convex and concave shapes, are designed. Distance functions are adopted that test the convergence performance of a CMOEA. In LIR-CMOP5-14, the distance functions are multiplied by a scale factor, which increases difficulty of convergence. The detailed definitions of LIR-CMOPs are listed in the Appendix.In this test suite, four test problems, including LIR-CMOP1-4, have large infeasible regions. Fig. <ref>(a)-(d) plot the feasible regions of LIR-CMOP1-4, respectively. It can be seen that the feasible regions of these test instances are very small. In other words, there are a number of large infeasible regions.LIR-CMOP5 and LIR-CMOP6 have convex and concave PFs, respectively, as shown in Fig. <ref>(e)-(f) , and their PFs are the same as those of their unconstrained counterparts. The PFs of LIR-CMOP5 and LIR-CMOP6 can be achieved by a MOEA without any constraint-handling mechanisms.In order to expand the test scope, LIR-CMOP7 and LIR-CMOP8 are designed. For these two test instance, their unconstrained PFs are located in the infeasible regions, and their PFs are situated on their constraint boundaries. Thus, a MOEA without constraint-handling methods cannot find the real PFs for LIR-CMOP7 and LIR-CMOP8, which are shown in Fig. <ref>(g)-(h).LIR-CMOP9-12 have two different types of constraints. The first type creates large infeasible regions as shown in the black ellipses in Fig. <ref>(i)-(l). The second type creates difficulty in the entire objective space, as it divides the PFs of LIR-CMOP9-12 into a number of disconnected segments. For LIR-CMOP9-10, their PFs are a part of their unconstrained PFs, and for LIR-CMOP11-12, their PFs are situated on their constraint boundaries.In the LIR-CMOP test suite, CMOPs with three objectives are also designed. Two CMOPs, including LIR-CMOP13 and LIR-CMOP14, have three objectives as shown in Fig. <ref> (a)-(b) . The PF of LIR-CMOP13 is the same as that of its unconstrained counterpart. The PF of LIR-CMOP14 is located on the boundaries of its constraints.§ EXPERIMENTAL STUDY §.§ Experimental Settings To evaluate the performance of the proposed MOEA/D-IEpsilon, four other CMOEAs (MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D), with differential evolution (DE) crossover, are tested on LIR-CMOP1-14. The detailed parameters of these five CMOEAs are listed as follows:* Mutation probability Pm = 1/n (n is the number of decision variables) and its distribution index is set to 20. CR = 1.0, f = 0.5.* Population size: N = 300. Neighborhood size: T = 30.* Stopping condition: each algorithm runs for 30 times independently, and stops when 300,000 function evaluations are reached.* Probability of selecting individuals in the neighborhood: δ = 0.9.* The maximal number of solutions replaced by a child: nr = 2.* Parameter setting in MOEA/D-IEpsilon: T_c = 800, α = 0.95, τ = 0.1 and θ = 0.05 N.* Parameter setting in MOEA/D-Epsilon: T_c = 800, cp = 2, and θ = 0.05 N. * Parameter setting in MOEA/D-SR: S_r = 0.05.§.§ Performance MetricTo measure the performance of MOEA/D-IEpsilon, C-MOEA/D, MOEA/D-CDP, MOEA/D-SR and MOEA/D-Epsilon, two commonly used metrics–the inverted generation distance (IGD) (<cit.>) and the hypervolume (<cit.>) are adopted. The definition of IGD is shown next.* Inverted Generational Distance (IGD):The IGD metric reflects the performance regarding convergence and diversity simultaneously. The detailed definition is as follows:IGD(P^*,A) = ∑_y^* ∈ P^*d(y^*,A)/| P^* | d(y^*,A) = min_y ∈ A{√(∑_i = 1 ^m (y^*_i - y_i)^2)}where P^* is a set of representative solutions in the ideal PF, A is an approximate PF achieved by a CMOEA. The value of IGD denotes the distance between P^* and A. For CMOPs with two objectives, 1000 points are sampled uniformly from the true PF to construct P^*. (Note that this measure cannot be used if the true Pareto front is unknown, so it is used primarily for benchmarking purposes.) For CMOPs with three objectives, 10000 points are sampled uniformly from the PF to constitute P^*. It is worth noting that a smaller value of IGD represents better performance with regards to both diversity and convergence.* Hypervolume (HV):HV reflects the closeness of the non-dominated set achieved by a CMOEA to the real PF. The larger HV means that the corresponding non-dominated set is closer to the true PF.HV(S)=VOL(⋃_x∈ S [f_1(x),z_1^r]× ...[f_m(x),z_m^r] )where VOL(·) is the Lebesgue measure, 𝐳^r=(z_1^r,...,z_m^r)^T is a reference point in the objective space. For a LIR-CMOP, the reference point is placed at 1.2 times the distance to the nadir point of the true PF. It is worth noting that a larger value of HV represents better performance regarding both diversity and convergence. §.§ Discussion of Experiments§.§.§ Performance comparison on LIR-CMOP test suiteThe statistical results of the IGD values on LIR-CMOP1-14 achieved by five CMOEAs in 30 independent runs are listed in Table <ref>. As discussed in Section <ref>, LIR-CMOP1-4 have large infeasible regions in the entire search space. For these four test instances, MOEA/D-IEpsilon is significantly better than the other four tested CMOEAs in term of the IGD metric. Fig. <ref>(a)-(b) shows the final populations achieved by each CMOEA with the best IGD values during the 30 runs on LIR-CMOP1 and LIR-CMOP4. It is clear that MOEA/D-IEpsilon has the best performance regarding diversity among the five CMOEAs under test.LIR-CMOP5-12 have large infeasible regions, as discussed in Section <ref>. It can be observed that MOEA/D-IEpsilon is significantly better than the other four tested CMOEAs on NCMOP5-12. The final populations achieved by each CMOEA on LIR-CMOP9 and LIR-CMOP11 with the best IGD values are plotted in Fig. <ref>(c)-(d). For LIR-CMOP9, MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D only achieve a part of the real PF. However, MOEA/D-IEpsilon can obtain the whole real PF. Thus, MOEA/D-IEpsilon performs better than the other four CMOEAs in terms of diversity. For LIR-CMOP11, the proposed method MOEA/D-IEpsilon can achieve the whole PF. However, the other four CMOEAs do not converge to the whole PF. Thus, MOEA/D-IEpsilon has better convergence performance than the other four CMOEAs. For three-objective test instances (LIR-CMOP13 and LIR-CMOP14), MOEA/D-IEpsilon is also significantly better than the other four CMOEAs. Table <ref> shows the results of the HV values of LIR-CMOP1-14 achieved by five CMOEAs in 30 independent runs. It is clear that MOEA/D-IEpilon is significantly better than the other four CMOEAs on all of the fourteen test instances in terms of the HV metric. §.§.§ Analysis of Experimental Results From the above performance comparison on the fourteen test instances LIR-CMOP1-14, it is clear that MOEA/D-IEpsilon has better diversity and convergence performance than the other four decomposition-based CMOEAs on these fourteen test instances. A common feature of these test instances is that each of them has a number of large infeasible regions, which demonstrates that the proposed epsilon constraint-handling method can deal with the large infeasible regions very well using its automatic adjustment of the epsilon level. § ROBOT GRIPPER OPTIMIZATIONTo verify the capability of MOEA/D-IEpsilon to solve real world optimization problems, a robot gripper optimization problem with two conflicting objectives and eight constraints is explored. §.§ Definition of the robot gripper optimizationThe robot gripper optimization problem is defined in (<cit.>). Five objectives are formulated in these papers. The robot gripper optimization problem considered in this paper has two conflicting objectives and eight constraints. The geometrical structure of the gripper is plotted in Fig. <ref>. The robot gripper optimization problem considered in this paper is defined as follows:f_1(𝐱) = P/min_zF_k(𝐱,z)f_2(x)=a+b+c+e+lc_1(𝐱) =Y_min-y(𝐱,Z_max) ≥ 0c_2(𝐱) = y(𝐱,Z_max) ≥ 0 c_3(𝐱) = y(𝐱,0)-Y_max≥ 0 c_4(𝐱) = Y_G-y(𝐱,0) ≥ 0 c_5(𝐱) = (a+b)^2-l^2-e^2 ≥ 0 c_6(𝐱) = (l-Z_max)^2+(a-e)^2 ≥ b^2 c_7(𝐱) = l-Z_max≥ 0 c_8(𝐱) = min F_k(𝐱,z)-F_G ≥ 0where 𝐱=[a, b, c, e, l, f, δ]^T has seven decision variables, and each variable is shown in Fig. <ref>. The range of each decision variable is as follows: 10mm≤ a≤150mm, 10mm≤ b≤150mm, 100mm≤ c≤200mm, 0mm≤ e≤ 50mm, 10mm≤ f≤ 150mm, 100mm≤ l≤ 300mm and 1.0≤δ≤ 3.14. Two rules are applied to fix the value of f, and they are defined as follows:Rule1: if (a<4b and c<a+b) then f=2e+10 Rule2: if (a<4b and c>a+b) then f=e+50According to the geometric analysis, the gripping force F_k in Fig. <ref> can be defined as follows:F_k=Pbsin(α+β)/2ccosα. The displacement of the gripper end is defined as follows: y(𝐱,z)=2[e+f+c+sin(β+δ)].where g=√((l-z)^2+e^2)+ϕ, α=arccos(a^2+g^2-b^2/2ag), β=arccos(b^2+g^2-a^2/2bg)-ϕ,ϕ=arctane/l-z and z denotes a dynamic displacement of the gripper actuator in the range of 0 to 100 mm.The first objective f_1(x) represents a force transmission ratio between the actuating force P and the minimum gripping force min F_k(𝐱,z). We prefer to transform more actuating force into the gripper force. Thus, this objective should be minimized.The second objective f_2(x) is the sum of all elements of the robot gripper. It is relevant to the weight of the robot gripper, and minimizing f_2(x) can lead to a lightweight design.To study the distribution of solutions in the objective space for the robot gripper optimization problem, 3,000,000 solutions are generated, where 1,500,000 solutions are generated randomly, and the other 1,500,000 solutions are generated by MOEA/D-IEpsilon. In Fig. <ref>, we can observe that the robot gripper optimization problem has large infeasible regions (RFS=0.1396), which can be solved well by the proposed method MOEA/D-IEpsilon according to our previous analysis. To verify this hypothesis, MOEA/D-IEpislon and the other four decomposition-based CMOEAs are tested on the robot gripper optimization problems. §.§ Experimental study§.§.§ Experimental settingsTo solve the robot gripper optimization problem and evaluate the performance of the proposed MOEA/D-IEpsilon, five decomposition-based CMOEAs, including MOEA/D-IEpsilon, MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D with the differential evolution (DE) crossover, are tested on the robot gripper optimization problem. The detailed parameters of these five CMOEAs are the same as listed in Section <ref> except for the number of function evaluations. In the case of the robot gripper optimization problem, each CMOEA stops when 600,000 function evaluations are reached. As the ideal PF of the gripper optimization problem is not known in advance, we use only the hypervolume metric (<cit.>) to measure the performance of the five tested CMOEAs. In the robot gripper optimization case, the reference point z^r = [5,800]^T. §.§.§ Analysis of experiments Table <ref> shows the statistical results of HV values of MOEA/D-IEpsilon and the other four CMOEAs on the robot gripper optimization problem. It is clear that MOEA/D-IEpsilon is significantly better than the other four CMOEAs. To further demonstrate the superiority of the proposed method MOEA/D-IEpsilon, the non-dominated solutions achieved by each CMOEA during the 30 independent runs are plotted in Fig. <ref>(a)-(e). The box plot of HV values of the five CMOEAs is shown in Fig. <ref>(f). From Fig. <ref>, we see that MOEA/D-IEpsion has better performance than the other four CMOEAs.In order to verify the correctness of the optimization results of the robot gripper optimization problem, three representative individuals (A, B and C) are selected from the non-dominated solutions achieved by MOEA/D-IEpsilon as shown in Fig. <ref>. The configurations of the robot gripper mechanism at each point are also plotted in Fig. <ref>. To measure the minimum gripping force min_z F_k(𝐱,z), a spring with a large stiffness coefficient is set vertically at the end of the robot gripper during the simulation process. The spring force is regarded as the gripping force when the robot gripper is balanced by the spring. The simulation tool is ADAMS 2013, and the stiffness coefficient of the spring is 10^13 N/m.Table <ref> shows the simulation results of the minimum gripping force min_z F_k(𝐱,z) with three different configurations of the robot gripper. The relative errors between the theoretical gripping forces and the simulated gripping forces are less than 0.1 %. Thus, we can conclude that the optimization results of the robot gripper optimization problem achieved by MOEA/D-IEpsilon are correct.§ CONCLUSION This paper proposes an improved epsilon constraint-handling method embedded in the framework of MOEA/D. A new CMOEA named MOEA/D-IEpsilon has been proposed. The comprehensive experimental results indicate that MOEA/D-IEpsilon has the ability to cross the large infeasible regions. Compared with the other four decomposition-based CMOEAs including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and C-MOEA/D, MOEA/D-IEpsilon has following advantages:* The performance of MOEA/D-IEpsilon is not sensitive to the initial epsilon value.* MOEA/D-IEpsilon has the ability to explore the feasible and infeasible regions simultaneously during the evolutionary process.* MOEA/D-IEpsilon utilizes the feasible ratio of the current population to dynamically balance the exploration between the feasible regions and infeasible regions. It keeps a good balance of the searching between infeasible and feasible regions.* MOEA/D-IEpsilon is suitable for solving CMOPs with large infeasible regions. In terms of CMOPs, a new set of CMOPs named LIR-CMOP1-14 was designed and presented in this paper. A common feature of these test instances is that they have large infeasible regions. The experimental results show that MOEA/D-IEpsion is significantly better than the other four CMOEAs on this test suite. Thus, we hypothesize that MOEA/D-IEpsilon is better than the other four CMOEAs in solving CMOPs with large infeasible regions, in general. To demonstrate the capacity of MOEA/D-IEpsilon to solve real engineering problems, a robot gripper optimization problem with two conflicting objectives and eight constraints was used as a test problem. The experimental results also demonstrated that MOEA/D-IEpsilon outperformed the other four CMOEAs.Proposed further work includes studying new constraint-handling mechanisms to solve CMOPs with different types of difficulty. One possible way is to collect more information about the working population, and utilize such information to guide a CMOEA to select appropriate constraint-handling methods in different evolutionary stages. This work was supported in part by the National Natural Science Foundation of China (NSFC) under grant 61300159, 61473241 and 61332002, by the Natural Science Foundation of Jiangsu Province of China under grant BK20130808, by the Project of Internation as well as Hongkong，Macao&Taiwan Science and Technology Cooperation Innovation Platform in Universities in Guangdong Province under grant 2015KGJH2014, by China Postdoctoral Science Foundation under grant 2015M571751, by the Science and Technology Planning Project of Guangdong Province of China under grant 2013B011304002, by Educational Commission of Guangdong Province of China under grant 2015KGJHZ014, by the Fundamental Research Funds for the Central Universities of China under grant NZ2013306, and by the Guangdong High-Level University Project “Green Technologies” for Marine Industries.§ APPENDIXIn this section, the detailed definitions of LIR-CMOP1-14 are listed in Table <ref>. § COMPLIANCE WITH ETHICAL STANDARDS Conflict of Interest The authors declare that they have no conflict of interest.Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors. spbasic | http://arxiv.org/abs/1707.08767v1 | {
"authors": [
"Zhun Fan",
"Wenji Li",
"Xinye Cai",
"Han Huang",
"Yi Fang",
"Yugen You",
"Jiajie Mo",
"Caimin Wei",
"Erik Goodman"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170727075931",
"title": "An Improved Epsilon Constraint-handling Method in MOEA/D for CMOPs with Large Infeasible Regions"
} |
Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ======================================================================================================================================== enumi.* Université de Strasbourg, CNRS, Laboratoire d'Innovation Thérapeutique (LIT), UMR7200, Labex MEDALIS, 67000, Strasbourg, France* Structural Biophysics Group, School of Optometry and Vision Sciences, Cardiff University, United Kingdom* CASC4DE Le Lodge, 20, Avenue du Neuhof, 67100 Strasbourg, France* Institut de Génétique et de Biologie Moléculaire et Cellulaire (IGBMC), INSERM U596, CNRS UMR 7104, Université de Strasbourg, Illkirch-Graffenstaden, France 1cm§ ABSTRACT Liquid state NMR is a powerful tool for the analysis of complex mixtures of unknown molecules. This capacity has been used in many analytical approaches: metabolomics, identification of active compounds in natural extracts, characterization of species, and such studies require the acquisition of many diverse NMR measurements on series of samples.While acquisition can easily be performed automatically, the number of NMR experiments involved in these studies increases very rapidly and this data avalanche requires to resort to automatic processing and analysis.We present here a program that allows the autonomous, unsupervised processing of a large corpus of 1D, 2D and DOSY experiments from a series of samples acquired in different conditions. The program provides all the signal processing steps, as well as peak-picking and bucketing of 1D and 2D spectra, the program and its components are fully available. In an experiment mimicking the search of an active species in natural extract, we use it for the automatic detection of small amounts of artemisin added to a series of plant extracts, and for the generation of the spectral fingerprint of this molecules.This program called Plasmodesma is a novel tool which should be useful to decipher complex mixtures, particularly in the discovery of biologically active natural products from plants extracts, but can also in drug discovery or metabolomics studies.§ INTRODUCTION Liquid state NMR is a powerful tool for the analysis of mixtures containing unknown molecules. All species in the solution display their NMR spectra, with a signal intensity proportional to their relative concentrations, provided that slow tumbling rates or relaxation agents do not hide the lines by fast relaxation processes. This capacity has been used in many analytical approaches: metabolomics, identification of active compounds in natural extracts, characterization of species1–5. See references (6), (7), and (8) for recent reviews.Such studies require the acquisition of many diverse NMR measurements on series of samples. Modern NMR spectrometers allow sequential actions (introduction of the sample, probe tuning, acquisition) in order to produce automatically the corresponding 1D and 2D data sets.Unfortunately, if the acquisition is quite easily performed, the access to the final informations is less straightforward: signal processing (Fourier transform, phasing, baseline correction, peak detection) and finally spectrum interpretation are not trivial tasks. Moreover, the use of 2D spectra implies more complex steps and additional tasks such as reduction of t1-noise and t1-ridges, or the determination of contour levels for display.In the case of metabolomics studies, or natural extracts screening, the number of NMR experiments increases very rapidly and this data avalanche requires to resort to automatic processing. While metabolomics are aimed at measuring precisely the amount of well-known compounds, and to quantify precisely their variations from sample to sample, the identification of an active molecule in a natural extracts starts with its detection and then its characterization of an unknown compond or eventually a family of related species.In this article, we present the specific development of a computer program allowing the autonomous, unsupervised processing of a large corpus of 1D and 2D experiments from a series of samples acquired in different conditions.Results obtained using this program on series of complex natural extracts highlight the time saving and the efficiency increase regarding classical “hand-made” processing of raw data.§ SOFTWARE DEVELOPMENTS§.§ Plasmodesma The program Plasmodesma9 developped for this project relies on the SPIKE library for most of its operation10. It is intended to process autonomously a large series of different spectra originated from different samples, obtained in varying conditions. This analytical process involves the handling of a complex set of NMR experiments (1D and 2D homo- or hetero-nuclear spectra), at the end, a spectral report summarizing the analysis is expected, containing all figures, peak and bucket lists for each sample.The current work is based on SPIKE (Spectrometry Processing Innovative KErnel)a comprehensive software on which the current work is based, is a comprehensive software library aimed at the processing and analysis of Fourier transform spectroscopies. It provides basic functionalities such as apodization, Fourier transforms, phasing, peak-picking, line-fitting, baseline correction as well as more advanced tools. It is easily extensible through a plug-in mechanism. SPIKE combines the use of a parallel multi processor approach to a low memory footprint, thus insuring rapid processing with an optimal use of the computer hardware. Moreover, SPIKE allows the efficient handling and visualization of very large data-sets limited only by disk space. SPIKE is a continuation of the previous Gifa and NPK11,12 NMR processing softwares, and was developed to include other Fourier transform spectroscopies, in particular FT Mass spectrometry (Orbitrap and FT-ICR) and 2D-FT-ICR13–15. For portability reasons and ease of development, the program is written in Python, and relies on external libraries such as , , , and .16–20 §.§ Principle of operations The program Plasmodesma operates without any human interaction. When applied to a folder, all NMR files are imported, processed, and a global report is generated for the totality of the analysis. All the processing and analysis steps are optimized depending on the acquisition parameters found in the data-sets, either 1D or 2D data. No other input is required. The 1D and 2D experiments are processed sequentially, The data are apodized, Fourier transformed, and the baseline corrected, 1D spectra are also automatically phased. Additionally, an efficient denoising step21 is performed on the 2D experiments, in order to reduce the t1-noise. The F1 Fourier transform step of the 2D data-sets is performed depending on the spectral type and on the acquisition protocol. The calibration is then determined precisely from the reference signal (in this case, 0 ppm for the TMS). A peak-picking and bucket analysis are then performed (see below). Peak lists and bucket lists are generated as csv files for each experiment. Finally, figures of each spectrum is created, with and without peaks displayed. A final report that contains all acquisition and processing parameters is generated (see S.I. S1). §.§ Specific developments Some functions used by Plasmodesma have been developed specifically for this analysis, and were implemented as SPIKE plug-ins. AutophasingIn the context of metabonomics and screening studies, the possibility to detect and quantify precisely the intensity of vanishing small peaks is paramount. The phase of a 1D spectrum, if set slightly off, may have a strong impact on the possibility to detect small signals, in particular if they are close to a large one. Errors of only a few degrees introduce bias resulting to too low or too high quantization, as well as shifts of the maximum. Automatically acquired natural extract spectra are usually difficult to phase because of strong solvent lines and other artifacts present in the spectra. The improvement of the simple but robust automatic phasing procedure developed in NPK12 contributes efficiently to resolve this problem. The principle is to minimize the negative wing of the 1D spectrum, by performing a grid search first on 0th order (frequency independent) alone, then on both 0th and 1st order (frequency dependent) corrections, the larger peak being used as the 1st order pivot. An automatic baseline correction (see below) is performed at each correction step, and an optional inwater mode allows to ignore the central spectral zone.§.§.§ Autobaseline A flat baseline is also a requisite for correct analysis, and a specific plugin as been developed in this respect. We developed a new approach, which relies on an iterative statistical treatment on the signal split into pieces of constant length, and fitting the baseline by piecewise linear segments. The fit is based on the use of a linear regression minimizing the ℓ_p(x) = ( ∑(|x|^p ) )^1/p norm of the difference. A rough estimate of the spectral baseline is first generated using p=1 on each pieces. Then, the estimate is iteratively improved by removing that part of the signal above the current baseline approximation, and using p=3 for fitting. This method guarantees baselines that stick well to the signal avoiding spurious oscillations that higher-order polynomials or splines may produce.§.§.§ Bucketing Bucketing is an important operation in the processing pipeline. It consists in computing the area under the spectrum over small spectral segments which cover the whole spectral width. The segments should be large enough to blur the small discrepancies that appear from one sample to another, while preserving the resolutive power of the spectra. A bucket size of 0.01 ppm was used for 1D 1H spectra.Bucketing also reduces the size of the data that will be submitted to statistical analysis. This is of foremost importance in the analysis of 2D spectra, which routinely contains millions of points. The reduction of 2D datasets to tractable sizes in statistical tools requires nevertheless bucket sizes on the order of 0.03 to 0.05 ppm in 1H spectroscopy and to 1.0 ppm in 13C. Such sizes are certainly too large to capture all the details contained in the 2D spectra. One solution to this difficulty could be to use segments of varying size, however we rather chose to enrich the information by adding to the area of each bucket, additional information. For each bucket, computed over 1D or 2D spectra, the coordinates of the bucket center and its size in pixel were stored, along with the area information computed as the mean over the bucket, and enriched with the values of the min and max points, and the standard deviation of data over the bucket.§.§.§ Processing of DOSY experiment DOSY spectra are extremely efficient in deciphering complex mixtures, and have been used in many different work (see Mahrous et al7 and reference therein). They require a specific processing for the analysis of the exponential decays observed along the indirect dimension of the 2D spectrum. In this work this specific processing was performed by using the recently introduced PALMA algorithm22 that implements a rapid Inverse Laplace Transform analysis, using a hybrid constraint, maximizing the entropy while minimizing the ℓ_1 norm of the reconstructed spectrum. This algorithm was developed using the SPIKE library, so it was particularly easy to insert it into the processing pipe-line. As a consequence, they are systematically processed and a peak list and an adapted bucket list is also generated.§.§.§ Report Finally, a concise report is produced as a csv file (see S.I. S2). The report contain all the important parameters related to data acquisition and processing. They are finally displayed, as rendered using the pandas python library18. §.§ Analysis Given a set of 1D and 2D NMR raw experiments, the approach described above is able to produce, in full automation and without any human interaction, a set of correctly processed spectra, along with complete peak lists and enriched bucket lists.The artifacts observed in the spectra, such as antidiagonals, t1 noise and ridges, etc. were corrected on the bucket list. On modern spectrometers these artifacts are usually at a low intensity, however as the purpose here is to detect species at low concentration, and their presence is detrimental.The 2D bucket list is corrected for remaining t1 noise and t1 ridges for each column in the matrix, by setting to null all buckets below twice the median value of the considered column. The bucket list originated from symmetric spectra, such COSY and TOCSY, were further corrected for departure of this symmetry by setting symmetrical buckets to the minimum value of the pair. These two operations have the effect of preserving the most significant buckets, without loosing the weak spectral areas.§ M&M Chemicals. Artemisinin 98% was purchased in Sigma-Aldrich and deuterated methanol (10 x 0.75mL) in Eurisotop (Saint Aubin, France).Algae collection and identification. The algae Sargassum muticum was collected in June 2006 in Cap Lévy (Manche), France. Taxonomic determination was performed by Dr A-M. Rusig and a voucher specimen was deposited in the Herbarium of the University of Caen. Extraction was realized as in Vonthron-Sénécheau et al.23.Samples preparation. Five samples containing 10 mg of S. muticum hydroalcoholic extract were prepared. An artemisinin DMSO solution at 3 mg/mL was added to the samples to obtain a final concentration of artemisinin of 0.2, 0.3, 0.4 and 2.7 mg/ml in NMR tubes, as summarized in Table 1. All samples were lyophilized and dissolved in 750 µL of methanol d4, and put in 5 mm NMR tubes. The NMR tubes were spun with a small bench centrifuge to help sedimentation of insoluble parts, and placed in the NMR sample changer.[]@rccccc@ Samples preparationSample n° 1 2 3 4 5Sample n° 1 2 3 4 5S. muticum extract 10 mg 10 mg 10 mg 10 mg 10 mgadded artemisinin 0 mg 0.15 mg 0.24 mg 0.32 mg 2 mgA sample of pure artemisinin was prepared in methanol d4 and studied by NMR.§.§.§ NMR spectroscopy Acquisitions were performed on a Bruker Avance-III spectrometer operating at 700 MHz, and equipped with a TCI cryo probe and a standard Bac60 sample changer. Each sample was automatically inserted into the spectrometer, tuned and shimmed after a stabilization delay of 120 seconds. All experiments were automatically run on each sample, the whole sequence being programmed using a TopSpin macro (see E.S.I S2). Spectral parameters (π / 2 pulses, receiver gain…) were optimized on one sample and used for the whole series without further check.Spectral widthes were set to 12 ppm in 1H and to 150 ppm in 13C. 1D spectra were acquired on 64 scans, 16384 points, and a relaxation delay of 1.5 sec, for a total time of 3 minutes. COSY experiments were performed with 8 scans, with 512 increments of 4096 points each, for a total acquisition time of 2 hours. TOCSY experiments were performed with 8 scans, with 400 increments of 4096 points each, and using a DISPSI-2 mixing sequence of 80 msec duration. TOCSY acquisition time was 1 hour 40 minutes. DOSY experiments were performed with 32 scans, with 50 increments of 4096 points each, for a total acquisition time of 50 minutes. HSQC experiments were performed with 4 scans, with 512 increments of 2048 points each, for a total acquisition time of 1 hour. HMBC experiments were performed with 48 scans, with 400 increments of 4096 points each, for a total acquisition time of 10 hours.The complete acquisition time for one sample, including sample injection and tuning, took about 16 hours. The five samples were acquired in one continuous run.The artemisinin sample was studied by NMR: Artemisinin: 1H NMR (CD3OD, 700 MHz) δ 0.99 (3H, d, J = 6.2 Hz, 6-CH3), 1.16 (3H, d, J = 7.2 Hz, 9-CH3), 1.38 (3H, s, 3-CH3), 2.08 (1H, ddd, H4), 2.40 (1H, ddd, H4), 2.01 (1H, m, H5), 1.47 (1H, m, H5), 1.38 (1H, m, H5a), 1.52 (1H, m, H6), 1.09 (1H, m, H7), 1.77 (1H, m, H7), 1.17 (1H, m, H8), 1.86 (1H, m, H8), 1.82 (1H, m, H8a), 3.31 (1H, dq, H9), 6.03 (1H, dq, H12), 13C NMR (CD3OD, 700 MHz) δ 106.7 (C, C3), 25.2 (CH3, C3), 36.6 (CH2, C4), 25.7 (CH2, C5), 51.2 (CH, C5a), 38.1 (CH, C6), 19.9 (CH3, C6), 34.6 (CH2, C7), 24.0 (CH2, C8), 45.6 (CH, C8a), 34.0 (CH, C9), 12.7 (CH3, C9), 81.0 (CH, C12a), 95.5 (CH, C12).§.§.§ Data Processing Spectral Processing was integrally performed using the Plasmodesma program presented here. The program is written in python, and is compatible both with python 2 and python 3. It is based on the SPIKE library10 and the DOSY processing were performed using the PALMA approach22 embedded in SPIKE as a plugin.Complete processing took 96 minutes on a MacOs machine, running the python anaconda distribution 4.2 from Continuum Analytics (Austin, TX).Statistical Analysis. The bucket lists and peak lists produced by the Plasmodesma run were analyzed with a python script based on the pandas library, using the Jupyter notebook environment.The Plasmodesma program, along with examples, experimental data related to the artemisin series, Jupyter notebooks presenting the data analysis, and the specific SPIKE plugins are freely available at https://github.com/delsuc/plasmodesma repository.§ RESULTS To mimic the presence of a bioactive molecule at different concentrations in complex mixtures, crude plant extracts were supplemented at different concentrations with artemisinin, a naturally occurring and structurally known sesquiterpene lactone, and five different samples were prepared. All five samples were placed in the sample changer and NMR data were acquired in an automatic manner, after an initial tuning of the first sample. §.§ Data Processing The raw data-sets were processed as described above, and the peak lists and bucket lists generated. Figures <ref> and <ref> show an example of the result of such a processing.The bucketing procedure is used to summarize the spectral content, and by reducing the size of the data to handle, to ease further statistical analysis. However, it can be seen in Figure <ref> that t1-noise and other spectral artifacts are present in particular in the standard deviation analysis, which enhances the local signal variations. It appears that corruption of buckets from spectral artifacts appear more deleterious in 2D spectroscopy than in classical 1D. For this reason, the areas and standard deviation values of the bucket list were subjected to the simple procedures described above. The first step consists in setting to a null value all values below a certain threshold computed from the median over the vertical column of the considered bucket. This procedure allows to remove a large part of the noise, and to only retain the peaks separated above the threshold. The threshold level adapted for each column permit to efficiently clean the strong t1-noise stripes, while preserving weak peaks located in less crowded regions. In a second step, homonuclear experiments a symmetrization procedure can be applied, it was done here by simply taking the smaller of the two values related by symmetry. This procedure is much simpler and more robust on bucket lists than on real spectra, as the bucketing has already homogenized the spectral axes and produced squared buckets. This procedure was applied on the area and the standard deviation values of the bucket list.Figure <ref> shows the result of each cleaning steps. It can clearly be seen that this procedure, allows an improvement of the quality of data and a better compatibility with automatic analysis. §.§ Data Analysis The cleaned bucket lists can be efficiently used for detection of the spectral features varying from spectrum to another. This can be done on any 1D or 2D spectra: Figure <ref> shows the result on the analysis of the COSY spectrum.Obviously, the positions of the signals of the artemisinin spectrum is detected and separated from the constant background, even though the background is of much larger intensity. Here the original spectrum is not genuinely recovered, not only because some signals are missing, but principally because of the loss of the intensities. However, the generated spectral pattern can be used to extract chemical shifts and topologies, and recognize a molecular pattern, which can be used as a fingerprint. The same result cannot be obtained directly from the spectrum, and is efficient because the bucketing standardizes the spectra, the standard deviation measures the fluctuation rather the intensity. Finally the cleaning operation smooth out the random fluctuations which otherwise would hamper the direct comparison to operate.The procedure above is not very sensitive, and the samples with lower level of added artemisin could not be processed efficiently. A second procedure was tested by taking the ratio of the bucket standard deviation values. The results are shown in Figure <ref>: despite a low level of concentration (few hundred micrograms of artemisin in 10 mg of crude material), the spectral fingerprint is recovered. In this case the diagonal of the homonuclear spectrum is not recovered, this does not have a strong imapct, as it can be fully infered from the off-diagonal fingerprint dots.§.§.§ Linear regression This first approach take spectra two by two, and can be used on homonuclear spectra, as shown here, and also on heteronuclear ones. Using the whole set of spectra at once requires to have an additional information, eventually imprecise, on the amount of active material in each sample. In this case, signals coming from the studied molecule are expected to be proportional to its concentration, and this property can be exploited to separate those signals, varying along with the concentration value, to the other signal, uncorrelated with it. This was performed by using thelibrary24, a generic tool for machine learning, written in python with full interoperability with python, Jupyter and SPIKE. Thefunction and the Recursive Feature Elimination tool were used, and applied on the the bucket lists area values (see S.I. for detailed operation).These tools allow to select a small subset of parameters which best correlate with the estimated concentration. The selected features are then supposed to define a spectral fingerprint in a manner equivalent with the previous approach, but with a quantitative aspect this time. As the whole spectrum series is used, it is expected to produce better results. Results are shown in Figure <ref> for the HSQC spectra obtained on the 4 samples presenting the lowest concentration. It can be seen that the HSQC spectrum of artemisinin is extracted from the complex spectrum of the mixture. The main artifacts observed in the finger print are associated with the solvent lines (here water, methanol and DMSO)§ DISCUSSION The measurement of NMR spectra over series of complex samples, and their analysis, is a common procedure in screening or activity studies. The acquisition part is usually well covered through the use of sample changers and associated softwares, with the eventual help of companion programs, allowing an optimized set-up25. Here, we extend the set of tools for these studies to the possibility of automatic processing and statistical analysis of the set of 1D and 2D spectra. There are already basic tools which allow to perform the first processing steps of the data, however, they usually rely either on preset parameters values (phase corrections, window function) or crude estimate of the optimum parameters (baseline correction). In contrast, the program Plasmodesma presented here, works in an autonomous manner, without any user interaction, relying on a small set of preset global parameters. It is able to autonomously process 1D, 2D, and DOSY experiments, processing parameters are optimized either from the experiment types (window function) or optimized automatically on the data (phase correction, baseline). In addition, advanced methods are used for the denoising of 2D spectra or the analysis of DOSY experiments. Finally, the program generates spectra and peak lists and bucket lists for all spectra, as well as reports on the data and the analysis.The use of the SPIKE library10, a generalist processing library for NMR and other Fourier spectroscopies, allowed a rapid development of the program, as well as the use of advance tools. Not relying on parameters previously by an operator allow to process directly after the measure, and use the result of the processing as a token for the quality of the acquisition. The processing step being more rapid than the acquisition, it is perfectly possible to repeat the processing along the series of measurements, to monitor the advance and quality of the current experiment.The series of spectra are further algorithmically analyzed using machine-learning inspired approaches. However, each 2D spectrum is typically composed of several million of data points, and this size hampers the possibility to algorithmically compare efficiently several spectra For instance the series of spectra generated in this study represents more than 20 million points overall, and some pre-conditioning of the data is required. For this reason, the automatic analysis of the spectra is here principally performed on the bucket lists, which provided a reduced but faithful representation of the spectrum. Many artifacts such as antidiagonals, t1 noise and ridges, are also present. These artifacts are of rather low intensity, however as the purpose here is the detection of compounds at low concentration, their presence is detrimental, and we chose to correct them on the bucket list rather than on the original spectra. This smoothing and spectral normalization afforded by the bucketing operation allows optimal spectral corrections and makes comparison between spectra obtained from different samples easier. Finally, access to quantities such as standard deviation of the signal, min and max values, allows a finer description of the spectra.From this material, a spectral fingerprint of the searched molecule could be first determined from a two-by-two comparison of spectra with a presence/absence of the searched compound, In this case the approach consisting in comparing by ratio the standard deviation of buckets from spectra of COSY type showed to be able to detect and extract the spectral features of the compound even at low concentrations. Linear regression over the whole series of spectra was also used to generate a faithful fingerprint. One step regression as well as recursive feature selection were used, and both proved to be efficient in extracting a spectral fingerprint for both homonuclear and heteronuclear experiments (see figure <ref> and S.I. S2).Each experiment types can be used for the determination of the fingerprint, and COSY type and HSQC type experiments were explored. It is possible to perform the same analysis on a concatenation of all experiments, however such an approach did not provide a correct result, probably because of the heterogeneity of the different spectra types.§ CONCLUSION The program developed in this work represents an efficient alternative for the autonomous processing of a series of NMR data (1D and 2D) and contributes efficiently to the discovery of structurally unknown molecules present in natural extracts, without any chromatographic separation. It fully exploits the NMR technique as a fingerprinting technique: complete 2D NMR fingerprint of the compound is recovered through differential analysis performed both by comparison of local variation in the spectra or by linear regression between signal intensity and the concentrations of the natural product in the sample.The extended bucketing procedure allows a strong reduction of the size of the data, while preserving a large part of the molecular information present in the original spectra. Basic machine learning approaches were used to analyze this compressed but rich information, and proved sufficient to readily extract the spectral fingerprint of the unknown molecule, either from spectral comparison, or by handling of the whole spectral series at once.Plasmodesma is a novel tool which should be useful to decipher complex mixtures, particularly in the discovery of biologically active natural products from plants extracts, but can also in drug discovery or metabolomics studies.§ ACKNOWLEDGMENTS The authors are very grateful to Labex Medalis and Région Alsace for a fellowship (LM), and Europe for an Erasmus fellowship (PM). We are also grateful to A-M.Rusig for the collect and the identification of the algal material, J.Viéville for help in the NMR set-up, and G.Bret for help in the statistical analysis. We acknowledge Wikimedia26 for the S.muticum picture used in the Graphical Abstract.§ E.S.I. * S1 Processing of the artemisinin series* S2 Analysis of the artemisinin series § REFERENCEStocsectionReferencesrefs ref-Bakiri2017 [1] Bakiri, A.; Hubert, J.; Reynaud, R.; Lanthony, S.; Harakat, D.; Renault, J.-H.; Nuzillard, J.-M. J Nat Prod. 2017, 80 (5), 1387–1396.ref-Dabrosca2017 [2] D'Abrosca, B.; Lavorgna, M.; Scognamiglio, M.; Russo, C.; Graziani, V.; Piscitelli, C.; Fiorentino, A.; Isidori, M. Food Chem Toxicol. 2017.ref-Abdelsalam2017 [3] Abdelsalam, A.; Mahran, E.; Chowdhury, K.; Boroujerdi, A.; El-Bakry, A. Physiol Mol Biol Plants. 2017, 23 (2), 369–383.ref-Hubert2014 [4] Hubert, J.; Nuzillard, J.-M.; Purson, S.; Hamzaoui, M.; Borie, N.; Reynaud, R.; Renault, J.-H. Anal Chem. 2014, 86 (6), 2955–2962.ref-Oettl2014 [5] Oettl, S.-K.; Hubert, J.; Nuzillard, J.-M.; Stuppner, H.; Renault, J.-H.; Rollinger, J.-M. Anal Chim Acta. 2014, 846, 60–67.ref-Larive:2014vp [6] Larive, C. 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Analyst 2017, 142 (5), 772–779.ref-Vonthron:2011 [23] Vonthron-Sénécheau, C.; Kaiser, M.; Devambez, I.; Vastel, A.; Mussio, I.; Rusig, A.-M. Mar. Drugs 2011, 9, 922–933.ref-scikit-learn [24] Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E. Journal of Machine Learning Research 2011, 12, 2825–2830.ref-Clos:2013 [25] Clos, L. J.; Jofre, M. F.; Ellinger, J. J.; Westler, W. M.; Markley, J. L. Metabolomics 2013, 9 (3), 558–563.ref-wiki:SMuticum [26] Commons, W. File:Sargassum muticum yendo fensholt 1955 lamiot wimmereuxhautsdefrance estran juillet 2016a9.jpg — wikimedia commons, the free media repository, 2016.§ SUPP.INFO 1 [pages=-]SI1.pdf§ SUPP.INFO 2Supp Info 2 can be found at https://github.com/delsuc/plasmodesma/blob/master/Analysis.ipynb | http://arxiv.org/abs/1707.08805v1 | {
"authors": [
"Laure Margueritte",
"Petar Markov",
"Lionel Chiron",
"Jean-Philippe Starck",
"Catherine Vonthron-Sénécheau",
"Mélanie Bourjot",
"Marc-André Delsuc"
],
"categories": [
"physics.chem-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170727100126",
"title": "Automatic differential analysis of NMR experiments in complex samples"
} |
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